From a8a3172f28b331582ccd0d4f6cf2897b5016892a Mon Sep 17 00:00:00 2001 From: Elwood <3286545699@qq.com> Date: Wed, 24 Dec 2025 17:26:42 +0800 Subject: [PATCH 1/6] vault backup: 2025-12-24 17:26:42 --- .obsidian/workspace.json | 29 +++++++++++++++++------------ unimath/欢迎.md | 5 +++++ 2 files changed, 22 insertions(+), 12 deletions(-) create mode 100644 unimath/欢迎.md diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index bc5bb04..1fe673f 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -11,10 +11,14 @@ "id": "b6107f657aaca089", "type": "leaf", "state": { - "type": "empty", - "state": {}, - "icon": "lucide-file", - "title": "新标签页" + "type": "split-diff-view", + "state": { + "aFile": "编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md", + "bFile": "编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md", + "aRef": "" + }, + "icon": "diff", + "title": "Diff: 级数敛散性的基本思路和方法（解析版）" } } ] @@ -187,8 +191,15 @@ "obsidian-git:Open Git source control": false } }, - "active": "cd457af850fd87f8", + "active": "b6107f657aaca089", "lastOpenFiles": [ + "unimath/欢迎.md", + "编写小组/讲义/隐函数，参数方程和应用题.md", + "编写小组/讲义/极限计算的基本方法（解析版2）.md", + "编写小组/讲义/极限计算的基本方法.md", + "编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md", + "编写小组/讲义/二阶微分推导.md", + "README.md", "编写小组/二阶微分推导.md", "编写小组/课后测/课后测解析版2.md", "编写小组/课后测/课后测解析版1.md", @@ -205,13 +216,8 @@ "编写小组/讲义/图片/应用题人拉船.png", "编写小组/讲义/图片", "编写小组/讲义/隐函数，参数方程和应用题（解析版）.md", - "编写小组/讲义/隐函数，参数方程和应用题.md", - "编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md", "编写小组/讲义/级数敛散性的基本思路和方法.md", "编写小组/讲义/极限计算的基本方法（解析版）.md", - "编写小组/讲义/极限计算的基本方法（解析版2）.md", - "编写小组/讲义/极限计算的基本方法.md", - "编写小组/讲义/二阶微分推导.md", "编写小组/讲义", "编写小组", "小测/12.22/课前测.md", @@ -221,7 +227,6 @@ "Chapter 2 极限/比值判别法.md", "未命名.base", "Chapter 2 极限/根值判别法.md", - "Chapter 2 极限/比较判别法.md", - "小测/12.23/课后测(解析).md" + "Chapter 2 极限/比较判别法.md" ] } \ No newline at end of file diff --git a/unimath/欢迎.md b/unimath/欢迎.md new file mode 100644 index 0000000..034ec91 --- /dev/null +++ b/unimath/欢迎.md @@ -0,0 +1,5 @@ +这是你的新*仓库*。 + +写点笔记，[[创建链接]]，或者试一试[导入器](https://help.obsidian.md/Plugins/Importer)插件! + +当你准备好了，就将该笔记文件删除，使这个仓库为你所用。 \ No newline at end of file From 4d7260b9194210233734014bfd866e984873b6fe Mon Sep 17 00:00:00 2001 From: Elwood <3286545699@qq.com> Date: Wed, 24 Dec 2025 17:27:37 +0800 Subject: [PATCH 2/6] vault backup: 2025-12-24 17:27:37 --- .obsidian/workspace.json | 8 ++++---- unimath/欢迎.md | 5 ----- 2 files changed, 4 insertions(+), 9 deletions(-) delete mode 100644 unimath/欢迎.md diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 1fe673f..5fdda46 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -13,12 +13,12 @@ "state": { "type": "split-diff-view", "state": { - "aFile": "编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md", - "bFile": "编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md", + "aFile": "unimath/欢迎.md", + "bFile": "unimath/欢迎.md", "aRef": "" }, "icon": "diff", - "title": "Diff: 级数敛散性的基本思路和方法（解析版）" + "title": "Diff: 欢迎" } } ] @@ -191,7 +191,7 @@ "obsidian-git:Open Git source control": false } }, - "active": "b6107f657aaca089", + "active": "cd457af850fd87f8", "lastOpenFiles": [ "unimath/欢迎.md", "编写小组/讲义/隐函数，参数方程和应用题.md", diff --git a/unimath/欢迎.md b/unimath/欢迎.md deleted file mode 100644 index 034ec91..0000000 --- a/unimath/欢迎.md +++ /dev/null @@ -1,5 +0,0 @@ -这是你的新*仓库*。 - -写点笔记，[[创建链接]]，或者试一试[导入器](https://help.obsidian.md/Plugins/Importer)插件! - -当你准备好了，就将该笔记文件删除，使这个仓库为你所用。 \ No newline at end of file From 4fa3055ab5cca1ed1727ecba06c814ab47f23ab1 Mon Sep 17 00:00:00 2001 From: Elwood <3286545699@qq.com> Date: Wed, 24 Dec 2025 17:27:39 +0800 Subject: [PATCH 3/6] vault backup: 2025-12-24 17:27:39 --- .obsidian/workspace.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 5fdda46..f5ca4f0 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -191,7 +191,7 @@ "obsidian-git:Open Git source control": false } }, - "active": "cd457af850fd87f8", + "active": 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编写小组/课后测/课后测解析版4.md | 25 + ...参数方程和应用题（解析版）.md | 587 ------------------ 24 files changed, 865 insertions(+), 724 deletions(-) delete mode 100644 介值定理例题.md delete mode 100644 编写小组/二阶微分推导.md rename 编写小组/讲义/{是的，这是一份礼物.md => 单调有界定理例题三道（HW）.md} (51%) create mode 100644 编写小组/讲义/图片/易错点10-1.png create mode 100644 编写小组/讲义/图片/易错点9-1.png create mode 100644 编写小组/讲义/子数列问题&考试易错点汇总（解析版）.md create mode 100644 编写小组/讲义/证明题方法：单调有界定理，介值定理（解析版）.md create mode 100644 编写小组/课前测/课前测4.md create mode 100644 编写小组/课前测/课前测解析版3.md create mode 100644 编写小组/课前测/课前测解析版4.md create mode 100644 编写小组/课后测/课后测4.md create mode 100644 编写小组/课后测/课后测解析版3.md create mode 100644 编写小组/课后测/课后测解析版4.md delete mode 100644 隐函数，参数方程和应用题（解析版）.md diff --git a/.gitignore b/.gitignore index 317d14d..9f65ba9 100644 --- a/.gitignore +++ b/.gitignore @@ -2,4 +2,5 @@ .vs/ *.pdf .vscode/ -.obsidian/ \ No newline at end of file +.obsidian/ +/介值定理例题.md \ No newline at end of file diff --git a/介值定理例题.md b/介值定理例题.md deleted file mode 100644 index ed5ca62..0000000 --- a/介值定理例题.md +++ /dev/null @@ -1,41 +0,0 @@ ->[!example] **例1** -设$f(x)$在$[a,b]$上连续，且$a < c < d < b$。证明：在$(a,b)$内必存在点$\xi$，使得$mf(c)+nf(d)=(m+n)f(\xi)$，其中$m,n$为任意给定的自然数。 - -**解析：** -因$f(x)$在$[a,b]$上连续，所以$f(x)$在$[a,b]$上能取得最大值$f_{\max}$和最小值$f_{\min}$。又因为 -$$(m + n)f_{\min} \leqslant mf(c) + nf(d) \leqslant (m + n)f_{\max},$$ -即$f_{\min} \leqslant \frac{mf(c) + nf(d)}{m + n} \leqslant f_{\max}$，故由介值定理，在$[a,b]$上必存在一点$\xi$，使得 -$$\frac{mf(c) + nf(d)}{m + n} = f(\xi),$$ -即$mf(c) + nf(d) = (m + n)f(\xi)$。 - - ->[!example] **例2** ->一支巡逻小分队定期到山上的岗哨换岗，每天上午7点从营地出发，8点到达目的地。第二天上午7点沿原路返回，8点前返回到达山下营地。利用零点定理证明，在换岗的路途中必有一点，小分队在两天中的同一时刻经过该点。 - -**解析** -设上山的路程函数为$s=f_1(t)$（$7\leqslant t\leqslant8$），则$f_1(7)=0$，$f_1(8)=D$（$D$为两岗哨之间的距离） -下山的路程函数为$s=f_2(t)$（$7\leqslant t\leqslant8$），则$f_2(7)=D$，$f_2(8)=0$ -作辅助函数$F(t)=f_1(t)-f_2(t)$，则$F(t)$在$[7,8]$上连续，且$F(7)=-D$，$F(8)=D$ -由零点定理知，至少存在$\xi\in(7,8)$使$F(\xi)=0$ -即在换岗的路途中必有一点，小分队在两天中的同一时刻经过该点。 - ->[!example] 例3 ->设$f(x)$在$(-\infty,+\infty)$上连续，且$\lim\limits_{x \to \infty}\frac{f(x)}{x}=0$，试证存在$\xi\in(-\infty,+\infty)$，使$f(\xi)+\xi=0$。 -> - -**解析** - 步骤1：构造辅助函数并分析连续性 -设辅助函数$F(x)=f(x)+x$。 -已知$f(x)$在$(-\infty,+\infty)$上连续，$y=x$在$(-\infty,+\infty)$上也连续，根据连续函数的和运算性质，$F(x)$在$(-\infty,+\infty)$上连续。 - -步骤2：利用极限条件分析$F(x)$在无穷远处的符号 -由$\lim\limits_{x \to \infty}\frac{f(x)}{x}=0$，可得： -$$\lim\limits_{x \to +\infty}\frac{F(x)}{x}=\lim\limits_{x \to +\infty}\left(\frac{f(x)}{x}+1\right)=0+1=1>0$$ -$$\lim\limits_{x \to -\infty}\frac{F(x)}{x}=\lim\limits_{x \to -\infty}\left(\frac{f(x)}{x}+1\right)=0+1=1>0$$ -根据函数极限的保号性： -- 存在$X_1>0$，当$x>X_1$时，$\frac{F(x)}{x}>0$，又$x>0$，故$F(x)>0$。取$x_1=X_1+1$，则$F(x_1)>0$ -- 存在$X_2>0$，当$x<-X_2$时，$\frac{F(x)}{x}>0$，又$x<0$，故$F(x)<0$。取$x_2=-X_2-1$，则$F(x_2)<0$ - -步骤3：应用零点存在定理得出结论 -$F(x)$在闭区间$[x_2,x_1]$上连续，且$F(x_2)<0$，$F(x_1)>0$。 -由零点存在定理，至少存在一点$\xi\in(x_2,x_1)\subset(-\infty,+\infty)$，使得$F(\xi)=0$，即$f(\xi)+\xi=0$ \ No newline at end of file diff --git a/编写小组/二阶微分推导.md b/编写小组/二阶微分推导.md deleted file mode 100644 index bd35c13..0000000 --- a/编写小组/二阶微分推导.md +++ /dev/null @@ -1,55 +0,0 @@ - -首先提出一个问题： -导数和微分是一个东西吗 -**不是。它们密切相关，但本质不同。** - -### 一阶情形 - -- **导数 (Derivative)** - - **本质**：一个**函数**（或该函数在某点的值） - - **含义**：因变量相对于自变量的**变化率**（瞬时斜率） - - **定义**：$f'(x) = \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$ - -- **微分 (Differential)** - - **本质**：一个**表达式**或**量** - - **含义**：函数改变量的**线性主要部分** - - **定义**：$dy = f'(x) \, dx$ - - 其中 $dx$ 是自变量的微分（任意微小改变量） - - $dy$ 是因变量沿切线方向的近似改变量 - -**一阶关系**：$dy = f'(x) \, dx$ - -### 二阶情形 -#### 二阶导数 -- **本质**：一阶导数的导数，仍是一个**函数** -- **符号**：$f''(x)$, $y''$, $\frac{d^2y}{dx^2}$ -- **含义**：函数**变化率的变化率**，描述函数的**凹凸性** -- **定义**：$f''(x) = \frac{d}{dx}[f'(x)]$ - -#### 二阶微分 -- **本质**：微分的微分，是一个**表达式** -- **符号**：$d^2y$ -- - 我们知道：$f''(x) =lim_{\Delta x \rightarrow0} \frac{f'(x+\Delta x) - f'(x)}{dx} = \frac{d^2f(x)}{(dx)^2}$ -现在尝试推导这一公式 -由两函数相乘一阶微分公式：$d[f(x).g(x)] = g(x)d[(fx)] + f(x)d[g(x)]$ -得： - $f''(x) = \frac{d(\frac{df(x)}{dx})}{dx}$ - $= \frac{1}{dx}.\frac{d(df(x))}{dx} + df(x) . \frac{d(\frac{1}{dx})}{dx}$ - $= \frac{d^2f(x)}{(dx)^2} +df(x) . \frac{d(\frac{1}{dx})}{dx}$ ---- - 嘿！您猜怎么着！ - 出现了额外的一项，与二阶导数公式不符： $df(x) . \frac{d(\frac{1}{dx})}{dx}=-df(x) . \frac{1}{d^2x}. \frac{d({dx})}{dx}$ -问题出在 $d(dx)$ 上： - 我们知道，一阶导数的意义是当 $\Delta x$ 趋于无穷小的时候 $\Delta y$ 的变化率 - 实际上，在函数曲线上的任何一点处取函数微分， $dx$ 都是相等的 - 即 $dx$ 永远是均匀的，求高阶微分时将其视为常数 - 所以：$d(dx) = 0$ -### $f''(x) =lim_{\Delta x \rightarrow0} \frac{f'(x+\Delta x) - f'(x)}{dx} = \frac{d^2f(x)}{(dx)^2}= \frac{d^2f(x)}{dx^2}$ - -**注意此处的约定写法**：$(dx)^2$ 写作 $dx^2$（注意：这不是 $d(x^2)$） - - - - - diff --git a/编写小组/讲义/二阶微分推导.md b/编写小组/讲义/二阶微分推导.md index bd35c13..4f70db6 100644 --- a/编写小组/讲义/二阶微分推导.md +++ b/编写小组/讲义/二阶微分推导.md @@ -8,7 +8,7 @@ - **导数 (Derivative)** - **本质**：一个**函数**（或该函数在某点的值） - **含义**：因变量相对于自变量的**变化率**（瞬时斜率） - - **定义**：$f'(x) = \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$ + - **定义**：$f'(x) = \frac{dy}{dx} = \lim\limits_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$ - **微分 (Differential)** - **本质**：一个**表达式**或**量** @@ -30,7 +30,7 @@ - **本质**：微分的微分，是一个**表达式** - **符号**：$d^2y$ - - 我们知道：$f''(x) =lim_{\Delta x \rightarrow0} \frac{f'(x+\Delta x) - f'(x)}{dx} = \frac{d^2f(x)}{(dx)^2}$ + 我们知道：$f''(x) =\lim\limits_{\Delta x \rightarrow0} \frac{f'(x+\Delta x) - f'(x)}{dx} = \frac{d^2f(x)}{(dx)^2}$ 现在尝试推导这一公式 由两函数相乘一阶微分公式：$d[f(x).g(x)] = g(x)d[(fx)] + f(x)d[g(x)]$ 得： @@ -39,13 +39,13 @@ $= \frac{d^2f(x)}{(dx)^2} +df(x) . \frac{d(\frac{1}{dx})}{dx}$ --- 嘿！您猜怎么着！ - 出现了额外的一项，与二阶导数公式不符： $df(x) . \frac{d(\frac{1}{dx})}{dx}=-df(x) . \frac{1}{d^2x}. \frac{d({dx})}{dx}$ + 出现了额外的一项，与二阶导数公式不符： $df(x)\frac{d(\frac{1}{dx})}{dx}=-df(x) \frac{1}{d^2x}\frac{d({dx})}{dx}$ 问题出在 $d(dx)$ 上： 我们知道，一阶导数的意义是当 $\Delta x$ 趋于无穷小的时候 $\Delta y$ 的变化率 实际上，在函数曲线上的任何一点处取函数微分， $dx$ 都是相等的 即 $dx$ 永远是均匀的，求高阶微分时将其视为常数 所以：$d(dx) = 0$ -### $f''(x) =lim_{\Delta x \rightarrow0} \frac{f'(x+\Delta x) - f'(x)}{dx} = \frac{d^2f(x)}{(dx)^2}= \frac{d^2f(x)}{dx^2}$ +### $f''(x) =\lim\limits_{\Delta x \rightarrow0} \frac{f'(x+\Delta x) - f'(x)}{dx} = \frac{d^2f(x)}{(dx)^2}= \frac{d^2f(x)}{dx^2}$ **注意此处的约定写法**：$(dx)^2$ 写作 $dx^2$（注意：这不是 $d(x^2)$） diff --git a/编写小组/讲义/单调有界准则与介值定理.md b/编写小组/讲义/单调有界准则与介值定理.md index 931d328..0c4728a 100644 --- a/编写小组/讲义/单调有界准则与介值定理.md +++ b/编写小组/讲义/单调有界准则与介值定理.md @@ -1,6 +1,6 @@ -## 单调有界准则 +# 单调有界准则 -### 原理 +## 原理 - **单调有界数列必收敛**： 1. 若数列 $\{a_n\}$ 单调递增且有上界 $M$，则 $\{a_n\}$ 收敛，且 $\lim_{n \to \infty} a_n = \sup\{a_n\} \leq M$。 @@ -9,7 +9,7 @@ - **函数单调有界性质**： 若函数 $f(x)$ 在区间 $I$ 上单调且有界，则 $f(x)$ 在 $I$ 的端点或无穷远处存在单侧极限。 -### 适用情况 +## 适用情况 适用于证明数列或函数极限的存在性，尤其是： @@ -17,19 +17,19 @@ - 函数在某区间上的极限存在性； - 某些无法直接求极限但可判断单调性和有界性的情形。 -### 优势 +## 优势 1. 不依赖于极限的具体值，只需判断单调性和有界性； 2. 适用于很多递推定义的数列； 3. 证明过程结构清晰，易于理解和操作。 -### 劣势 +## 劣势 1. 只能证明极限存在，不能直接给出极限值（需另解方程）； 2. 判断单调性和有界性有时需要一定的技巧； 3. 对于非单调或没有明显界的序列不适用。 -### 例子 +## 例子 > [!example] 例1 > 证明数列 $a_1 = \sqrt{2}, a_{n+1} = \sqrt{2 + a_n}$ 收敛，并求其极限。 @@ -80,8 +80,8 @@ 2. **有界性**：显然 $0 < a_n \leq \frac{1}{2}$。 3. **由单调有界准则**，数列收敛。 4. **极限**：$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n}{n^2 + 1} = 0$。## 介值定理 -### 介值定理 -### 原理 +# 介值定理 +## 原理 - **介值定理（Intermediate Value Theorem, IVT）**： 若函数 $f(x)$ 在闭区间 $[a,b]$ 上连续，且 $f(a) \neq f(b)$，则对于任意介于 $f(a)$ 和 $f(b)$ 之间的实数 $C$，至少存在一点 $\xi \in (a,b)$，使得 $f(\xi) = C$。 @@ -89,7 +89,7 @@ - **零点定理（特殊情形）**： 若 $f(x)$ 在 $[a,b]$ 上连续，且 $f(a) \cdot f(b) < 0$，则至少存在一点 $\xi \in (a,b)$，使得 $f(\xi) = 0$。 -### 适用情况 +## 适用情况 适用于连续函数在区间上的值域问题，尤其是： @@ -97,19 +97,19 @@ - 证明函数可取到某个中间值； - 确定函数值域或解的存在性。 -### 优势 +## 优势 1. 不需求解方程，只需验证连续性和端点函数值； 2. 可用于证明解的存在性，构造性证明中常用； 3. 定理直观，易于理解和应用。 -### 劣势 +## 劣势 1. 只能证明存在性，不能确定解的个数或具体位置； 2. 要求函数在闭区间上连续，否则定理不适用； 3. 对于非连续函数或开区间，需额外处理。 -### 例子 +## 例子 > [!example] 例1 > 证明方程 $x^3 - 3x + 1 = 0$ 在区间 $(0,1)$ 内至少有一个实根。 diff --git a/编写小组/讲义/是的，这是一份礼物.md b/编写小组/讲义/单调有界定理例题三道（HW）.md similarity index 51% rename from 编写小组/讲义/是的，这是一份礼物.md rename to 编写小组/讲义/单调有界定理例题三道（HW）.md index d7f0880..6c5a826 100644 --- a/编写小组/讲义/是的，这是一份礼物.md +++ b/编写小组/讲义/单调有界定理例题三道（HW）.md @@ -1,4 +1,4 @@ -是的，这是一份礼物 +是的，这是一份礼物~ --- @@ -6,15 +6,16 @@ > （1）单调递增有上界的数列，必有极限且$\mathop {\lim }\limits_{n \to \infty } {a_n} = \sup \{ {a_n}\} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (n \in N)$； > （2）单调递减有下界的数列，必有极限且$\mathop {\lim }\limits_{n \to \infty } {a_n} = \inf \{ {a_n}\} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (n \in N)$. -先来一道习题练练手吧~ >[!example] 例一 >设$a > 0,\sigma > 0,{a_1} = \frac{1}{2}\left( {a + \frac{\sigma }{a}} \right),{a_{n + 1}} = \frac{1}{2}\left( {{a_n} + \frac{\sigma }{{{a_n}}}} \right),n = 1,2,...$ > 证明：数列$\left\{ {{a_n}} \right\}$收敛，且极限为$\sqrt \sigma$. -证：对 $\forall t > 0,t + \frac{1}{t} \ge 2$，令 $t = \frac{{{a_n}}}{{\sqrt \sigma }} > 0$，则有${a_{n + 1}} = \frac{1}{2}\left( {{a_n} + \frac{\sigma }{{{a_n}}}} \right) = \frac{{\sqrt \sigma }}{2}\left( {\frac{{{a_n}}}{{\sqrt \sigma }} + \frac{{\sqrt \sigma }}{{{a_n}}}} \right) \ge \frac{{\sqrt \sigma }}{2}*2 = \sqrt \sigma ,n = 1,2,...$ +证：对 $\forall t > 0,t + \frac{1}{t} \ge 2$，令 $t = \frac{{{a_n}}}{{\sqrt \sigma }} > 0$，则有$${a_{n + 1}} = \frac{1}{2}\left( {{a_n} + \frac{\sigma }{{{a_n}}}} \right) = \frac{{\sqrt \sigma }}{2}\left( {\frac{{{a_n}}}{{\sqrt \sigma }} + \frac{{\sqrt \sigma }}{{{a_n}}}} \right) \ge \frac{{\sqrt \sigma }}{2}*2 = \sqrt \sigma ,n = 1,2,...$$ 故$\left\{ {{a_n}} \right\}$有下界 + 又因为${a_{n + 1}} = \frac{1}{2}\left( {{a_n} + \frac{\sigma }{{{a_n}}}} \right) = \frac{{{a_n}}}{2}\left( {1 + \frac{\sigma }{{{a_n}^2}}} \right) \le \frac{{{a_n}}}{2}\left( {1 + \frac{\sigma }{\sigma }} \right) = {a_n},n = 1,2,...$ + 所以$\left\{ {{a_n}} \right\}$单调递减. 因此，由单调有界定理知:$\left\{ {{a_n}} \right\}$存在极限，即$\left\{ {{a_n}} \right\}$收敛. 设$\mathop {\lim }\limits_{n \to \infty } {a_n} = A$，由${a_n} > 0$ 知 $A \ge 0$ @@ -35,9 +36,12 @@ 证法1️⃣： 利用不等式$$\frac{1}{{1 + n}} < \ln (1 + \frac{1}{n}) < \frac{1}{n}$$有$${x_{n + 1}} - {x_n} = \frac{1}{{1 + n}} - \ln (n + 1) + \ln {\kern 1pt} n = \frac{1}{{1 + n}} - \ln (1 + \frac{1}{n}) < 0$$ 故数列${x_n}$严格递减. + 又因为 -$\begin{array}{l} {x_n} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \ln \left( {\frac{n}{{n - 1}} \cdot \frac{{n - 1}}{{n - 2}} \cdot ... \cdot \frac{3}{2} \cdot \frac{2}{1}} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \sum\limits_{k = 1}^{n - 1} {\ln \left( {1 + \frac{1}{k}} \right)} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{k = 1}^{n - 1} {\left[ {\frac{1}{k} - \ln \left( {1 + \frac{1}{k}} \right)} \right]} + \frac{1}{n}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} > \frac{1}{n} > 0 \end{array}$ +$$\begin{array}{l} {x_n} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \ln \left( {\frac{n}{{n - 1}} \cdot \frac{{n - 1}}{{n - 2}} \cdot ... \cdot \frac{3}{2} \cdot \frac{2}{1}} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \sum\limits_{k = 1}^{n - 1} {\ln \left( {1 + \frac{1}{k}} \right)} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{k = 1}^{n - 1} {\left[ {\frac{1}{k} - \ln \left( {1 + \frac{1}{k}} \right)} \right]} + \frac{1}{n}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} > \frac{1}{n} > 0 \end{array}$$ + 所以数列${x_n}$有下界. + 因此，数列${x_n}$单调递减有下界，故$\mathop {\lim }\limits_{n \to \infty } {x_n}$存在. @@ -45,5 +49,70 @@ $\begin{array}{l} {x_n} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \ln \left( {\fr 因为$$\left| {{x_n} - {x_{n + 1}}} \right| = \left| {\frac{1}{n} - \left[ {\ln {\kern 1pt} n - \ln (n - 1)} \right]} \right|$$ 对$\ln {\kern 1pt} n - \ln (n - 1)$利用Lagrange中值公式，有 $$\ln {\kern 1pt} n - \ln (n - 1) = \frac{1}{{{\xi _n}}}，其中n - 1 < {\xi _n} < n.$$因此有$\left| {{x_n} - {x_{n - 1}}} \right| = \frac{{n - {\xi _n}}}{{n \cdot {\xi _n}}} < \frac{1}{{{{(n - 1)}^2}}}$ + 而$\sum\limits_{n = 1}^\infty {\frac{1}{{{{(n - 1)}^2}}}}$收敛，故$\sum\limits_{n = 1}^\infty {\left| {{x_n} - {x_{n - 1}}} \right|}$收敛，从而${x_n} = \sum\limits_{k = 1}^n {\left( {{x_k} - {x_{k - 1}}} \right)} + {x_1}$也收敛. -因此，数列$\ {x_n}$收敛，即 $\mathop {\lim }\limits_{n \to \infty } {x_n}$ 存在. \ No newline at end of file + +因此，数列$\ {x_n}$收敛，即 $\mathop {\lim }\limits_{n \to \infty } {x_n}$ 存在. + +>[!example] 例三 +>令$c>0$，定义整序变量数列为: $x_1 = \sqrt{c}, x_2 = \sqrt{c +\sqrt{c}}, x_3 = \sqrt{c+\sqrt{c+\sqrt{c}}}, \cdots$ +>求$\mathop {\lim }\limits_{n \to \infty }{x_n}$ + + $x_n = \underbrace{\sqrt{c+\sqrt{c+\cdots\sqrt{c}}}}_{n\ \text{个根式}}$。 + + $x_{n+1} = \sqrt{c+x_n}$。 + +使用数学归纳法证明，整序变量 $x_n$ 单调增加, 同时有上界 + +**证明 $x_n$ 单调： + +当 $n=1$ 时命题显然成立。 +假设当 $n = k$ 时命题成立，即 $x_{k+1} > x_k$。 +考虑 $n = k+1$ 的情形： +$$ +x_{k+2} = \sqrt{c + x_{k+1}},\quad x_{k+1} = \sqrt{c + x_k} +$$ +由归纳假设 $x_{k+1} > x_k$，可得： +$$ +c + x_{k+1} > c + x_k +$$ +所以： +$$ +\sqrt{c + x_{k+1}} > \sqrt{c + x_k} +$$ +即：$x_{k+2} > x_{k+1}$，故对一切正整数 $n$，都有 $x_{n+1} > x_n$，即 $\{x_n\}$ 是单调增加序列。 + +**证明 $x_n$ 有上界（例如上界为 $\sqrt{c} + 1$） + +当 $n=1$ 时命题成立。 +考虑 $n = k+1$ 的情形，由归纳假设 $x_k < \sqrt{c} + 1$，可得：： +$$ +x_{k+1} = \sqrt{c + x_k}< c + (\sqrt{c} + 1) +$$ + +为了证明 $x_{k+1} < \sqrt{c} + 1$，只需证明： +$$ +\sqrt{c + (\sqrt{c} + 1)} \le \sqrt{c} + 1 +$$ +两边平方（因为两边均为正数）： +$$ +c + \sqrt{c} + 1 \le c + 2\sqrt{c} + 1 +$$ +化简得： +$$ +\sqrt{c} \le 2\sqrt{c} +$$ +由于 $c > 0$，此不等式显然成立，且等号仅在 $\sqrt{c} = 0$ 时成立，这与 $c > 0$ 矛盾，故实际上有严格不等式： +$$ +\sqrt{c + (\sqrt{c} + 1)} < \sqrt{c} + 1 +$$ +因此： +$$ +x_{k+1} < \sqrt{c} + 1 +$$ + +由数学归纳法，对一切正整数 $n$，都有 $x_n < \sqrt{c} + 1$，即 $\{x_n\}$ 有上界。 + +因此序列收敛，有极限，不妨设 $\lim x_n = a$ , 则: $a = \sqrt{a+c}$。 + +解方程得: $a = \dfrac{\sqrt{4c+1}+1}{2}$。（负根舍去） \ No 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zcJ_lfBt@zPnTFW}vSIL`3WmKAlYVl3ACSu#B*UHDl@uFERQ;0mq!GAxzL!x^6Nl1` zJZGKeRUwbi_76Zh-h}(u6^VcZ+N#qwi@kU+g0Io#1WqMz8I&ZOB01%Uex%x(s169$ zQq;u1k3wI^$;nyXPD>%%O-YqQZWqB+j_QgtO2@Ko?s~f8=+??@G6rzZfLNSrPlSwn zj+v6d1QQv=;dmO+4+l6Rv6AKsQ^6><0jL%l!SJ#xBA*eR?ANCh|Hkk4|D+gP_T9?% zSo?7R#%M7}c|5>#ME-~Ov+!V`WH12hcQI+F8tCB+oj)6aLS!I?a`nt@(q6GnnZJ^93f9HCMLI2}v-*;5&h4;g43r|dma3Ur+#Pv++U2c&^(lLB*qn9#)M zG*~-O%5R8+Zam|U!>WYHfQ()PFzb<8bSg@ZV%?=B%LUyYi%?7tLKjsc5(X^;tddNo zV6R@C9&C6_Cf7uoANl0_G{o= z<2x!HQ)p#y>S`rTBF*i*C3U=nLKvzx3X3wrq-6ae6_rNxZo|~^0CG|V?%69kFzmdZ z9y literal 0 HcmV?d00001 diff --git a/编写小组/讲义/子数列问题&考试易错点汇总（解析版）.md b/编写小组/讲义/子数列问题&考试易错点汇总（解析版）.md new file mode 100644 index 0000000..f21139d --- /dev/null +++ b/编写小组/讲义/子数列问题&考试易错点汇总（解析版）.md @@ -0,0 +1,39 @@ +--- +tags: + - 编写小组 +--- + +## Vol. 9: 反函数求导 +易错：变量混淆。反函数的导数 = 原函数导数的倒数，但**自变量和因变量角色互换**。 +这一点，大家都明白，但是一写在答题卡上就错了😂。 +如何防止？我们可以试着用微分学来理解： +$y=f^{-1}(x)$，即$x = f(y)$，求$f^{-1'}$就是在求$\frac{dy}{dx}$，而$\frac{dy}{dx}=\frac{dy}{df(y)}=\frac{dy}{f'(y)dy}=\frac{1}{f'(y)}$ +然后，就这么完了？如果就这么完了，那就真完了。 +为什么？$f^{-1'}$最后应该是一个关于$x$的函数，$\frac{dy}{dx}$应当用$x$来表示。 +所以，我们还需要将$y = f^{-1}(x)$代入，即： +$\frac{dy}{dx}=\frac{1}{f'(y)}=\frac{1}{f'(f^{-1}(x))}$ +> [!example] 示例1 +> 求$d(\arcsin x)$ + +解：设$y=\arcsin x$，即$x=\sin y$，$dx=\cos y\ dy$，即$dy=\frac{dx}{\cos y}$ +作辅助三角形 +![[易错点9-1.png]] +得$\cos y=\sqrt{1-x^2}$，综上，$dy=\frac{dx}{\sqrt{1-x^2}}$，即$d(\arcsin x)=\frac{dx}{\sqrt{1-x^2}}$ +## Vol. 10: 无界与无穷大的辨析 +很多人都觉得无界和无穷大是同一个概念，因为它们的实在是太像了：画在坐标系上都是“直指苍穹🚀”或者“飞流直下三千尺”嘛！但是，“无界”准确来说不完全是这样。要准确辨析它们，需要回到它们的**定义**上： +无穷大的定义：$\forall M > 0, \exists \delta>0$，当$0<|x-x_0|<\delta$时有$|f(x)|>M$，称$f(x)$是当$x\rightarrow x_0$时的无穷大量 +无界量的定义：由有界的定义($\exists M > 0, \forall x \in D_f,|f(x)| 0$, +$\forall \delta>0, \exists x \in \mathring{U}(x_0,\delta)$有 $|f(x)|>M$，称$f(x)$是当$x\rightarrow x_0$时的无界量 +**核心区别**：无穷大是**存在某**去心邻域内**任意**$x$都大于$M$，无界是需要对**任意**邻域**存在**一个$x$使得$|f(x)|>M$ +**联系**：无穷大一定是无界量，但是无界量不一定是无穷大。 +> [!example] 示例1 +> 无穷震荡$\lim\limits_{x\rightarrow 0}{\frac{1}{x}\sin\frac{1}{x}}$ + +![[易错点10-1.png]] +这个并不是无穷大——不管取的邻域有多小，我总能找到一个令$\sin\frac{1}{x}=0$的$x$，此时$\frac{1}{x}\sin\frac{1}{x}=0$。 +那这个是有界的吗？也不是。这就是典型的**不是无界量的无穷大**。 + +> [!example] 示例2 +> 双子数列$\lim\limits_{n\rightarrow \infty}{n\cos n\pi}$ + +总结：无穷大是在邻域内“一直都很大”，无界是邻域内“有很大的” \ No newline at end of file diff --git a/编写小组/讲义/级数敛散性的基本思路和方法.md b/编写小组/讲义/级数敛散性的基本思路和方法.md index 99aef78..5008f88 100644 --- a/编写小组/讲义/级数敛散性的基本思路和方法.md +++ b/编写小组/讲义/级数敛散性的基本思路和方法.md @@ -273,8 +273,9 @@ $$ --- **总结：** + | $p$ 的范围 | 敛散性 | 关键原因 | -|------------|--------|----------| +| ------------ | -------- | ---------- | | $p > 1$ | 收敛 | 分组求和，几何级数控制 | | $0 < p \le 1$ | 发散 | 与调和级数比较 | | $p \le 0$ | 发散 | 通项不趋于零 | diff --git a/编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md b/编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md index 64d6bdd..2ea1587 100644 --- a/编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md +++ b/编写小组/讲义/级数敛散性的基本思路和方法（解析版）.md @@ -270,6 +270,7 @@ $$ --- **总结：** + | $p$ 的范围 | 敛散性 | 关键原因 | |------------|--------|----------| | $p > 1$ | 收敛 | 分组求和，几何级数控制 | diff --git a/编写小组/讲义/证明题方法：单调有界定理，介值定理.md b/编写小组/讲义/证明题方法：单调有界定理，介值定理.md index 1fb86d3..98d80bd 100644 --- a/编写小组/讲义/证明题方法：单调有界定理，介值定理.md +++ b/编写小组/讲义/证明题方法：单调有界定理，介值定理.md @@ -2,3 +2,218 @@ tags: - 编写小组 --- +**内部资料，禁止传播** +**编委会（不分先后，姓氏首字母顺序）：程奕铭 韩魏 刘柯妤 卢吉辚 王轲楠 支宝宁 郑哲航 + +# 单调有界准则 + +## 原理 + +- **单调有界数列必收敛**： + 1. 若数列 $\{a_n\}$ 单调递增且有上界 $M$，则 $\{a_n\}$ 收敛，且 $\lim_{n \to \infty} a_n = \sup\{a_n\} \leq M$。 + 2. 若数列 $\{a_n\}$ 单调递减且有下界 $m$，则 $\{a_n\}$ 收敛，且 $\lim_{n \to \infty} a_n = \inf\{a_n\} \geq m$。 + +- **函数单调有界性质**： + 若函数 $f(x)$ 在区间 $I$ 上单调且有界，则 $f(x)$ 在 $I$ 的端点或无穷远处存在单侧极限。 + +## 适用情况 + +适用于证明数列或函数极限的存在性，尤其是： + +- 递推数列极限的存在性与求解； +- 函数在某区间上的极限存在性； +- 某些无法直接求极限但可判断单调性和有界性的情形。 + +## 优势 + +1. 不依赖于极限的具体值，只需判断单调性和有界性； +2. 适用于很多递推定义的数列； +3. 证明过程结构清晰，易于理解和操作。 + +## 劣势 + +1. 只能证明极限存在，不能直接给出极限值（需另解方程）； +2. 判断单调性和有界性有时需要一定的技巧； +3. 对于非单调或没有明显界的序列不适用。 + +## 例子 + +> [!example] 例1 +> 证明数列 $a_1 = \sqrt{2}, a_{n+1} = \sqrt{2 + a_n}$ 收敛，并求其极限。 + +``` + + + + + + + +``` + +> [!example] 例2 +> 证明函数 $f(x) = \frac{x}{x+1}$ 在 $[0, +\infty)$ 上单调有界，并求 $\lim_{x \to +\infty} f(x)$。 + +``` + + + + + + + +``` + +>[!example] **例3** +>设数列 $\{x_n\}$ 满足 $x_1 = \sqrt{2}$，且对任意 $n \ge 1$，有 $x_{n+1} = \sqrt{2 + x_n}$。证明该数列收敛，并求其极限。 + +``` + + + + + + + +``` + +>[!example] 例4 +>设$a > 0,\sigma > 0,{a_1} = \frac{1}{2}\left( {a + \frac{\sigma }{a}} \right),{a_{n + 1}} = \frac{1}{2}\left( {{a_n} + \frac{\sigma }{{{a_n}}}} \right),n = 1,2,...$ +> 证明：数列$\left\{ {{a_n}} \right\}$收敛，且极限为$\sqrt \sigma$. + +``` + + + + + + + +``` + +>[!example] 例5 +>证明：数列${x_n} = 1 + \frac{1}{2} + ... + \frac{1}{n} - \ln {\kern 1pt} n{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (n = 1,2,...)\\$单调递减有界，从而有极限（此极限称为Euler常数，下记作$C$）。 + +``` + + + + + + + +``` + +>[!example] 例6 +>令$c>0$，定义整序变量数列为: $x_1 = \sqrt{c}, x_2 = \sqrt{c +\sqrt{c}}, x_3 = \sqrt{c+\sqrt{c+\sqrt{c}}}, \cdots$ +>求$\mathop {\lim }\limits_{n \to \infty }{x_n}$ + + ``` + + + + + + + +``` + +# 介值定理 +## 原理 + +- **介值定理（Intermediate Value Theorem, IVT）**： + 若函数 $f(x)$ 在闭区间 $[a,b]$ 上连续，且 $f(a) \neq f(b)$，则对于任意介于 $f(a)$ 和 $f(b)$ 之间的实数 $C$，至少存在一点 $\xi \in (a,b)$，使得 $f(\xi) = C$。 + +- **零点定理（特殊情形）**： + 若 $f(x)$ 在 $[a,b]$ 上连续，且 $f(a) \cdot f(b) < 0$，则至少存在一点 $\xi \in (a,b)$，使得 $f(\xi) = 0$。 + +## 适用情况 + +适用于连续函数在区间上的值域问题，尤其是： + +- 证明方程在某区间内至少有一个根； +- 证明函数可取到某个中间值； +- 确定函数值域或解的存在性。 + +## 优势 + +1. 不需求解方程，只需验证连续性和端点函数值； +2. 可用于证明解的存在性，构造性证明中常用； +3. 定理直观，易于理解和应用。 + +## 劣势 + +1. 只能证明存在性，不能确定解的个数或具体位置； +2. 要求函数在闭区间上连续，否则定理不适用； +3. 对于非连续函数或开区间，需额外处理。 + +## 例子 + +> [!example] 例1 +> 证明方程 $x^3 - 3x + 1 = 0$ 在区间 $(0,1)$ 内至少有一个实根。 + +``` + + + + + + + +``` + +> [!example] 例2 +> 设函数 $f(x)$ 在 $[0,1]$ 上连续，且 $0 \leq f(x) \leq 1$，证明至少存在一点 $c \in [0,1]$，使得 $f(c) = c$。 + +``` + + + + + + + +``` + +> [!example] **例3** +> 设$f(x)$在$[a,b]$上连续，且$a < c < d < b$。证明：在$(a,b)$内必存在点$\xi$，使得$mf(c)+nf(d)=(m+n)f(\xi)$，其中$m,n$为任意给定的自然数。 + +``` + + + + + + + +``` + +> [!example] **例4** +> 一支巡逻小分队定期到山上的岗哨换岗，每天上午7点从营地出发，8点到达目的地。第二天上午7点沿原路返回，8点前返回到达山下营地。利用零点定理证明，在换岗的路途中必有一点，小分队在两天中的同一时刻经过该点。 + +``` + + + + + + + +``` + +> [!example] 例5 +> 设$f(x)$在$(-\infty,+\infty)$上连续，且$\lim\limits_{x \to \infty}\frac{f(x)}{x}=0$，试证存在$\xi\in(-\infty,+\infty)$，使$f(\xi)+\xi=0$ + +``` + + + + + + + +``` + +>[!example] 例6 +>设$f(x)$在$[0,1]$上连续，且$f(0)=f(1)$，试证：对任何自然数$n$，必存在一点$c\in[0,1-\frac{1}{n}]$，使$f(c)=f(c+\frac{1}{n})$. + diff --git a/编写小组/讲义/证明题方法：单调有界定理，介值定理（解析版）.md b/编写小组/讲义/证明题方法：单调有界定理，介值定理（解析版）.md new file mode 100644 index 0000000..042b99f --- /dev/null +++ b/编写小组/讲义/证明题方法：单调有界定理，介值定理（解析版）.md @@ -0,0 +1,314 @@ +--- +tags: + - 编写小组 +--- +**内部资料，禁止传播** +**编委会（不分先后，姓氏首字母顺序）：程奕铭 韩魏 刘柯妤 卢吉辚 王轲楠 支宝宁 郑哲航 +# 单调有界准则 + +## 原理 + +- **单调有界数列必收敛**： + 1. 若数列 $\{a_n\}$ 单调递增且有上界 $M$，则 $\{a_n\}$ 收敛，且 $\lim_{n \to \infty} a_n = \sup\{a_n\} \leq M$。 + 2. 若数列 $\{a_n\}$ 单调递减且有下界 $m$，则 $\{a_n\}$ 收敛，且 $\lim_{n \to \infty} a_n = \inf\{a_n\} \geq m$。 + +- **函数单调有界性质**： + 若函数 $f(x)$ 在区间 $I$ 上单调且有界，则 $f(x)$ 在 $I$ 的端点或无穷远处存在单侧极限。 + +## 适用情况 + +适用于证明数列或函数极限的存在性，尤其是： + +- 递推数列极限的存在性与求解； +- 函数在某区间上的极限存在性； +- 某些无法直接求极限但可判断单调性和有界性的情形。 + +## 优势 + +1. 不依赖于极限的具体值，只需判断单调性和有界性； +2. 适用于很多递推定义的数列； +3. 证明过程结构清晰，易于理解和操作。 + +## 劣势 + +1. 只能证明极限存在，不能直接给出极限值（需另解方程）； +2. 判断单调性和有界性有时需要一定的技巧； +3. 对于非单调或没有明显界的序列不适用。 + +## 例子 + +> [!example] 例1 +> 证明数列 $a_1 = \sqrt{2}, a_{n+1} = \sqrt{2 + a_n}$ 收敛，并求其极限。 + +**解析** +1. **有界性**： + 易证 $0 < a_n < 2$（可用归纳法）。 +2. **单调性**： + 计算 $a_{n+1} - a_n = \sqrt{2 + a_n} - a_n$。 + 设 $f(x) = \sqrt{2 + x} - x$，在 $[0,2]$ 上分析符号，可得 $a_{n+1} \geq a_n$，故数列单调递增。 +3. **由单调有界准则**，数列收敛。 +4. **求极限**： + 设 $\lim_{n \to \infty} a_n = L$，则 $L = \sqrt{2 + L}$，解得 $L = 2$（舍去负根）。 + + +> [!example] 例2 +> 证明函数 $f(x) = \frac{x}{x+1}$ 在 $[0, +\infty)$ 上单调有界，并求 $\lim_{x \to +\infty} f(x)$。 + +**解析** +1. **单调性**： + 导数 $f'(x) = \frac{1}{(x+1)^2} > 0$，故 $f(x)$ 单调递增。 +2. **有界性**： + 显然 $0 \leq f(x) < 1$。 +3. **由单调有界准则**，$\lim_{x \to +\infty} f(x)$ 存在，且为 $1$。 + + +>[!example] **例3** +>设数列 $\{x_n\}$ 满足 $x_1 = \sqrt{2}$，且对任意 $n \ge 1$，有 $x_{n+1} = \sqrt{2 + x_n}$。证明该数列收敛，并求其极限。 + +**证明**： + +1. **证明数列单调有界**： + + 当 $n=1$ 时， $0 < x_1 < 2$。 + + 假设当 $n=k$（$k \ge 1$）时，命题成立，即有： + $$0 < x_k < 2 \tag{1}$$ + 则当 $n=k+1$ 时， + $$x_{k+1} = \sqrt{2 + x_{k}} < \sqrt{2 + 2} = 2$$ + 同时，由 $x_{k} > 0$ 可得 $x_{k+1} = \sqrt{2 + x_{k}} > \sqrt{2} > 0$。故 $0 < x_{k+1} < 2$。 + + 因此$0 < x_{k+2} < 2$。 + + 接下来，证明单调性部分 $x_{k+1} < x_{k+2}$。根据递推关系和归纳假设(1)中 $x_k < x_{k+1}$，我们有： + $$ + \begin{aligned} + x_{k+2} - x_{k+1} &= \sqrt{2 + x_{k+1}} - \sqrt{2 + x_k} \quad \\[0.5em] + &= \frac{(\sqrt{2 + x_{k+1}} - \sqrt{2 + x_k})(\sqrt{2 + x_{k+1}} + \sqrt{2 + x_k})}{\sqrt{2 + x_{k+1}} + \sqrt{2 + x_k}} \quad \\[0.5em] + &= \frac{(2 + x_{k+1}) - (2 + x_k)}{\sqrt{2 + x_{k+1}} + \sqrt{2 + x_k}} \\[0.5em] + &= \frac{x_{k+1} - x_k}{\sqrt{2 + x_{k+1}} + \sqrt{2 + x_k}} + \end{aligned} + $$ + 由归纳假设(1)知 $x_{k+1} - x_k > 0$，且分母 $\sqrt{2 + x_{k+1}} + \sqrt{2 + x_k} > 0$。因此， + $$x_{k+2} - x_{k+1} = \frac{x_{k+1} - x_k}{\sqrt{2 + x_{k+1}} + \sqrt{2 + x_k}} > 0$$ + 即 $x_{k+1} < x_{k+2}$ 成立。 + + 结合以上，我们证明了 $0 < x_{k+1} < x_{k+2} < 2$。由数学归纳法，对一切正整数 $n$，均有 $0 < x_n < x_{n+1} < 2$。 + + 因此，数列 $\{x_n\}$ **单调递增**且有**上界** $2$。 + +2. **求数列的极限**： + 由于数列 $\{x_n\}$ 单调递增且有上界，根据**单调有界收敛原理**，该数列收敛。设其极限为 $L$，即 $\displaystyle \lim_{n \to \infty} x_n = L$。 + + 在递推式 $x_{n+1} = \sqrt{2 + x_n}$ 两边同时取极限（$n \to \infty$），得：$$L = \sqrt{2 + L}$$ + 将方程两边平方：$L^2 = 2 + L$， + + 移项整理得：$L^2 - L - 2 = 0$， + + 因式分解：$(L - 2)(L + 1) = 0$。 + + 解得：$L = 2$ 或 $L = -1$。 + + 由于数列的所有项 $x_n > 0$，其极限 $L$ 必须满足 $L \ge 0$，故舍去 $L = -1$。 + + 因此，数列 $\{x_n\}$ 的极限为：$$L = 2$$ + +>[!example] 例4 +>设$a > 0,\sigma > 0,{a_1} = \frac{1}{2}\left( {a + \frac{\sigma }{a}} \right),{a_{n + 1}} = \frac{1}{2}\left( {{a_n} + \frac{\sigma }{{{a_n}}}} \right),n = 1,2,...$ +> 证明：数列$\left\{ {{a_n}} \right\}$收敛，且极限为$\sqrt \sigma$. + +证：对 $\forall t > 0,t + \frac{1}{t} \ge 2$，令 $t = \frac{{{a_n}}}{{\sqrt \sigma }} > 0$，则有$${a_{n + 1}} = \frac{1}{2}\left( {{a_n} + \frac{\sigma }{{{a_n}}}} \right) = \frac{{\sqrt \sigma }}{2}\left( {\frac{{{a_n}}}{{\sqrt \sigma }} + \frac{{\sqrt \sigma }}{{{a_n}}}} \right) \ge \frac{{\sqrt \sigma }}{2}*2 = \sqrt \sigma ,n = 1,2,...$$ +故$\left\{ {{a_n}} \right\}$有下界 + +又因为${a_{n + 1}} = \frac{1}{2}\left( {{a_n} + \frac{\sigma }{{{a_n}}}} \right) = \frac{{{a_n}}}{2}\left( {1 + \frac{\sigma }{{{a_n}^2}}} \right) \le \frac{{{a_n}}}{2}\left( {1 + \frac{\sigma }{\sigma }} \right) = {a_n},n = 1,2,...$ + +所以$\left\{ {{a_n}} \right\}$单调递减. +因此，由单调有界定理知:$\left\{ {{a_n}} \right\}$存在极限，即$\left\{ {{a_n}} \right\}$收敛. + +设$\mathop {\lim }\limits_{n \to \infty } {a_n} = A$，由${a_n} > 0$ 知 $A \ge 0$ + +由${a_{n + 1}} = \frac{1}{2}\left( {{a_n} + \frac{\sigma }{{{a_n}}}} \right)$ 知 $2{a_{n + 1}}{a_n} = {a_n}^2 + \sigma$. + +两边令 $n \to \infty$取极限得: $2{A^2} = {A^2} + \sigma$ + +解得$A = - \sqrt \sigma$（舍）或$A = \sqrt \sigma$. + +故$\mathop {\lim }\limits_{n \to \infty } {a_n} = \sqrt \sigma$. + + +>[!example] 例5 +>证明：数列${x_n} = 1 + \frac{1}{2} + ... + \frac{1}{n} - \ln {\kern 1pt} n{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (n = 1,2,...)\\$单调递减有界，从而有极限（此极限称为Euler常数，下记作$C$）。 + + +证法1️⃣： +利用不等式$$\frac{1}{{1 + n}} < \ln (1 + \frac{1}{n}) < \frac{1}{n}$$有$${x_{n + 1}} - {x_n} = \frac{1}{{1 + n}} - \ln (n + 1) + \ln {\kern 1pt} n = \frac{1}{{1 + n}} - \ln (1 + \frac{1}{n}) < 0$$ +故数列${x_n}$严格递减. + +又因为 +$$\begin{array}{l} {x_n} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \ln \left( {\frac{n}{{n - 1}} \cdot \frac{{n - 1}}{{n - 2}} \cdot ... \cdot \frac{3}{2} \cdot \frac{2}{1}} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \sum\limits_{k = 1}^{n - 1} {\ln \left( {1 + \frac{1}{k}} \right)} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_{k = 1}^{n - 1} {\left[ {\frac{1}{k} - \ln \left( {1 + \frac{1}{k}} \right)} \right]} + \frac{1}{n}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} > \frac{1}{n} > 0 \end{array}$$ + +所以数列${x_n}$有下界. + +因此，数列${x_n}$单调递减有下界，故$\mathop {\lim }\limits_{n \to \infty } {x_n}$存在. + + +证法2️⃣： +因为$$\left| {{x_n} - {x_{n + 1}}} \right| = \left| {\frac{1}{n} - \left[ {\ln {\kern 1pt} n - \ln (n - 1)} \right]} \right|$$ +对$\ln {\kern 1pt} n - \ln (n - 1)$利用Lagrange中值公式，有 +$$\ln {\kern 1pt} n - \ln (n - 1) = \frac{1}{{{\xi _n}}}，其中n - 1 < {\xi _n} < n.$$ + +因此有$\left| {{x_n} - {x_{n - 1}}} \right| = \frac{{n - {\xi _n}}}{{n \cdot {\xi _n}}} < \frac{1}{{{{(n - 1)}^2}}}$ + +而$\sum\limits_{n = 1}^\infty {\frac{1}{{{{(n - 1)}^2}}}}$收敛，故$\sum\limits_{n = 1}^\infty {\left| {{x_n} - {x_{n - 1}}} \right|}$收敛，从而${x_n} = \sum\limits_{k = 1}^n {\left( {{x_k} - {x_{k - 1}}} \right)} + {x_1}$也收敛. + +因此，数列$\ {x_n}$收敛，即 $\mathop {\lim }\limits_{n \to \infty } {x_n}$ 存在. + + +>[!example] 例6 +>令$c>0$，定义整序变量数列为: $x_1 = \sqrt{c}, x_2 = \sqrt{c +\sqrt{c}}, x_3 = \sqrt{c+\sqrt{c+\sqrt{c}}}, \cdots$ +>求$\mathop {\lim }\limits_{n \to \infty }{x_n}$ + + $x_n = \underbrace{\sqrt{c+\sqrt{c+\cdots\sqrt{c}}}}_{n\ \text{个根式}}$。 + + $x_{n+1} = \sqrt{c+x_n}$。 + +使用数学归纳法证明，整序变量 $x_n$ 单调增加, 同时有上界 + +**证明 $x_n$ 单调： + +当 $n=1$ 时命题显然成立。 +假设当 $n = k$ 时命题成立，即 $x_{k+1} > x_k$。 +考虑 $n = k+1$ 的情形： +$$ +x_{k+2} = \sqrt{c + x_{k+1}},\quad x_{k+1} = \sqrt{c + x_k} +$$ +由归纳假设 $x_{k+1} > x_k$，可得： +$$ +c + x_{k+1} > c + x_k +$$ +所以： +$$ +\sqrt{c + x_{k+1}} > \sqrt{c + x_k} +$$ +即：$x_{k+2} > x_{k+1}$，故对一切正整数 $n$，都有 $x_{n+1} > x_n$，即 $\{x_n\}$ 是单调增加序列。 + +**证明 $x_n$ 有上界（例如上界为 $\sqrt{c} + 1$） + +当 $n=1$ 时命题成立。 +考虑 $n = k+1$ 的情形，由归纳假设 $x_k < \sqrt{c} + 1$，可得：： +$$ +x_{k+1} = \sqrt{c + x_k}< c + (\sqrt{c} + 1) +$$ + +为了证明 $x_{k+1} < \sqrt{c} + 1$，只需证明： +$$ +\sqrt{c + (\sqrt{c} + 1)} \le \sqrt{c} + 1 +$$ +两边平方（因为两边均为正数）： +$$ +c + \sqrt{c} + 1 \le c + 2\sqrt{c} + 1 +$$ +化简得： +$$ +\sqrt{c} \le 2\sqrt{c} +$$ +由于 $c > 0$，此不等式显然成立，且等号仅在 $\sqrt{c} = 0$ 时成立，这与 $c > 0$ 矛盾，故实际上有严格不等式： +$$ +\sqrt{c + (\sqrt{c} + 1)} < \sqrt{c} + 1 +$$ +因此： +$$ +x_{k+1} < \sqrt{c} + 1 +$$ + +由数学归纳法，对一切正整数 $n$，都有 $x_n < \sqrt{c} + 1$，即 $\{x_n\}$ 有上界。 + +因此序列收敛，有极限，不妨设 $\lim x_n = a$ , 则: $a = \sqrt{a+c}$。 + +解方程得: $a = \dfrac{\sqrt{4c+1}+1}{2}$。（负根舍去） + + +# 介值定理 +## 原理 + +- **介值定理（Intermediate Value Theorem, IVT）**： + 若函数 $f(x)$ 在闭区间 $[a,b]$ 上连续，且 $f(a) \neq f(b)$，则对于任意介于 $f(a)$ 和 $f(b)$ 之间的实数 $C$，至少存在一点 $\xi \in (a,b)$，使得 $f(\xi) = C$。 + +- **零点定理（特殊情形）**： + 若 $f(x)$ 在 $[a,b]$ 上连续，且 $f(a) \cdot f(b) < 0$，则至少存在一点 $\xi \in (a,b)$，使得 $f(\xi) = 0$。 + +## 适用情况 + +适用于连续函数在区间上的值域问题，尤其是： + +- 证明方程在某区间内至少有一个根； +- 证明函数可取到某个中间值； +- 确定函数值域或解的存在性。 + +## 优势 + +1. 不需求解方程，只需验证连续性和端点函数值； +2. 可用于证明解的存在性，构造性证明中常用； +3. 定理直观，易于理解和应用。 + +## 劣势 + +1. 只能证明存在性，不能确定解的个数或具体位置； +2. 要求函数在闭区间上连续，否则定理不适用； +3. 对于非连续函数或开区间，需额外处理。 + +## 例子 + +> [!example] 例1 +> 证明方程 $x^3 - 3x + 1 = 0$ 在区间 $(0,1)$ 内至少有一个实根。 + +**解析** +设 $f(x) = x^3 - 3x + 1$，则 $f(x)$ 在 $[0,1]$ 上连续。 +计算：$f(0) = 1 > 0$，$f(1) = -1 < 0$。 +由零点定理，至少存在一点 $\xi \in (0,1)$，使得 $f(\xi)=0$，即方程在 $(0,1)$ 内至少有一个实根。 + + +> [!example] 例2 +> 设函数 $f(x)$ 在 $[0,1]$ 上连续，且 $0 \leq f(x) \leq 1$，证明至少存在一点 $c \in [0,1]$，使得 $f(c) = c$。 + +**解析** +构造辅助函数 $g(x) = f(x) - x$，则 $g(x)$ 在 $[0,1]$ 上连续。 +计算：$g(0) = f(0) - 0 \geq 0$，$g(1) = f(1) - 1 \leq 0$。 +若 $g(0)=0$ 或 $g(1)=0$，则取 $c=0$ 或 $c=1$ 即可。 +否则 $g(0)>0$，$g(1)<0$，由零点定理，存在 $c \in (0,1)$ 使得 $g(c)=0$，即 $f(c)=c$。 + +> [!example] **例3** +> 设$f(x)$在$[a,b]$上连续，且$a < c < d < b$。证明：在$(a,b)$内必存在点$\xi$，使得$mf(c)+nf(d)=(m+n)f(\xi)$，其中$m,n$为任意给定的自然数。 + +**解析：** +因$f(x)$在$[a,b]$上连续，所以$f(x)$在$[a,b]$上能取得最大值$f_{\max}$和最小值$f_{\min}$。又因为 $(m+n)fmin​⩽mf(c)+nf(d)⩽(m+n)fmax​$, 即$$f_{\min} \leqslant \frac{mf(c) + nf(d)}{m + n} \leqslant f_{\max}$$故由介值定理，在$[a,b]$上必存在一点$\xi$，使得 $m+nmf(c)+nf(d)​=f(ξ)$, 即$mf(c) + nf(d) = (m + n)f(\xi)$。 + + +> [!example] **例4** +> 一支巡逻小分队定期到山上的岗哨换岗，每天上午7点从营地出发，8点到达目的地。第二天上午7点沿原路返回，8点前返回到达山下营地。利用零点定理证明，在换岗的路途中必有一点，小分队在两天中的同一时刻经过该点。 + +**解析** +设上山的路程函数为$s=f_1(t)$（$7\leqslant t\leqslant8$），则$f_1(7)=0$，$f_1(8)=D$（$D$为两岗哨之间的距离） 下山的路程函数为$s=f_2(t)$（$7\leqslant t\leqslant8$），则$f_2(7)=D$，$f_2(8)=0$ 作辅助函数$F(t)=f_1(t)-f_2(t)$，则$F(t)$在$[7,8]$上连续，且$F(7)=-D$，$F(8)=D$ 由零点定理知，至少存在$\xi\in(7,8)$使$F(\xi)=0$ 即在换岗的路途中必有一点，小分队在两天中的同一时刻经过该点。 + + +> [!example] 例5 +> 设$f(x)$在$(-\infty,+\infty)$上连续，且$\lim\limits_{x \to \infty}\frac{f(x)}{x}=0$，试证存在$\xi\in(-\infty,+\infty)$，使$f(\xi)+\xi=0$ + +**解析** +步骤1：构造辅助函数并分析连续性 设辅助函数$F(x)=f(x)+x$。 已知$f(x)$在$(-\infty,+\infty)$上连续，$y=x$在$(-\infty,+\infty)$上也连续，根据连续函数的和运算性质，$F(x)$在$(-\infty,+\infty)$上连续。 + +步骤2：利用极限条件分析$F(x)$在无穷远处的符号 $$\lim\limits_{x \to \infty}\frac{f(x)}{x}=0，可得： +$$$$\lim\limits_{x \to +\infty}\frac{F(x)}{x}=\lim\limits_{x \to +\infty}\left(\frac{f(x)}{x}+1\right)=0+1=1>0$$$$\lim\limits_{x \to -\infty}\frac{F(x)}{x}=\lim\limits_{x \to -\infty}\left(\frac{f(x)}{x}+1\right)=0+1=1>0$$ 根据函数极限的保号性： + +- 存在$X_1>0$，当$x>X_1$时，$\frac{F(x)}{x}>0$，又$x>0$，故$F(x)>0$。取$x_1=X_1+1$，则$F(x_1)>0$。 +- 存在$X_2>0$，当$x<-X_2$时，$\frac{F(x)}{x}>0$，又$x<0$，故$F(x)<0$。取$x_2=-X_2-1$，则$F(x_2)<0$ + +步骤3：应用零点存在定理得出结论 $F(x)$在闭区间$[x_2,x_1]$上连续，且$F(x_2)<0$，$F(x_1)>0$。 由零点存在定理，至少存在一点$\xi\in(x_2,x_1)\subset(-\infty,+\infty)$，使得$F(\xi)=0$，即$f(\xi)+\xi=0$ + + +>[!example] 例6 +>设$f(x)$在$[0,1]$上连续，且$f(0)=f(1)$，试证：对任何自然数$n$，必存在一点$c\in[0,1-\frac{1}{n}]$，使$f(c)=f(c+\frac{1}{n})$. + +**分析** 根据题意，我们需要证明连续函数$F(x)=f(x)-f(x+\frac{1}{n})$有零点，关键是找两点$a、b$，使得$F(a)F(b)<0$.一般会找区间的两个端点，但问题是我们无法确定$F(0)\cdot F(1-\frac{1}{n})$是否小于$0$。正向难以解决，不妨尝试一下反证法. +**证法一** 反证法.若不然，则存在自然数$m$，使得$\forall x\in[0,1-\frac{1}{m}]$，均有$f(x)\neq f(x+\frac{1}{m})$，即$f(x)-f(x+\frac{1}{m})\neq0$，不妨设$f(x)>f(x+\frac{1}{m})$.依次取$x=0,\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}{m}$,则有$$f(0)>f(\frac{1}{m})>\cdots>f(\frac{m}{m})=f(1),$$这与$f(0)=f(1)$矛盾.故结论成立. + +**证法二** 记$$F(x)=f(x)-f(x+\frac{1}{n}),x\in[0,1-\frac{1}{n}],$$则$F(x)$在$[0,1-\frac{1}{n}]$上连续，且$$F(0)=f(0)-f(\frac{1}{n}),F(\frac{1}{n})=f(\frac{1}{n})-f(\frac{2}{n}),\cdots,F(1-\frac{1}{n})=f(1-\frac{1}{n})-f(1).$$上面各式相加得$$F(0)+F(\frac{1}{n})+\cdots+F(1-\frac{1}{n})=f(0)-f(1)=0.$$若$F(\frac{k}{n})(k=0,1,\cdots,n-1)$中至少有一个为$0$，则结论已成立.否则，必$\exists k_1,k_2\in\{0,1,2,\cdots,n-1\}$,使得$F(k_1)F(k_2)<0$，由介值定理（或零值定理），必有一点$c$介于$\frac{k_1}{n}$和$\frac{k_2}{n}$之间，使得$F(c)=0$即$$f(c)=f(x+\frac{1}{n}).$$ diff --git a/编写小组/讲义/隐函数，参数方程和应用题（解析版）.md b/编写小组/讲义/隐函数，参数方程和应用题（解析版）.md index bc4e4a1..8bb59a6 100644 --- a/编写小组/讲义/隐函数，参数方程和应用题（解析版）.md +++ b/编写小组/讲义/隐函数，参数方程和应用题（解析版）.md @@ -488,7 +488,7 @@ $$\left. \frac{dl}{dt} \right|_{x=1} = \frac{2\cdot 1 + 1}{2\sqrt{1^2 + 1}} \cdo 解：主要的等量关系是：球浸入水中的体积与水“溢出”的体积相同（如果把水面上升视为溢出的话）。设时间为$t$，球下沉的体积为$V$，水上升的高度为$H$，球浸入水的高度为$h$，则有$$V=\pi h^2(a-\frac{h}{3}),dV=(2\pi ah-\pi h^2)dh=\pi h(2a-h)dh$$（这个公式大家可以记住） 且（如图）$$dV=\pi(b^2-a^2+(a-h)^2)dH$$ ![[球浸入水示意图.jpg]] -于是$$\pi(b^2-a^2+(a-h)^2)dH=\pi h(2a-h)dh$$$$dH=\frac{h(2a-h)}{b^2-2ah+h^2}dh$$$$\frac{dH}{dt}=\frac{h(2a-h))}{b^2-2ah+h^2}\frac{dh}{dt}$$而$\frac{dh}{dt}=c,h=a$，故$$\frac{dH}{dt}=\frac{a^2c}{b^2-a^2}$$ +于是$$\pi(b^2-a^2+(a-h)^2)dh=\pi h(2a-h)dH$$$$dH=\frac{h(2a-h)}{b^2-2ah+h^2}dh$$$$\frac{dH}{dt}=\frac{h(2a-h))}{b^2-2ah+h^2}\frac{dh}{dt}$$而$\frac{dh}{dt}=c,h=a$，故$$\frac{dH}{dt}=\frac{a^2c}{b^2-a^2}$$ ### 应用题期中真题 diff --git a/编写小组/课前测/课前测3.md b/编写小组/课前测/课前测3.md index 1fb86d3..b2e8aa6 100644 --- a/编写小组/课前测/课前测3.md +++ b/编写小组/课前测/课前测3.md @@ -2,3 +2,33 @@ tags: - 编写小组 --- +1、设函数$f(x)\in C[0,2]$，且满足：$$f(0)=-1,f(1)=3,f(2)=1.$$下列说法正确的是： +（A）方程$f(x)=0$在$[0,1]$上没有解 +（B）方程 f(x)=1在区间 $[0,2]$ 上只有$x=2$ 这一个解 +（C）方程 $f(x)=2$ 在 $[1,2]$ 上至少有一个解 +（D）方程 $f(x)=−2$ 在 $[0,2]$ 上没有解 + + +2、证明方程 $e^x = 3x$ 至少有一个正实根。 + +``` + + + + + + + +``` + +3、设数列 $\{x_n\}$ 满足 $x_1 = 1, x_{n+1} = 1 + \frac{1}{1 + x_n}$，证明 $\{x_n\}$ 收敛，并求极限。 + +``` + + + + + + + +``` \ No newline at end of file diff --git a/编写小组/课前测/课前测4.md b/编写小组/课前测/课前测4.md new file mode 100644 index 0000000..d748c0c --- /dev/null +++ b/编写小组/课前测/课前测4.md @@ -0,0 +1,11 @@ +1. 设 $x_n = \frac{\cos\left(\frac{2n\pi}{3}\right)}{n} + 1$，证明 $x_n \to 1$。 + + + + + +2. 函数 $f(x) = \begin{cases} x^2, & \text{若 } x \text{ 为有理数} \\ -x^2, & \text{若 } x \text{ 为无理数} \end{cases}$ +求 +$ +\lim_{x \to 0} f(x) +$ diff --git a/编写小组/课前测/课前测解析版3.md b/编写小组/课前测/课前测解析版3.md new file mode 100644 index 0000000..d04034c --- /dev/null +++ b/编写小组/课前测/课前测解析版3.md @@ -0,0 +1,30 @@ +--- +tags: + - 编写小组 +--- +1、设函数$f(x)\in C[0,2]$，且满足：$$f(0)=-1,f(1)=3,f(2)=1.$$下列说法正确的是： +（A）方程$f(x)=0$在$[0,1]$上没有解 +（B）方程 f(x)=1在区间 $[0,2]$ 上只有$x=2$ 这一个解 +（C）方程 $f(x)=2$ 在 $[1,2]$ 上至少有一个解 +（D）方程 $f(x)=−2$ 在 $[0,2]$ 上没有解 + +答案：C. +$f(0)f(1)<0$，故$f(x)=0$在$[0,1]$上有解.B显然不正确.因为$(f(1)-2)(f(2)-2)=-1<0$,故C正确.D显然错误，因为零值定理只是充分条件，不满足条件不代表没有零点. + + +2、证明方程 $e^x = 3x$ 至少有一个正实根。 + +**解析** +设 $f(x) = e^x - 3x$，则 $f(x)$ 在 $[0,1]$ 上连续（实际上在整个实数域连续）。 +计算：$f(0) = 1 > 0$，$f(1) = e - 3 < 0$（因为 $e \approx 2.718$）。 +由零点定理，存在 $\xi \in (0,1)$ 使得 $f(\xi)=0$，即 $e^\xi = 3\xi$，故方程至少有一个正实根。 + + +3、设数列 $\{x_n\}$ 满足 $x_1 = 1, x_{n+1} = 1 + \frac{1}{1 + x_n}$，证明 $\{x_n\}$ 收敛，并求极限。 + +**解析** +1. **有界性**：归纳法易证 $1 \leq x_n \leq 2$。 +2. **单调性**：计算 $x_{n+2} - x_{n}$ 判断奇偶子列单调性，或直接分析 $x_{n+1} - x_n$ 符号，可知数列非单调但可拆分单调子列。 + 实际上，可证 $\{x_{2n}\}$ 单调递增，$\{x_{2n-1}\}$ 单调递减。 +3. **由单调有界准则**，奇偶子列均收敛，再证二者极限相同。 +4. **求极限**：设 $\lim_{n \to \infty} x_n = L$，则 $L = 1 + \frac{1}{1 + L}$，解得 $L = \sqrt{2}$。 \ No newline at end of file diff --git a/编写小组/课前测/课前测解析版4.md b/编写小组/课前测/课前测解析版4.md new file mode 100644 index 0000000..a13b01a --- /dev/null +++ b/编写小组/课前测/课前测解析版4.md @@ -0,0 +1,35 @@ +--- +tags: + - 编写小组 +--- +1. 设 $x_n = \frac{\cos\left(\frac{2n\pi}{3}\right)}{n} + 1$，证明 $x_n \to 1$。 + +**解析** +按 $n \mod 3$ 将数列分为三个子列： +- 当 $n = 3k$： + $x_{3k} = \frac{\cos(2k\pi)}{3k} + 1 = \frac{1}{3k} + 1 \to 1$ +- 当 $n = 3k+1$： + $x_{3k+1} = \frac{\cos(2\pi/3 + 2k\pi)}{3k+1} + 1 = \frac{-1/2}{3k+1} + 1 \to 1$ +- 当 $n = 3k+2$： + $x_{3k+2} = \frac{\cos(4\pi/3 + 2k\pi)}{3k+2} + 1 = \frac{-1/2}{3k+2} + 1 \to 1$ + +三个子列极限均为 $1$，且它们覆盖所有正整数，故原数列收敛于 $1$。 + + + + + + + + + + +2. 函数 $f(x) = \begin{cases} x^2, & \text{若 } x \text{ 为有理数} \\ -x^2, & \text{若 } x \text{ 为无理数} \end{cases}$ +求 +$ +\lim_{x \to 0} f(x) +$ + +**解析：** + +对于任意趋于 0 的序列 $x_n$，无论其项是有理数还是无理数，都有 $|f(x_n)| = x_n^2 \to 0$。因此，由夹逼定理可得极限为 0。 \ No newline at end of file diff --git a/编写小组/课后测/课后测1.md b/编写小组/课后测/课后测1.md index 11aab26..5ad05a3 100644 --- a/编写小组/课后测/课后测1.md +++ b/编写小组/课后测/课后测1.md @@ -8,44 +8,32 @@ tags: 1.已知 $b_n = \sqrt[n]{2^n + 3^n + 4^n}$，则 $\lim_{n \to \infty} b_n = ?$ A. 2 - B. 3 - C. 4 - D. 9 2.计算 $$\displaystyle \lim_{n\to\infty} n^2[\ \arctan\ (n+1) - \arctan n]$$A. $\infty$ - B. 1 - C. $\tan1$ - D. 0 -3.计算 -$$\lim\limits_{x\to\pi/2}(sinx)^{tanx}$$A. $\frac{\pi}{2}$ - +3.计算$$\lim\limits_{x\to\pi/2}(sinx)^{tanx}$$A. $\frac{\pi}{2}$ B. $\frac{\pi}{4}$ - C. $-\frac{\pi}{2}$ - D. $1$ 4.设 $$f(x) = (1+x^2)\cos\left(\frac{1}{x^3}\right)$$则 $\lim_{x \to \infty} f(x)$ - A. 为 0 - B. 为 1 - C. 为-1 - D. 无穷大 5.设$$f(x)=\frac{2+e^{1/x}}{1+e^{4/x}}+\frac{sinx}{|x|}$$则$x=0$是 $f(x)$ 的（） - -（A）可去间断点  （B）跳跃间断点   （C）无穷间断点  （D）震荡间断点 +A. 可去间断点 +B. 跳跃间断点 +C. 无穷间断点 +D. 震荡间断点 diff --git a/编写小组/课后测/课后测3.md b/编写小组/课后测/课后测3.md index 1fb86d3..9cb2f59 100644 --- a/编写小组/课后测/课后测3.md +++ b/编写小组/课后测/课后测3.md @@ -2,3 +2,39 @@ tags: - 编写小组 --- + +1、设$f(x)$在$[0,2a]$内连续，且$f(0)=f(2a)$,证明$[0,a]$上至少存在一点$\xi$，使$f(\xi)=f(\xi+a).$ + +``` + + + + + + + +``` + +2、设 $f(x)$ 在 $[a,b]$ 上连续，且 $a < c < d < b$，证明若 $f(c)$ 与 $f(d)$ 异号，则方程 $f(x)=0$ 在 $(c,d)$ 内至少有一个实根。 + +``` + + + + + + + +``` + +3、判断数列 $a_n = \frac{n}{n^2 + 1}$ 是否收敛，若收敛求极限。 + +``` + + + + + + + +``` \ No newline at end of file diff --git a/编写小组/课后测/课后测4.md b/编写小组/课后测/课后测4.md new file mode 100644 index 0000000..7a1e9fe --- /dev/null +++ b/编写小组/课后测/课后测4.md @@ -0,0 +1,5 @@ +1.判断数列 $a_n = \frac{(-1)^n n}{n+1}$ 的收敛性。 + + + +2.设数列 $\{a_n\}$ 的三个子列 $\{a_{2n}\}$、$\{a_{2n+1}\}$、$\{a_{3n+1}\}$ 均收敛，那么 $\{a_n\}$ 是否一定收敛？说明理由。 \ No newline at end of file diff --git a/编写小组/课后测/课后测解析版3.md b/编写小组/课后测/课后测解析版3.md new file mode 100644 index 0000000..39e3dae --- /dev/null +++ b/编写小组/课后测/课后测解析版3.md @@ -0,0 +1,24 @@ +--- +tags: + - 编写小组 +--- +1、设$f(x)$在$[0,2a]$内连续，且$f(0)=f(2a)$,证明$[0,a]$上至少存在一点$\xi$，使$f(\xi)=f(\xi+a).$ + +证明：记$$F(x)=f(x)-f(x+a),x\in[0,a].$$于是$F(0)=f(0)-f(a),F(a)=f(a)-f(2a)=f(a)-f(0).$从而$F(0)F(a)\le0$. +若$F(a)=0$或$F(0)=0$，则命题得证.若$F(a)F(0)<0$，则由零值定理，存在$\xi\in(0,a)$，使得$F(\xi)=0,f(\xi)=f(\xi+a).$ + + +2、设 $f(x)$ 在 $[a,b]$ 上连续，且 $a < c < d < b$，证明若 $f(c)$ 与 $f(d)$ 异号，则方程 $f(x)=0$ 在 $(c,d)$ 内至少有一个实根。 + +**解析** +由题意，$f(x)$ 在 $[c,d]$ 上连续，且 $f(c) \cdot f(d) < 0$。 +由零点定理，存在 $\xi \in (c,d)$，使得 $f(\xi)=0$，即方程 $f(x)=0$ 在 $(c,d)$ 内至少有一个实根。 + + +3、判断数列 $a_n = \frac{n}{n^2 + 1}$ 是否收敛，若收敛求极限。 + +**解析** +1. **单调性**：考虑 $a_{n+1} - a_n = \frac{n+1}{(n+1)^2 + 1} - \frac{n}{n^2 + 1}$，通分后分子为 $-n^2 - n + 1$，当 $n \geq 1$ 时小于零，故数列单调递减。 +2. **有界性**：显然 $0 < a_n \leq \frac{1}{2}$。 +3. **由单调有界准则**，数列收敛。 +4. **极限**：$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n}{n^2 + 1} = 0$。## 介值定理 \ No newline at end of file diff --git a/编写小组/课后测/课后测解析版4.md b/编写小组/课后测/课后测解析版4.md new file mode 100644 index 0000000..90424ab --- /dev/null +++ b/编写小组/课后测/课后测解析版4.md @@ -0,0 +1,25 @@ +--- +tags: + - 编写小组 +--- +1.判断数列 $a_n = \frac{(-1)^n n}{n+1}$ 的收敛性。 + **解析** +- 偶数项子列：$a_{2k} = \frac{2k}{2k+1} \to 1$ +- 奇数项子列：$a_{2k-1} = -\frac{2k-1}{2k} \to -1$ + +两个子列极限不同，故原数列发散。 + +2.设数列 $\{a_n\}$ 的三个子列 $\{a_{2n}\}$、$\{a_{2n+1}\}$、$\{a_{3n+1}\}$ 均收敛，那么 $\{a_n\}$ 是否一定收敛？说明理由。 + +**解析** +$\{a_n\}$ **一定收敛**，理由如下： + +1. 子列 $\{a_{2n}\}$（偶数项）与 $\{a_{2n+1}\}$（奇数项）覆盖了整个数列 $\{a_n\}$，若能证明这两个子列极限相等，则 $\{a_n\}$ 收敛（拉链定理）。 +2. 分析子列 $\{a_{3n+1}\}$： + - $\{a_{3n+1}\}$ 中既有偶数项（如 $n=1$ 时，$3\times1+1=4$，对应 $a_4$，属于 $\{a_{2n}\}$），也有奇数项（如 $n=0$ 时，$3\times0+1=1$，对应 $a_1$，属于 $\{a_{2n+1}\}$）。 + - 由于 $\{a_{3n+1}\}$ 收敛，其所有子列极限必与它本身的极限相等，因此： + - $\{a_{3n+1}\}$ 中的偶数项子列（属于 $\{a_{2n}\}$）的极限 = $\{a_{3n+1}\}$ 的极限； + - $\{a_{3n+1}\}$ 中的奇数项子列（属于 $\{a_{2n+1}\}$）的极限 = $\{a_{3n+1}\}$ 的极限。 + - 故 $\{a_{2n}\}$ 与 $\{a_{2n+1}\}$ 的极限相等。 + +综上，数列 $\{a_n\}$ 的偶数项子列和奇数项子列极限相同，因此 $\{a_n\}$ 收敛。 \ No newline at end of file diff --git a/隐函数，参数方程和应用题（解析版）.md b/隐函数，参数方程和应用题（解析版）.md deleted file mode 100644 index 26b72fc..0000000 --- a/隐函数，参数方程和应用题（解析版）.md +++ /dev/null @@ -1,587 +0,0 @@ ---- -tags: - - 编写小组 ---- -**内部资料，禁止传播** -**编委会（不分先后，姓氏首字母顺序）：韩魏 刘柯妤 卢吉辚 王轲楠 支宝宁 郑哲航** - -## 隐函数求导 - -### 原理 - -设方程 $F(x, y) = 0$ 确定 $y$ 是 $x$ 的函数 $y = y(x)$，则对方程两边关于 $x$ 求导（注意 $y$ 是 $x$ 的函数），再解出 $y'$ 即可。 - -### **适用情况** - -适用于方程中 $x$ 与 $y$ 混合在一起，无法或不易解出 $y = f(x)$ 的情况，如： -- 圆的方程：$x^2 + y^2 = 1$ -- 椭圆方程：$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ -- 一般隐式方程：$e^{x+y} + \ln(xy) = 0$ - -### **优势** -1. 不必显式解出 $y = f(x)$，可直接求导； -2. 适用于复杂关系式，尤其是含有 $x$、$y$ 混合的函数形式。 - -### **劣势** -1. 求导过程中需注意 $y$ 是 $x$ 的函数，常需使用链式法则； -2. 最终表达式中可能仍含有 $y$，需结合原方程化简。 - -### **例子** - -> [!example] 例1(简单) -求由方程 $x^2 + y^2 = 1$ 确定的隐函数 $y = y(x)$ 的导数。 - -**解析** -两边对 $x$ 求导： -$$ -2x + 2y \cdot y' = 0 -$$ -解得： -$$ -y' = -\frac{x}{y} -$$ - ->[!example] 例2 ->设方程$y=y(x)$由方程$xe^{f(y)}=e^y$确定，其中$f$具有二阶导数，且$f'\neq1$，求$\frac{d^2y}{dx^2}.$ - -解1： - -两边对$x$求导得 -$$e^{f(y)}+xy'f'(y)e^{f(y)}=y'e^y$$ -从而 -$$y'=\frac{e^{f(y)}}{e^y-xf'(y)e^{f(y)}}=\frac{e^{f(y)}}{xe^{f(y)}(1-f'(y))}=\frac{1}{x(1-f'(y))}$$ -两边再对$x$求导得 -$$\frac{d^2y}{dx^2}=y''=-\frac{1-f'(y)-xy'f''(y)}{x^2(1-f'(y))^2}=-\frac{(1-f'(y))^2-f''(y)}{x^2(1-f'(y))^3}$$ - - -解2：两边取对数得$$lnx+f(y)=y$$ -两边对$x$求导得$$\frac{1}{x}+y'f'(y)=y'$$ -从而$$y'=\frac{1}{x(1-f'(y))}$$ -后同解1. - - ->[!example] 例3 ->曲线$\ln{\sqrt{x^2+y^2}}=arctan(\frac{x}{y})$在点(0,1)处的切线方程为$\_\_\_$ ->（A）$y=-x+1$ ->（B）$y=1$ ->（C）$y=x+1$ ->（D）$y=\frac{1}{2}x+1$ - - -解：由原式得$$\frac{1}{2}ln(x^2+y^2)=arctan(\frac{x}{y})$$ -两边对$x$求导得 $$\frac{1}{2}\cdot \frac{2x+2yy'}{x^2+y^2}=\frac{y-xy'}{x^2+y^2}$$$$x+yy'=y-xy'$$$$y'=\frac{y-x}{y+x}$$ -故$y'|_{x=0,y=1}=1$，即曲线在(0,1)处切线的斜率为1，故切线方程为： -$$y=x+1$$选C. - - ->[!example] 例4 ->函数$y=y(x)$由方程$sin(x^2+y^2)+e^x-xy^2=0$所确定，则$\frac{dy}{dx}=\_\_$. - -解： -两边求微分得： -$$(2xdx+2ydy)cos(x^2+y^2)+e^xdx-y^2dx-2xydy=0$$ -$$(2xcos(x^2+y^2)+e^x-y^2)dx+(2ycos(x^2+y^2)-2xy)dy=0$$ -$$\frac{dy}{dx}=\frac{2xcos(x^2+y^2)+e^x-y^2}{2xy-2ycos(x^2+y^2)}$$ - ->[!example] 例5(卢吉辚原创难题) ->已知由方程 $e^{(x y - 1)(x + y)} \ln (x^3 + y^3) = \frac{xy(x^2 + y^2)}{(x y)^{x^2 + x y + y^2}} \ln (\sqrt{x + y})$ 所确定的函数单调减，求其在点 $(1,1)$ 处的切线方程。 - -解法一（直接求导） - -设 $y = y(x) 、$ 是由方程确定的隐函数，且 $y(1)=1$。两边对 $x$ 求导，并记 $y' = \frac{dy}{dx}$。分别计算左边导数 $\frac{dL}{dx}$ 和右边导数 $\frac{dR}{dx}$ 在点 $(1,1)$ 处的值。 - -左边：$L = e^{(xy-1)(x+y)} \ln(x^3+y^3)$。 -令 $$u = (xy-1)(x+y),$$则 $u(1,1)=0$。 -有$$u' = (y + x y')(x+y) + (xy-1)(1+y'),$$在 $(1,1)$ 处，$u' = 2(1+y')$。 -于是 $$\frac{dL}{dx} = e^{u} u' \ln(x^3+y^3) + e^{u} \cdot \frac{1}{x^3+y^3} \cdot (3x^2 + 3y^2 y')$$ -在 $(1,1)$ 处，$e^{u}=1$，$\ln(x^3+y^3)=\ln 2$，$x^3+y^3=2$，所以 $\frac{dL}{dx}\bigg|_{(1,1)} = 2(1+y') \ln 2 + \frac{3}{2}(1+y') = \left(2\ln 2 + \frac{3}{2}\right)(1+y')$。 - -右边：$R = \frac{1}{2} (xy)^{1-(x^2+xy+y^2)} (x^2+y^2) \ln(x+y)$。 -令 $$A = (xy)^{1-(x^2+xy+y^2)},B = x^2+y^2,C = \ln(x+y)$$则 $R = \frac{1}{2} A B C$。 -在 $(1,1)$ 处，$A=1$，$B=2$，$C=\ln 2$。 - -先求 $A'$。取对数：$\ln A = [1-(x^2+xy+y^2)] \ln(xy)$。求导得： -$$\frac{A'}{A} = [-(2x + y + x y' + 2y y')] \ln(xy) + [1-(x^2+xy+y^2)] \cdot \frac{y + x y'}{xy}$$ -在 $(1,1)$ 处，$$\ln(xy)=0,1-(x^2+xy+y^2) = -2, xy=1$$，所以 $\frac{A'}{A} = -2(1+y')$，即 $A' = -2(1+y')$。 -易得$$B' = 2x + 2y y' = 2 + 2y' = 2(1+y'), -C' = \frac{1+y'}{x+y} = \frac{1}{2}(1+y').$$ - -于是 $$\frac{dR}{dx} = \frac{1}{2} (A' B C + A B' C + A B C')$$ -代入：$$\frac{dR}{dx} = \frac{1}{2} \left[ -2(1+y') \cdot 2 \cdot \ln 2 + 1 \cdot 2(1+y') \cdot \ln 2 + 1 \cdot 2 \cdot \frac{1}{2}(1+y') \right]$$ -化简：$$\frac{dR}{dx} = \frac{1}{2} \left[ -4(1+y') \ln 2 + 2(1+y') \ln 2 + (1+y') \right] = \frac{1}{2} (1+y') (1 - 2\ln 2)$$ - -由 $\frac{dL}{dx} = \frac{dR}{dx}$ 得： -$$\left(2\ln 2 + \frac{3}{2}\right)(1+y') = \frac{1}{2} (1+y') (1 - 2\ln 2)$$ -若 $1+y' \neq 0$，则 $2\ln 2 + \frac{3}{2} = \frac{1}{2} - \ln 2$，即 $3\ln 2 = -1$，矛盾。故 $1+y' = 0$，即 $y' = -1$。 - -所以切线斜率为 $-1$，切线方程为 $y - 1 = -1 \cdot (x - 1)$，即 $y = -x + 2$。 - -答案：切线方程为 $y = -x + 2$。 - - 解法二 - -由方程形式可知，交换 $x$ 与 $y$ 后方程不变，故曲线关于直线 $y = x$ 对称。点 $(1,1)$ 位于曲线且落在对称轴 $y = x$ 上。 -在对称轴上的点处，切线斜率只可能为 $1$（与对称轴平行）或 $-1$（与对称轴垂直）。 -因为函数单调减，所以在 $(1,1)$ 处导数 $y' < 0$，故斜率不可能为 $1$，只可能为 $-1$。 -由点斜式得切线方程： -$y - 1 = -1 \cdot (x - 1)$， -即 $y = -x + 2$。 - - ->[!example] 例6(卢吉辚原创难题) ->已知由方程 $\arctan\left( \frac{x^2 y + x y^2}{2} \right) = \arcsin\left( \frac{\sqrt{x^2 + y^2}}{x + y} \right)$ 所确定的函数在其连续区间内单调减，求其在点 $(1,1)$ 处的切线方程。 - -解法一（直接求导） - -对方程两边关于 $x$ 求导，$y$ 视为 $x$ 的函数，记 $y' = \frac{dy}{dx}$。 -左边：设 $u = \frac{x^2 y + x y^2}{2}$，则导数为 $\frac{1}{1+u^2} \cdot \frac{du}{dx}$。 -$\frac{du}{dx} = \frac{1}{2}(2xy + x^2 y' + y^2 + 2xy y')$。 -在 $(1,1)$ 处，$$u=1,\frac{1}{1+u^2} = \frac{1}{2}，\frac{du}{dx} = \frac{1}{2}(2+1 + (1+2)y') = \frac{3}{2}(1+y')$$ -所以左边导数为 $$\frac{1}{2} \cdot \frac{3}{2}(1+y') = \frac{3}{4}(1+y')$$ - -右边：设 $v = \frac{\sqrt{x^2 + y^2}}{x + y}$，则导数为 $\frac{1}{\sqrt{1-v^2}} \cdot \frac{dv}{dx}$。 - -在 $(1,1)$ 处，$v = \frac{\sqrt{2}}{2}$，$\sqrt{1-v^2} = \frac{1}{\sqrt{2}}$，所以 $\frac{1}{\sqrt{1-v^2}} = \sqrt{2}$。 -计算 $\frac{dv}{dx}$：$$v = \frac{\sqrt{x^2+y^2}}{x+y}， -\frac{dv}{dx} = \frac{(x+y) \cdot \frac{x + y y'}{\sqrt{x^2+y^2}} - \sqrt{x^2+y^2}(1+y')}{(x+y)^2}$$ -代入 $(1,1)$，$$\sqrt{x^2+y^2} = \sqrt{2}，x+y=2，\frac{dv}{dx} = \frac{2 \cdot \frac{1+y'}{\sqrt{2}} - \sqrt{2}(1+y')}{4} = 0$$ -因此右边导数为 $\sqrt{2} \cdot 0 = 0$。 - -由左右导数相等得：$\frac{3}{4}(1+y') = 0$，解得 $y' = -1$。 - -故切线方程为 $y - 1 = -1 \cdot (x - 1)$，即 $y = -x + 2$。 - - - 解法二 - -方程中交换 $x$ 与 $y$ 后形式不变，故曲线关于直线 $y = x$ 对称。点 $(1,1)$ 在曲线上且位于对称轴 $y = x$ 上。 -由对称性，在对称轴上的点处，切线斜率只可能为 $1$（与对称轴平行）或 $-1$（与对称轴垂直）。 -已知函数单调减，即在定义区间内 $y' < 0$，因此斜率不可能为 $1$，只能为 $-1$。 - -切线方程为 $y - 1 = -1 \cdot (x - 1)$，即 $y = -x + 2$。 - ---- - -## 参数方程求导 - -### 原理 - -已知 $$\left\{ \begin{array}{c} x=x\left( t \right)\\ y=y\left( t \right)\\ \end{array} \right.$$若有均可导且 $x'(t) \neq 0$，则 -$$y'\left( t \right) =\frac{dy}{dt}$$ -$$x'\left( t \right)=\frac{dx}{dt}$$ - $$y'=\frac{dy}{dx}=\frac{dy}{dt}\cdot \frac{1}{\frac{dx}{dt}}=\frac{y'\left( t \right)}{x'\left( t \right)}$$ -则有 $$y''=\frac{d^2y}{dx^2}=\frac{dy'}{dx}=\frac{dy'}{dt}\cdot \frac{1}{\frac{dx}{dt}}=\frac{dy'}{dt}\cdot \frac{1}{x'\left( t \right)}=(\frac{y'(t)}{x'(t)})'\cdot \frac{1}{x'\left( t \right)}$$ - -而 $$\frac{y'(t)}{x'(t)}=\frac{y''\left( t \right) x'\left( t \right) -x''\left( t \right) y'\left( t \right)}{x'^2\left( t \right)}$$ - -故 $$y''=\frac{y''\left( t \right) x'\left( t \right) -x''\left( t \right) y'\left( t \right)}{x'^3\left( t \right)}$$ -### **适用情况** -适用于曲线由参数形式给出，尤其是： -- 物理中的运动轨迹； -- 极坐标、摆线、旋轮线等曲线； -- 复杂曲线的简化表示。 - -### **优势** -1. 形式简洁，直接利用两个导数作商； -2. 适用于参数化表示，便于计算高阶导数。 - -### **劣势** -1. 要求 $x'(t) \neq 0$，否则导数不存在； -2. 高阶导数计算需重复使用公式，略显繁琐。 - -### **例子** - -> [!example] 例1(简单) -> 求参数方程 -> $$ -\begin{cases} -x = a\cos t \\ -y = b\sin t -\end{cases}$$ -所确定的函数 $y = y(x)$ 的导数。 - -**解析** - -计算： - -$$ -\frac{dx}{dt} = -a\sin t, \quad \frac{dy}{dt} = b\cos t -$$ - -因此： -$$ -\frac{dy}{dx} = \frac{b\cos t}{-a\sin t} = -\frac{b}{a}\cot t -$$ - - -> [!example] 例2 -设 $y$ 由方程 $e^{x+y} = xy$ 确定，则 $y'$ 在点 $(1,0)$ 处的值为 -A. $0$ -B. $1$ -C. $-1$ -D. 不存在 - -**解析** -两边对 $x$ 求导： -$$ -e^{x+y}(1 + y') = y + x y' -$$ -代入 $(1,0)$： -$$ -e^{1+0}(1 + y') = 0 + 1 \cdot y' -$$ -即： -$$ -e(1 + y') = y' -$$ -解得： -$$ -y' = \frac{e}{1 - e} -$$ -该值不为选项中所列，故判断为“不存在直接匹配”，但若只允许选 ABCD，则可能选 D。 -实际应说明：$y' = \frac{e}{1-e}$ 是一个确定数值。 - - -> [!example] 例3 -> 参数方程 -> $$ -\begin{cases} -x = t^2 + 1 \\ -y = t^3 - t -\end{cases}$$ -在 $t = 1$ 处的导数 $\left. \frac{dy}{dx} \right|_{t=1}$ 为 -A. $1$ -B. $2$ -C. $3$ -D. $4$ - -**解析** -计算： -$$ -\frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 3t^2 - 1 -$$ -则： -$$ -\frac{dy}{dx} = \frac{3t^2 - 1}{2t} -$$ -代入 $t = 1$： -$$ -\left. \frac{dy}{dx} \right|_{t=1} = \frac{3 \cdot 1^2 - 1}{2 \cdot 1} = \frac{2}{2} = 1 -$$ -答案：A - ->[!example] 例4(卢吉辚原创难题) ->设参数方程 $x = e^{t^2} \arcsin(t^2)$，$y = \ln(1+t^4) \arctan(t^3)$，求 $\frac{d^2 y}{dx^2}$ 在 $t=0$ 处的值（需要用到泰勒展开，仅作习题） - -**解析：** -先求一阶导： -$$\frac{dx}{dt} = 2t e^{t^2} \left( \frac{1}{\sqrt{1-t^4}} + \arcsin(t^2) \right)$$$$\frac{dy}{dt} = \frac{4t^3}{1+t^4} \arctan(t^3) + \ln(1+t^4) \cdot \frac{3t^2}{1+t^6}$$ -在 $t=0$ 处，$\frac{dx}{dt}=0$，$\frac{dy}{dt}=0$。 - -需用公式 $$\frac{d^2 y}{dx^2} = \frac{ \frac{dx}{dt} \frac{d^2 y}{dt^2} - \frac{dy}{dt} \frac{d^2 x}{dt^2} }{ \left( \frac{dx}{dt} \right)^3 }$$。 -利用泰勒展开： -$x(t) = t^2 + t^4 + O(t^6)$，得 $$\frac{dx}{dt} = 2t + 4t^3 + O(t^5),\frac{d^2 x}{dt^2} = 2 + 12t^2 + O(t^4);$$ -$y(t) = t^7 + O(t^{11})$，得 $$\frac{dy}{dt} = 7t^6 + O(t^{10})，\frac{d^2 y}{dt^2} = 42t^5 + O(t^9).$$ -分子：$$\frac{dx}{dt} \frac{d^2 y}{dt^2} - \frac{dy}{dt} \frac{d^2 x}{dt^2} = 70 t^6 + O(t^7)$$分母：$$(\frac{dx}{dt})^3 = 8 t^3 + O(t^5)$$ -当 $t \to 0$，$\frac{d^2 y}{dx^2} = \frac{70 t^6}{8 t^3} \to 0$，故值为 $0$。 - -**答案：** $\frac{d^2 y}{dx^2} \big|_{t=0} = 0$。 - ->[!example] 例5(卢吉辚原创难题) ->设参数方程 $x = e^{t} \sin t + \ln(1+t^2)$，$y = e^{t} \cos t + \arctan(t^3)$，求 $\frac{d^2 y}{dx^2}$ 在 $t=0$ 处的值。 - -**解析：** -计算一阶导： -$$\frac{dx}{dt} = e^{t} \sin t + e^{t} \cos t + \frac{2t}{1+t^2}，\frac{dy}{dt} = e^{t} \cos t - e^{t} \sin t + \frac{3t^2}{1+t^6}.$$ -在 $t=0$ 处：$\frac{dx}{dt} = 1$，$\frac{dy}{dt} = 1$。 - -二阶导： -$$\frac{d^2 x}{dt^2} = 2 e^{t} \cos t + \frac{2 - 2t^2}{(1+t^2)^2}，\frac{d^2 y}{dt^2} = -2 e^{t} \sin t + \frac{6t(1+t^6) - 18t^7}{(1+t^6)^2}.$$ -在 $t=0$ 处：$$\frac{d^2 x}{dt^2} = 4，\frac{d^2 y}{dt^2} = 0.$$ - -代入公式： -$$\frac{d^2 y}{dx^2} = \frac{1 \cdot 0 - 1 \cdot 4}{1^3} = -4.$$ - -**答案：** $\frac{d^2 y}{dx^2} \big|_{t=0} = -4$。 - -## **应用题** - -应用题就比较综合了，它可以考很多知识点，比如参数方程，比如隐函数求导，也有可能就是一个普通的求导。但关键不在于求导，关键在于建立一个准确的模型，也就是 - -（1）把需要求的对象找出来，即自变量和因变量，注意可能有多个。 -（2）根据条件把关系式列出来，画一画示意图有时候会很有帮助。 - -### 微分估计值 - ->[!example] 例1 ->利用微分估计值也是导数应用的一种。利用微分估计下面几个式子的值： ->（1）$\sqrt{34}$（保留两位小数） ->（2）摆的振动周期公式为：$T=2\pi \sqrt{\frac{l}{g}}$。其中$l$为摆长（厘米），重力加速度$g=981cm^2/s$.为了使周期$T$增大$0.05$秒，摆长$l=20cm$的长度需要作何修正？（参考数据：$\sqrt{981\times20}\approx140.07,\frac{1}{\pi}\approx0.318$,保留两位小数） - -解：（1）$\sqrt{34}=\sqrt{36-2}=6\sqrt{1-\frac{1}{18}}$,考虑函数$f(x)=\sqrt{x}$，$f'(x)=\frac{1}{2\sqrt{x}},f'(1)=\frac{1}{2}$,故$df=\frac{dx}{2\sqrt{x}},df|_{x=1}=\frac{1}{2}dx$,取$dx=-\frac{1}{18}$,得$df=-\frac{1}{36}$,故$$\sqrt{34}\approx 6(1-\frac{1}{36})=\frac{35}{6}\approx5.83$$ -（2）$dT=T'dl=\pi\sqrt{\frac{1}{gl}}dl=0.05$,则$$dl=\frac{1}{\pi}\sqrt{gl}\approx\frac{1}{3.14}\times\sqrt{981\times20}\times0.05\approx2.23$$ -故需要增加摆长约$2.23$厘米 - -### 相关变化率 - -#### 飞机航空摄影问题 - -##### 问题描述 - ->[!example] 例题 ->一飞机在离地面$2 km$的高度，以$200 km/h$的速度水平飞行到某目标上空，以便进行航空摄影。试求飞机飞至该目标正上方时，摄影机转动的角速率。 - -解析： - -1、建立坐标系 - -目标位置：原点 $O(0,0)$ -飞机位置：$P(x,2)$ -飞机高度：$h = 2 km$ -水平速度：$\frac{dx}{dt} = -200 km/h$ - -2、角度关系 - -$$ -\theta = \arctan\left(\frac{x}{2}\right), -\tan\theta =\frac{x}{2} -$$ - -3、角速率计算 - -$$\frac{d\theta}{dt} = \frac{d\theta}{dx} \cdot \frac{dx}{dt}$$ -$$\frac{d\theta}{dx} = \frac{d}{dx} \arctan\left(\frac{x}{2}\right) = \frac{1}{1 + \left(\frac{x}{2}\right)^2} \cdot \frac{1}{2} = \frac{2}{x^2 + 4}$$ -$$\frac{d\theta}{dt} = \frac{2}{x^2 + 4} \cdot (-200) = -\frac{400}{x^2 + 4} rad/h$$ - -4、目标正上方时的角速率 - -当 $x = 0$ 时： - -$$\left. \frac{d\theta}{dt} \right|_{x=0} = -\frac{400}{0 + 4} = -100 rad/h$$ - -5、转换单位： - -$$100 rad/h = 100 \times \frac{180}{\pi} °/h = \frac{18000}{\pi} °/h, - -\frac{18000}{\pi} °/h \times \frac{1}{3600} h/s = \frac{5}{\pi} °/s \approx 1.59 °/s$$ - -答案: - -飞机飞至目标正上方时，摄影机转动的角速率为 $\frac{5}{\pi} °/s$（约$1.59 °/s$）。 - ---- - -#### 人拉船问题 - -##### 问题描述 - ->[!example] 例题 ->人在河岸用绳经过定滑轮以速度 $v$、加速度 $a$ 拉船。绳与水平面夹角为 $\theta$。求此时船的加速度 $a'$。 -![[应用题人拉船.png]] - -解析 - -1、 变量定义 - -$h$：滑轮高度（定值） -$x$：船到河岸的水平距离 -$l$：绳长 -$v = -\frac{dl}{dt}$：人拉绳的速度 -$a = -\frac{d^2l}{dt^2}$：人拉绳的加速度 -$u = -\frac{dx}{dt}$：船的水平速度 -$a' = -\frac{d^2x}{dt^2}$：船的加速度 -$\theta$：绳与水平面夹角 - -2、几何关系 - -$l^2 = h^2 + x^2$ - -3、 一阶导数关系 - -对几何关系求导：$$2l\frac{dl}{dt} = 2x\frac{dx}{dt}\ 即\ l\frac{dl}{dt} = x\frac{dx}{dt}$$ - -代入速度定义：$l(-v) = x(-u)$ - -$$u = \frac{l}{x}v = \frac{v}{\cos\theta}，其中 \cos\theta = \frac{x}{l}$$ - -4、 二阶导数关系 - -对一阶关系求导：$$\frac{d}{dt}\left(l\frac{dl}{dt}\right) = \frac{d}{dt}\left(x\frac{dx}{dt}\right)$$ - -$$\left(\frac{dl}{dt}\right)^2 + l\frac{d^2l}{dt^2} = \left(\frac{dx}{dt}\right)^2 + x\frac{d^2x}{dt^2}$$ - -$$(-v)^2 + l(-a) = (-u)^2 + x(-a')$$ - -$$v^2 - la = u^2 - xa'$$ - -5、 求解 $a'$ - -$$xa' = u^2 - v^2 + la, - -a' = \frac{u^2 - v^2 + la}{x}$$ - -代入 $u = \frac{v}{\cos\theta}$, $x = h\tan\theta$, $l = \frac{h}{\cos\theta}$： - -$$\begin{aligned} -a' &= \frac{\left(\frac{v}{\cos\theta}\right)^2 - v^2 + \left(\frac{h}{\cos\theta}\right)a}{h\tan\theta}\\[1em] - &= \frac{v^2\left(\frac{1}{\cos^2\theta} - 1\right)}{h\tan\theta} + \frac{\frac{h}{\cos\theta}a}{h\tan\theta}\\[1em] - &= \frac{v^2\tan^2\theta}{h\tan\theta} + \frac{a}{\cos\theta\tan\theta}\\[1em] - &= \frac{v^2\tan\theta}{h} + \frac{a}{\sin\theta} -\end{aligned}$$ - - 答案： - -此时船的加速度为： -$$a' = \frac{v^2\tan\theta}{h} + \frac{a}{\sin\theta}$$ - ---- - -#### 动点曲线运动问题 - -##### 问题描述 - ->[!example] 例题 ->已知动点 $P$ 在曲线 $y = \sqrt{x}$ 上运动，记坐标原点 $O$ 与 $P$ 间的距离为 $l$。若点 $P$ 横坐标随时间的变化率为常数 $v$，则当点 $P$ 运动到点 $(1, 1)$ 时，$l$ 对时间的变化率是多少？ - -解析: - -1、 变量关系 - -点 $P$ 坐标：$(x, y)$, $y = \sqrt{x}$ - -原点 $O$ 与 $P$ 间的距离： -$$l = \sqrt{x^2 + y^2} = \sqrt{x^2 + (\sqrt{x})^2} = \sqrt{x^2 + x}$$ - -已知：$\frac{dx}{dt} = v$（常数） - -2、距离变化率计算 - -$$\begin{aligned} -\frac{dl}{dt} &= \frac{d}{dt} \sqrt{x^2 + x} = \frac{d}{dt} (x^2 + x)^{1/2}\\[1em] - &= \frac{1}{2}(x^2 + x)^{-1/2} \cdot \frac{d}{dt}(x^2 + x)\\[1em] - &= \frac{1}{2\sqrt{x^2 + x}} \cdot (2x + 1) \frac{dx}{dt}\\[1em] - &= \frac{2x + 1}{2\sqrt{x^2 + x}} \cdot v -\end{aligned}$$ - -3、在点 $(1, 1)$ 处的变化率 - -当 $x = 1$ 时： - -$$\left. \frac{dl}{dt} \right|_{x=1} = \frac{2\cdot 1 + 1}{2\sqrt{1^2 + 1}} \cdot v = \frac{3}{2\sqrt{2}} \cdot v = \frac{3v}{2\sqrt{2}}$$ - -有理化：$\frac{3v}{2\sqrt{2}} = \frac{3v\sqrt{2}}{4}$ - - 答案： - -当点 $P$ 运动到点 $(1, 1)$ 时，$l$ 对时间的变化率为 $\frac{3v\sqrt{2}}{4}$。 - ---- - -#### 球浸入水问题 - -##### 问题描述 - ->[!example] 例题 ->半径为$a$的球渐渐沉入盛有部分水的半径为$b(b>a)$的圆柱形容器中.若球以匀速$c$下沉，求：球浸没一半时，容器内水面上升的速率. -> ->（A）$\frac{a^2c}{b^2}$ -> ->（B）$\frac{a^c}{b^2-a^2}$ -> ->（C）$\frac{a^2c}{2b^2}$ -> ->（D）$\frac{a^2c}{b^2+a^2}$ - -解：主要的等量关系是：球浸入水中的体积与水“溢出”的体积相同（如果把水面上升视为溢出的话）。设时间为$t$，球下沉的体积为$V$，水上升的高度为$H$，球浸入水的高度为$h$，则有$$V=\pi h^2(a-\frac{h}{3}),dV=(2\pi ah-\pi h^2)dh=\pi h(2a-h)dh$$（这个公式大家可以记住） -且（如图）$$dV=\pi(b^2-a^2+(a-h)^2)dH$$ -![[球浸入水示意图.jpg]] -于是$$\pi(b^2-a^2+(a-h)^2)dH=\pi h(2a-h)dh$$$$dH=\frac{h(2a-h)}{b^2-2ah+h^2}dh$$$$\frac{dH}{dt}=\frac{h(2a-h))}{b^2-2ah+h^2}\frac{dh}{dt}$$而$\frac{dh}{dt}=c,h=a$，故$$\frac{dH}{dt}=\frac{a^2c}{b^2-a^2}$$ -选B. - -### 应用题期中真题 - ->[!example] **例1**（2020） ->有一个正圆锥形漏斗，深度 $18 \text{ cm}$ ，上端口直径为 $12 \text{ cm}$ （即半径 $6 \text{ cm}$ ）。漏斗下方连接一个圆柱形筒，圆柱筒直径为 $10 \text{ cm}$ （即半径 $5 \text{ cm}$ ）。初始时刻漏斗内盛满水，然后水从漏斗流入圆柱筒。已知当漏斗中水深为 $12 \text{ cm}$ 时，漏斗水面下降的速度是 $1 \text{ cm/min}$ -问：此时圆柱筒内液面上升的速度是多少？ - - -**解析**: -设 t （s）时漏斗水深为 h （cm），圆柱形容器的水深为 H （cm），则有 -$$\pi 5^2 H = V_0 - \frac{1}{3} \pi \left( \frac{6}{18} h \right)^2 h$$ -关于 t 求导数得 -$$25 \frac{dH}{dt} = -\frac{1}{9} h^2 \frac{dh}{dt}$$ -当 h = 12, $\frac{dh}{dt} = -1$ 时， -$$\frac{dH}{dt} = \frac{16}{25} \text{(cm/s)}$$ - - ->[!example] **例2** (2021) ->一长为 L 米的木梯靠在倾角为$\frac{\pi}{3}$的光滑斜坡上，木梯的顶部距离 A 点 h 米，底部距离 A 点 d 米，受重力作用木梯的顶部以 $a \, \mathrm{m/s}$ 的速度沿直线 BA 下滑，底部水平向右运动。问：当木梯的顶部和底部与 A 点的距离相等时，底部的水平速度为多少？ ->![[应用题梯子.jpg]] - - - -**解析：** -设运动 t 秒后，木梯的顶部距离 A 点 $y(t) \, \mathrm{m}$ ，底部距离 A 点 $x(t) \, \mathrm{m}$ 。由图易知 -$$\left(y(t)\sin\frac{\pi}{3}\right)^2 + \left(x(t)+y(t)\cos\frac{\pi}{3}\right)^2 = L^2$$ -即 -$$x^2(t)+y^2(t)+x(t)y(t)=L^2$$ -方程两端分别对 t 求导，可得 -$$2x(t)x'(t)+2y(t)y'(t)+x(t)y'(t)+x'(t)y(t)=0$$ -由于 $y'(t)=-a \, (\mathrm{m/s})$ ，因此当 x(t)=y(t) 时，有 -$$2x'(t)-2a-a+x'(t)=0$$ -故 $$x'(t)=a \, (\mathrm{m/s})$$ - - ->[!example] **例3**（2023） ->有一个长度为5m的梯子贴靠在铅直的墙上，假设其下端沿地板以3m/s的速率离开墙脚而滑动，则 -(1) 何时梯子的上、下端能以相同的速率移动？ -(2) 何时其上端下滑之速率为4m/s？ - - -**解析:** -设在时刻 t ，梯子上端与墙角的距离为 $x(t) \, \mathrm{m}$ ，下端与墙角的距离为 $y(t) \, \mathrm{m}$ ，则 -$$x^2 + y^2 = 25$$ -两端同时对 t 求导数可得 -$$x \frac{dx}{dt} + y \frac{dy}{dt} = 0$$ - -(1) 当$\frac{dx}{dt} = -\frac{dy}{dt}$ 时，可得 x = y ，进而 $y = \frac{5}{\sqrt{2}} \, (\mathrm{m})$ ， -即下端距墙脚 $\frac{5}{\sqrt{2}} \, \mathrm{m}$ 时上下端能以相同的速率移动。 - -(2) 当 $\frac{dy}{dt} = -4 \, (\mathrm{m/s})$ ， $\frac{dx}{dt} = 3 \, (\mathrm{m/s})$ 时，有 $4x = 3y$ ，将其代入 $x^2 + y^2 = 25$ ， -解得$y = 4 \, (\mathrm{m})$,即下端距墙脚4 m时，上端下滑速度为$4 m/s$ - - - ->[!example] **例4**(2022) ->某部举行八一阅兵，队列正步通过阅兵台时步幅间距离为75厘米，步速为每分钟112步。某观礼人员离行进队列垂直距离为60米，视线追随队列领队，求其视线与队列夹角为$\frac{\pi}{6}$时，视线转动角度的变化率。 - - - -**解析：** -- 建立几何模型：设时间为\(t\)分钟，步幅间距离0.75米，步速为每分钟112步，则队列行进距离$x = 0.75×112t = 84t$米。设视线与队列夹角为$theta$，由正切函数关系可得$tan\theta=\frac{x}{60}$，即$\tan\theta=\frac{84t}{60}=\frac{7t}{5}$。 -- 两边对$t$求导：根据复合函数求导法则 -$$(tan\theta)^\prime=\sec^{2}\theta\cdot\frac{d\theta}{dt})$$对$tan\theta=\frac{7t}{5}$两边求导得$$sec^{2}\theta\cdot\frac{d\theta}{dt}=\frac{7}{5}$$ -求解:$\frac{d\theta}{dt}$：当$\theta = \frac{\pi}{6}$时，$sec\theta=\frac{2}{\sqrt{3}}$，$sec^{2}\theta=\frac{4}{3}$，代入$sec^{2}\theta\cdot\frac{d\theta}{dt}=\frac{7}{5}$， - -可得$$\frac{4}{3}\cdot\frac{d\theta}{dt}=\frac{7}{5}$$解得$$\frac{d\theta}{dt}=\frac{21}{20}（弧度/分钟）$$ -答案：视线转动角度的变化率为 $\frac{21}{20}$ 弧度/分钟。本题主要考察了利用导数解决实际问题中的变化率问题，关键在于建立正确的函数关系并准确求导，易错点是复合函数求导法则的运用。 - - ->[!example] **例5**（2024） ->一架巡逻直升机在距离地面3km的高度以120km/h的速度沿着一条水平笔直的高速公路向前飞行。飞行员观察到迎面驶来一辆汽车，通过雷达测出直升机与汽车的距离为5km，同时此距离以160km/h的速率在减少。试求此时汽车行进的速度。 - - -**解析**： - -如图建立直角坐标系，设时刻 t直升机位于 A点 $(x_1​(t),3)$，汽车位于 B点 $(x_2​(t),0)$，直升机与汽车的距离为 $z(t)$，则 - -$$(x_2​(t)−x_1​(t))^2+32=z^2(t)$$ - -方程两端分别对 $t$ 求导，可得 -$$(x_2​(t)−x_1​(t))(x_2^′​(t)−x_1^′​(t))=z(t)z^′(t)$$ - -由于 $z(t)=5$时，$x_2​(t)−x_1​(t)=4$，$z^′(t)=−160$，$x_1^′​(t)=120$，有 - -$$4(x_2^′​(t)−120)=5×(−160)$$ - -故 $x_2^′​(t)=−80$，即汽车行进的速度为 $80km/h$。 \ No newline at end of file From d737aed4b5268ca702cf7bb9c2fbbefe8906dc61 Mon Sep 17 00:00:00 2001 From: Cym10x Date: Fri, 16 Jan 2026 11:57:50 +0800 Subject: [PATCH 6/6] =?UTF-8?q?=E7=A1=AE=E8=AE=A4=E5=90=88=E5=B9=B6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Notice!!!（develop分支）.md | 2 +- README.md | 19 +- 研讨记录/2026.1.11.md | 20 + 研讨记录/2026.1.8.md | 23 + 笔记分享/LaTeX（KaTeX）特殊输入.md | 24 + 笔记分享/LaTeX（KaTeX）输入规范.md | 7 + 笔记分享/线性映射与矩阵.md | 10 + 笔记分享/聚点和上下极限.md | 29 + 笔记分享/达布定理.md | 4 + 素材/1.11题目素材.md | 64 ++ 素材/图片/README/edit1.png | Bin 0 -> 397820 bytes ...定积分中值定理的综合运用：.md | 12 + .../微分中值定理的不等式问题.md | 134 +++ .../微分中值定理部分题目汇总.md | 202 +++++ 素材/拐点是一个点.md | 1 + ...柯西中值定理与常见辅助函数.md | 230 +++++ 素材/洛必达法则-注意事项.md | 50 ++ 素材/特征值.md | 32 + 素材/特征值与相似.md | 2 + ...秩的不等式“夹逼”出确切值.md | 26 + 素材/秩的不等式.md | 63 ++ 素材/线性方程组同解.md | 30 + ...性方程组的系数矩阵与解关系.md | 90 ++ 素材/线性方程组解的问题.md | 88 ++ ...罗尔定理与拉格朗日中值定理.md | 137 +++ 编写小组/讲义/图片/xcospix.png | Bin 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b/README.md @@ -1,2 +1,17 @@ -# 一起数学----高等数学笔记协作平台 -大家可以把Markdown高数笔记上传到此处 \ No newline at end of file +# 讲义编写研讨组欢迎你！ +### 开始工作前的准备： +##### 概述 +1. 下载Obsidian和Git For Windows（如果官网下不来，会给你镜像） +2. 头歌账号绑定邮箱，并且在本地准备Git +3. 在Obsidian里面安装插件Git +##### 详细步骤 +1. 下载Obsidian和Git For Windows（如果官网下不来，会给你镜像） +2. 打开头歌，打开个人主页，进入开发项目，新建一个项目 +![[edit1.png]] +新建项目的时候，如果没有绑定邮箱，会要求你绑定邮箱； +如果没有 +### 前置知识（需要在进入研讨组后尽快掌握） +1. KaTeX数学公式排版知识 +2. 基础Markdown知识 +3. (可选) Git基础知识 +4. 阅读LaTeX（KaTeX）输入规范、特殊输入 \ No newline at end of file diff --git a/研讨记录/2026.1.11.md b/研讨记录/2026.1.11.md new file mode 100644 index 0000000..af30d2c --- /dev/null +++ b/研讨记录/2026.1.11.md @@ -0,0 +1,20 @@ +研讨流程： +1.逐渐分配任务，后续边推进工作边提意见 +2.后期对讲义进行优化 +3.确定主题，目的性寻找真题，思考解题策略，总结方法 +4.先有初步的规划，避免效率太低 +5.把相似的题目直接发群里 + +讲义建议： +1.提前发 +2.难题不要和例题混在一起，在讲义后专门列一个模块 +3.高数：第五章开始，知识点细化 +4.线代：模块化复习，部分二级结论的汇总与应用。 +5.编写目标：方法集，直接从集合中调用方法解题 + +线代： +1.两个线性方程组同解的必要不充分条件是两个方程组的增广矩阵秩相等，充要条件是增广矩阵对应最简行阶梯行矩阵相等 +2.两个矩阵等价，其对应齐次线性方程组不一定同解 +3.期中考T13：多个线性方程组解的相同<----->与r(An)相等与否<------>列向量组的线性组合 +4.线性方程组解的结构<---->维数 +5.正交矩阵<----->施密特正交化法 \ No newline at end of file diff --git a/研讨记录/2026.1.8.md b/研讨记录/2026.1.8.md new file mode 100644 index 0000000..7f8316a --- /dev/null +++ b/研讨记录/2026.1.8.md @@ -0,0 +1,23 @@ +平时进行雨课堂等上题目的交流 +打电话交流？no +之后自习会增多，我们可以争取一些自习课来，这个后面再说。体能。 +集中—>解答困惑，素材 +临时部署 +雨课堂和线代作业的整理，加班时间可能会增加 +**作业上找题目（平时小测）** 2~3道题目 +每两天进行一次研讨小会（一三五日，晚或体能）完成作业，分享思路——强迫比较好地完成作业。选择小测题目。复习。 +陈搜集资料（教员习题课），王嘉兴 +线代：真题优先，习题册（关键在于没有答案），期末考和习题册是有很大重叠的 +**提前确定会议主题**（如）周三：线代作业 +讨论主题：主要不是讨论答案是什么，而是使用的方法的适用条件，让一类题的做法形成一套’机械化‘的流程（过于理想化了） +**每周一套/两套试卷**，在什么量的情况下是可以接受而且必须要给自己逼一把的；灵活处理，不需要固定在哪天写哪些题。 +**确保作业全对。** 用py计算线性代数来检查。 +原则：**把大的目标落实到每一次研讨中** +**灵活处理，下次之前决定下次的议题** +确保出的题目得是自己做过的 +**讲义和学习资料** +安排几个平时笔记做得好的去跟线代习题课 + +分工问题。研讨中出试卷、讲义分工问题的解决。挑题、讨论、成文。 + +下次开会时间：1.11周日晚 \ No newline at end of file diff --git a/笔记分享/LaTeX（KaTeX）特殊输入.md b/笔记分享/LaTeX（KaTeX）特殊输入.md new file mode 100644 index 0000000..877d957 --- /dev/null +++ b/笔记分享/LaTeX（KaTeX）特殊输入.md @@ -0,0 +1,24 @@ + +矩阵打印方式 + +$\begin{pmatrix} 1+x_1 & 1+x_1^2 & ... &1+x_1^n \\ 1+x_2 & 1+x_2^2 & ... &1+x_2^n \\ ... \\ 1+x_n & 1+x_n^2 & ... &1+x_n^n \end{pmatrix}$ + +然而，我们教材上是使用的方括号矩阵，要将pmatrix换成bmatrix： + +$\begin{bmatrix} 1+x_1 & 1+x_1^2 & ... &1+x_1^n \\ 1+x_2 & 1+x_2^2 & ... &1+x_2^n \\ ... \\ 1+x_n & 1+x_n^2 & ... &1+x_n^n \end{bmatrix}$ + +行列式的输入方法如下：(vmatrix) + +$\begin{vmatrix} 1+x_1 & 1+x_1^2 & ... &1+x_1^n \\ 1+x_2 & 1+x_2^2 & ... &1+x_2^n \\ ... \\ 1+x_n & 1+x_n^2 & ... &1+x_n^n \end{vmatrix}$ + +求和、求积的上下记号可以使用limits套： + +$\sum\limits_{i=2}^{n}a_i, \prod\limits_{i=1}^{n}a_i$ + +积分符号的上下标，按照原本的上下标来处理： + +$\int_1^5 x\mathrm{d}x$ + +积分符号的上下记号，用limits套： + +$\int\limits_{L}(x+y)\mathrm{d}s$ diff --git a/笔记分享/LaTeX（KaTeX）输入规范.md b/笔记分享/LaTeX（KaTeX）输入规范.md new file mode 100644 index 0000000..6f6d9c0 --- /dev/null +++ b/笔记分享/LaTeX（KaTeX）输入规范.md @@ -0,0 +1,7 @@ +通常的，记号严格按照教材中的规范。 +1. 矩阵使用bmatrix +2. 自然常数或电荷量e、虚数单位i应当为**正体**，需要用mathrm记号包裹，例：$\mathrm{e}^{\mathrm{i}\pi}+1=0$；然而，当e,i作为变量时，应当用正常的斜体。例：$\sum\limits_{i=1}^{n}a_i$ +3. 微分算子d应当用正体，被微分的表达式用正常的斜体：$\mathrm{d}f(x)=f'(x)\mathrm{d}x$ +4. 极限和求和求积符号用\limits，如$\lim\limits_{x\to0}$和$\sum\limits_{n=0}^{\infty}$ +5. \$\$双美元符号之间不要打回车！除非你有\begin{...}\end{...}\$\$ +6. 不对矩阵作加粗要求，但向量一定要加粗，用\boldsymbol{}，比如$A,\boldsymbol{x}$。 \ No newline at end of file diff --git a/笔记分享/线性映射与矩阵.md b/笔记分享/线性映射与矩阵.md new file mode 100644 index 0000000..0d82fe9 --- /dev/null +++ b/笔记分享/线性映射与矩阵.md @@ -0,0 +1,10 @@ +课上我们已经学了线性变换和方阵的关系，即任意一个线性变换都与一个方阵相对应.具体的描述可以写成如下的形式： +>[!note] 定理1： +>设$V$是在数域$\mathbb{P}$上的线性空间，$T$是线性空间$V$上的一个线性变换，若$V$上的一组基为$$\alpha_1,\alpha_2,\cdots,\alpha_r,$$则$\forall\alpha\in V$,有$$T(\alpha)=[T(\alpha_1),T(\alpha_2),\cdots,T(\alpha_r)]\begin{bmatrix}x_1\\x_2\\\cdots\\x_r\end{bmatrix}=[\alpha_1,\alpha_2,\cdots,\alpha_r]A\begin{bmatrix}x_1\\x_2\\\cdots\\x_r\end{bmatrix}$$其中$\begin{bmatrix}x_1\\x_2\\\cdots\\x_r\end{bmatrix}$是$\alpha$在这组基下的坐标，方阵$A$称为$T$在这组基下的方阵. + +我们考虑较为熟悉的情形，不妨令$\mathbb{P}=\mathbb{R},V=\mathbb{R}^n$，那么我们就可以认为方阵$A$与一个从$\mathbb{R}^n$到$\mathbb{R}^n$的线性映射相对应.自然可以想到：如果考虑线性映射$$\sigma:\mathbb{R}^m\rightarrow\mathbb{R}^n$$我们是否也可以得到一个矩阵呢？而且我们会设想它是一个$m\times n$或者$n\times m$的矩阵.为使表述简单，我们取标准正交基$$\varepsilon_i=\begin{bmatrix}0\\\vdots\\1\\\vdots\\0\end{bmatrix}\in\mathbb{R}^m,其中1在第i行.$$设$$\sigma(\varepsilon_i)=\mathcal{A}_i\in\mathbb{R}^n.$$取$\alpha\in\mathbb{R}^m$,设$\alpha=\begin{bmatrix}a_1 \\a_2\\ \vdots \\a_m\end{bmatrix}=a_1\varepsilon_1+a_2+\varepsilon_2+\cdots+a_m\epsilon_m$,则$$\sigma(\alpha)=\sigma(a_1\varepsilon_1+a_2\varepsilon_2+\cdots+a_m\varepsilon_m)=a_1\sigma(\varepsilon_1)+a_2\sigma(\varepsilon_2)+\cdots+a_m\sigma(\varepsilon_m)$$带入得$$\sigma(\alpha)=a_1\mathcal{A}_1+a_2\mathcal{A}_2+\cdots+a_m\mathcal{A}_m=\begin{bmatrix}\mathcal{A}_1\ \mathcal{A}_2\ \cdots\ \mathcal{A}_m\end{bmatrix}\begin{bmatrix}a_1\\a_2\\\vdots\\a_m\end{bmatrix}$$也就是说，对$\alpha$作用$\sigma$等价于对$\alpha$的坐标左乘一个$n$行$m$列的矩阵$$A=\begin{bmatrix}\mathcal{A}_1\ \mathcal{A}_2\ \cdots\ \mathcal{A}_m\end{bmatrix},$$也就是说，一个线性映射对应一个矩阵.反过来，一个矩阵能否对应一个从$\mathbb{R}^m$到$\mathbb{R}^n$的线性映射呢？ +取$A\in\mathbb{R}^{n\times m}$,设$A=[\mathcal{A}_1\ \mathcal{A}_2\ \cdots\ \mathcal{A}_m\ ]$,取线性映射$\sigma:\mathbb{R}^m\to\mathbb{R}^n$，满足$$\sigma(\varepsilon_i)=\mathcal{A_i}$$于是若$\alpha=\begin{bmatrix}a_1\\a_2\\\vdots\\a_m\end{bmatrix}\in\mathbb{R}^m$,有$$\begin{aligned}\sigma(\alpha)&=\sigma(a_1\varepsilon_1+a_2\varepsilon_2+\cdots+a_m\varepsilon_m)\\&=a_1\sigma(\varepsilon_1)+a_2\sigma(\varepsilon_2)+\cdots+a_m\sigma(\varepsilon_m)\\&=a_1\mathcal{A}_1+a_2\mathcal{A}_2+\cdots+a_m\mathcal{A}_m\\&=A\begin{bmatrix}a_1\\a_2\\\vdots\\a_m\end{bmatrix}\\&=A\alpha\end{aligned}$$于是一个矩阵也对应一个线性映射.从而有如下定理： +>[!note] 定理2： +>从$\mathbb{R}^m$到$\mathbb{R}^n$的线性映射全体与$\mathbb{R}^{n\times m}$中的矩阵一一对应. + +由于每个线性空间都能由其基与一个向量空间构成同构，我们就能凭借上面这个定理把任意线性空间中的一个线性映射与一个矩阵一一对应起来. \ No newline at end of file diff --git a/笔记分享/聚点和上下极限.md b/笔记分享/聚点和上下极限.md new file mode 100644 index 0000000..7ab39d9 --- /dev/null +++ b/笔记分享/聚点和上下极限.md @@ -0,0 +1,29 @@ +## **聚点和上下极限** +此知识点仅作为拓展了解，不要求掌握。 +### 一、聚点 +聚点是研究实数性质的重要概念，这里先下定义： +>[!note] **定义：** +>对于一集合$I\subset\mathbb{R}$，如果存在一点$x_0\in \mathbb{R}$,使得$\forall \delta>0$，有$\overset{\circ}{U}(x_0,\delta)\cap I\neq\varnothing$,则称$x_0$是$I$的一个**聚点**. + +**注：** +1、$x_0$未必是$I$中的一个点； +2、一个集合$I$存在聚点$x_0$与以下两个命题等价： + （1）$\forall \delta>0$,在$U(x_0,\delta)$中有$I$中的无穷多个点； + （2）存在$I$中互异的点组成的数列$\{x_n\}$使得$\lim\limits_{n\to\infty}x_n=x_0$; +3、若$x_0\in I$，但它不是$I$的一个聚点，则称之为孤立点. + +对聚点有如下定理： +>[!note] 聚点原理 +>$\mathbb{R}$的任何一个有界无穷子集至少有一个聚点. + +证明略，有兴趣的可以自己去查阅任何数学分析教材。需要指出的是，它与单调有界原理是等价的。实际上，有四五个与之等价的命题，它们之间互相等价，一般会取一个（一般是戴德金分割定理或者确界原理）作为公理，推出另外几个。它们描述的都是实数的连续性。 + +### **二、上下极限** +我们都知道有左右极限，但其实函数和数列都有一个上下极限。这里给出数列上下极限的定义，有兴趣的也同样可以去找数学分析教材。 +对数列$\{x_n\}$，定义$$l_n=\inf\{x_n,x_{n+1},\cdots\} , h_n=\sup\{x_n,x_{n+1},\cdots\}$$其中$\inf$和$\sup$分别表示下确界和上确界.显然有$$l_1\le l_2\le\cdots l_{n}\le\cdots\le h_n\le\cdots h_2\le\cdots h_1.$$于是数列$\{l_n\}$和$\{h_n\}$是单调有界数列，由单调有界原理，它们都存在极限，且$$\lim\limits_{n\to\infty}l_n=\sup\{l_n\},\lim\limits_{n\to\infty}h_n=\inf\{h_n\}.$$记$$\overset{\_\_\_\_}{\lim\limits_{n\to\infty}}x_n=\lim\limits_{n\to\infty}h_n,\underset{n\to\infty}{\underline{\lim}}x_n=\lim\limits_{n\to\infty}l_n$$ +分别为数列$\{x_n\}$的上、下极限。容易证明，$\{x_n\}$极限存在等价于其上下极限都存在且相等。 + + + + + diff --git a/笔记分享/达布定理.md b/笔记分享/达布定理.md new file mode 100644 index 0000000..8c1c529 --- /dev/null +++ b/笔记分享/达布定理.md @@ -0,0 +1,4 @@ +>[!note] 定理： +>如果函数$f(x)$在区间$[a,b]$上可导，则其导函数$f'(x)$在$[a,b]$上有介值性质，即若$f(x)$在$[a,b]$上的值域为$[m,M]$,则$\forall \xi\in[m,M]$，总$\exists \eta\in[a,b],$有$\xi=f'(\eta)$. + +**证明**：若$m=M$，结论显然成立.若$m0.$$由零值定理，$\exists \eta\in(x_1,x_2) \subset(a,b),g'(\eta)=0\implies f'(\eta)=\xi$，证毕. \ No newline at end of file diff --git a/素材/1.11题目素材.md b/素材/1.11题目素材.md new file mode 100644 index 0000000..35c8c84 --- /dev/null +++ b/素材/1.11题目素材.md @@ -0,0 +1,64 @@ +设 $A, B$ 为 $n$ 阶矩阵，则 +(A) $\mathrm{rank}\begin{bmatrix}A&AB\end{bmatrix}=\mathrm{rank}A$ +(B) $\mathrm{rank}\begin{bmatrix}A&BA\end{bmatrix}=\mathrm{rank}A$ +(C) $\mathrm{rank}\begin{bmatrix}A&B\end{bmatrix}=\max{\{\mathrm{rank}A,\mathrm{rank}B\}}$ +(D) $\mathrm{rank}\begin{bmatrix}A&B\end{bmatrix}=\mathrm{rank}\begin{bmatrix}A^T&B^T\end{bmatrix}$ + +（[[1.10线代限时练]]）设 $A$ 是 $m\times n$ 实矩阵， $\beta \neq 0$ 是 $m$ 维实列向量，证明： + (1) $\mathrm{rank}A=\mathrm{rank}(A^\mathrm{T}A)$ . + (2) 线性方程组 $A^\mathrm{T}Ax = A^\mathrm{T}\beta$ 有解. + +设 $A$ 是 $m\times n$ 矩阵， $B$ 是 $n\times m$ 矩阵，$E$ 是 $m$ 阶单位矩阵，若 $AB=E$，则 +(A) $\mathrm{rank}A=m,\mathrm{rank}A=m$ +(B) $\mathrm{rank}A=m,\mathrm{rank}A=n$ +(C) $\mathrm{rank}A=n,\mathrm{rank}A=m$ +(D) $\mathrm{rank}A=n,\mathrm{rank}A=n$ + +已知 $A,B,C,D$ 都是 $4$ 阶非零矩阵，且 $ABCD=O$，如果 $|BC|\ne 0$，记 $\mathrm{rank}A+\mathrm{rank}B+\mathrm{rank}C+\mathrm{rank}A=r$，则 $r$ 的最大值是 +(A) $11$ +(B) $12$ +(C) $13$ +(D) $14$ + +已知 $A,B,C$ 都是 $n$ 阶非零矩阵，且 $ABC=O$，$E$ 是 $n$ 阶单位矩阵，记 $\begin{bmatrix}O&A\\BC&E\end{bmatrix}\begin{bmatrix}AB&C\\O&E\end{bmatrix},\begin{bmatrix}E&AB\\AB&O\end{bmatrix}$ 的秩分别是 $r_1,r_2,r_3$，则 +(A) $r_1 \le r_2 \le r_3$ +(B) $r_1 \le r_3 \le r_2$ +(C) $r_3 \le r_1 \le r_2$ +(D) $r_2 \le r_1 \le r_3$ + +设 $A,B$ 都是 $n$ 阶矩阵，求证：$\mathrm{rank}(AB-E) \le \mathrm{rank}(A-E)+\mathrm{rank}(B-E)$ + +设 $A$ 为 $n$ 阶矩阵，$1v65L8eaq&uV}Rk{SEM7p~qq*bJ(1w^E!nRFwP64E8zA|ZM1 zxz@YCz0bM!-`~7kE}2Z87|$5@{i{Kknu;7D9u*z}fgn_nfA|D}z?VQEuy5gD!8;Fc zcB;Vt(43ygNg;~*sMp{N3=2tRNd%%a2LH?$6TZf^m)CJZAZS8S|DnY!?U^AEmpKX# zCDq*xH>Z8{)m>)9FUGzj9!&MwvwZ%X$j@smS?SYC+Sn{0^!Fld-18>vkUva7t7xX2;A_F zJ+~M4e~P3(XiYK}NOxM^7uvaflixPw#>4;hCGD?Y>;L+Hds&v4AU^zW-*_EDfRz#U zzkSED%jo}?A9)jXlfRGEK>ok)h=uEW391_5zBFIbQX{u|!d48W`~J=#YIuN%GHY@J*ZD>;E1w z%YoQ#Qye_JK;1&~*sZ2F_N>UP%$8^;-HT1V`N!0%5hX0=ji*@(evuEz^ z?*D#R94=}1;2@6uKUYh2?YjT}yl^@8ga_~83VE8meSGK>oWA|nDAK0&Ui){1;P1hV z#g!H8Bqc@zCDhf*Gsf-(-v0OQsowqH2b5uqZSLt&{O97Z{Hv$MW&t0#Hy|G!Hd{^y8PEz7;hp&t9o3aGmtM9ftyK78oB#2Exv 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+>设函数$f(x)$在闭区间$[0,2]$上可导，且$\large{\int}_{0}^{1}f(x)\mathrm{d}x=0$.证明：至少存在一点$\xi\in(0,2)$，使得$f'(\xi)=\frac{2}{2-\xi}f(\xi)$ + diff --git a/素材/微分中值定理的不等式问题.md b/素材/微分中值定理的不等式问题.md new file mode 100644 index 0000000..730737f --- /dev/null +++ b/素材/微分中值定理的不等式问题.md @@ -0,0 +1,134 @@ + +## 微分中值定理证明不等式的要点归纳 + +### 1. **识别不等式结构** + - 若不等式形如 $f(b) - f(a)$ 与 $b-a$ 的关系，或含有函数值差与自变量差之商，可考虑**拉格朗日中值定理**。 + + +### 2. **选择合适定理与辅助函数** + - **拉格朗日定理**：常用于"单函数"差值型不等式，构造 $f(x)$ 使 $f'(\xi)$ 出现在不等式中。 + - **柯西定理**：适用于"双函数"比值型不等式，构造 $f(x), g(x)$ 使 $\frac{f'(\xi)}{g'(\xi)}$ 出现。 + - **辅助函数构造**：常借助常见函数如 $\ln x, e^x, x^n, \arctan x, \sin x, \cos x$ 等，通过求导形式匹配目标。 + +### 3. **利用导数单调性估计中值** + - 应用中值定理得到含 $\xi$ 的表达式后，可以通过函数极值的求法求出其最大最小值进行比较 + + +## 例一 +设 $e < a < b < e^2$，证明： +$$ +\ln^2 b - \ln^2 a > \frac{4}{e^2}(b-a). +$$ + +**证明**： +考虑函数 $f(x) = \ln^2 x$，则 $f(x)$ 在 $[a,b]$ 上连续，在 $(a,b)$ 内可导。由拉格朗日中值定理，存在 $\xi \in (a,b)$，使得 +$$ +\frac{\ln^2 b - \ln^2 a}{b-a} = f'(\xi) = \frac{2\ln \xi}{\xi}. +$$ +令 $g(x) = \dfrac{2\ln x}{x}$，求导得 +$$ +g'(x) = \frac{2(1-\ln x)}{x^2}. +$$ +当 $x > e$ 时，$\ln x > 1$，故 $g'(x) < 0$，即 $g(x)$ 在 $(e, +\infty)$ 上单调递减。 +由于 $e < a < \xi < b < e^2$，所以 +$$ +g(\xi) > g(e^2) = \frac{2\ln e^2}{e^2} = \frac{4}{e^2}. +$$ +因此 +$$ +\frac{\ln^2 b - \ln^2 a}{b-a} > \frac{4}{e^2}, +$$ +即 +$$ +\ln^2 b - \ln^2 a > \frac{4}{e^2}(b-a). +$$ +证毕。 +## 例2 +设 $a > e$，$0 < x < y < \dfrac{\pi}{2}$，证明： +$$ +a^y - a^x > (\cos x - \cos y) \cdot a^x \ln a. +$$ + +**证明**： +令 $f(t) = a^t$，则 $f(t)$ 在 $[x, y]$ 上连续，在 $(x, y)$ 内可导。由拉格朗日中值定理，存在 $\xi \in (x, y)$，使得 +$$ +\frac{a^y - a^x}{\cos x - \cos y} = \frac{a^\xi \ln a}{\sin \xi } +$$ + $$\frac{a^\xi \ln a}{\sin \xi }>a^\xi \ln a>a^x \ln a$$ + 证毕 +## 例3 +证明：当 $x>0$ 时， +$$ +\frac{\arctan x}{\ln(1+x)} \leq \frac{1+\sqrt{2}}{2}. +$$ + +**证明**： +考虑函数 $f(t) = \arctan t$ 与 $g(t) = \ln(1+t)$，两者在 $[0, x]$ 上连续，在 $(0, x)$ 内可导，且 $g'(t) = \frac{1}{1+t} \neq 0$。由柯西中值定理，存在 $\xi \in (0, x)$，使得 +$$ +\frac{\arctan x}{\ln(1+x)} = \frac{f(x) - f(0)}{g(x) - g(0)} = \frac{f'(\xi)}{g'(\xi)} = \frac{1/(1+\xi^2)}{1/(1+\xi)} = \frac{1+\xi}{1+\xi^2}. +$$ +令 $\phi(\xi) = \dfrac{1+\xi}{1+\xi^2}$，则 +$$ +\phi'(\xi) = \frac{(1+\xi^2) - (1+\xi) \cdot 2\xi}{(1+\xi^2)^2} = \frac{1 - 2\xi - \xi^2}{(1+\xi^2)^2} = \frac{2 - (1+\xi)^2}{(1+\xi^2)^2}. +$$ +令 $\phi'(\xi) = 0$，得 $(1+\xi)^2 = 2$，因 $\xi > 0$，故 $\xi = \sqrt{2} - 1$。 +当 $0 < \xi < \sqrt{2} - 1$ 时，$\phi'(\xi) > 0$；当 $\xi > \sqrt{2} - 1$ 时，$\phi'(\xi) < 0$。 +因此 $\phi(\xi)$ 在 $\xi = \sqrt{2} - 1$ 处取得最大值： +$$ +\phi(\sqrt{2} - 1) = \frac{1 + (\sqrt{2} - 1)}{1 + (\sqrt{2} - 1)^2} = \frac{\sqrt{2}}{1 + (3 - 2\sqrt{2})} = \frac{\sqrt{2}}{4 - 2\sqrt{2}} = \frac{\sqrt{2}}{2(2 - \sqrt{2})}. +$$ +化简： +$$ +\frac{\sqrt{2}}{2(2 - \sqrt{2})} = \frac{\sqrt{2}}{2} \cdot \frac{1}{2 - \sqrt{2}} = \frac{\sqrt{2}}{2} \cdot \frac{2 + \sqrt{2}}{2} = \frac{\sqrt{2}(2 + \sqrt{2})}{4} = \frac{2\sqrt{2} + 2}{4} = \frac{1 + \sqrt{2}}{2}. +$$ +于是对任意 $\xi > 0$，有 $\phi(\xi) \leq \dfrac{1+\sqrt{2}}{2}$，从而 +$$ +\frac{\arctan x}{\ln(1+x)} \leq \frac{1+\sqrt{2}}{2}, \quad x > 0. +$$ +等号在 $\xi = \sqrt{2} - 1$ 时成立，即存在 $x > 0$ 使等号成立。证毕。 + +## 例3 + +(1) 证明：存在 $\theta \in (0, 1)$ 使得 $\ln(1+x) - \ln\left(1+\frac{x}{2}\right) = \frac{x}{2+(1+\theta)x}, \, x > 0$; +(2) 证明不等式 +$$ +\left(1+\frac{1}{n}\right)^{n+1} < e\left(1+\frac{1}{2n}\right), +$$ +其中 $n$ 为正整数。 + +## 解答 + +**证明** +（1）对 $x > 0$ 定义函数 $f(t) = \ln(1+t), t \in \left[\frac{x}{2}, x\right]$, +由拉格朗日中值定理知：存在 $\theta \in (0, 1)$ 使得 + +$$ +\begin{aligned} +f(x) - f\left(\frac{x}{2}\right) &= \ln(1+x) - \ln\left(1+\frac{x}{2}\right) \\ +&= \frac{1}{1+\frac{x}{2}+\theta} \cdot \frac{x}{2} \\ +&= \frac{x}{2+(1+\theta)x}. +\end{aligned} +$$ + +（2）不等式两边取对数，可知仅证明 $(n+1)\ln\left(1+\frac{1}{n}\right) < 1 + \ln\left(1+\frac{1}{2n}\right)$ 即可。 + +令 $F(x) = x + x\ln\left(1+\frac{x}{2}\right) - (x+1)\ln(1+x), x \geq 0$，则由（1）知 + +$$ +\begin{aligned} +F'(x) &= 1 + \frac{\frac{x}{2}}{1+\frac{x}{2}} + \ln\left(1+\frac{x}{2}\right) - 1 - \ln(1+x) \\ +&= \frac{\frac{x}{2}}{1+\frac{x}{2}} - \left[\ln(1+x) - \ln\left(1+\frac{x}{2}\right)\right] \\ +&= \frac{\frac{x}{2}}{1+\frac{x}{2}} - \frac{\frac{x}{2}}{1+(1+\theta)\frac{x}{2}} \\ +&= \frac{\frac{x}{2}}{1+\frac{x}{2}} - \frac{\frac{x}{2}}{1+\frac{x}{2}} = 0. +\end{aligned} +$$ + +因此 $F(x) > F(0) = 0, x > 0$。即 $(x+1)\ln(1+x) < x + x\ln\left(1+\frac{x}{2}\right), x > 0$。 + +令 $x = \frac{1}{n}$，则有 $(n+1)\ln\left(1+\frac{1}{n}\right) < 1 + \ln\left(1+\frac{1}{2n}\right)$。因此对任意正整数 $n$ 有不等式 + +$$ +\left(1+\frac{1}{n}\right)^{n+1} < e\left(1+\frac{1}{2n}\right) +$$ + +成立。 \ No newline at end of file diff --git a/素材/微分中值定理部分题目汇总.md b/素材/微分中值定理部分题目汇总.md new file mode 100644 index 0000000..5b1bf51 --- /dev/null +++ b/素材/微分中值定理部分题目汇总.md @@ -0,0 +1,202 @@ + +>[!example] 例1 +设 $f(x)$ 在 $[a, b]$ 上连续，在 $(a, b)$ 内可导，且 $0 < a < b$，试证存在 $\xi, \eta \in (a, b)$，使得 $$f'(\xi) = \frac{a + b}{2\eta} f'(\eta).$$ + +**解析**： +本题结论中含有两个不同的中值 $\xi$ 和 $\eta$，且涉及两个不同的函数形式。可考虑分别对 $f(x)$ 和 $g(x)=x^2$ 在 $[a,b]$ 上应用柯西中值定理： +由柯西中值定理，存在 $\eta \in (a,b)$，使得 +$$ +\frac{f(b)-f(a)}{b^2-a^2} = \frac{f'(\eta)}{2\eta} +$$ +整理得 +$$ +\frac{f(b)-f(a)}{b-a} = \frac{a+b}{2\eta} f'(\eta) +$$ +再对 $f(x)$ 在 $[a,b]$ 上应用拉格朗日中值定理，存在 $\xi \in (a,b)$，使得 +$$ +\frac{f(b)-f(a)}{b-a} = f'(\xi) +$$ +比较两式即得结论。 + +--- + +>[!example] 例2 +设函数 $f(x)$ 在 $[0,3]$ 上连续，在 $(0,3)$ 内可导，且 $f(0) + f(1) + f(2) = 3$，$f(3) = 1$，试证必存在 $\xi \in (0, 3)$，使 $f'(\xi) = 0$。 + +**解析**： +由介值定理，$f(x)$ 在 $[0,2]$ 上的平均值为 $\frac{f(0)+f(1)+f(2)}{3} = 1$，又 $f(3)=1$，由连续函数介值定理，存在 $c \in [0,2]$，使得 $f(c)=1$，则在 $[c,3]$ 上，$f(c)=f(3)=1$，由罗尔定理存在 $\xi \in (c,3) \subset (0,3)$，使 $f'(\xi)=0$。 + +--- + +>[!example] 例3 +设 $f(x)$ 在区间 $[0,1]$ 上连续，在 $(0,1)$ 内可导，且 $f(0) = f(1) = 0$，$f(1/2) = 1$，试证： +(1)存在 $\eta \in (1/2, 1)$，使得 $f(\eta) = \eta$； +(2)对任意实数 $\lambda$，必存在 $\xi \in (0, \eta)$，使得 $f'(\xi) - \lambda [f(\xi) - \xi] = 1$。 + +**解析**： +(1) 令 $g(x)=f(x)-x$，则 $g(1/2)=1-1/2=1/2>0$，$g(1)=0-1=-1<0$，由零点定理，存在 $\eta \in (1/2,1)$，使 $g(\eta)=0$，即 $f(\eta)=\eta$。 +(2) 令 $h(x)=e^{-\lambda x}[f(x)-x]$，则 $h(0)=0$，$h(\eta)=0$，由罗尔定理，存在 $\xi \in (0,\eta)$，使 $h'(\xi)=0$，即 +$$ +e^{-\lambda \xi}[f'(\xi)-1] - \lambda e^{-\lambda \xi}[f(\xi)-\xi] = 0 +$$ +整理得 $f'(\xi) - \lambda [f(\xi) - \xi] = 1$。 + +--- + +## 5.2 微分中值定理及其应用 + +>[!example] 例1 +设函数 $f(x)$ 在 $(-1,1)$ 内可微，且 +$$f(0) = 0, \quad |f'(x)| \leq 1,$$证明：在 $(-1,1)$ 内，$|f(x)| < 1$。 + +**解析**： +对任意 $x \in (-1,1)$，由拉格朗日中值定理，存在 $\xi$ 介于 $0$ 与 $x$ 之间，使得 +$$ +f(x) - f(0) = f'(\xi)(x-0) +$$ +即 $f(x) = f'(\xi) x$。由于 $|f'(\xi)| \leq 1$，$|x| < 1$，故 $|f(x)| = |f'(\xi)| \cdot |x| < 1$。 + +--- + +>[!example] 例2 +设 $a_i \in \mathbb{R} (i = 0,1,2,\cdots,n)$，且满足 +$$a_0 + \frac{a_1}{2} + \frac{a_2}{3} + \cdots + \frac{a_n}{n+1} = 0，$$证明：方程 $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n = 0$ 在 $(0,1)$ 内至少有一个实根。 + +**解析**： +构造辅助函数 +$$ +F(x) = a_0x + \frac{a_1}{2}x^2 + \frac{a_2}{3}x^3 + \cdots + \frac{a_n}{n+1}x^{n+1} +$$ +则 $F(0)=0$，且由条件 $F(1)=0$。由罗尔定理，存在 $\xi \in (0,1)$，使 $F'(\xi)=0$，即 +$$ +a_0 + a_1\xi + a_2\xi^2 + \cdots + a_n\xi^n = 0 +$$ + +--- + +>[!example] 例3 +设函数 $f(x)$ 在 $[a,b]$ 上可导，且 +$$f(a) = f(b) = 0，\quad f'_+(a)f'_-(b) > 0，$$ +试证明 $f'(x) = 0$ 在 $(a,b)$ 内至少有两个根。 + +**解析**： +由导数极限定理及 $f'_+(a)f'_-(b) > 0$，知在 $a$ 右侧和 $b$ 左侧，$f(x)$ 的符号相同，不妨设 $f'_+(a)>0$，$f'_-(b)>0$。则在 $a$ 右侧附近 $f(x)>0$，在 $b$ 左侧附近 $f(x)>0$。由于 $f(a)=f(b)=0$，由极值点的费马定理，$f(x)$ 在 $(a,b)$ 内至少有一个极大值点，该点处导数为零。又因为 $f(x)$ 在 $[a,b]$ 上连续，在 $(a,b)$ 内可导，且 $f(a)=f(b)$，由罗尔定理至少存在一点 $c \in (a,b)$ 使 $f'(c)=0$。结合极大值点处的导数零点，可知至少有两个导数为零的点。 + +--- + +>[!example] 例4 +设 $f(x)$ 在 $[a, b]$ 上连续，在 $(a, b)$ 内二阶可导，又若 $f(x)$ 的图形与联结 $A(a, f(a))$，$B(b, f(b))$ 两点的弦交于点 $C(c, f(c))$ ($a \leq c \leq b$)，证明在 $(a, b)$ 内至少存在一点 $\xi$，使得 $f''(\xi) = 0$。 + +**解析**： +弦 $AB$ 的方程为 +$$ +y = f(a) + \frac{f(b)-f(a)}{b-a}(x-a) +$$ +由条件，$f(c) = f(a) + \frac{f(b)-f(a)}{b-a}(c-a)$。分别对 $f(x)$ 在 $[a,c]$ 和 $[c,b]$ 上应用拉格朗日中值定理，存在 $\xi_1 \in (a,c)$，$\xi_2 \in (c,b)$，使得 +$$ +f'(\xi_1) = \frac{f(c)-f(a)}{c-a} = \frac{f(b)-f(a)}{b-a} +$$ +$$ +f'(\xi_2) = \frac{f(b)-f(c)}{b-c} = \frac{f(b)-f(a)}{b-a} +$$ +故 $f'(\xi_1)=f'(\xi_2)$。再对 $f'(x)$ 在 $[\xi_1,\xi_2]$ 上应用罗尔定理，存在 $\xi \in (\xi_1,\xi_2) \subset (a,b)$，使 $f''(\xi)=0$。 + +--- + +>[!example] 例5（柯西中值定理例） +试证至少存在一点 $\xi \in (1, e)$，使 $\sin 1 = \cos \ln \xi$。 + +**解析**： +考虑函数 $f(x)=\sin(\ln x)$，$g(x)=\ln x$，在 $[1,e]$ 上应用柯西中值定理： +存在 $\xi \in (1,e)$，使得 +$$ +\frac{f(e)-f(1)}{g(e)-g(1)} = \frac{f'(\xi)}{g'(\xi)} +$$ +计算得 $f(e)=\sin 1$，$f(1)=0$，$g(e)=1$，$g(1)=0$，$f'(x)=\frac{\cos(\ln x)}{x}$，$g'(x)=\frac{1}{x}$，代入得 +$$ +\frac{\sin 1 - 0}{1-0} = \frac{\cos(\ln \xi)/\xi}{1/\xi} = \cos(\ln \xi) +$$ +即 $\sin 1 = \cos(\ln \xi)$。 + +--- + +## 练习 + +>[!example] Ex1 +设 $f(x)$ 在 $(a, b)$ 内可导，且 $f'(x) \neq 1$。试证明 $f(x)$ 在 $(a, b)$ 内至多只有一个不动点，即方程 $f(x) = x$ 在 $(a, b)$ 内至多只有一个实根。 + +**解析**： +反证法。假设存在两个不动点 $x_1 < x_2$，即 $f(x_1)=x_1$，$f(x_2)=x_2$。由拉格朗日中值定理，存在 $\xi \in (x_1,x_2)$，使得 +$$ +f'(\xi) = \frac{f(x_2)-f(x_1)}{x_2-x_1} = \frac{x_2-x_1}{x_2-x_1} = 1 +$$ +与 $f'(x) \neq 1$ 矛盾。故至多只有一个不动点。 + +--- + +>[!example] Ex2 +设 $f(x)$ 在 $[0, 1]$ 上具有二阶导数，且满足 +$$f(0) = 0, \, f(1) = 1, \, f\left(\frac{1}{2}\right) > \frac{1}{4}$$证明： +(1)至少存在一点 $\xi \in (0, 1)$，使得 $f''(\xi) < 2$； +(2)若对一切 $x \in (0, 1)$，有 $f''(x) \neq 2$，则当 $x \in (0, 1)$ 时，恒有 $f(x) > x^2$。 + +**解析**： +(1) 考虑函数 $g(x)=f(x)-x^2$，则 $g(0)=0$，$g(1)=0$，$g(1/2)=f(1/2)-1/4>0$。由极值点的费马定理及罗尔定理，$g(x)$ 在 $(0,1)$ 内存在极大值点 $\eta$，且 $g'(\eta)=0$，$g''(\eta) \leq 0$。即 $f'(\eta)=2\eta$，$f''(\eta) \leq 2$。若 $f''(\eta) < 2$，则取 $\xi=\eta$ 即可；若 $f''(\eta)=2$，则考虑在 $\eta$ 两侧应用拉格朗日中值定理，可找到另一个点 $\xi$ 使得 $f''(\xi)<2$。 +(2) 用反证法。假设存在 $x_0 \in (0,1)$ 使 $f(x_0) \leq x_0^2$，结合 $f(0)=0$，$f(1)=1$ 和 $f(1/2)>1/4$，利用连续性及中值定理可推出存在 $\xi$ 使 $f''(\xi)=2$，矛盾。 + +--- + +>[!example] Ex3 +若 $f(x)$ 可导，试证在其两个零点间一定有 $f(x) + f'(x)$ 的零点。 + +**解析**： +设 $a[!example] Ex4 +设 $f(x)$ 在 $[0,1]$ 连续，$(0,1)$ 可导，且 $f(1) = 0$，求证存在 $\xi \in (0,1)$ 使得 $nf(\xi) + \xi f'(\xi) = 0$。 + +**解析**： +设辅助函数 $\varphi(x) = x^n f(x)$，则 $\varphi(0)=0$，$\varphi(1)=0$。由罗尔定理，存在 $\xi \in (0,1)$，使得 $\varphi'(\xi)=0$，即 +$$ +n\xi^{n-1} f(\xi) + \xi^n f'(\xi) = 0 +$$ +两边除以 $\xi^{n-1}$ ($\xi>0$)，得 $nf(\xi) + \xi f'(\xi) = 0$。 + +--- + +>[!example] Ex5 +设 $f''(x) < 0$，$f(0) = 0$，证明对任意 $x_1 > 0, x_2 > 0$ 有 +$$f(x_1 + x_2) < f(x_1) + f(x_2)$$ + +**解析**： +不妨设 $0 < x_1 < x_2$。由拉格朗日中值定理： +$$ +f(x_1+x_2)-f(x_2) = f'(\xi_1)x_1, \quad \xi_1 \in (x_2, x_1+x_2) +$$ +$$ +f(x_1)-f(0) = f'(\xi_2)x_1, \quad \xi_2 \in (0, x_1) +$$ +于是 +$$ +f(x_1+x_2)-f(x_2)-f(x_1) = [f'(\xi_1)-f'(\xi_2)]x_1 +$$ +对 $f'(x)$ 在 $[\xi_2,\xi_1]$ 上应用中值定理，存在 $\xi \in (\xi_2,\xi_1)$，使 +$$ +f'(\xi_1)-f'(\xi_2) = f''(\xi)(\xi_1-\xi_2) < 0 +$$ +故 $f(x_1+x_2)-f(x_2)-f(x_1) < 0$，即 $f(x_1+x_2) < f(x_1)+f(x_2)$。 + +--- + +## 解题方法总结 +1. **含一个中值的等式或根的存在**：多用罗尔定理，可用原函数法找辅助函数。 +2. **结论涉及含中值的两个不同函数**：可考虑用柯西中值定理。 +3. **结论中含两个或两个以上的中值**：必须多次应用中值定理。 +4. **已知条件中含高阶导数**：多考虑用泰勒公式，有时也可考虑对导数用中值定理。 +5. **结论为不等式**：要注意适当放大或缩小的技巧。 \ No newline at end of file diff --git a/素材/拐点是一个点.md b/素材/拐点是一个点.md new file mode 100644 index 0000000..65cf990 --- /dev/null +++ b/素材/拐点是一个点.md @@ -0,0 +1 @@ +拐点一定要是点 \ No newline at end of file diff --git a/素材/柯西中值定理与常见辅助函数.md b/素材/柯西中值定理与常见辅助函数.md new file mode 100644 index 0000000..76332e8 --- /dev/null +++ b/素材/柯西中值定理与常见辅助函数.md @@ -0,0 +1,230 @@ +## **柯西中值定理** + +### **原理** + +设函数 $f(x)$ 和 $g(x)$ 满足以下条件： +1. 在闭区间 $[a, b]$ 上连续； +2. 在开区间 $(a, b)$ 内可导； +3. 对任意 $x \in (a, b)$，有 $g'(x) \neq 0$； +则在 $(a, b)$ 内至少存在一点 $\xi$，使得： + +$$ +\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(\xi)}{g'(\xi)} +$$ + +柯西中值定理的几何意义为：由参数方程 $(g(t), f(t))$ 表示的曲线，在两点间的割线斜率等于曲线上某点切线的斜率。 + +它与拉格朗日中值定理的关系为：当 $g(x) = x$ 时，柯西中值定理退化为拉格朗日中值定理。它是处理两个函数之间微分中值关系的通用形式。 + +### **适用条件** + +柯西中值定理的核心适用题型是**证明形如 $\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(\xi)}{g'(\xi)}$ 的等式成立**，以及处理**涉及两个中值点 $\xi, \eta$ 的问题**。 + +常见应用方向包括： +1. 直接证明存在性等式； +2. 通过函数配对，将目标等式转化为柯西中值定理的标准形式； +3. 处理“双中值问题”，常与拉格朗日中值定理结合使用。 + +### **例题** + +>[!example] 例1 +设 $f(x)$ 在 $[a, b]$ 上连续，在 $(a, b)$ 内可导，且 $a>0$。证明存在 $\xi \in (a, b)$，使得： +$$\frac{f(b)-f(a)}{b-a} = \xi f'(\xi) \cdot \frac{\ln(b/a)}{b-a}$$ + +**解析**： +将等式变形为： +$$ +\frac{f(b)-f(a)}{\ln b - \ln a} = \xi f'(\xi) +$$ +取 $g(x) = \ln x$，则 $g'(x) = \frac{1}{x} \neq 0$ 在 $(a, b)$ 内成立。 +对 $f(x)$ 与 $g(x)$ 应用柯西中值定理，存在 $\xi \in (a, b)$ 使得： +$$ +\frac{f(b)-f(a)}{\ln b - \ln a} = \frac{f'(\xi)}{1/\xi} = \xi f'(\xi) +$$ +整理即得所求。 + +--- + +>[!example] 例2 +设 $f(x)$ 在 $[a, b]$ 上连续，在 $(a, b)$ 内可导，证明存在不同的 $\xi, \eta \in (a, b)$，使得： +$$f'(\xi) = \frac{a+b}{2\eta} f'(\eta)$$ + +**解析**： +1. 对 $f(x)$ 与 $g(x) = \frac{x^2}{2}$ 应用柯西中值定理，存在 $\eta \in (a, b)$ 使得： + $$ + \frac{f(b)-f(a)}{(b^2 - a^2)/2} = \frac{f'(\eta)}{\eta} + $$ + 整理得： + $$ + f(b)-f(a) = \frac{b^2 - a^2}{2\eta} f'(\eta) + $$ + +2. 对 $f(x)$ 应用拉格朗日中值定理，存在 $\xi \in (a, b)$ 使得： + $$ + f(b)-f(a) = (b-a) f'(\xi) + $$ + +3. 联立两式，消去 $f(b)-f(a)$ 得： + $$ + (b-a) f'(\xi) = \frac{(b-a)(a+b)}{2\eta} f'(\eta) + $$ + 由于 $b-a \neq 0$，约去后即得： + $$ + f'(\xi) = \frac{a+b}{2\eta} f'(\eta) + $$ + +--- + +>[!example] 例3 +设 $0 < a < b$，证明存在 $\xi \in (a, b)$，使得： +$$f(b)-f(a) = \frac{3\xi^2}{a^2+ab+b^2} f'(\xi)(b-a)$$ + +**解析**： +将等式变形为： +$$ +\frac{f(b)-f(a)}{b^3 - a^3} = \frac{f'(\xi)}{3\xi^2} +$$ +取 $g(x) = x^3$，则 $g'(x) = 3x^2 \neq 0$ 在 $(a, b)$ 内成立。 +由柯西中值定理，存在 $\xi \in (a, b)$ 使得： +$$ +\frac{f(b)-f(a)}{b^3 - a^3} = \frac{f'(\xi)}{3\xi^2} +$$ +整理后即得所求。 + + + + +## **辅助函数的构造方法** + +### **原理** + +在证明与导数相关的等式或不等式时，常通过构造辅助函数，将原问题转化为对某个函数应用中值定理（如罗尔定理、拉格朗日定理等）。构造辅助函数的核心思想是：**将待证等式视为某个函数求导后的结果**。 + +### **常见构造类型** + +#### 1. 乘积型与商型 +若结论形如： +$$ +f'(\xi)g(\xi) + f(\xi)g'(\xi) = 0 +$$ +可构造辅助函数： +$$ +F(x) = f(x)g(x) +$$ +若结论形如： +$$ +f'(\xi)g(\xi) - f(\xi)g'(\xi) = 0 +$$ +可构造辅助函数： +$$ +F(x) = \frac{f(x)}{g(x)} \quad (g(x) \neq 0) +$$ + +#### 2. 含幂函数因子 +若结论形如： +$$ +n f(\xi) + \xi f'(\xi) = 0 +$$ +可构造辅助函数： +$$ +F(x) = x^n f(x) +$$ + +#### 3. 一阶线性微分结构 +若结论形如： +$$ +f'(\xi) + P(\xi)f(\xi) = 0 +$$ +可构造积分因子： +$$ +\mu(x) = e^{\int P(x)\mathrm{d}x} +$$ +并设辅助函数： +$$ +F(x) = \mu(x) f(x) +$$ + +#### 4. 对数型 +若结论形如： +$$ +\frac{f'(\xi)}{f(\xi)} = k +$$ +可构造辅助函数： +$$ +F(x) = \ln|f(x)| - kx +$$ + +#### 5. 常数变易法 +若结论形如： +$$ +f'(\xi) = \lambda f(\xi) +$$ +可构造辅助函数： +$$ +F(x) = e^{-\lambda x} f(x) +$$ + +--- + +### **例题** + +>[!example] 例1 +设 $f(x)$ 在 $[0, 1]$ 上连续，在 $(0, 1)$ 内可导，且 $f(0)=0$，$f(1)=1$。 +证明：存在 $\xi \in (0, 1)$ 使得$f'(\xi) = 2\xi f(\xi)$ + +**解析**： +将结论改写为： +$$ +f'(\xi) - 2\xi f(\xi) = 0 +$$ +属于一阶线性微分结构，其中 $P(x) = -2x$。 +积分因子为： +$$ +\mu(x) = e^{\int (-2x)\mathrm{d}x} = e^{-x^2} +$$ +构造辅助函数： +$$ +F(x) = e^{-x^2} f(x) +$$ +则 $F(0) = 0$，$F(1) = e^{-1}$。 +需进一步寻找另一个点 $c$ 使 $F(c)=0$，才可应用罗尔定理。通常需结合题目其他条件（如积分中值定理、零点定理等）找出该点。 + +--- + +>[!example] 例2 +设 $f(x)$ 在 $[a, b]$ 上三阶可导，且 $f(a) = f'(a) = f(b) = 0$。 +证明：存在 $\xi \in (a, b)$ 使得：$f'''(\xi) + k f''(\xi) = 0$ + +**解析**： +结论可写为： +$$ +\bigl[ e^{kx} f''(x) \bigr]' \big|_{x=\xi} = 0 +$$ +因此构造辅助函数： +$$ +H(x) = e^{kx} f''(x) +$$ +由条件可推知存在 $\eta_1, \eta_2 \in (a, b)$ 使 $f''(\eta_1) = f''(\eta_2) = 0$，从而 $H(\eta_1)=H(\eta_2)=0$。 +对 $H(x)$ 应用罗尔定理即得证。 + +--- + +>[!example] 例3 +设 $f(x)$ 在 $[0, 1]$ 上可导，且$f(1) = 2\int_0^{1/2} e^{1-x} f(x) dx$ +证明：存在 $\xi \in (0, 1)$ 使得：$f'(\xi) = (1-\xi) f(\xi)$ + +**解析**： +结论化为： +$$ +f'(\xi) - (1-\xi) f(\xi) = 0 +$$ +积分因子为： +$$ +\mu(x) = e^{\int (x-1) \mathrm{d}x} = e^{\frac{x^2}{2} - x} +$$ +构造辅助函数： +$$ +F(x) = e^{\frac{x^2}{2} - x} f(x) +$$ +利用题设积分条件与积分中值定理，可找到 $\eta \in (0, \frac{1}{2})$ 使 $F(\eta) = F(1)$，再对 $F(x)$ 应用罗尔定理即证。 + diff --git a/素材/洛必达法则-注意事项.md b/素材/洛必达法则-注意事项.md new file mode 100644 index 0000000..fee34d8 --- /dev/null +++ b/素材/洛必达法则-注意事项.md @@ -0,0 +1,50 @@ +--- +tags: + - 素材 +--- +## **洛必达法则证明：** +1.$对于在(a,a+\delta)上连续可导函数f(x),g(x)，且g'(x)\neq0.如果\lim\limits_{x\to a^+}f(x)=0,\lim\limits_{x\to a^+}g(x)=0,$$且\lim\limits_{x\to a^+}\frac{f'(x)}{g'(x)}存在(或为无穷)，则$$$\lim\limits_{x\to a^+}\frac{f(x)}{g(x)}=\lim\limits_{x\to a^+}\frac{f'(x)}{g'(x)}(或无穷)$$ +证明：令$$F(x)=\begin{cases}f(x),x\in(a,a+\delta) \\ 0,x=a\end{cases},G(x)=\begin{cases}g(x),x\in(a,a+\delta) \\ 0,x=a\end{cases},$$显然有$F(x),G(x)$在$[a,a+\delta)$上连续可导.$\forall x\in (a,a+\delta)$,由柯西中值定理，$\exists\xi\in(a,x)$，使得$$\frac{F(x)-F(a)}{G(x)-G(a)}=\frac{F'(\xi)}{G'(\xi)} \Rightarrow \frac{f(x)}{g(x)}=\frac{f'(\xi)}{g'(\xi)},$$令$\xi=a+\theta(x-a),\theta\in(0,1)$,故当$x\to a^+$时，$\xi\to a^+$，从而两边取极限得： +$$\lim\limits_{x\to a^+}\frac{f(x)}{g(x)}=\lim\limits_{\xi\to a^+}\frac{f'(\xi)}{g'(\xi)}=\lim\limits_{x\to a^+}\frac{f'(x)}{g'(x)}.$$得证. +2.类似地可以证明$x\to a^-,x\to a$的情况.若是$x\to +\infty$,则令$t=\frac{1}{x}$,则$t\to0^+$，从而$$\lim\limits_{x\to+\infty}\frac{f(x)}{g(x)}=\lim\limits_{t\to0+}\frac{f(1/t)}{g(1/t)}=\lim\limits_{t\to0^+}\frac{f'(1/t)\cdot(-\frac{1}{t^2})}{g'(1/t)\cdot(-\frac{1}{t^2})}=\lim\limits_{t\to0^+}\frac{f'(1/t)}{g'(1/t)}=\lim\limits_{x\to+\infty}\frac{f'(x)}{g'(x)};$$$x\to-\infty和x\to\infty$类似. +## **注意事项：** +1. 使用时需要在等号上方写明是用的什么类型的洛必达，$\frac{0}{0}$？$\frac{\infty}{\infty}$？ + 例如：$\lim\limits_{x\to\infty}\frac{\mathrm{e}^{2x}+1}{\mathrm{e}^{2x}-1} \overset{\frac{\infty}{\infty}}{=}\lim\limits_{x\to\infty}\frac{2\mathrm{e}^{2x}}{2\mathrm{e}^{2x}}=1$ + +2. 先化简，再使用洛必达 + 适当使用洛必达，不要一直用洛必达，有时用等价无穷小化简更方便$$\lim_{x \to 0} \frac{e^x-e^{-x}-2x}{\tan^{3}x}$$先利用等价无穷小，将 $\tan^{3}x$ 转化为 $x^3$ ，然后再使用洛必达 + +3. 在满足定理的某些情况下洛必达法则不能解决计算问题 + 下面的例子就会导致反复变为倒数$$\lim_{x \to +\infty} \frac{\sqrt{1+x^2}}{x}$$ + +4. 必须要求分子分母的导数极限都存在 + 例如$$\lim_{x \to +\infty} \frac{x+\sin x}{x}$$正确的办法是转换成两个极限相加。 + + +>[!warning] 不能使用洛必达的的情况： +>1、必须要是$\frac{0}{0}$或$\frac{\infty}{\infty}$形式的，其他形式必须转化成前两种形式才能使用洛必达法则； +>2、必须要$\lim\frac{f'(x)}{g'(x)}$存在才能使用； +>3、一些特殊的函数可能会无法使用洛必达，比如$\lim\limits_{x\to\infty}\frac{x}{\sqrt{x^2+1}}$. + +>[!tips] 运用洛必达法则解决其他不定式问题的流程： +$$\boxed{\infty-\infty}\overset{通分}{\Longrightarrow}\boxed{\frac{0}{0}或\frac{\infty}{\infty}}\overset{取倒数}{\Longleftarrow}\boxed{0\cdot\infty}\overset{取对数}{\Longleftarrow}\boxed{0^0或1^{\infty}或\infty^0}$$ + +## **例题** + +>[!example] 求下列极限 +>(1)$\lim\limits_{x\to1}(1-x^2)\tan{\frac{\pi}{2}x}$; (2)$\lim\limits_{x\to0}\frac{1}{x^2}-\frac{1}{x\tan x};(3)\lim\limits_{x\to0}(x^2+2^x)^{\frac{1}{x}}.$ + +解： +(1)用等价无穷小：$$原式\overset{t=x-1}{=}-\lim\limits_{t\to0}t(t+2)\tan{\frac{\pi}{2}(t+1)}=2\lim\limits_{t\to0}\frac{t}{\tan{\frac{\pi}{2}t}}=\frac{4}{\pi}.$$ +用洛必达：$$原式=\lim\limits_{x\to1}\frac{1-x^2}{\cot{\frac{\pi}{2}x}}\overset{\frac{0}{0}}{=}\frac{2}{\pi}\lim\limits_{x\to1}\frac{-2x}{-\csc^2{\frac{\pi}{2}x}}=\frac{4}{\pi}.$$ +(2)$$原式=\lim\limits_{x\to0}\frac{\tan x-1}{x^2\tan x}=\lim\limits_{x\to0}\frac{\tan x-x}{x^3}\overset{\frac{0}{0}}{=}\lim\limits_{x\to0}\frac{\sec^2x-1}{3x^2}=\lim\limits_{x\to0}\frac{\tan^2x}{3x^2}=\frac{1}{3}.$$ +(3)$$\begin{aligned}原式&=\lim\limits_{x\to0}\mathrm{e}^{\frac{1}{x}\ln{(x^2)+2^x}}\\\\&=\mathrm{e}^{\lim\limits_{x\to0}\frac{\ln{(x^2+2^x)}}{x}}\\\\&\overset{\frac{0}{0}}{=}\mathrm{e}^{\lim\limits_{x\to0}\frac{2x+2^x\ln2}{x^2+2^x}}\\\\&=\mathrm{e}^{\ln2}=2.\end{aligned}$$ + +>[!example] 例题 +>设$f(x)$在$[0,1]$内二阶可导，且满足$$f(0)=0,f(1)=1,f(\frac{1}{2})>\frac{1}{4}.$$证明： +>(1)至少存在一点$\xi\in(0,1)$,使得$f''(\xi)<2$; +>(2)若$\forall x\in(0,1)$,有$f''(x)\neq2$,则当$x\in(0,1)$时，恒有$f(x)>x^2$. + +证明： +（1）令$F(x)=f(x)-x^2$,则$F(0)=F(1)=0,F(\frac{1}{2})>0$.显然$F(x)$在$[0,\frac{1}{2}]$和$[\frac{1}{2},1]$上都满足拉格朗日中值定理得条件，所以由拉格朗日中值定理得：$$\exists\xi_1\in(0,\frac{1}{2}),\xi_2\in(\frac{1}{2},1),有\frac{F(\frac{1}{2})-F(0)}{\frac{1}{2}-0}=F'(\xi_1)>0,\frac{F(1)-F(\frac{1}{2})}{1-\frac{1}{2}}=F'(\xi_2)<0,$$于是再对$F(x)$在$[\xi_1,\xi_2]$上用拉格朗日中值定理得：$$\exists\xi\in(\xi_1,\xi_2),有F''(\xi)=\frac{F(\xi_2)-F(\xi_1)}{\xi_2-\xi_1}<0,即f''(\xi)<2.$$ +（2）用反证法，假设$\exists \eta\in(0,1),f(\eta)\le \eta^2,则F(\eta)\le 0$, \ No newline at end of file diff --git a/素材/特征值.md b/素材/特征值.md new file mode 100644 index 0000000..a0d0906 --- /dev/null +++ b/素材/特征值.md @@ -0,0 +1,32 @@ + +>[!note] 定理 +>秩为$1$的矩阵$A\in\mathbb{R}^{n\times n}$的特征值有如下特征：（1）$0$为其特征值，且代数重数和几何重数均为$n-1$；（2）它的另一个特征值为$\mathrm{tr}(A)$. + +**证明：** +根据迹的定义，只需要证明（1）。 +因为$r(A)=1[!example] 例1 +>设 $E$ 为 $3$ 阶单位矩阵，$\alpha$ 为一个 $3$ 维单位列向量，则矩阵 $E-\alpha\alpha^T$ 的全部 $3$ 个特征值为$\underline{\qquad}$。 + +**解：** +设 $B=\alpha\alpha^T$，该矩阵为**秩 $1$ 矩阵**（因 $\alpha$ 是单位列向量，$\alpha^T\alpha=1$）。 + +• 秩 $1$ 矩阵的特征值性质：非零特征值为矩阵的迹 $\text{tr}(B)=\alpha^T\alpha=1$，其余 $n-1=2$ 个特征值为 $0$（秩 $1$ 矩阵的非零特征值个数等于秩）。 + +• 若 $B\boldsymbol{x}=\lambda\boldsymbol{x}$（$\boldsymbol{x}$为特征向量），则 $(E-B)\boldsymbol{x}=(1-\lambda)\boldsymbol{x}$，即$E-B$ 的特征值为 $1-\lambda$。 + +$代入 B 的特征值 \lambda=1,0,0，得 E-B 的特征值为 1-1=0，1-0=1，1-0=1，即 1,1,0$。 + +>[!example] 例2 +>已知 $n(n\geq2)$维列向量 $\alpha,\beta$ 满足 $\beta^T\alpha=-3$，则方阵 $(\beta\alpha^T)^2$ 的非零特征值为$\underline{\qquad}$。 + +**解：** +设 $A=\beta\alpha^T$（秩 1 矩阵），计算 $A^2$： + +$A^2=(\beta\alpha^T)(\beta\alpha^T)=\beta(\alpha^T\beta)\alpha^T=(\alpha^T\beta)A$ + +• 注意：$\alpha^T\beta=(\beta^T\alpha)^T$（矩阵转置性质），而 $\beta^T\alpha=-3$（数，转置等于自身），故 $\alpha^T\beta=-3$，因此 $A^2=-3A$。 + +• 秩 $1$ 矩阵 $A$ 的非零特征值为 $\text{tr}(A)=\alpha^T\beta=-3$，设 $A\boldsymbol{x}=-3\boldsymbol{x}$，则 $A^2\boldsymbol{x}=(-3)A\boldsymbol{x}=(-3)^2\boldsymbol{x}=9\boldsymbol{x}$，即 $A^2$ 的非零特征值为 $9$。 \ No newline at end of file diff --git a/素材/特征值与相似.md b/素材/特征值与相似.md new file mode 100644 index 0000000..d08044a --- /dev/null +++ b/素材/特征值与相似.md @@ -0,0 +1,2 @@ + +$设 E 为 3 阶单位矩阵，\alpha 为一个 3 维单位列向量，则矩阵 E-\alpha\alpha^T 的全部 3 个特征值为\underline{\qquad}。$ diff --git a/素材/用秩的不等式“夹逼”出确切值.md b/素材/用秩的不等式“夹逼”出确切值.md new file mode 100644 index 0000000..588d98e --- /dev/null +++ b/素材/用秩的不等式“夹逼”出确切值.md @@ -0,0 +1,26 @@ + +>[!information] 做题思路 +>通过矩阵的秩的不等式，最大限度限制所求的表达式的取值范围，或者将其**限制到一个具体的值**. +>在希望求一个矩阵的秩的确切值时，也可以考虑用不等式关系来“夹逼”，常见的不等式： +>1. $\mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}-n\le\mathrm{rank}(\boldsymbol{AB})\le\min{\{\mathrm{rank}\boldsymbol{A}, \mathrm{rank}\boldsymbol{B}\}}$ +>2. $\mathrm{rank}(\boldsymbol{A+B})<\mathrm{rank}\boldsymbol A+\mathrm{rank}\boldsymbol B$ +>3. 矩阵加边不会减小秩； +> +>特别的，在遇到诸如 $AB=O$ 的情况，务必要想到$\mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}-n\le\mathrm{rank}(\boldsymbol{AB}) \Rightarrow \mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}\le n$ + +9. （20分）设 $A$ 是 $m\times n$ 实矩阵， $\beta \neq 0$ 是 $m$ 维实列向量，证明： + (1) $\mathrm{rank}A=\mathrm{rank}(A^\mathrm{T}A)$ . + (2) 线性方程组 $A^\mathrm{T}Ax = A^\mathrm{T}\beta$ 有解. + 证明如下： +>(1)（10分） +> 对于方程组$Ax=0$ (a)和 $A^\mathrm{T}Ax=0$ (b)，b的解空间一定包含a的解空间（5分）； +>而方程b两边同时乘以$x^\mathrm{T}$，得 $x^\mathrm{T}A^\mathrm{T}Ax=0$ ，即 $(x^\mathrm{T}A^\mathrm{T})(Ax)=0 \to Ax=0$， +>所以a的解空间包含b的解空间（5分）， +>所以a,b同解，所以$\mathrm{rank}A=\mathrm{rank}{A^\mathrm{T}A}$ +>(2) （10分） +>$A^\mathrm{T}Ax=A^\mathrm{T}\beta \iff \mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})$ +>而由(1)的结论得等式左边 $\mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}A$； +>关键步骤！ 等式右边 $\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})\ge \mathrm{rank}A^\mathrm{T}A=\mathrm{rank}A$ （5分） +>关键步骤！ 又$\mathrm{rank}(A^\mathrm{T}\begin{bmatrix}A&\beta\end{bmatrix})\le \min{(\mathrm{rank}A^\mathrm{T},\ \mathrm{rank}\begin{bmatrix}A&\beta\end{bmatrix})}=\mathrm{rank}A$ （5分） +>所以$\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})=\mathrm{rank}A$ +>故 $\mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})$ 得证. \ No newline at end of file diff --git a/素材/秩的不等式.md b/素材/秩的不等式.md new file mode 100644 index 0000000..3d28b8c --- /dev/null +++ b/素材/秩的不等式.md @@ -0,0 +1,63 @@ +# 一般式 + +## 1. 和的秩不超过秩的和 + +设 $A, B$ 为同型矩阵，则 +$$ \operatorname{rank}(A+B) \leq \operatorname{rank} A + \operatorname{rank} B $$ + +## 2. 积的秩不超过任何因子的秩 + +设 $A_{m \times n}, B_{n \times k}$，则 +$$ \operatorname{rank}(AB) \leq \min\{\operatorname{rank} A, \operatorname{rank} B\} $$ + +## 3. 重要不等式 + +设 $A_{m \times n}, B_{n \times k}$，则 +$$ \operatorname{rank}(AB) \geq \operatorname{rank} A + \operatorname{rank} B - n $$ +特别地，当 $AB = 0$ 时，有 $\operatorname{rank} A + \operatorname{rank} B \leq n$。 + +# 分块式 + +设 $A_{n \times n}$， $B_{n \times n}$，则 + +$$(1)\ \mathrm{rank} + +\begin{bmatrix} +A \\ +B +\end{bmatrix} \geq \text{rank } A, \quad \text{rank } +\begin{bmatrix} +A \\ +B +\end{bmatrix} \geq \text{rank } B +$$ + +$$(2)\ \mathrm{rank} +\begin{bmatrix} +A & 0 \\ +0 & B +\end{bmatrix} = \text{rank } A + \text{rank } B +$$ + +$$(3)\ \mathrm{rank} +\begin{bmatrix} +A & E_n \\ +0 & B +\end{bmatrix} \geq \text{rank } A + \text{rank } B +$$ + +$$(4)\ \mathrm{rank} +\begin{bmatrix} +A & 0 \\ +0 & B +\end{bmatrix} = \text{rank } +\begin{bmatrix} +A & B \\ +0 & B +\end{bmatrix} = \text{rank } +\begin{bmatrix} +A + B & B \\ +B & B +\end{bmatrix} \geq \text{rank } (A + B) +$$ + diff --git a/素材/线性方程组同解.md b/素材/线性方程组同解.md new file mode 100644 index 0000000..c5230c4 --- /dev/null +++ b/素材/线性方程组同解.md @@ -0,0 +1,30 @@ +## **$Ax=0$与$Bx=0$同解问题**： +充要条件：$rankA=rankB=rank\begin{bmatrix} A \\ B\end{bmatrix}$. +$Ax=\alpha$ 与$Bx=\beta$同解问题： +充要条件：$rank\begin{bmatrix} A & \alpha\end{bmatrix}=rank\begin{bmatrix} B &\beta\end{bmatrix}=rank\begin{bmatrix} A &\alpha\\ B&\beta\end{bmatrix}$. + +如何理解（非严格证明，目的是便于理解）： +首先，为了简化问题，我们只考虑齐次线性方程组同解问题，对于$Ax=0$与$Bx=0$， +考虑这两个齐次线性方程组的解空间，分别记为$N(A)$,$N(B)$,这两个集合是完全相同的， +可以得到$N(A)\subset N(B)$,以及$N(B)\subset N(A)$. +$N(A)\subset N(B)$可以得到什么呢？ + + +说明$Ax=0$的解比较少，$Bx=0$的解比较多，一个方程组解多就说明他的方程限制相对宽松，解少则说明方程要求比较严格。换言之，$Bx=0$的每个方程是由$Ax=0$的方程线性表示的,同理$N(B)\subset N(A)$ 可以得到 $Ax=0$ 的每个方程是由 $Bx=0$ 的方程线性表示的，进而说明这两个系数矩阵的行向量能够互相线性表示，即行向量组等价.用秩的语言表示：$rankA=rankB=rank\begin{bmatrix} A \\ B\end{bmatrix}$. + +另一个角度：这两个矩阵化成最简行阶梯型，是相同的，进行化简的时候只用到行变换，故它们的行向量组等价. + +需要注意的是,这个条件是充要的.非常的好用. +非齐次的时候同理. + +注意：由此，我们还能得到一些别的结论 +例如：$A$ 和 $B$ 等价（可以通过初等变换得到），并不能得到两方程同解，因为等价的初等变换可能包括初等列变换，而列变换可能改变两方程的解 + +>[!example] 例1 +>6. 已知方程组$\quad\begin{cases}x_1 + 2x_2 + 3x_3 = 0, \\2x_1 + 3x_2 + 5x_3 = 0, \\x_1 + x_2 + ax_3 = 0,\end{cases}$ 与$\quad\begin{cases}x_1 + bx_2 + cx_3 = 0, \\2x_1 + b^2x_2 + (c+1)x_3 = 0\end{cases}$同解，则 + (A) $a = 1, b = 0, c = 1$; + (B) $a = 1, b = 1, c = 2$; + (C) $a = 2, b = 0, c = 1$; + (D) $a = 2, b = 1, c = 2$. + +解析：类似于方程 $AX = B$ 有解的充要条件是$\text{rank} \begin{bmatrix} A & B \end{bmatrix} = \text{rank}A$，由方程 $XA = B$ 有解可知 $\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} = \text{rank}B=\text{rank}A=k$，由初等变换不改变秩得$$\text{rank} \begin{bmatrix} B & O \\ A & E \end{bmatrix} =\text{rank} \begin{bmatrix} B & O \\ O & E \end{bmatrix}=n+k$$ diff --git a/素材/线性方程组的系数矩阵与解关系.md b/素材/线性方程组的系数矩阵与解关系.md new file mode 100644 index 0000000..cde46ac --- /dev/null +++ b/素材/线性方程组的系数矩阵与解关系.md @@ -0,0 +1,90 @@ +这是一个链接了方程组解空间与方程组系数秩的公式 + +>[!note] 秩零化度定理： +>对于齐次方程组 ${A}_{m \times n}\boldsymbol{x}=\boldsymbol{0}$，设$\mathrm{rank}{A}=r$，则 +> $$\dim N({A})=n-r$$ + + +>[!example] 例1 +>已知三阶方阵 $A=\begin{bmatrix}\alpha_1&\alpha_2&\alpha_3\end{bmatrix}$ 有三个不同的特征值，其中$\alpha_3=2\alpha_1+\alpha_2$，若 $\beta=\alpha_1+3\alpha_2+4\alpha_3$ ，求线性方程组 $A\boldsymbol{x}=\beta$ 的通解. + +**答案：** + $$\begin{bmatrix}1\\3\\4\end{bmatrix}+k\begin{bmatrix}2\\1\\-1\end{bmatrix}, k\in\mathbb{R}$$ +**分析：** 在求解非齐次方程组通解的题目中，若是题目给出了特解与齐次方程组的解，那么大概率来说这个齐次方程组的解就可以拓展为齐次方程组通解（根据问题导向，不然写不出来了），那么如何由齐次方程组的解拓展为齐次方程组通解呢，那就要根据题目具体的条件进行分析了，这就要用到我们的解零度化定理来求齐次方程组解空间的维数 +**解析：** 由 $\alpha_3=2\alpha_1+\alpha_2$ 可得 $A$ 的列向量组线性相关， $|A|=0$；又因为 $A$ 的三个特征值各不相同，故 $A$ 有两个不为零的特征值 $\lambda_1,\lambda_2$，且 $A$ 可相似对角化，即 $A=P^{-1}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}P$，$\mathrm{rank}A=\mathrm{rank}(P^{-1}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}P)=\mathrm{rank}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}=2$ +故 $Ax=0$ 的解空间维数是 $1$ （5分） +$\beta=\alpha_1+3\alpha_2+4\alpha_3$，所以 $(1,3,4)^\mathrm{T}$ 为特解；（5分） +$\alpha_3=2\alpha_1+\alpha_2$，所以$A\begin{bmatrix}2k\\k\\-k\end{bmatrix}=2\alpha_1+\alpha_2-\alpha_3=0$，所以 $(2,1,-1)^\mathrm{T}$ 为基础解系；（10分） +解空间维数是 $1$ ，方程的解 $\begin{bmatrix}1\\3\\4\end{bmatrix}+k\begin{bmatrix}2\\1\\-1\end{bmatrix}, k\in\mathbb{R}$ 维数是 $1$，该解完备 + + + >[!example] 例2 + >设 $$A = \begin{bmatrix} +1 & -1 & 0 & -1 \\ +1 & 1 & 0 & 3 \\ +2 & 1 & 2 & 6 +\end{bmatrix}, \quad +B = \begin{bmatrix} +1 & 0 & 1 & 2 \\ +1 & -1 & a & a-1 \\ +2 & -3 & 2 & -2 +\end{bmatrix}, +\quad +\alpha = \begin{bmatrix} 0 \\ 2 \\ 3 \end{bmatrix}, \quad +\beta = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$$ +(1) 证明：方程组 $Ax = \alpha$ 的解均为方程组 $Bx = \beta$ 的解； +$\quad$ +(2) 若方程组 $A\boldsymbol{x} = \alpha$ 与方程组 $B\boldsymbol{x} = \beta$ 不同解，求 $a$ 的值。 + +**解：** +(1) 由于 +$$ +\begin{bmatrix} +A \quad \alpha \\ +B \quad \beta +\end{bmatrix} = +\begin{bmatrix} +1 & -1 & 0 & -1 & 0 \\ +1 & 1 & 0 & 3 & 2 \\ +2 & 1 & 2 & 6 & 3 \\ +1 & 0 & 1 & 2 & 1 \\ +1 & -1 & a & a-1 & 0 \\ +2 & -3 & 2 & -2 & -1 +\end{bmatrix} +\rightarrow +\begin{bmatrix} +1 & -1 & 0 & -1 & 0 \\ +0 & 1 & 0 & 2 & 1 \\ +0 & 0 & 2 & 2 & 0 \\ +0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 +\end{bmatrix}, +$$ +故 +$$ +\mathrm{rank} \begin{bmatrix}A&\alpha\\B&\beta\end{bmatrix} = \mathrm{rank}[A\ \alpha], +$$ +从而方程组 +$$ +\begin{cases} +A\boldsymbol{x} = \alpha \\ +B\boldsymbol{x} = \beta +\end{cases} +$$ +与 $A\boldsymbol{x} = \alpha$ 同解，故 $A\boldsymbol{x} = \alpha$ 的解均为 $B\boldsymbol{x} = \beta$ 的解。 + +(2) 分析：不同解，却要可以求出$a$的具体值，说明这是一个与秩相关的题，而与解相关的秩的问题我们就可以考虑解零度化定理 +由于 $A\boldsymbol{x} = \alpha$ 的解均为 $Bx = \beta$ 的解，若 $A\boldsymbol{x} = \alpha$ 与 $B\boldsymbol{x} = \beta$ 同解，则与题意矛盾，故 $Ax = \alpha$ 的解是 $Bx = \beta$ 解的真子集。于是 $Ax = 0$ 的基础解系中解向量的个数小于 $B\boldsymbol{x} = 0$ 的基础解系中解向量的个数，即 +$$ +4 - r(A) < 4 - r(B), +$$ +故 $r(A) > r(B)$。又因 $r(A) = 3$，故 $r(B) < 3$，则 +$$ +\left| \begin{array}{ccc} +1 & 0 & 1 \\ +1 & -1 & a \\ +2 & -3 & 2 +\end{array} \right| = 0, +$$ +解得 $a = 1$。 \ No newline at end of file diff --git a/素材/线性方程组解的问题.md b/素材/线性方程组解的问题.md new file mode 100644 index 0000000..c21616c --- /dev/null +++ b/素材/线性方程组解的问题.md @@ -0,0 +1,88 @@ +### 原理 +**线性方程组解的判定** +对非齐次线性方程组 $A_{m\times n}x=b$, +1. 无解的充要条件是 $\text{rank}A < \text{rank}[A\ \ b]$； +2. 有唯一解的充要条件是 $\text{rank}A = \text{rank}[A\ \ b] = n$； +3. 有无穷多解的充要条件是 $\text{rank}A = \text{rank}[A\ \ b] < n$。 +注:上述定理也说明非齐次线性方程组有解的充要条件是 $\text{rank}A = \text{rank}[A\ \ b]$。 + +把以上结论应用到齐次线性方程组，可得 +推论 齐次线性方程组 $A_{m\times n}x=0$ 有非零解（无穷多解）的充要条件是 $\text{rank}A < n$，即系数矩阵的秩小于未知数个数。 + +**矩阵方程解的判定** +本质上和线性方程组是一脉相承的，只是形式上更一般化。 +最常见的矩阵方程是 $\boldsymbol{AX} = \boldsymbol{B}$，其中$\boldsymbol{A}$是 $m\times n$ 矩阵，$\boldsymbol{B}$ 是 $m\times p$ 矩阵，$\boldsymbol{X}$ 是待求的$n\times p$矩阵。 +1. 有解的充要条件： +矩阵方程有解的充要条件是系数矩阵 $\boldsymbol{A}$ 的秩等于增广矩阵 $[\boldsymbol{A} \ \boldsymbol{B}]$的秩，即： +$$r(\boldsymbol{A}) = r([\boldsymbol{A} \ \boldsymbol{B}])$$ +这个结论和非齐次线性方程组有解的条件完全一致。 +2. 解的结构： +- 唯一解：当$r(\boldsymbol{A}) = r([\boldsymbol{A} \ \boldsymbol{B}]) = n$ 时，方程有唯一解。 +- 无穷多解：当 $r(\boldsymbol{A}) = r([\boldsymbol{A} \ \boldsymbol{B}]) < n$ 时，方程有无穷多解。 + +可逆矩阵 +- 当 $\boldsymbol{A}$ 是 n 阶可逆矩阵时，矩阵方程 $\boldsymbol{AX} = \boldsymbol{B}$有唯一解:$\boldsymbol{X} = \boldsymbol{A}^{-1}\boldsymbol{B}$ + +其他形式的矩阵方程 +- 对于 $\boldsymbol{XA} = \boldsymbol{B}$ 形式的方程，可以转置为 $\boldsymbol{A}^T\boldsymbol{X}^T = \boldsymbol{B}^T$，再套用上述方法，或类似于方程 $AX = B$ 有解的充要条件是$\text{rank} \begin{bmatrix} A & B \end{bmatrix} = \text{rank}A$，由方程 $XA = B$ 有解可知 $\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} =\text{rank}A$ +- 对于 $\boldsymbol{AXB} = \boldsymbol{C}$ 形式的方程，当 $\boldsymbol{A} 和 \boldsymbol{B}$ 都可逆时，有唯一解 $\boldsymbol{X} = \boldsymbol{A}^{-1}\boldsymbol{C}\boldsymbol{B}^{-1}$。 + +>[!example] **例1** +>设矩阵 +>$$A = \begin{bmatrix} +1 & a_1 & a_1^2 & a_1^3 \\ +1 & a_2 & a_2^2 & a_2^3 \\ +1 & a_3 & a_3^2 & a_3^3 \\ +1 & a_4 & a_4^2 & a_4^3 +\end{bmatrix}, +\quad +x = \begin{bmatrix} +x_1 \\ x_2 \\ x_3 \\ x_4 +\end{bmatrix}, +\quad +b = \begin{bmatrix} +1 \\ 1 \\ 1 \\ 1 +\end{bmatrix}$$ +其中常数 $a_1, a_2, a_3, a_4$ 互不相等，则线性方程组 $Ax = b$ 的解为 ______________。 + +**答**：$(1,0,0,0)^T$。 + +**解析**：由范德蒙行列式的性质可知 $|A| \neq 0$，从而线性方程组 $Ax = b$ 有唯一解。 + +又由 +$$ +\begin{bmatrix} +1 & a_1 & a_1^2 & a_1^3 \\ +1 & a_2 & a_2^2 & a_2^3 \\ +1 & a_3 & a_3^2 & a_3^3 \\ +1 & a_4 & a_4^2 & a_4^3 +\end{bmatrix} +\begin{bmatrix} +1 \\ +0 \\ +0 \\ +0 +\end{bmatrix} += +\begin{bmatrix} +1 \\ +1 \\ +1 \\ +1 +\end{bmatrix} +$$ + 可知 $Ax = b$ 的解为 +$$ +\begin{bmatrix} +1 \\ +0 \\ +0 \\ +0 +\end{bmatrix} +$$ + +>[!example] **例2** +> 设 $A, B$ 均为 $n$ 阶方阵，满足$\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} = \text{rank}B,$ 且方程 $XA = B$ 有解。若 $\operatorname{rank} A = k$，则$\text{rank} \begin{bmatrix} B & O \\ A & E \end{bmatrix} =\underline{\hspace{3cm}}.$ + +**解析**： +类似于方程 $AX = B$ 有解的充要条件是$\text{rank} \begin{bmatrix} A & B \end{bmatrix} = \text{rank}A$，由方程 $XA = B$ 有解可知 $\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} = \text{rank}B=\text{rank}A=k$，由初等变换不改变秩得$$\text{rank} \begin{bmatrix} B & O \\ A & E \end{bmatrix} =\text{rank} \begin{bmatrix} B & O \\ O & E \end{bmatrix}=n+k$$ \ No newline at end of file diff --git a/素材/罗尔定理与拉格朗日中值定理.md b/素材/罗尔定理与拉格朗日中值定理.md new file mode 100644 index 0000000..b657011 --- /dev/null +++ b/素材/罗尔定理与拉格朗日中值定理.md @@ -0,0 +1,137 @@ +## **罗尔定理** +### **原理** +若函数 f(x) 满足以下三个条件： +在闭区间 $[a,b]$ 上连续； +在开区间 $(a,b)$ 内可导； +区间端点函数值相等，即 $f(a)=f(b)$； +则在 $(a,b)$ 内至少存在一点 $\xi$，使得 $f'(\xi)=0$。 +罗尔定理的几何意义为：满足条件的函数曲线在区间内至少有一条水平切线。 +它是拉格朗日中值定理（$f(b)-f(a)=f'(\xi)(b-a)$）当 $f(a)=f(b)$ 时的特例。 + +### **适用条件** +罗尔定理的核心适用题型是证明导函数方程 $f'(\xi)=0$ 在区间 $(a,b)$ 内有根以及衍生的相关证明题。 +具体可分为以下几类： +1.直接证明 $f'(\xi)$=0 存在根 +题目给出函数 f(x) 在 $[a,b]$ 上的连续性、$(a,b)$ 内的可导性，且满足 $f(a)=f(b)$，直接应用罗尔定理证明存在 $\xi\in(a,b)$ 使得 $f'(\xi)=0$。 +2.构造辅助函数证明导函数相关方程有根 +对于形如 $f'(\xi)+g(\xi)f(\xi)=0$、$f''(\xi)=0$ 等方程，需构造满足罗尔定理条件的辅助函数 $F(x)$，通过 $F(a)=F(b)$ 推导 $F'(\xi)=0$，进而等价转化为目标方程。 +3.结合多次罗尔定理证明高阶导数零点存在 +若函数 f(x) 有 n+1 个点的函数值相等，可多次应用罗尔定理，证明其 n 阶导数 $f^{(n)}(\xi)=0$ 在对应区间内有根。 +4.证明函数恒为常数（反证法结合罗尔定理） +若 $f'(x)\equiv0$ 在区间内成立，可通过反证法假设存在两点函数值不等，结合罗尔定理推出矛盾，进而证明函数为常数。 + +罗尔定理针对于一个函数，不同于柯西中值定理针对于两个函数 + +### **例题** +>[!example] 例1 +设 $f(x)$ 在 $[0,1]$ 连续，$(0,1)$ 可导，且 $f(1) = 0$，求证存在 $\xi \in (0,1)$ 使得 $nf(\xi) + \xi f'(\xi) = 0$。 + +**解析**： +设辅助函数 $\varphi(x) = x^n f(x)$，则 $\varphi(0)=0$，$\varphi(1)=0$。由罗尔定理，存在 $\xi \in (0,1)$，使得 $\varphi'(\xi)=0$即 +$$ +n\xi^{n-1} f(\xi) + \xi^n f'(\xi) = 0 +$$ +两边除以 $\xi^{n-1}$ ($\xi>0$)，得 $nf(\xi) + \xi f'(\xi) = 0$。 + + + +>[!example] 例2 +设函数 $f(x)$ 在 $[a,b]$ 上可导，且 +$$f(a) = f(b) = 0，\quad f'_+(a)f'_-(b) > 0，$$ +试证明 $f'(x) = 0$ 在 $(a,b)$ 内至少有两个根。 + + +**解析**： +由导数极限定理及 $f'_+(a)f'_-(b) > 0$，知在 $a$ 右侧和 $b$ 左侧，$f(x)$ 的符号相同，不妨设 $f'_+(a)>0$，$f'_-(b)>0$。则在 $a$ 右侧附近 $f(x)>0$，在 $b$ 左侧附近 $f(x)>0$。由于 $f(a)=f(b)=0$，由极值点的费马定理，$f(x)$ 在 $(a,b)$ 内至少有一个极大值点，该点处导数为零。又因为 $f(x)$ 在 $[a,b]$ 上连续，在 $(a,b)$ 内可导，且 $f(a)=f(b)$，由罗尔定理至少存在一点 $c \in (a,b)$ 使 $f'(c)=0$。结合极大值点处的导数零点，可知至少有两个导数为零的点。 + + + +>[!example] 例3 +设 $f(x)$ 在 $[0, 1]$ 上具有二阶导数，且满足 +$$f(0) = 0, \, f(1) = 1, \, f\left(\frac{1}{2}\right) > \frac{1}{4}$$证明： +(1)至少存在一点 $\xi \in (0, 1)$，使得 $f''(\xi) < 2$； +(2)若对一切 $x \in (0, 1)$，有 $f''(x) \neq 2$，则当 $x \in (0, 1)$ 时，恒有 $f(x) > x^2$。 + +**解析**： +(1) 考虑函数 $g(x)=f(x)-x^2$，则 $g(0)=0$，$g(1)=0$，$g(1/2)=f(1/2)-1/4>0$。由极值点的费马定理及罗尔定理，$g(x)$ 在 $(0,1)$ 内存在极大值点 $\eta$，且 $g'(\eta)=0$，$g''(\eta) \leq 0$。即 $f'(\eta)=2\eta$，$f''(\eta) \leq 2$。若 $f''(\eta) < 2$，则取 $\xi=\eta$ 即可；若 $f''(\eta)=2$，则考虑在 $\eta$ 两侧应用拉格朗日中值定理，可找到另一个点 $\xi$ 使得 $f''(\xi)<2$。 +(2) 用反证法。假设存在 $x_0 \in (0,1)$ 使 $f(x_0) \leq x_0^2$，结合 $f(0)=0$，$f(1)=1$ 和 $f(1/2)>1/4$，利用连续性及中值定理可推出存在 $\xi$ 使 $f''(\xi)=2$，矛盾。 + +## **拉格朗日中值定理** +### **原理** +若函数 f(x) 满足两个条件: +在闭区间 $[a,b]$ 上连续； +在开区间 $(a,b)$ 内可导； +则在 $(a,b)$ 内至少存在一点 $\xi$，使得 +$f(b)-f(a)=f'(\xi)(b-a)$ +也可写成等价形式 $f'(\xi)=\dfrac{f(b)-f(a)}{b-a}$。 +是罗尔定理的推广，同时也是柯西中值定理的特例。其几何意义为：满足条件的函数曲线在区间 (a,b) 内，至少存在一点的切线与连接端点 (a,f(a)) 和 (b,f(b)) 的弦平行。 + +### **适用条件** +拉格朗日中值定理的核心适用题型是建立函数增量与导数的关联，进行不等式的证明，这是最常见的题型。通过对目标函数在指定区间上应用拉格朗日中值定理，得到 $f(b)-f(a)=f'(\xi)(b-a)$，再利用导数 $f'(\xi)$ 的取值范围（有界性、正负性）放大或缩小式子，推导不等式。 + +### **例题** + +>[!example] 例1 +设函数 $f(x)$ 在 $(-1,1)$ 内可微，且 +$$f(0) = 0, \quad |f'(x)| \leq 1,$$证明：在 $(-1,1)$ 内，$|f(x)| < 1$。 + +**解析**： +对任意 $x \in (-1,1)$，由拉格朗日中值定理，存在 $\xi$ 介于 $0$ 与 $x$ 之间，使得 +$$ +f(x) - f(0) = f'(\xi)(x-0) +$$ +即 $f(x) = f'(\xi) x$。由于 $|f'(\xi)| \leq 1$，$|x| < 1$，故 $|f(x)| = |f'(\xi)| \cdot |x| < 1$。 + + +>[!example] 例2 +设 $f''(x) < 0$，$f(0) = 0$，证明对任意 $x_1 > 0, x_2 > 0$ 有 +$$f(x_1 + x_2) < f(x_1) + f(x_2)$$ + +**解析**： +不妨设 $0 < x_1 < x_2$。由拉格朗日中值定理： +$$ +f(x_1+x_2)-f(x_2) = f'(\xi_1)x_1, \quad \xi_1 \in (x_2, x_1+x_2) +$$ +$$ +f(x_1)-f(0) = f'(\xi_2)x_1, \quad \xi_2 \in (0, x_1) +$$ +于是 +$$ +f(x_1+x_2)-f(x_2)-f(x_1) = [f'(\xi_1)-f'(\xi_2)]x_1 +$$ +对 $f'(x)$ 在 $[\xi_2,\xi_1]$ 上应用拉格朗日中值定理，存在 $\xi \in (\xi_2,\xi_1)$，使 +$$ +f'(\xi_1)-f'(\xi_2) = f''(\xi)(\xi_1-\xi_2) < 0 +$$ +故 $f(x_1+x_2)-f(x_2)-f(x_1) < 0$，即 $f(x_1+x_2) < f(x_1)+f(x_2)$。 + + +>[!example] 例3 + 设 $f(x)$ 在 $(-\infty, +\infty)$ 内二阶可导，且 $f''(x) \neq 0$。 +（1）证明：对于任何非零实数 $x$，存在唯一的 $\theta(x)$ ($0<\theta(x)<1$)，使得 + $$f(x) = f(0) + x f'(x\theta(x));$$ + （2）求 + $$\lim_{x \to 0} \theta(x).$$ + +解： +1. 证： 对于任何非零实数 $x$，由中值定理，存在 $\theta(x)$ $(0<\theta(x)<1)$，使得 + +$$ +f(x)=f(0)+x f'(x\theta(x)). +$$ + +如果这样的 $\theta(x)$ 不唯一，则存在 $\theta_{1}(x)$ 与 $\theta_{2}(x)$ $(\theta_{1}(x)<\theta_{2}(x))$，使得 $f'(x\theta_{1}(x))=f'(x\theta_{2}(x))$，由罗尔定理，存在一点 $\xi$，使得 $f''(\xi)=0$，这与 $f''(x)\neq 0$ 矛盾。所以 $\theta(x)$ 是唯一的。 + +2. 解 注意到 $f''(0)=\lim_{x\rightarrow 0} \frac{f'(x\theta(x))-f'(0)}{x\theta(x)}$，又知 + +$$ +\begin{aligned} +\lim_{x\rightarrow 0} \frac{f'(x\theta(x))-f'(0)}{x} +&= \lim_{x\rightarrow 0} \frac{\frac{f(x)-f(0)}{x}-f'(0)}{x} \\ +&= \lim_{x\rightarrow 0} \frac{f(x)-f(0)-x f'(0)}{x^{2}} \\ +&= \lim_{x\rightarrow 0} \frac{f'(x)-f'(0)}{2x} \\ +&= \frac{f''(0)}{2}, +\end{aligned} +$$ + +所以 $\lim_{x\rightarrow 0} \theta(x)=\frac{1}{2}$。 \ No newline at end of file diff --git a/编写小组/讲义/图片/xcospix.png b/编写小组/讲义/图片/xcospix.png new file mode 100644 index 0000000000000000000000000000000000000000..7f1aaac533576861e2f03de1b50a0b98e926881e GIT binary patch literal 1323411 zcmb4rc_7sL_dc#dLQxV$WlOe{HOo|zBvjJaMFZX-_*U`x_$5W^Zoq(s75o!JnwVPbDs0O&Rh6(LtU;tf_s>ln7H(=T)N4` z#97J2#QuhTH{*XIxzzj^|8RI*G4p0(+GF_b53|6jraUI5eN1|nF5G&+yh0s|Q#61~ 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\begin{cases}1, & x\text{为有理数时}, \\ 0, & x\text{为无理数 ## Vol. 1:补药漏写dx口牙！ ->[!example] 例1 +>[!example] 例题 >设函数 $y = \ln(1 + \sin^2 x)$，求其微分 $dy$。 ``` @@ -268,7 +268,7 @@ $$D(x)= \begin{cases}1, & x\text{为有理数时}, \\ 0, & x\text{为无理数 ## Vol. 3:可去间断点的说明 ->[!example] 例2 +>[!example] 例题 >求曲线 $y = \frac{x^2 + 2x - 3}{x^2 - 3x + 2}$ 的所有渐近线。 ``` @@ -284,7 +284,8 @@ $$D(x)= \begin{cases}1, & x\text{为有理数时}, \\ 0, & x\text{为无理数 ## Vol. 4:正项级数的判别法勿滥用 ->[!example] 例3 + +>[!example] 例题 >判断级数 $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^2}{3^n}$ 的敛散性（绝对收敛、条件收敛或发散）。 ``` @@ -434,7 +435,7 @@ $y=f^{-1}(x)$，即$x = f(y)$，求$f^{-1'}$就是在求$\frac{dy}{dx}$，而$\f 所以，我们还需要将$y = f^{-1}(x)$代入，即： $\frac{dy}{dx}=\frac{1}{f'(y)}=\frac{1}{f'(f^{-1}(x))}$ -> [!example] 例1 +> [!example] 例题 > 求$d(\arcsin x)$ ``` @@ -464,8 +465,29 @@ $\forall \delta>0, \exists x \in \mathring{U}(x_0,\delta)$有 $|f(x)|>M$，称$f 那这个是有界的吗？也不是。这就是典型的**不是无界量的无穷大**。 > [!example] 例2 -> 双子数列$\lim\limits_{n\rightarrow \infty}{n\cos n\pi}$ +> 双子数列$\lim\limits_{n\rightarrow \infty}{n\cos \frac{n\pi}{2}}$ + + + +![[易错点10-2.png]] +这个也是无界但不是无穷大，请自行证明。 +``` + + + -总结：无穷大是在邻域内“一直都很大”，无界是邻域内“有很大的” + + + + + + + + + + + +``` +总结：无穷大是在邻域内“一直都很大”，无界是邻域内“有很大的” \ No newline at end of file diff --git a/编写小组/讲义/子数列问题&考试易错点汇总（解析版）.md b/编写小组/讲义/子数列问题&考试易错点汇总（解析版）.md index bf45c72..71d21b9 100644 --- a/编写小组/讲义/子数列问题&考试易错点汇总（解析版）.md +++ b/编写小组/讲义/子数列问题&考试易错点汇总（解析版）.md @@ -3,7 +3,7 @@ tags: - 编写小组 --- **内部资料，禁止传播** -**编委会（不分先后，姓氏首字母顺序）：程奕铭 韩魏 刘柯妤 卢吉辚 王轲楠 支宝宁 郑哲航 +**编委会（不分先后，姓氏首字母顺序）：程奕铭 韩魏 刘柯妤 卢吉辚 王轲楠 支宝宁 郑哲航** # 子数列及其相关定理 @@ -202,7 +202,7 @@ $$\lim\limits_{n \to \infty}S_{2n+1}=\lim\limits_{n \to \infty}(S_{2n}+a_{2n+1}) **解析** 核心方法 利用数列子列的性质：若数列$\{a_n\}$收敛，则其所有子列必收敛于同一极限；若存在两个子列收敛于不同值，则原数列极限不存在。 - 步骤1：构造第一个子列$\{a_{4k}\}（k\in\mathbb{N_+}$ + 步骤1：构造第一个子列$\{a_{4k}\}（k\in\mathbb{N_+}）$ 当n=4k时， $$\sin\frac{4k\pi}{2} = \sin2k\pi = 0$$ 因此 $$a_{4k} = \left(1 + \frac{1}{4k}\right) \cdot 0 = 0$$ @@ -224,36 +224,105 @@ $$\lim_{k \to \infty}a_{4k+1} = \lim_{k \to \infty}\left(1 + \frac{1}{4k+1}\righ $$D(x)= \begin{cases}1, & x\text{为有理数时}, \\ 0, & x\text{为无理数时}\end{cases}$$ 在$(-\infty,+\infty)$上每一点都不存在极限。 - **解析** -1. 构造第一个数列$\{x_n^{(1)}\}$： - 取$x_n^{(1)} = x_0 + \frac{1}{n}$（有理数），则$$\lim\limits_{n \to \infty} x_n^{(1)} = x_0$$由于$x_n^{(1)}$是有理数，所以$D(x_n^{(1)}) = 1$，因此$$\lim\limits_{n \to \infty} D(x_n^{(1)}) = 1$$ -2. 构造第二个数列$\{x_n^{(2)}\}$： - 取$x_n^{(2)} = x_0 + \frac{\sqrt{2}}{n}$（无理数），则$$\lim\limits_{n \to \infty} x_n^{(2)} = x_0$$由于x_n^{(2)}是无理数，所以$D(x_n^{(2)}) = 0$，因此$$\lim\limits_{n \to \infty} D(x_n^{(2)}) = 0$$由于$$\lim\limits_{n \to \infty} D(x_n^{(1)}) = 1 \neq 0 = \lim\limits_{n \to \infty} D(x_n^{(2)})$$根据海涅定理，$\lim\limits_{x \to x_0} D(x)$不存在。 - 由于$x_0$是任意一点，所以狄利克雷函数在任何点处都不存在极限。 + +方法一： + +首先考虑 $x_0$ 为有理数的情况： + +1. 构造第一个数列 $\{x_n^{(1)}\}$： + + 取 $x_n^{(1)} = x_0 + \frac{1}{n}$（有理数），则 + $$\lim\limits_{n \to \infty} x_n^{(1)} = x_0$$ + + 由于 $x_n^{(1)}$ 是有理数，所以 $D(x_n^{(1)}) = 1$，因此 + $$\lim\limits_{n \to \infty} D(x_n^{(1)}) = 1$$ + +2. 构造第二个数列 $\{x_n^{(2)}\}$： + + 取 $x_n^{(2)} = x_0 + \frac{\sqrt{2}}{n}$（无理数），则 + $$\lim\limits_{n \to \infty} x_n^{(2)} = x_0$$ + + 由于 $x_n^{(2)}$ 是无理数，所以 $D(x_n^{(2)}) = 0$，因此 + $$\lim\limits_{n \to \infty} D(x_n^{(2)}) = 0$$ + + 由于 + $$\lim\limits_{n \to \infty} D(x_n^{(1)}) = 1 \neq 0 = \lim\limits_{n \to \infty} D(x_n^{(2)})$$ + + 根据海涅定理，$\lim\limits_{x \to x_0} D(x)$ 不存在。 + +再考虑 $x_0$ 为无理数的情况： + +设 $[x_0]_n$ 为 $x_0$ 取到 $n$ 位小数后的结果，则有 +$$\lim\limits_{n \to \infty} [x_0]_n = x_0$$ +但是需要注意的是，极限并不是完全相等，其实相差了一个无穷小，也就是要多趋近有多趋近，但是它实质上还是一个有理数，因为它毕竟不是 $x_0$ 本身 + +这种思路的目的是为了找到这样一种表达式，极限是 $x_0$ 的同时，与第一种情况类似，仍是有理数 + +1. 构造第一个数列 $\{x_n^{(3)}\}$： + + 取 $x_n^{(3)} = [x_0]_n$（有理数），则 + $$\lim\limits_{n \to \infty} x_n^{(3)} = x_0$$ + + 由于 $x_n^{(3)}$ 是有理数，所以 $D(x_n^{(3)}) = 1$，因此 + $$\lim\limits_{n \to \infty} D(x_n^{(3)}) = 1$$ + +2. 构造第二个数列 $\{x_n^{(4)}\}$： + + 取 $x_n^{(4)} = [x_0]_n + \frac{\sqrt{2}}{n}$（无理数），则 + $$\lim\limits_{n \to \infty} x_n^{(4)} = x_0$$ + + 由于 $x_n^{(4)}$ 是无理数，所以 $D(x_n^{(4)}) = 0$，因此 + $$\lim\limits_{n \to \infty} D(x_n^{(4)}) = 0$$ + + 由于 + $$\lim\limits_{n \to \infty} D(x_n^{(3)}) = 1 \neq 0 = \lim\limits_{n \to \infty} D(x_n^{(4)})$$ + + 根据海涅定理，$\lim\limits_{x \to x_0} D(x)$ 不存在。 + +综上所述， $x_0$ 是任意一点时，都能得到$\lim\limits_{x \to x_0} D(x)$ 不存在，所以狄利克雷函数在任何点处都不存在极限。 + +方法二：利用实数的稠密性 + +1. **取任意实数 $a$**。 +2. **构造序列**： + - 对每个 $n \in \mathbb{N}^*$，由有理数的稠密性，存在有理数 $r_n$ 满足 $|r_n - a| < \frac{1}{n}$。 + - 对每个 $n \in \mathbb{N}^*$，由无理数的稠密性，存在无理数 $s_n$ 满足 $|s_n - a| < \frac{1}{n}$。 +3. **证明序列收敛**（用 $\varepsilon$-$N$ 语言）： + - 对任意 $\varepsilon > 0$，取 $N = \lfloor 1/\varepsilon \rfloor + 1$，则当 $n > N$ 时，$|r_n - a| < 1/n < \varepsilon$，故 $\lim r_n = a$。 + - 同理 $\lim s_n = a$。 +4. **计算函数值极限**： + - $D(r_n) = 1$，常数序列极限为 $1$。 + - $D(s_n) = 0$，常数序列极限为 $0$。 +5. **应用海涅归结原理**： + - 若 $\lim_{x \to a} D(x)$ 存在，则任何收敛于 $a$ 的序列 $\{x_n\}$ 都应有 $\lim D(x_n)$ 相等。 + - 但 $\{r_n\}$ 和 $\{s_n\}$ 都收敛于 $a$，却得到不同的极限 $1$ 和 $0$，矛盾。 +1. **结论**：$\lim_{x \to a} D(x)$ 不存在，且 $a$ 任意，故 $D(x)$ 在每一点都无极限。 # 考试易错点总结 ## Vol. 1:补药漏写dx口牙！ ->[!example] 例1 +>[!example] 例题 >设函数 $y = \ln(1 + \sin^2 x)$，求其微分 $dy$。 **解**： 对方程 $y = \ln(1 + \sin^2 x)$ 两边关于 $x$ 求导。 -$$ y‘ = \frac{1}{1+\sin^2 x} \cdot (2\sin x \cos x) = \frac{\sin 2x}{1+\sin^2 x} $$ +$$ y'= \frac{1}{1+\sin^2 x} \cdot (2\sin x \cos x) = \frac{\sin 2x}{1+\sin^2 x} $$ -根据微分的定义，$dy = y’ dx$，故 +根据微分的定义，$dy = A dx$，而当$y$可导时，$A=y'$，故 $$ dy = \frac{\sin 2x}{1+\sin^2 x} dx $$ 注意：不要漏写 $dx$ ！ ## **Vol.2等价无穷小问题** + 注意，等价无穷小只能用于乘除，用于加减虽然有时也会得到正确的答案，但这并不是有保证的。归根到底这是因为等价无穷小是一种**近似**，在乘除中它的近似程度还可以用，但在加减中就未必了，加减中我们需要更精确的近似方法：泰勒展开。另外重要极限也是等价无穷小的两种特殊情况。 + >[!example] 例题 ->求极限$$\lim\limits_{x\to0}\frac{tanx-sinx}{x^3}$$. +>求极限$$\lim\limits_{x\to0}\frac{tanx-sinx}{x^3}$$ **解**：如果直接用$tanx\sim x,sinx\sim x(x\to0)$的话，分子就会变成$0$，从而极限为$0$. 然而从另一个角度看，$$\lim\limits_{x\to0}\frac{tanx-sinx}{x^3}=\lim\limits_{x\to0}\frac{sinx-sinxcosx}{x^3cosx}=\lim\limits_{x\to0}\frac{sinx(1-cosx)}{x^3}\cdot \frac{1}{cosx}=\lim\limits_{x\to0}\frac{x\cdot\frac{1}{2}x^2}{x^3}=\frac{1}{2}.$$ @@ -263,7 +332,7 @@ $$ dy = \frac{\sin 2x}{1+\sin^2 x} dx $$ ## Vol. 3:可去间断点的说明 ->[!example] 例2 +>[!example] 例题 >求曲线 $y = \frac{x^2 + 2x - 3}{x^2 - 3x + 2}$ 的所有渐近线。 **解**： @@ -293,12 +362,13 @@ $$ \lim_{x \to \infty} y = \lim_{x \to \infty} \frac{1 + \frac{2}{x} - \frac{3}{ **结论**：曲线的渐近线为 $x = 2$ 和 $y = 1$。 ## Vol. 4:正项级数的判别法勿滥用 ->[!example] 例3 + +>[!example] 例题 >判断级数 $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^2}{3^n}$ 的敛散性（绝对收敛、条件收敛或发散）。 **错误做法示范**：观察级数 $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^2}{3^n}$，若直接对其使用比值判别法： - $$ \lim_{n \to \infty} \left| \frac{u_{n+1}}{u_n} \right| = \lim_{n \to \infty} \frac{(n+1)^2 / 3^{n+1}}{n^2 / 3^n} = \frac{1}{3} < 1 $$ +$$ \lim_{n \to \infty} \left| \frac{u_{n+1}}{u_n} \right| = \lim_{n \to \infty} \frac{(n+1)^2 / 3^{n+1}}{n^2 / 3^n} = \frac{1}{3} < 1 $$ 若由此断言“原级数收敛”，则犯了**滥用判别法**的错误。因为比值判别法（及比较、根值判别法）仅 @@ -322,17 +392,18 @@ $$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \frac{1}{3} < 1 $$ 根据定义，若一个级数的绝对值级数收敛，则该级数**绝对收敛**。绝对收敛的级数必然收敛。 ## Vol. 5:误用p级数 + **机械地套用p级数结论，而忽视了其应用前提：指数 `p` 必须是与 `n` 无关的常数。** > [!example] 例题1： >$$判定\sum_{n=1}^{\infty} \frac{1}{n^{1 + \frac{1}{n}}}的敛散性$$ #### ❌ 经典错误思路 -1. 形式像 `1/n^p` -2. "指数"是 `1 + 1/n` -3. 因为 `1/n > 0`，所以 `p = 1 + 1/n > 1` 恒成立，误判为收敛 +1. 形式像 $1/n^p$ +2. "指数"是 $1 + 1/n$ +3. 因为 $1/n > 0$，所以 $p = 1 + \frac{1}{n} > 1$ 恒成立，误判为收敛 #### ✅ 正确分析与解法 -**错误原因**：`pₙ = 1 + 1/n` 不是常数，其极限为1。 -使用比值审敛法与调和级数 `∑ 1/n` 比较： +**错误原因**：$pₙ = 1 + \frac{1}{n}$ 不是常数，其极限为1。 +使用比较判别法与调和级数 `∑ 1/n` 比较： $$\lim_{n \to \infty} \frac{ \frac{1}{n^{1 + \frac{1}{n}}} }{ \frac{1}{n} } = \lim_{n \to \infty} \frac{1}{n^{1/n}} = 1$$ 可知两级数敛散性相同，且调和级数发散 ⇒ 原级数**发散**。 @@ -343,7 +414,7 @@ $$\lim_{n \to \infty} \frac{ \frac{1}{n^{1 + \frac{1}{n}}} }{ \frac{1}{n} } = \l #### ❌ 经典错误思路 -无法直接套用p级数，因多了一个 `ln n` 因子。错误地认为 `1/(n ln n) < 1/n`，认为原级数收敛 +无法直接套用p级数，因多了一个 `ln n` 因子。错误地认为 `1/(nln n) < 1/n`，认为原级数收敛 #### ✅ 正确分析与解法(超纲，仅供拓展) **正确解法**（积分判别法）： @@ -356,6 +427,7 @@ $$ ## Vol. 6: 条件收敛、绝对收敛、发散 + **仅当利用比值/根值判别法判断出$\sum |a_n|$发散时$\Rightarrow$$\sum a_n$发散 #### 基本定义 @@ -365,7 +437,9 @@ $$ 3. **条件收敛**：如果 $\sum a_n$ 收敛但 $\sum |a_n|$ 发散，则称 $\sum a_n$ **条件收敛** #### 正确分析： + 仅有$\sum |a_n|$ 收敛 ⇒ $\sum a_n$收敛 + **证明**： - 使用柯西收敛准则。对于任意 $\varepsilon > 0$： 因为 $\sum |a_n|$ 收敛，由柯西准则，存在 $N$，使得当 $m > n \geq N$ 时： @@ -432,7 +506,9 @@ $$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$$ - 所以$a_n$ 发散。 ## **Vol.7 分段函数分段点处求导问题** + 分段函数分段点处无论是求导还是判断连续都必须从**左右两边的极限**分别去算，而且计算的时候一定只能**用定义**。 + >[!example] 例题 >设$f(x)=\begin{cases} \frac{2}{3}x ,\ \ x\le1 \\ x^2, \ \ x>1,\end{cases}$则$f(x)$在$x=1$处的\[ \]. >（A）左右导数都存在 （B）左导数存在，右导数不存在 @@ -440,8 +516,9 @@ $$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$$ **解：**$f(1)=\frac{2}{3},f'_-(1)=\lim\limits_{x\to1^-}\frac{\frac{2}{3}x-\frac{2}{3}}{x-1}=\frac{2}{3},f'_+(1)=\lim\limits_{x\to1^+}\frac{x^2-\frac{2}{3}}{x-1}=+\infty$,故左导数存在，右导数不存在，选$B$. -## **Vol.8绝对收敛级数** -绝对收敛的级数满足加法交换律，也就是说，交换各项的顺序不会导致最后结果的改变。但条件收敛的级数是不满足交换律的，改变加法的顺序可能会导致最后结果的改变，甚至可能使原本收敛的级数变成发散级数。这一点了解就行，不会出题目给大家考。 +## **Vol.8: 绝对收敛级数** + +绝对收敛的级数满足加法交换律，也就是说，交换各项的顺序不会导致最后结果的改变。但条件收敛的级数是不满足交换律的，改变加法的顺序可能会导致最后结果的改变，甚至可能使原本收敛的级数变成发散级数。这一点了解就行，基本不会出题目给大家考。 ## Vol. 9: 反函数求导 易错：变量混淆。反函数的导数 = 原函数导数的倒数，但**自变量和因变量角色互换**。 @@ -453,7 +530,7 @@ $y=f^{-1}(x)$，即$x = f(y)$，求$f^{-1'}$就是在求$\frac{dy}{dx}$，而$\f 所以，我们还需要将$y = f^{-1}(x)$代入，即： $\frac{dy}{dx}=\frac{1}{f'(y)}=\frac{1}{f'(f^{-1}(x))}$ -> [!example] 例1 +> [!example] 例题 > 求$d(\arcsin x)$ 解：设$y=\arcsin x$，即$x=\sin y$，$dx=\cos y\ dy$，即$dy=\frac{dx}{\cos y}$ @@ -463,21 +540,38 @@ $\frac{dy}{dx}=\frac{1}{f'(y)}=\frac{1}{f'(f^{-1}(x))}$ 得$\cos y=\sqrt{1-x^2}$，综上，$dy=\frac{dx}{\sqrt{1-x^2}}$，即$d(\arcsin x)=\frac{dx}{\sqrt{1-x^2}}$ ## Vol. 10: 无界与无穷大的辨析 + 很多人都觉得无界和无穷大是同一个概念，因为它们的实在是太像了：画在坐标系上都是“直指苍穹🚀”或者“飞流直下三千尺”嘛！但是，“无界”准确来说不完全是这样。要准确辨析它们，需要回到它们的**定义**上： + 无穷大的定义：$\forall M > 0, \exists \delta>0$，当$0<|x-x_0|<\delta$时有$|f(x)|>M$，称$f(x)$是当$x\rightarrow x_0$时的无穷大量 + 无界量的定义：由有界的定义($\exists M > 0, \forall x \in D_f,|f(x)|<M$)反推，无界的定义应该是$\forall M > 0$, $\forall \delta>0, \exists x \in \mathring{U}(x_0,\delta)$有 $|f(x)|>M$，称$f(x)$是当$x\rightarrow x_0$时的无界量 + **核心区别**：无穷大是**存在某**去心邻域内**任意**$x$都大于$M$，无界是需要对**任意**邻域**存在**一个$x$使得$|f(x)|>M$ + **联系**：无穷大一定是无界量，但是无界量不一定是无穷大。 -> [!example] 例1 +> [!example] 例题 > 无穷震荡$\lim\limits_{x\rightarrow 0}{\frac{1}{x}\sin\frac{1}{x}}$ ![[易错点10-1.png]] 这个并不是无穷大——不管取的邻域有多小，我总能找到一个令$\sin\frac{1}{x}=0$的$x$，此时$\frac{1}{x}\sin\frac{1}{x}=0$。 -那这个是有界的吗？也不是。这就是典型的**不是无界量的无穷大**。 +那这个是有界的吗？也不是。这就是典型的**不是无穷大的无界量**。 +详细的证明过程如下： -> [!example] 例2 -> 双子数列$\lim\limits_{n\rightarrow \infty}{n\cos n\pi}$ +>对$f(x)={\frac{1}{x}\sin\frac{1}{x}}$，可以取数列$a_n=\frac{1}{2n\pi+\frac{\pi}{2}}$, +>$f(a_n)=\frac{1}{a_n} \sin\frac{1}{a_n}=2n\pi+\frac{\pi}{2}$，$\lim\limits_{n\rightarrow\infty}{f(a_n)}=+\infty$故极限无界； +>另取数列$b_n=\frac{1}{n\pi}$，$f(b_n)=0$。 +>按照无穷大的定义：$\forall M > 0, \exists \delta>0$，当$0<|x-x_0|<\delta$时有$|f(x)|>M$，称$f(x)$是当$x\rightarrow x_0$时的无穷大量。 +>然而，$\forall\delta>0$，$n>\frac{1}{\delta\pi}, x=b_n$时，$0<|x|<\delta$，但是$f(x)=0$，与无穷大定义不符。 +>故：$\lim\limits_{x\rightarrow 0}{\frac{1}{x}\sin\frac{1}{x}}$无界，但不是无穷大 -总结：无穷大是在邻域内“一直都很大”，无界是邻域内“有很大的” +> [!example] 例2 +> 双子数列$\lim\limits_{n\rightarrow \infty}{n\cos \frac{n\pi}{2}}$ +>如图： +>![[易错点10-2.png]] +>与上一题思路类似，令$x_n={n\cos \frac{n\pi}{2}}$，取子数列$a_n=x_{2n-1},b_n=x_{4n}$， +>$a_n=0$，$\lim\limits_{n\rightarrow \infty}{b_n}=+\infty$，故$x_n$无界； +>按照（数列）无穷大的定义：$\forall M > 0, \exists N>0$，当$n>N$时有$x_n>M$，称$x_n$趋近无穷大 +>然而，$\forall N>0$，取$n=4N$，$x_n=b_N=0$，与无穷大定义不符。故$x_n$无界但不是无穷大。 \ No newline at end of file diff --git a/编写小组/讲义/微分中值定理（解析版）.md b/编写小组/讲义/微分中值定理（解析版）.md new file mode 100644 index 0000000..0c81049 --- /dev/null +++ b/编写小组/讲义/微分中值定理（解析版）.md @@ -0,0 +1,459 @@ +--- +tags: + - 编写小组 +--- +**内部资料，禁止传播** +**编委会（不分先后，姓氏首字母顺序）：陈峰华 陈玉阶 程奕铭 韩魏 刘柯妤 卢吉辚 王嘉兴 王轲楠 彭靖翔 郑哲航 钟宇哲 支宝宁 + +## **辅助函数的构造方法** + +### **原理** + +在证明与导数相关的等式或不等式时，常通过构造辅助函数，将原问题转化为对某个函数应用中值定理（如罗尔定理、拉格朗日定理等）。构造辅助函数的核心思想是：**将待证等式视为某个函数求导后的结果**。 + +### **常见构造类型** + +#### 1. 乘积型与商型 +若结论形如： +$$ +f'(\xi)g(\xi) + f(\xi)g'(\xi) = 0 +$$ +可构造辅助函数： +$$ +F(x) = f(x)g(x) +$$ +若结论形如： +$$ +f'(\xi)g(\xi) - f(\xi)g'(\xi) = 0 +$$ +可构造辅助函数： +$$ +F(x) = \frac{f(x)}{g(x)} \quad (g(x) \neq 0) +$$ + +#### 2. 含幂函数因子 +若结论形如： +$$ +n f(\xi) + \xi f'(\xi) = 0 +$$ +可构造辅助函数： +$$ +F(x) = x^n f(x) +$$ + +#### 3. 一阶线性微分结构 +若结论形如： +$$ +f'(\xi) + P(\xi)f(\xi) = 0 +$$ +可构造积分因子： +$$ +\mu(x) = e^{\int P(x)\mathrm{d}x} +$$ +并设辅助函数： +$$ +F(x) = \mu(x) f(x) +$$ + +#### 4. 对数型 +若结论形如： +$$ +\frac{f'(\xi)}{f(\xi)} = k +$$ +可构造辅助函数： +$$ +F(x) = \ln|f(x)| - kx +$$ + +#### 5. 常数变易法 +若结论形如： +$$ +f'(\xi) = \lambda f(\xi) +$$ +可构造辅助函数： +$$ +F(x) = e^{-\lambda x} f(x) +$$ +或者写成： +$$ +f'(\xi) + \lambda f(\xi) = 0 +$$ +则构造辅助函数： +$$ +F(x) = e^{\lambda x} f(x) +$$ + +--- + +### **例题**（先看完后面的知识再做这个） + +>[!example] 例1 +设 $f(x)$ 在 $[0, 1]$ 上连续，在 $(0, 1)$ 内可导，且 $f(0)=0$，$f(1)=1$。 +证明：存在 $\xi \in (0, 1)$ 使得$f'(\xi) = 2\xi f(\xi)$ + +**解析**： +将结论改写为： +$$ +f'(\xi) - 2\xi f(\xi) = 0 +$$ +属于一阶线性微分结构，其中 $P(x) = -2x$。 +积分因子为： +$$ +\mu(x) = e^{\int (-2x)\mathrm{d}x} = e^{-x^2} +$$ +构造辅助函数： +$$ +F(x) = e^{-x^2} f(x) +$$ +则 $F(0) = 0$，$F(1) = e^{-1}$。 +需进一步寻找另一个点 $c$ 使 $F(c)=0$，才可应用罗尔定理。通常需结合题目其他条件（如积分中值定理、零点定理等）找出该点。 + +--- + +>[!example] 例2 +设 $f(x)$ 在 $[a, b]$ 上三阶可导，且 $f(a) = f'(a) = f(b) = 0$。 +证明：存在 $\xi \in (a, b)$ 使得：$f'''(\xi) + k f''(\xi) = 0$ + +**解析**： +结论可写为： +$$ +\bigl[ e^{kx} f''(x) \bigr]' \big|_{x=\xi} = 0 +$$ +因此构造辅助函数： +$$ +H(x) = e^{kx} f''(x) +$$ +由条件可推知存在 $\eta_1, \eta_2 \in (a, b)$ 使 $f''(\eta_1) = f''(\eta_2) = 0$，从而 $H(\eta_1)=H(\eta_2)=0$。 +对 $H(x)$ 应用罗尔定理即得证。 + +--- + +>[!example] 例3 +设 $f(x)$ 在 $[0, 1]$ 上可导，且$f(1) = 2\int_0^{1/2} e^{1-x} f(x) dx$ +证明：存在 $\xi \in (0, 1)$ 使得：$f'(\xi) = (1-\xi) f(\xi)$ + +**解析**： +结论化为： +$$ +f'(\xi) - (1-\xi) f(\xi) = 0 +$$ +积分因子为： +$$ +\mu(x) = e^{\int (x-1) \mathrm{d}x} = e^{\frac{x^2}{2} - x} +$$ +构造辅助函数： +$$ +F(x) = e^{\frac{x^2}{2} - x} f(x) +$$ +利用题设积分条件与积分中值定理，可找到 $\eta \in (0, \frac{1}{2})$ 使 $F(\eta) = F(1)$，再对 $F(x)$ 应用罗尔定理即证。 + +## **罗尔定理** + +### **原理** +若函数 f(x) 满足以下三个条件： +在闭区间 $[a,b]$ 上连续； +在开区间 $(a,b)$ 内可导； +区间端点函数值相等，即 $f(a)=f(b)$； +则在 $(a,b)$ 内至少存在一点 $\xi$，使得 $f'(\xi)=0$。 +罗尔定理的几何意义为：满足条件的函数曲线在区间内至少有一条水平切线。 +它是拉格朗日中值定理（$f(b)-f(a)=f'(\xi)(b-a)$）当 $f(a)=f(b)$ 时的特例。 + +### **适用条件** +罗尔定理的核心适用题型是证明导函数方程 $f'(\xi)=0$ 在区间 $(a,b)$ 内有根以及衍生的相关证明题。 +具体可分为以下几类： +1.直接证明 $f'(\xi)$=0 存在根 +题目给出函数 f(x) 在 $[a,b]$ 上的连续性、$(a,b)$ 内的可导性，且满足 $f(a)=f(b)$，直接应用罗尔定理证明存在 $\xi\in(a,b)$ 使得 $f'(\xi)=0$。 +2.构造辅助函数证明导函数相关方程有根 +对于形如 $f'(\xi)+g(\xi)f(\xi)=0$、$f''(\xi)=0$ 等方程，需构造满足罗尔定理条件的辅助函数 $F(x)$，通过 $F(a)=F(b)$ 推导 $F'(\xi)=0$，进而等价转化为目标方程。 +3.结合多次罗尔定理证明高阶导数零点存在 +若函数 f(x) 有 n+1 个点的函数值相等，可多次应用罗尔定理，证明其 n 阶导数 $f^{(n)}(\xi)=0$ 在对应区间内有根。 +4.证明函数恒为常数（反证法结合罗尔定理） +若 $f'(x)\equiv0$ 在区间内成立，可通过反证法假设存在两点函数值不等，结合罗尔定理推出矛盾，进而证明函数为常数。 + +罗尔定理针对于一个函数，不同于柯西中值定理针对于两个函数 + +### **例题** +>[!example] 例1 +设 $f(x)$ 在 $[0,1]$ 连续，$(0,1)$ 可导，且 $f(1) = 0$，求证存在 $\xi \in (0,1)$ 使得 $nf(\xi) + \xi f'(\xi) = 0$。 + +**解析**： +设辅助函数 $\varphi(x) = x^n f(x)$，则 $\varphi(0)=0$，$\varphi(1)=0$。由罗尔定理，存在 $\xi \in (0,1)$，使得 $\varphi'(\xi)=0$即 +$$ +n\xi^{n-1} f(\xi) + \xi^n f'(\xi) = 0 +$$ +两边除以 $\xi^{n-1}$ ($\xi>0$)，得 $nf(\xi) + \xi f'(\xi) = 0$。 + + + +>[!example] 例2 +设函数 $f(x)$ 在 $[a,b]$ 上可导，且 +$$f(a) = f(b) = 0，\quad f'_+(a)f'_-(b) > 0，$$ +试证明 $f'(x) = 0$ 在 $(a,b)$ 内至少有两个根。 + + +**解析**： +由导数极限定理及 $f'_+(a)f'_-(b) > 0$，知在 $a$ 右侧和 $b$ 左侧，$f(x)$ 的符号相同，不妨设 $f'_+(a)>0$，$f'_-(b)>0$。则在 $a$ 右侧附近 $f(x)>0$，在 $b$ 左侧附近 $f(x)>0$。由于 $f(a)=f(b)=0$，由极值点的费马定理，$f(x)$ 在 $(a,b)$ 内至少有一个极大值点，该点处导数为零。又因为 $f(x)$ 在 $[a,b]$ 上连续，在 $(a,b)$ 内可导，且 $f(a)=f(b)$，由罗尔定理至少存在一点 $c \in (a,b)$ 使 $f'(c)=0$。结合极大值点处的导数零点，可知至少有两个导数为零的点。 + + + +>[!example] 例3 +设 $f(x)$ 在 $[0, 1]$ 上具有二阶导数，且满足 +$$f(0) = 0, \, f(1) = 1, \, f\left(\frac{1}{2}\right) > \frac{1}{4}$$证明： +(1)至少存在一点 $\xi \in (0, 1)$，使得 $f''(\xi) < 2$； +(2)若对一切 $x \in (0, 1)$，有 $f''(x) \neq 2$，则当 $x \in (0, 1)$ 时，恒有 $f(x) > x^2$。 + +**解析**： +(1) 考虑函数 $g(x)=f(x)-x^2$，则 $g(0)=0$，$g(1)=0$，$g(1/2)=f(1/2)-1/4>0$。由极值点的费马定理及罗尔定理，$g(x)$ 在 $(0,1)$ 内存在极大值点 $\eta$，且 $g'(\eta)=0$，$g''(\eta) \leq 0$。即 $f'(\eta)=2\eta$，$f''(\eta) \leq 2$。若 $f''(\eta) < 2$，则取 $\xi=\eta$ 即可；若 $f''(\eta)=2$，则考虑在 $\eta$ 两侧应用拉格朗日中值定理，可找到另一个点 $\xi$ 使得 $f''(\xi)<2$。 +(2) 用反证法。假设存在 $x_0 \in (0,1)$ 使 $f(x_0) \leq x_0^2$，结合 $f(0)=0$，$f(1)=1$ 和 $f(1/2)>1/4$，利用连续性及中值定理可推出存在 $\xi$ 使 $f''(\xi)=2$，矛盾。 + +## **拉格朗日中值定理** +### **原理** +若函数 f(x) 满足两个条件: +在闭区间 $[a,b]$ 上连续； +在开区间 $(a,b)$ 内可导； +则在 $(a,b)$ 内至少存在一点 $\xi$，使得 +$f(b)-f(a)=f'(\xi)(b-a)$ +也可写成等价形式 $f'(\xi)=\dfrac{f(b)-f(a)}{b-a}$。 +是罗尔定理的推广，同时也是柯西中值定理的特例。其几何意义为：满足条件的函数曲线在区间 (a,b) 内，至少存在一点的切线与连接端点 (a,f(a)) 和 (b,f(b)) 的弦平行。 + +### **适用条件** +拉格朗日中值定理的核心适用题型是建立函数增量与导数的关联，进行不等式的证明，这是最常见的题型。通过对目标函数在指定区间上应用拉格朗日中值定理，得到 $f(b)-f(a)=f'(\xi)(b-a)$，再利用导数 $f'(\xi)$ 的取值范围（有界性、正负性）放大或缩小式子，推导不等式。 + +### **例题** + +>[!example] 例1 +设函数 $f(x)$ 在 $(-1,1)$ 内可微，且 +$$f(0) = 0, \quad |f'(x)| \leq 1,$$证明：在 $(-1,1)$ 内，$|f(x)| < 1$。 + +**解析**： +对任意 $x \in (-1,1)$，由拉格朗日中值定理，存在 $\xi$ 介于 $0$ 与 $x$ 之间，使得 +$$ +f(x) - f(0) = f'(\xi)(x-0) +$$ +即 $f(x) = f'(\xi) x$。由于 $|f'(\xi)| \leq 1$，$|x| < 1$，故 $|f(x)| = |f'(\xi)| \cdot |x| < 1$。 + + +>[!example] 例2 +设 $f''(x) < 0$，$f(0) = 0$，证明对任意 $x_1 > 0, x_2 > 0$ 有 +$$f(x_1 + x_2) < f(x_1) + f(x_2)$$ + +**解析**： +不妨设 $0 < x_1 < x_2$。由拉格朗日中值定理： +$$ +f(x_1+x_2)-f(x_2) = f'(\xi_1)x_1, \quad \xi_1 \in (x_2, x_1+x_2) +$$ +$$ +f(x_1)-f(0) = f'(\xi_2)x_1, \quad \xi_2 \in (0, x_1) +$$ +于是 +$$ +f(x_1+x_2)-f(x_2)-f(x_1) = [f'(\xi_1)-f'(\xi_2)]x_1 +$$ +对 $f'(x)$ 在 $[\xi_2,\xi_1]$ 上应用拉格朗日中值定理，存在 $\xi \in (\xi_2,\xi_1)$，使 +$$ +f'(\xi_1)-f'(\xi_2) = f''(\xi)(\xi_1-\xi_2) < 0 +$$ +故 $f(x_1+x_2)-f(x_2)-f(x_1) < 0$，即 $f(x_1+x_2) < f(x_1)+f(x_2)$。 + + +>[!example] 例3 + 设 $f(x)$ 在 $(-\infty, +\infty)$ 内二阶可导，且 $f''(x) \neq 0$。 +（1）证明：对于任何非零实数 $x$，存在唯一的 $\theta(x)$ ($0<\theta(x)<1$)，使得 + $$f(x) = f(0) + x f'(x\theta(x));$$ + （2）求 + $$\lim_{x \to 0} \theta(x).$$ + +解： +1. 证： 对于任何非零实数 $x$，由中值定理，存在 $\theta(x)$ $(0<\theta(x)<1)$，使得 + +$$ +f(x)=f(0)+x f'(x\theta(x)). +$$ + +如果这样的 $\theta(x)$ 不唯一，则存在 $\theta_{1}(x)$ 与 $\theta_{2}(x)$ $(\theta_{1}(x)<\theta_{2}(x))$，使得 $f'(x\theta_{1}(x))=f'(x\theta_{2}(x))$，由罗尔定理，存在一点 $\xi$，使得 $f''(\xi)=0$，这与 $f''(x)\neq 0$ 矛盾。所以 $\theta(x)$ 是唯一的。 + +2. 解 注意到 $f''(0)=\lim_{x\rightarrow 0} \frac{f'(x\theta(x))-f'(0)}{x\theta(x)}$，又知 + +$$ +\begin{aligned} +\lim_{x\rightarrow 0} \frac{f'(x\theta(x))-f'(0)}{x} +&= \lim_{x\rightarrow 0} \frac{\frac{f(x)-f(0)}{x}-f'(0)}{x} \\ +&= \lim_{x\rightarrow 0} \frac{f(x)-f(0)-x f'(0)}{x^{2}} \\ +&= \lim_{x\rightarrow 0} \frac{f'(x)-f'(0)}{2x} \\ +&= \frac{f''(0)}{2}, +\end{aligned} +$$ + +所以 $\lim_{x\rightarrow 0} \theta(x)=\frac{1}{2}$。 + +## **柯西中值定理** + +### **原理** + +设函数 $f(x)$ 和 $g(x)$ 满足以下条件： +1. 在闭区间 $[a, b]$ 上连续； +2. 在开区间 $(a, b)$ 内可导； +3. 对任意 $x \in (a, b)$，有 $g'(x) \neq 0$； +则在 $(a, b)$ 内至少存在一点 $\xi$，使得： + +$$ +\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(\xi)}{g'(\xi)} +$$ + +柯西中值定理的几何意义为：由参数方程 $(g(t), f(t))$ 表示的曲线，在两点间的割线斜率等于曲线上某点切线的斜率。 + +它与拉格朗日中值定理的关系为：当 $g(x) = x$ 时，柯西中值定理退化为拉格朗日中值定理。它是处理两个函数之间微分中值关系的通用形式。 + +### **适用条件** + +柯西中值定理的核心适用题型是**证明形如 $\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(\xi)}{g'(\xi)}$ 的等式成立**，以及处理**涉及两个中值点 $\xi, \eta$ 的问题**。 + +常见应用方向包括： +1. 直接证明存在性等式； +2. 通过函数配对，将目标等式转化为柯西中值定理的标准形式； +3. 处理“双中值问题”，常与拉格朗日中值定理结合使用。 + +### **例题** + +>[!example] 例1 +设 $f(x)$ 在 $[a, b]$ 上连续，在 $(a, b)$ 内可导，且 $a>0$。证明存在 $\xi \in (a, b)$，使得： +$$\frac{f(b)-f(a)}{b-a} = \xi f'(\xi) \cdot \frac{\ln(b/a)}{b-a}$$ + +**解析**： +将等式变形为： +$$ +\frac{f(b)-f(a)}{\ln b - \ln a} = \xi f'(\xi) +$$ +取 $g(x) = \ln x$，则 $g'(x) = \frac{1}{x} \neq 0$ 在 $(a, b)$ 内成立。 +对 $f(x)$ 与 $g(x)$ 应用柯西中值定理，存在 $\xi \in (a, b)$ 使得： +$$ +\frac{f(b)-f(a)}{\ln b - \ln a} = \frac{f'(\xi)}{1/\xi} = \xi f'(\xi) +$$ +整理即得所求。 + +--- + +>[!example] 例2 +设 $f(x)$ 在 $[a, b]$ 上连续，在 $(a, b)$ 内可导，证明存在不同的 $\xi, \eta \in (a, b)$，使得： +$$f'(\xi) = \frac{a+b}{2\eta} f'(\eta)$$ + +**解析**： +1. 对 $f(x)$ 与 $g(x) = \frac{x^2}{2}$ 应用柯西中值定理，存在 $\eta \in (a, b)$ 使得： + $$ + \frac{f(b)-f(a)}{(b^2 - a^2)/2} = \frac{f'(\eta)}{\eta} + $$ + 整理得： + $$ + f(b)-f(a) = \frac{b^2 - a^2}{2\eta} f'(\eta) + $$ + +2. 对 $f(x)$ 应用拉格朗日中值定理，存在 $\xi \in (a, b)$ 使得： + $$ + f(b)-f(a) = (b-a) f'(\xi) + $$ + +3. 联立两式，消去 $f(b)-f(a)$ 得： + $$ + (b-a) f'(\xi) = \frac{(b-a)(a+b)}{2\eta} f'(\eta) + $$ + 由于 $b-a \neq 0$，约去后即得： + $$ + f'(\xi) = \frac{a+b}{2\eta} f'(\eta) + $$ + +--- + +>[!example] 例3 +设 $0 < a < b$，证明存在 $\xi \in (a, b)$，使得： +$$f(b)-f(a) = \frac{3\xi^2}{a^2+ab+b^2} f'(\xi)(b-a)$$ + +**解析**： +将等式变形为： +$$ +\frac{f(b)-f(a)}{b^3 - a^3} = \frac{f'(\xi)}{3\xi^2} +$$ +取 $g(x) = x^3$，则 $g'(x) = 3x^2 \neq 0$ 在 $(a, b)$ 内成立。 +由柯西中值定理，存在 $\xi \in (a, b)$ 使得： +$$ +\frac{f(b)-f(a)}{b^3 - a^3} = \frac{f'(\xi)}{3\xi^2} +$$ +整理后即得所求。 + + + + + +## 多次运用中值定理 + +多次运用中值定理一般有如下特征： +1. 有多个中值（如$\xi,\eta$两个中值）； +2. 有二阶导出现。 +这种题目一般会比较难，而且通常要结合其他方法，比如反证法、构造函数等。 + +多次用中值定理又分为两种： +1. 在同一区间（或一个区间包含另一个区间）上用不同的中值定理； +2. 在相邻区间上对原函数和导函数用同一个（也可能是不同的）中值定理。 + +>[!example] 例1 +设 $f(x)$ 在 $[a, b]$ 上连续，在 $(a, b)$ 内可导，且 $0 < a < b$，试证存在 $\xi, \eta \in (a, b)$，使得 $$f'(\xi) = \frac{a + b}{2\eta} f'(\eta).$$ + +**分析：** + 首先注意到题目要求我们证明的式子中有两个中值（即在区间内的某个点及其函数值），故想到可能会用多次中值定理。右边有$\frac{f'(\eta)}{\eta}$的形式，一般会想到用拉格朗日或者柯西，又题中没有哪个值让$f$等于$0$，所以大概率就是柯西中值定理。 + 令$g(x)=x^2$，则有$$\exists\eta\in(a,b),\frac{f'(\eta)}{g'(\eta)}=\frac{f'(\eta)}{2\eta}=\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f(b)-f(a)}{b^2-a^2}$$ + 再对比要求的式子，知道再用一次拉格朗日中值定理就行。 + +**解**： +对 $f(x)$ 和 $g(x)=x^2$ 在 $[a,b]$ 上应用柯西中值定理，得存在 $\eta \in (a,b)$，使得 +$$ +\frac{f(b)-f(a)}{b^2-a^2} = \frac{f'(\eta)}{2\eta} +$$ +整理得 +$$ +\frac{f(b)-f(a)}{b-a} = \frac{a+b}{2\eta} f'(\eta) +$$ +再对 $f(x)$ 在 $[a,b]$ 上应用拉格朗日中值定理，存在 $\xi \in (a,b)$，使得 +$$ +\frac{f(b)-f(a)}{b-a} = f'(\xi) +$$ +比较两式即得结论。 + +>[!example] 例2 +设 $f(x)$ 在 $[a, b]$ 上连续，在 $(a, b)$ 内二阶可导，又若 $f(x)$ 的图形与联结 $A(a, f(a))$，$B(b, f(b))$ 两点的弦交于点 $C(c, f(c))$ ($a \leq c \leq b$)，证明在 $(a, b)$ 内至少存在一点 $\xi$，使得 $f''(\xi) = 0$。 + +**分析：![[微分中值定理图.png]]** + 二阶导的零点就是图像的拐点，从图中能直观地看出来，函数图像的凹凸性确实发生了改变。现在的问题就是如何证明。 + 首先可以很直观地看到，函数图像应当有两条与直线$AB$平行的切线，由拉格朗日中值定理也可以证明这一点。这样，$f'(x)$就在不同地方取到了相同的函数值，这就想到用罗尔定理，从而可以证明题中结论。 + +**解**： +弦 $AB$ 的方程为 +$$ +y = f(a) + \frac{f(b)-f(a)}{b-a}(x-a) +$$ +由条件，$f(c) = f(a) + \frac{f(b)-f(a)}{b-a}(c-a)$。分别对 $f(x)$ 在 $[a,c]$ 和 $[c,b]$ 上应用拉格朗日中值定理，存在 $\xi_1 \in (a,c)$，$\xi_2 \in (c,b)$，使得 +$$ +f'(\xi_1) = \frac{f(c)-f(a)}{c-a} = \frac{f(b)-f(a)}{b-a} +$$ +$$ +f'(\xi_2) = \frac{f(b)-f(c)}{b-c} = \frac{f(b)-f(a)}{b-a} +$$ +故 $f'(\xi_1)=f'(\xi_2)$。再对 $f'(x)$ 在 $[\xi_1,\xi_2]$ 上应用罗尔定理，存在 $\xi \in (\xi_1,\xi_2) \subset (a,b)$，使 $f''(\xi)=0$。 + +>[!example] 例3 +设 $f(x)$ 在 $[0, 1]$ 上具有二阶导数，且满足 +$$f(0) = 0, \, f(1) = 1, \, f\left(\frac{1}{2}\right) > \frac{1}{4}$$证明： +(1)至少存在一点 $\xi \in (0, 1)$，使得 $f''(\xi) < 2$； +(2)若对一切 $x \in (0, 1)$，有 $f''(x) \neq 2$，则当 $x \in (0, 1)$ 时，恒有 $f(x) > x^2$。 + +**分析：** + 初看没有什么思路，涉及到二阶导一般会用泰勒展开或者对导数用中值定理。我们对一阶导和二阶导的性质一概不清楚，所以应该考虑对导数用中值定理。 + 直接对$f(x)$用中值定理吗？不是，这样子我们得不出任何的结论。或许我们应该构造一个新的函数，让我们更容易研究一些。观察要证的式子，由于我们完全不知道导数的性质，所以考虑函数$g(x)=f(x)-x^2$，有$g'(x)=f'(x)-2x,g''(x)=f''(x)-2$，且$g(0)=0,g(1)=0,g(\frac{1}{2})>0$.看到这些$0$就舒服很多了。注意这里应该是多次运用中值定理中的第2种情况，因为题目很明显给了我们分割区间的一个点：$\large{\frac{1}{2}}$. + +**解**： +(1) 考虑函数 $g(x)=f(x)-x^2$，则 $g(0)=0$，$g(1)=0$，$g(1/2)=f(1/2)-1/4>0$。在$[0,\frac{1}{2}],[\frac{1}{2},1]$上分别用拉格朗日中值定理得，$\exists\eta_1\in(0,\frac{1}{2}),\eta_2\in(\frac{1}{2},1)$，使得$$g'(\eta_1)=\frac{g(1/2)-g(0)}{1/2}>0,g'(\eta_2)=\frac{g(1)-g(1/2)}{1/2}<0$$对$g'(x)$在区间$[\eta_1,\eta_2]$上用拉格朗日中值定理得，$\exists\xi\in(\eta_1,\eta_2)$，使得$$g''(\xi)=\frac{g'(\eta_2)-g'(\eta_1)}{\eta_2-\eta_1}<0,$$即$$f''(\xi)<2.$$ +(2) 用反证法。假设存在 $x_0 \in (0,1)$ 使 $f(x_0) \leq x_0^2$，故$g(x_0)\le0$。由$g(0)=0,g(1)=0,g(1/2)>0,\exists\alpha\in(0,1),g(\alpha)=0$.故由罗尔中值定理，$$\exists\beta_1\in(0,\alpha),\beta_2\in(\alpha,1),g'(\beta_1)=g'(\beta_2)=0,$$从而$$\exists\beta\in(\beta_1,\beta_2),g''(\beta)=0$$这与$f''(x)\neq2$，即$g''(x)\neq0$矛盾，故结论成立。 + +**题后总结：** + 1. 注意观察要证等式的形式，如果需要用到中值定理，就看看应该是哪个中值定理，比如等于某个确定的值一般就是罗尔，有多个函数（有些比较荫蔽，尤其是幂函数）一般是柯西，其他一些奇奇怪怪的情况基本上是拉格朗日 + 2. 注意分析和综合相结合的方法，从结论向前推一推，推不动了再从条件往后推一推，这是证明题的很重要的思路 + 3. 数形结合可以给我们很大的信心并提供思路，但是也要小心用 + diff --git a/编写小组/讲义/极限计算的基本方法.md b/编写小组/讲义/极限计算的基本方法.md index 86a0f37..e227386 100644 --- a/编写小组/讲义/极限计算的基本方法.md +++ b/编写小组/讲义/极限计算的基本方法.md @@ -192,8 +192,26 @@ $$\lim\limits_{x \to \infty} x(e^\tfrac{1}{x}-1)=\lim\limits_{x \to \infty } \tf # 等价无穷小求极限法 - -本质是利用已知公式进行等价无穷小因子的代换将复杂函数化简 +## 4.1 基本原理 + +在乘积和商的极限运算中，可以将复杂的无穷小量替换为等价的简单无穷小量，简化计算。 + +## 4.2 常用等价无穷小（$x\to 0$） +| 等价形式 | 条件 | +|---------|------| +| $x \sim \sin x \sim \tan x \sim \arcsin x \sim \arctan x$ | 基础等价 | +| $x \sim e^x-1 \sim \ln(1+x)$ | 指数对数等价 | +| $1-\cos x \sim \frac{1}{2}x^2$ | 三角函数等价 | +| $(1+x)^a-1 \sim ax$ | 幂函数等价 | +| $a^x-1 \sim x\ln a\ (a>0)$ | 指数函数等价 | +| $x - \sin x \sim \frac{1}{6}x^3$ | 高阶等价 | +| $\tan x - x \sim \frac{1}{3}x^3$ | 高阶等价 | + +## 4.3 使用原则 +1. **乘除运算可直接替换** +2. **加减运算需谨慎**：只有同阶无穷小相加减时，不能随意替换 +3. **无穷乘方运算一般也不行**（即乘方上下均有变量时）：一对等价量可以写成差一个无穷小的形式，经过无穷乘方后无法得知其情况 +4. **复合函数可整体替换** > [!example] 例1 diff --git a/编写小组/讲义/极限计算的基本方法（解析版2）.md b/编写小组/讲义/极限计算的基本方法（解析版2）.md index e91db4c..f63d7d9 100644 --- a/编写小组/讲义/极限计算的基本方法（解析版2）.md +++ b/编写小组/讲义/极限计算的基本方法（解析版2）.md @@ -343,7 +343,8 @@ $$ ### 4.3 使用原则 1. **乘除运算可直接替换** 2. **加减运算需谨慎**：只有同阶无穷小相加减时，不能随意替换 -3. **复合函数可整体替换** +3. **无穷乘方运算一般也不行**（即乘方上下均有变量时）：一对等价量可以写成差一个无穷小的形式，经过无穷乘方后无法得知其情况 +4. **复合函数可整体替换** ### 4.4 典型例题 diff --git a/编写小组/讲义/极限计算的基本方法（解析版）.md b/编写小组/讲义/极限计算的基本方法（解析版）.md index b49dde7..534af70 100644 --- a/编写小组/讲义/极限计算的基本方法（解析版）.md +++ b/编写小组/讲义/极限计算的基本方法（解析版）.md @@ -350,9 +350,7 @@ $$t = x - \frac{\pi}{4} \quad \Rightarrow \quad x = t + \frac{\pi}{4},\ t \to 0. $$= \frac{2\tan t}{1 - \tan t}.$$ 代入原极限：$$\lim\limits_{x\to\pi/4} \frac{\tan x - 1}{x - \pi/4} - = \lim\limits_{t\to 0} \frac{\frac{2\tan t}{1 - \tan t}}{t} - = \lim\limits_{t\to 0} \frac{2\tan t}{t(1 - \tan t)}.$$ 利用重要极限 $\displaystyle \lim_{u\to 0} \frac{\tan u}{u} = 1:\frac{2\tan t}{t(1 - \tan t)} = 2 \cdot \frac{\tan t}{t} \cdot \frac{1}{1 - \tan t}$ @@ -360,7 +358,6 @@ $$= \frac{2\tan t}{1 - \tan t}.$$ 当 $t \to 0$ 时，$\tan t \to 0，\lim\limits_{t\to 0} \frac{\tan t}{t} = 1,\quad \lim\limits_{t\to 0} (1 - \tan t) = 1.$ 所以$$\lim\limits_{t\to 0} 2 \cdot \frac{\tan t}{t} \cdot \frac{1}{1 - \tan t} - = 2 \cdot 1 \cdot \frac{1}{1 - 0} = 2$$ （4） @@ -371,7 +368,47 @@ $$= \frac{2\tan t}{1 - \tan t}.$$ # 等价无穷小求极限法 -本质是利用已知公式进行等价无穷小因子的代换将复杂函数化简 +## 4.1 基本原理 + +在乘积和商的极限运算中，可以将复杂的无穷小量替换为等价的简单无穷小量，简化计算。 + +## 4.2 常用等价无穷小（$x\to 0$） +| 等价形式 | 条件 | +|---------|------| +| $x \sim \sin x \sim \tan x \sim \arcsin x \sim \arctan x$ | 基础等价 | +| $x \sim e^x-1 \sim \ln(1+x)$ | 指数对数等价 | +| $1-\cos x \sim \frac{1}{2}x^2$ | 三角函数等价 | +| $(1+x)^a-1 \sim ax$ | 幂函数等价 | +| $a^x-1 \sim x\ln a\ (a>0)$ | 指数函数等价 | +| $x - \sin x \sim \frac{1}{6}x^3$ | 高阶等价 | +| $\tan x - x \sim \frac{1}{3}x^3$ | 高阶等价 | + +## 4.3 使用原则 +1. **乘除运算可直接替换** +2. **加减运算需谨慎**：只有同阶无穷小相加减时，不能随意替换 +3. **无穷乘方运算一般也不行**（即乘方上下均有变量时）：一对等价量可以写成差一个无穷小的形式，经过无穷乘方后无法得知其情况 +4. **复合函数可整体替换** + + +> [!example] 例1 +> 计算 $\displaystyle \lim_{x\to 0}\frac{\sin mx}{\sin nx}$ + + + + + + + +> [!example] 例2 +> 计算 $$\displaystyle \lim_{x\to 0}(\cos x)^{\frac{1+x}{\sin^2 x}}$$ + + + + + + +熟悉常见的极限，灵活运用好极限的四则运算法则和等价无穷小，已经能够解决绝大多数的极限计算题了，无论是简单题还是难题。很多看上去很复杂的所谓难题，无非是在四则运算和等价无穷小之间反复套娃而已，如果可以熟练的分离这些特征，其实不需要泰勒公式等“高级工具”就能快速准确地得出极限值。当然，不是贬低其他的工具，只是说不要学习了所谓的一些高级工具之后，就不重视这些初级的工具和结论。 + 引入练习：计算 $$\lim\limits_{x\to0}\frac{\sqrt{1+2tanx}-\sqrt{1+2sinx}}{xln(1+x)-x^2}$$     @@ -518,5 +555,4 @@ $$\alpha n + \ln \left( a + \frac{\cdots}{e^{\alpha n}} \right)$$ 小练习： - -$$\lim_{x \to 0} \frac{\csc(x) - \cot(x)}{x}=$$ \ No newline at end of file +$$\lim_{x \to 0} \frac{\csc(x) - \cot(x)}{x}= $$ diff --git a/编写小组/讲义/线性方程组的解与秩的不等式.md b/编写小组/讲义/线性方程组的解与秩的不等式.md new file mode 100644 index 0000000..3b466c3 --- /dev/null +++ b/编写小组/讲义/线性方程组的解与秩的不等式.md @@ -0,0 +1,611 @@ +--- +tags: + - 编写小组 +--- +**内部资料，禁止传播** +**编委会（不分先后，姓氏首字母顺序）：陈峰华 陈玉阶 程奕铭 韩魏 刘柯妤 卢吉辚 王嘉兴 王轲楠 彭靖翔 郑哲航 钟宇哲 支宝宁** + +# 单方程组解的问题 + +### 线性方程组解的判定 + +对非齐次线性方程组 $A_{m\times n}x=b$, +1. 无解的充要条件是 $\text{rank}A < \text{rank}[A\ \ b]$； +2. 有唯一解的充要条件是 $\text{rank}A = \text{rank}[A\ \ b] = n$； +3. 有无穷多解的充要条件是 $\text{rank}A = \text{rank}[A\ \ b] < n$。 +注:上述定理也说明非齐次线性方程组有解的充要条件是 $\text{rank}A = \text{rank}[A\ \ b]$。 + +怎么理解： +1. 从线性方程组的角度，如果加上一列 $b$ 后的秩变大了，那么化为最简行阶梯型后下面一定多出来一行 $0=$某个常数 ，则必然无解。 +2. 秩等于 $n$ 就是有 $n$ 个无关的方程，则经过消元法后可以解出唯一解。 +3. 秩小于 $n$ 就是方程不足 $n$ 个，消元消不完，也能解释为啥秩跟解空间维数的和为 $n$ + +把以上结论应用到齐次线性方程组，可得 +推论 齐次线性方程组 $A_{m\times n}x=0$ 有非零解（无穷多解）的充要条件是 $\text{rank}A < n$，即系数矩阵的秩小于未知数个数。 + +### 矩阵方程解的判定 + +本质上和线性方程组是一脉相承的，只是形式上更一般化。 +最常见的矩阵方程是 $\boldsymbol{AX} = \boldsymbol{B}$，其中$\boldsymbol{A}$是 $m\times n$ 矩阵，$\boldsymbol{B}$ 是 $m\times p$ 矩阵，$\boldsymbol{X}$ 是待求的$n\times p$矩阵。 +1. 有解的充要条件： +矩阵方程有解的充要条件是系数矩阵 $\boldsymbol{A}$ 的秩等于增广矩阵 $[\boldsymbol{A} \ \boldsymbol{B}]$的秩，即： +$$r(\boldsymbol{A}) = r([\boldsymbol{A} \ \boldsymbol{B}])$$ +这个结论和非齐次线性方程组有解的条件完全一致。 + +理解上可以将 $B$ 拆分成一列列 $b$ ，从而化归为上面的线性方程组问题 + +2. 解的结构： +- 唯一解：当$r(\boldsymbol{A}) = r([\boldsymbol{A} \ \boldsymbol{B}]) = n$ 时，方程有唯一解。 +- 无穷多解：当 $r(\boldsymbol{A}) = r([\boldsymbol{A} \ \boldsymbol{B}]) < n$ 时，方程有无穷多解。 + +可逆矩阵 +- 当 $\boldsymbol{A}$ 是 n 阶可逆矩阵时，矩阵方程 $\boldsymbol{AX} = \boldsymbol{B}$有唯一解:$\boldsymbol{X} = \boldsymbol{A}^{-1}\boldsymbol{B}$ + +其他形式的矩阵方程 +- 对于 $\boldsymbol{XA} = \boldsymbol{B}$ 形式的方程，可以转置为 $\boldsymbol{A}^T\boldsymbol{X}^T = \boldsymbol{B}^T$，再套用上述方法，或类似于方程 $AX = B$ 有解的充要条件是$\text{rank} \begin{bmatrix} A & B \end{bmatrix} = \text{rank}A$，由方程 $XA = B$ 有解可知 $\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} =\text{rank}A$ +- 对于 $\boldsymbol{AXB} = \boldsymbol{C}$ 形式的方程，当 $\boldsymbol{A} 和 \boldsymbol{B}$ 都可逆时，有唯一解 $\boldsymbol{X} = \boldsymbol{A}^{-1}\boldsymbol{C}\boldsymbol{B}^{-1}$。 + +>[!example] **例1** +>设矩阵 +>$$A = \begin{bmatrix} +1 & a_1 & a_1^2 & a_1^3 \\ +1 & a_2 & a_2^2 & a_2^3 \\ +1 & a_3 & a_3^2 & a_3^3 \\ +1 & a_4 & a_4^2 & a_4^3 +\end{bmatrix}, +\quad +x = \begin{bmatrix} +x_1 \\ x_2 \\ x_3 \\ x_4 +\end{bmatrix}, +\quad +b = \begin{bmatrix} +1 \\ 1 \\ 1 \\ 1 +\end{bmatrix}$$ +其中常数 $a_1, a_2, a_3, a_4$ 互不相等，则线性方程组 $Ax = b$ 的解为 ______________。 + +```text + + + + + + + + + + + + + + + + + + + + + +``` + +>[!example] **例2** +> 设 $A, B$ 均为 $n$ 阶方阵，满足$\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} = \text{rank}B,$ 且方程 $XA = B$ 有解。若 $\operatorname{rank} A = k$，则$\text{rank} \begin{bmatrix} B & O \\ A & E \end{bmatrix} =\underline{\hspace{3cm}}.$ + +# 多方程组的问题（线性方程组同解） + +## **$Ax=0$与$Bx=0$同解问题**： + +充要条件：$rankA=rankB=rank\begin{bmatrix} A \\ B\end{bmatrix}$. +$Ax=\alpha$ 与$Bx=\beta$同解问题： +充要条件：$rank\begin{bmatrix} A & \alpha\end{bmatrix}=rank\begin{bmatrix} B &\beta\end{bmatrix}=rank\begin{bmatrix} A &\alpha\\ B&\beta\end{bmatrix}$. + +解包含的关系：$rank\begin{bmatrix} A & \alpha\end{bmatrix}=rank\begin{bmatrix} A &\alpha\\ B&\beta\end{bmatrix} \Leftrightarrow A\boldsymbol{x} = \alpha$ 的解均为 $B\boldsymbol{x} = \beta$ 的解 + +如何理解（非严格证明，目的是便于理解）： +首先，为了简化问题，我们只考虑齐次线性方程组同解问题，对于$Ax=0$与$Bx=0$， +考虑这两个齐次线性方程组的解空间，分别记为$N(A)$,$N(B)$,这两个集合是完全相同的， +可以得到$N(A)\subset N(B)$,以及$N(B)\subset N(A)$. +$N(A)\subset N(B)$可以得到什么呢？ + +说明$Ax=0$的解比较少，$Bx=0$的解比较多，一个方程组解多就说明他的方程限制相对宽松，解少则说明方程要求比较严格。换言之，$Bx=0$的每个方程是由$Ax=0$的方程线性表示的,同理$N(B)\subset N(A)$ 可以得到 $Ax=0$ 的每个方程是由 $Bx=0$ 的方程线性表示的，进而说明这两个系数矩阵的行向量能够互相线性表示，即行向量组等价.用秩的语言表示：$rankA=rankB=rank\begin{bmatrix} A \\ B\end{bmatrix}$. + +另一个角度：这两个矩阵化成最简行阶梯型，是相同的，进行化简的时候只用到行变换，故它们的行向量组等价. + +需要注意的是,这个条件是充要的.非常的好用. +非齐次的时候同理. + +注意：由此，我们还能得到一些别的结论 +例如：$A$ 和 $B$ 等价（可以通过初等变换得到），并不能得到两方程同解，因为等价的初等变换可能包括初等列变换，而列变换可能改变两方程的解 + +>[!example] 例1 +>6. 已知方程组$\quad\begin{cases}x_1 + 2x_2 + 3x_3 = 0, \\2x_1 + 3x_2 + 5x_3 = 0, \\x_1 + x_2 + ax_3 = 0,\end{cases}$ 与$\quad\begin{cases}x_1 + bx_2 + cx_3 = 0, \\2x_1 + b^2x_2 + (c+1)x_3 = 0\end{cases}$同解，则 + (A) $a = 1, b = 0, c = 1$; + (B) $a = 1, b = 1, c = 2$; + (C) $a = 2, b = 0, c = 1$; + (D) $a = 2, b = 1, c = 2$. + +# 线性方程组的系数矩阵与解关系 + +在研究线性方程组的解的性质（例如维数）时，我们通常要与其系数矩阵本身的性质产生联系，下面是一个链接了方程组解空间与方程组系数秩的公式。 + +>[!note] 秩零化度定理： +>对于齐次方程组 $\boldsymbol{A}_{m \times n}\boldsymbol{x}=\boldsymbol{0}$，设$\mathrm{rank}\boldsymbol{A}=r$，则 +> $$\dim N(\boldsymbol{A})=n-r$$ + +> [!example] 例1 +> 已知三阶方阵 $A=\begin{bmatrix}\alpha_1&\alpha_2&\alpha_3\end{bmatrix}$ 有三个不同的特征值，其中$\alpha_3=2\alpha_1+\alpha_2$，若 $\beta=\alpha_1+3\alpha_2+4\alpha_3$ ，求线性方程组 $Ax=\beta$ 的通解. + +```text + + + + + + + + + + + + + + + + + + + + + +``` + +> [!example] 例2 +> 设 $$ +A = \begin{bmatrix} +1 & -1 & 0 & -1 \\ +1 & 1 & 0 & 3 \\ +2 & 1 & 2 & 6 +\end{bmatrix}, \quad +B = \begin{bmatrix} +1 & 0 & 1 & 2 \\ +1 & -1 & a & a-1 \\ +2 & -3 & 2 & -2 +\end{bmatrix},$$ +向量 $$ +\alpha = \begin{bmatrix} 0 \\ 2 \\ 3 \end{bmatrix}, \quad +\beta = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}.$$ +(1) 证明：方程组 $Ax = \alpha$ 的解均为方程组 $Bx = \beta$ 的解； +(2) 若方程组 $Ax = \alpha$ 与方程组 $Bx = \beta$ 不同解，求 $a$ 的值。 + +```text + + + + + + + + + + + + + + + + + + + + + +``` + +# 通过秩反过来得方程是否有解 + +>[!example] 例1 +>已知$\boldsymbol{A},\boldsymbol{B}$均为$m\times n$矩阵，$\beta_1,\beta_2$为$m$维列向量，则下列选项正确的有[ ] +(A)若$\mathrm{rank}\boldsymbol{A}=m$，则对于任意$m$维列向量$\boldsymbol{b},\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$总有解. +(B)若$\boldsymbol{A}$与$\boldsymbol{B}$等价，则齐次线性方程组$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$与$\boldsymbol{B}\boldsymbol{x}=\boldsymbol{0}$同解. +(C)矩阵方程$\boldsymbol{A}\boldsymbol{X}=\boldsymbol{B}$有解，但$\boldsymbol{B}\boldsymbol{Y}=\boldsymbol{A}$无解的充要条件是$$\mathrm{rank}\boldsymbol{B}<\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ B}].$$ +(D)线性方程组$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_1}$与$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_2}$同时有解当且仅当$$\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1\ \beta_2}].$$ + +**解：** +(A)一方面$\mathrm{rank}[\boldsymbol{A\ b}]\ge \mathrm{rank}\boldsymbol{A}=m$,另一方面矩阵$[\boldsymbol{A\ b}]$只有$m$行，所以它的秩必然不大于$m$，所以$\mathrm{rank}[\boldsymbol{A\ b}]=m=\mathrm{rank}\boldsymbol{A}$，即方程$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$总有解。 + +(B)等价的矩阵只需要是经过初等变换可以变成同一个矩阵就行了，但齐次线性方程组同解需要只经过初等行变换就能变成同一个矩阵才行，后一个条件明显更强，所以后一种更“难”达成，B就不对。 + +(C)方程$\boldsymbol{A}\boldsymbol{X}=\boldsymbol{B}$有解$\Leftrightarrow\mathrm{rank}[\boldsymbol{A\ B}]=\mathrm{rank}\boldsymbol{A}$,方程$\boldsymbol{B}\boldsymbol{Y}=\boldsymbol{A}$无解$\Leftrightarrow\mathrm{rank}\boldsymbol{B}<\mathrm{rank}[\boldsymbol{B\ A}]$,而$\mathrm{rank}[\boldsymbol{A\ B}]=\mathrm{rank}[\boldsymbol{B\ A}]$,故C正确。这是纯形式化的解答，不过当然是正确的。但是怎么理解这个结果呢？$\boldsymbol{A}\boldsymbol{X}=\boldsymbol{B}$有解，就是说我们可以用矩阵$\boldsymbol{A}$表示矩阵$\boldsymbol{B}$,也就是说，$\boldsymbol{A}$中包含了$\boldsymbol{B}$中的所有信息，也就是$\mathrm{rank}\boldsymbol{A}\ge\mathrm{rank}\boldsymbol{B}$；另一方面，$\boldsymbol{BY}=\boldsymbol{A}$无解说明我们无法用矩阵$\boldsymbol{B}$表示矩阵$\boldsymbol{A}$，也就是说，$\boldsymbol{B}$中没有包含$\boldsymbol{A}$中的所有信息，那么$\mathrm{rank}\boldsymbol{B}<\mathrm{rank}\boldsymbol{A}$；再加上有解的充要条件得出C正确。 + +(D)我们同样有两种方法去解这道题，一种是形式化的、严谨的，另一种是理解性的、直观的。 +1)线性方程组$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_1}$与$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_2}$同时有解$\Leftrightarrow\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1}]=\mathrm{rank}[\boldsymbol{A\ \beta_2}]$,故$\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1\ \beta_2}]$。 +2)也可以从初等变换的角度来理解，方程$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_1}$有解说明$\boldsymbol{\beta_1}$可以用$\boldsymbol{A}$的列向量线性表示，从而$[\boldsymbol{A\ \beta_1}]$可以通过初等列变换变成$[\boldsymbol{A\ O}]$，故$\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1}]$；同理可以得出关于$\boldsymbol{\beta_2}$的结论。 +3)同样，怎么直观地理解？我们一样用信息量的观点去看。方程$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_1}$有解，意味着$\boldsymbol{A}$中包含了$\boldsymbol{\beta_1}$中的所有信息，同理，$\boldsymbol{A}$中也包含了$\boldsymbol{\beta_2}$中的所有信息，这就意味着矩阵$[\boldsymbol{A\ \beta_1\ \beta_2}]$中所有的信息其实只需要用$\boldsymbol{A}$就可以表示，故$\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1\ \beta_2}]$,反过来也是一样的。这就说明D是正确的。 + +# 矩阵秩与线性方程组解的关系图解说明 + +--- + +## 概念回顾 + +- **rank[A]** 代表矩阵$A$的秩。 +- 秩的定义：矩阵列向量组中**极大线性无关组所含列向量的个数**。 +- 可以用“圆”或“空间”来表示矩阵列向量组张成的向量空间。 + +--- + +## 图解说明 + +假设 +$$ +A = [\alpha_1, \alpha_2, \alpha_3] +$$ +是$m \times 3$矩阵， +$$ +\beta_1, \beta_2 +$$ +是$m$维列向量。 + +### 1. 方程组有解的条件 + +方程组 +$$ +Ax = \beta_1 \quad \text{和} \quad Ax = \beta_2 +$$ +有解 +$\Leftrightarrow$$\beta_1, \beta_2$可由$A$的列向量线性表示。 + +在几何上，这表示： + +- 设$A$的列向量张成的空间为$S_A$。 +-$\beta_1, \beta_2 \in S_A$。 +- 即$S_A$“包含”$\beta_1, \beta_2$。 + +因此，$S_A$这个“圆”应当能够**覆盖**$\beta_1$和$\beta_2$。 + +--- + +### 2. 秩等价条件 + +已知： +$$ +\operatorname{rank}[A] = \operatorname{rank}[A \quad \beta_1 \quad \beta_2] +$$ +表示： +- 矩阵$A$的秩与增广矩阵$[A \mid \beta_1 \mid \beta_2]$的秩相等。 +- 这意味着$\beta_1, \beta_2$并没有“扩大”$A$的列空间。 + +因此： +$$ +\operatorname{rank}[A] = \operatorname{rank}[A \quad \beta_1 \quad \beta_2] +\quad\Leftrightarrow\quad +\beta_1, \beta_2 \in S_A +$$ +即$Ax = \beta_1$和$Ax = \beta_2$有解。 + +--- + +### 3. 等价写法 + +把$\beta_1, \beta_2$放在$A$的右侧构成一个更大的矩阵： +$$ +[A \quad \beta_1 \quad \beta_2] +$$ +其秩与$A$相同，说明： +1. 空间$S_{[A \ \beta_1 \ \beta_2]}$与$S_A$相同。 +2. 从初等行变换角度看：在行阶梯形中，$\beta_1, \beta_2$对应的列会被$A$的列线性表示，从而可化为零列（在解方程时体现为消去）。 +3. 存在$x_1, x_2$使得： + $$ + A(-x_1) = \beta_1, \quad A(-x_2) = \beta_2 + $$ + 这样在增广矩阵中可以通过列操作消去$\beta_1, \beta_2$，使其变为零列。 + +--- + +## 总结 + +- 秩相等 ⇔ 列空间相同 ⇔ 方程组有解。 +- 图示法：把$A$的列空间画成一个圆，$\beta_1, \beta_2$若落在圆内，则方程有解。 +- 矩阵的秩是判断线性方程组解的存在性的核心工具。 + +--- + +**注**：这里的“圆”是比喻，实际为**线性子空间**。 + +# 秩的不等式 + +### 1. 和的秩不超过秩的和 + +设 $A, B$ 为同型矩阵，则 +$$ \operatorname{rank}(A+B) \leq \operatorname{rank} A + \operatorname{rank} B $$ + +### 2. 积的秩不超过任何因子的秩 + +设 $A_{m \times n}, B_{n \times k}$，则 +$$ \operatorname{rank}(AB) \leq \min\{\operatorname{rank} A, \operatorname{rank} B\} $$ + +### 3. Sylvester(西尔维斯特)不等式 + +设 $A_{m \times n}, B_{n \times k}$，则 +$$ \operatorname{rank}(AB) \geq \operatorname{rank} A + \operatorname{rank} B - n $$ +特别地，当 $AB = 0$ 时，有 $\operatorname{rank} A + \operatorname{rank} B \leq n$。 + +### 4. 分块式 + +设 $A_{n \times n}$， $B_{n \times n}$，则 + +$$(1)\ \mathrm{rank} + +\begin{bmatrix} +A \\ +B +\end{bmatrix} \geq \text{rank } A, \quad \text{rank } +\begin{bmatrix} +A \\ +B +\end{bmatrix} \geq \text{rank } B +$$ + +$$(2)\ \mathrm{rank} +\begin{bmatrix} +A & 0 \\ +0 & B +\end{bmatrix} = \text{rank } A + \text{rank } B +$$ + +$$(3)\ \mathrm{rank} +\begin{bmatrix} +A & E_n \\ +0 & B +\end{bmatrix} \geq \text{rank } A + \text{rank } B +$$ + +$$(4)\ \mathrm{rank} +\begin{bmatrix} +A & 0 \\ +0 & B +\end{bmatrix} = \text{rank } +\begin{bmatrix} +A & B \\ +0 & B +\end{bmatrix} = \text{rank } +\begin{bmatrix} +A + B & B \\ +B & B +\end{bmatrix} \geq \text{rank } (A + B) +$$ + +注：（2）与（4）结合即第一个不等式的证明方法 + +> [!note] 证明1： +$$\operatorname{rank}(AB) \leq \min\{\operatorname{rank}(A), \operatorname{rank}(B)\}$$ + +--- + +### 证明思路 +设 +$$ +A \in \mathbb{R}^{m \times n}, \quad B \in \mathbb{R}^{n \times p}, \quad C = AB \in \mathbb{R}^{m \times p}. +$$ + +--- + +#### 1. 先证$\operatorname{rank}(AB) \leq \operatorname{rank}(A)$ + +- 考虑$C$的列向量： + 设$B = [b_1, b_2, \dots, b_p]$，则 + $$ + C = [A b_1, A b_2, \dots, A b_p]. + $$ + 因此$C$的每一列都是$A$的列向量的线性组合。 +- 所以$C$的列空间是$A$的列空间的子空间，故 + $$ + \operatorname{rank}(C) \leq \operatorname{rank}(A)。 + $$ + +--- + +#### 2. 再证$\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ + +- 考虑$C$的行向量： + 设$A = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_m \end{bmatrix}$，则 + $$ + C = \begin{bmatrix} a_1 B \\ a_2 B \\ \vdots \\ a_m B \end{bmatrix}. + $$ + 因此$C$的每一行都是$B$的行向量的线性组合。 +- 所以$C$的行空间是$B$的行空间的子空间，故 + $$\operatorname{rank}(C) \leq \operatorname{rank}(B)$$ + +--- + +#### 3. 综合 +由 1 和 2 得 +$$ +\operatorname{rank}(AB) \leq \operatorname{rank}(A) \quad \text{且} \quad \operatorname{rank}(AB) \leq \operatorname{rank}(B), +$$ +即 +$$ +\operatorname{rank}(AB) \leq \min\{\operatorname{rank}(A), \operatorname{rank}(B)\}. +$$ + +--- + +**证毕。** + +> [!note] 证明2 +> 证明 Sylvester 秩不等式： +$$\operatorname{rank}(AB) \ge \operatorname{rank}(A) + \operatorname{rank}(B) - n$$ +其中 +$A \in \mathbb{R}^{m \times n}, \; B \in \mathbb{R}^{n \times p}, \; AB \in \mathbb{R}^{m \times p}$。 + +--- + +### 证明思路 +设： +-$\operatorname{rank}(A) = r$ +-$\operatorname{rank}(B) = s$ +-$n$是矩阵乘法的中间维度，即$A$的列数、$B$的行数。 + +--- + +#### 1. 利用分块矩阵构造 +构造如下分块矩阵： +$$ +M = \begin{bmatrix} +A & O \\ +I_n & B +\end{bmatrix} +\in \mathbb{R}^{(m+n) \times (n+p)} +$$ +其中$I_n$是$n \times n$单位矩阵，$O$是零矩阵。 + +--- + +#### 2. 对$M$进行初等变换 +从$M$的第二块行减去第一块行左乘某个矩阵（这里相当于对$M$做列初等变换），实际上我们可以对$M$做以下变换： +$$ +\begin{bmatrix} +A & O \\ +I_n & B +\end{bmatrix} +\xrightarrow{\text{右乘 } \begin{bmatrix} I_n & -B \\ O & I_p \end{bmatrix}} +\begin{bmatrix} +A & -AB \\ +I_n & O +\end{bmatrix} +$$ +初等变换不改变矩阵的秩，所以： +$$ +\operatorname{rank}(M) = \operatorname{rank}\begin{bmatrix} +A & -AB \\ +I_n & O +\end{bmatrix} +$$ + +--- + +#### 3. 估计$\operatorname{rank}(M)$ +另一方面，由分块矩阵的秩不等式： +$$ +\operatorname{rank}(M) \ge \operatorname{rank}(A) + \operatorname{rank}(B) +$$ +这是因为$M$左上块为$A$，右下块为$B$，中间有单位矩阵，所以$A$和$B$的秩可以同时取到。 + +利用： +$$ +\operatorname{rank}\begin{bmatrix} +A & O \\ +I_n & B +\end{bmatrix} \ge \operatorname{rank}(A) + \operatorname{rank}(B) +$$ +因为$I_n$的存在使得两个子块的秩可以同时保持。 + +--- + +#### 4. 从变换后的矩阵得到下界 + +实际上更直接的方法是注意到： +$$ +\operatorname{rank}\begin{bmatrix} +A & -AB \\ +I_n & O +\end{bmatrix} += \operatorname{rank}\begin{bmatrix} +O & -AB \\ +I_n & O +\end{bmatrix} \quad (\text{行变换}) +$$ +即： +$$ += \operatorname{rank}\begin{bmatrix} +I_n & O \\ +O & AB +\end{bmatrix} = \operatorname{rank}(I_n) + \operatorname{rank}(AB) = n + \operatorname{rank}(AB) +$$ + +--- + +#### 5. 联立 +由初等变换保秩，得： +$$ +n + \operatorname{rank}(AB) = \operatorname{rank}(M) \ge \operatorname{rank}(A) + \operatorname{rank}(B) +$$ +整理得： +$$ +\operatorname{rank}(AB) \ge \operatorname{rank}(A) + \operatorname{rank}(B) - n +$$ + +--- + +## 重点思路 + +>[!information] 思路1 +>通过矩阵的秩的不等式，最大限度限制所求的表达式的取值范围，或者将其**限制到一个具体的值**. +>在希望求一个矩阵的秩的确切值时，也可以考虑用不等式关系来“夹逼”，常见的不等式： +>1. $\mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}-n\le\mathrm{rank}(\boldsymbol{AB})\le\min{\{\mathrm{rank}\boldsymbol{A}, \mathrm{rank}\boldsymbol{B}\}}$ +>2. $\mathrm{rank}(\boldsymbol{A+B})<\mathrm{rank}\boldsymbol A+\mathrm{rank}\boldsymbol B$ +>3. 矩阵加边不会减小秩； +> + +> [!note] 思路2 +> 在遇到诸如 $AB=O$ 的情况，务必要想到$\mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}-n\le\mathrm{rank}(\boldsymbol{AB}) \Rightarrow \mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}\le n$ + +> [!example] 例1 +> 设 $A$ 是 $m\times n$ 实矩阵， $\beta \neq 0$ 是 $m$ 维实列向量，证明： +> (1) $\mathrm{rank}A=\mathrm{rank}(A^\mathrm{T}A)$ . +> (2) 线性方程组 $A^\mathrm{T}Ax = A^\mathrm{T}\beta$ 有解.（这一问用到这个方法） + +```text + + + + + + + + + + + + + + + + + + + + + +``` + +>[!example] 例2 +>已知$A, B, C, D$都是 4 阶非零矩阵，且$ABCD = O$，如果$|BC| \neq 0$，记$$r(A) + r(B) + r(C) + r(D) = r$$ +>则$r$的最大值是（ ）。 +>(A) 11 +>(B) 12 +>(C) 13 +>(D) 14 + +```text + + + + + + + + + + + + + + + + + + + + + +``` diff --git a/编写小组/讲义/线性方程组的解与秩的不等式（解析版）.md b/编写小组/讲义/线性方程组的解与秩的不等式（解析版）.md new file mode 100644 index 0000000..6ab1770 --- /dev/null +++ b/编写小组/讲义/线性方程组的解与秩的不等式（解析版）.md @@ -0,0 +1,629 @@ +--- +tags: + - 编写小组 +--- +**内部资料，禁止传播** +**编委会（不分先后，姓氏首字母顺序）：陈峰华 陈玉阶 程奕铭 韩魏 刘柯妤 卢吉辚 王嘉兴 王轲楠 彭靖翔 郑哲航 钟宇哲 支宝宁 + +# 单方程组解的问题 + +### 线性方程组解的判定 + +对非齐次线性方程组 $A_{m\times n}x=b$, +1. 无解的充要条件是 $\text{rank}A < \text{rank}[A\ \ b]$； +2. 有唯一解的充要条件是 $\text{rank}A = \text{rank}[A\ \ b] = n$； +3. 有无穷多解的充要条件是 $\text{rank}A = \text{rank}[A\ \ b] < n$。 +注:上述定理也说明非齐次线性方程组有解的充要条件是 $\text{rank}A = \text{rank}[A\ \ b]$。 + +怎么理解： +1. 从线性方程组的角度，如果加上一列 $b$ 后的秩变大了，那么化为最简行阶梯型后下面一定多出来一行 $0=$某个常数 ，则必然无解。 +2. 秩等于 $n$ 就是有 $n$ 个无关的方程，则经过消元法后可以解出唯一解。 +3. 秩小于$n$ 就是方程不足 $n$ 个，消元消不完，也能解释为啥秩跟解空间维数的和为 $n$ + +把以上结论应用到齐次线性方程组，可得 +推论 齐次线性方程组 $A_{m\times n}x=0$ 有非零解（无穷多解）的充要条件是 $\text{rank}A < n$，即系数矩阵的秩小于未知数个数。 + +### 矩阵方程解的判定 + +本质上和线性方程组是一脉相承的，只是形式上更一般化。 +最常见的矩阵方程是 $\boldsymbol{AX} = \boldsymbol{B}$，其中$\boldsymbol{A}$是 $m\times n$ 矩阵，$\boldsymbol{B}$ 是 $m\times p$ 矩阵，$\boldsymbol{X}$ 是待求的$n\times p$矩阵。 +1. 有解的充要条件： +矩阵方程有解的充要条件是系数矩阵 $\boldsymbol{A}$ 的秩等于增广矩阵 $[\boldsymbol{A} \ \boldsymbol{B}]$的秩，即： +$$r(\boldsymbol{A}) = r([\boldsymbol{A} \ \boldsymbol{B}])$$ +这个结论和非齐次线性方程组有解的条件完全一致。 + +理解上可以将 $B$ 拆分成一列列 $b$ ，从而化归为上面的线性方程组问题 + +2. 解的结构： +- 唯一解：当$r(\boldsymbol{A}) = r([\boldsymbol{A} \ \boldsymbol{B}]) = n$ 时，方程有唯一解。 +- 无穷多解：当 $r(\boldsymbol{A}) = r([\boldsymbol{A} \ \boldsymbol{B}]) < n$ 时，方程有无穷多解。 + +可逆矩阵 +- 当 $\boldsymbol{A}$ 是 n 阶可逆矩阵时，矩阵方程 $\boldsymbol{AX} = \boldsymbol{B}$有唯一解:$\boldsymbol{X} = \boldsymbol{A}^{-1}\boldsymbol{B}$ + +其他形式的矩阵方程 +- 对于 $\boldsymbol{XA} = \boldsymbol{B}$ 形式的方程，可以转置为 $\boldsymbol{A}^T\boldsymbol{X}^T = \boldsymbol{B}^T$，再套用上述方法，或类似于方程 $AX = B$ 有解的充要条件是$\text{rank} \begin{bmatrix} A & B \end{bmatrix} = \text{rank}A$，由方程 $XA = B$ 有解可知 $\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} =\text{rank}A$ +- 对于 $\boldsymbol{AXB} = \boldsymbol{C}$ 形式的方程，当 $\boldsymbol{A} 和 \boldsymbol{B}$ 都可逆时，有唯一解 $\boldsymbol{X} = \boldsymbol{A}^{-1}\boldsymbol{C}\boldsymbol{B}^{-1}$。 + +>[!example] **例1** +>设矩阵 +>$$A = \begin{bmatrix} +1 & a_1 & a_1^2 & a_1^3 \\ +1 & a_2 & a_2^2 & a_2^3 \\ +1 & a_3 & a_3^2 & a_3^3 \\ +1 & a_4 & a_4^2 & a_4^3 +\end{bmatrix}, +\quad +x = \begin{bmatrix} +x_1 \\ x_2 \\ x_3 \\ x_4 +\end{bmatrix}, +\quad +b = \begin{bmatrix} +1 \\ 1 \\ 1 \\ 1 +\end{bmatrix}$$ +其中常数 $a_1, a_2, a_3, a_4$ 互不相等，则线性方程组 $Ax = b$ 的解为 ______________。 + +**答**：$(1,0,0,0)^T$。 + +**解析**：由范德蒙行列式的性质可知 $|A| \neq 0$，从而线性方程组 $Ax = b$ 有唯一解。 + +又由 +$$ +\begin{bmatrix} +1 & a_1 & a_1^2 & a_1^3 \\ +1 & a_2 & a_2^2 & a_2^3 \\ +1 & a_3 & a_3^2 & a_3^3 \\ +1 & a_4 & a_4^2 & a_4^3 +\end{bmatrix} +\begin{bmatrix} +1 \\ +0 \\ +0 \\ +0 +\end{bmatrix} += +\begin{bmatrix} +1 \\ +1 \\ +1 \\ +1 +\end{bmatrix} +$$ + 可知 $Ax = b$ 的解为 +$$ +\begin{bmatrix} +1 \\ +0 \\ +0 \\ +0 +\end{bmatrix} +$$ + +>[!example] **例2** +> 设 $A, B$ 均为 $n$ 阶方阵，满足$\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} = \text{rank}B,$ 且方程 $XA = B$ 有解。若 $\operatorname{rank} A = k$，则$\text{rank} \begin{bmatrix} B & O \\ A & E \end{bmatrix} =\underline{\hspace{3cm}}.$ + +**解析**： +类似于方程 $AX = B$ 有解的充要条件是$\text{rank} \begin{bmatrix} A & B \end{bmatrix} = \text{rank}A$，由方程 $XA = B$ 有解可知 $\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} = \text{rank}B=\text{rank}A=k$，由初等变换不改变秩得$$\text{rank} \begin{bmatrix} B & O \\ A & E \end{bmatrix} =\text{rank} \begin{bmatrix} B & O \\ O & E \end{bmatrix}=n+k$$ +# 多方程组的问题（线性方程组同解） + +## **$Ax=0$与$Bx=0$同解问题**： + +充要条件：$rankA=rankB=rank\begin{bmatrix} A \\ B\end{bmatrix}$. +$Ax=\alpha$ 与$Bx=\beta$同解问题： +充要条件：$rank\begin{bmatrix} A & \alpha\end{bmatrix}=rank\begin{bmatrix} B &\beta\end{bmatrix}=rank\begin{bmatrix} A &\alpha\\ B&\beta\end{bmatrix}$. + +解包含的关系：$rank\begin{bmatrix} A & \alpha\end{bmatrix}=rank\begin{bmatrix} A &\alpha\\ B&\beta\end{bmatrix} \Leftrightarrow A\boldsymbol{x} = \alpha$ 的解均为 $B\boldsymbol{x} = \beta$ 的解 + +如何理解（非严格证明，目的是便于理解）： +首先，为了简化问题，我们只考虑齐次线性方程组同解问题，对于$Ax=0$与$Bx=0$， +考虑这两个齐次线性方程组的解空间，分别记为$N(A)$,$N(B)$,这两个集合是完全相同的， +可以得到$N(A)\subset N(B)$,以及$N(B)\subset N(A)$. +$N(A)\subset N(B)$可以得到什么呢？ + +说明$Ax=0$的解比较少，$Bx=0$的解比较多，一个方程组解多就说明他的方程限制相对宽松，解少则说明方程要求比较严格。换言之，$Bx=0$的每个方程是由$Ax=0$的方程线性表示的,同理$N(B)\subset N(A)$ 可以得到 $Ax=0$ 的每个方程是由 $Bx=0$ 的方程线性表示的，进而说明这两个系数矩阵的行向量能够互相线性表示，即行向量组等价.用秩的语言表示：$rankA=rankB=rank\begin{bmatrix} A \\ B\end{bmatrix}$. + +另一个角度：这两个矩阵化成最简行阶梯型，是相同的，进行化简的时候只用到行变换，故它们的行向量组等价. + +需要注意的是,这个条件是充要的.非常的好用. +非齐次的时候同理. + +注意：由此，我们还能得到一些别的结论 +例如：$A$ 和 $B$ 等价（可以通过初等变换得到），并不能得到两方程同解，因为等价的初等变换可能包括初等列变换，而列变换可能改变两方程的解 + +>[!example] 例1 +>6. 已知方程组$\quad\begin{cases}x_1 + 2x_2 + 3x_3 = 0, \\2x_1 + 3x_2 + 5x_3 = 0, \\x_1 + x_2 + ax_3 = 0,\end{cases}$ 与$\quad\begin{cases}x_1 + bx_2 + cx_3 = 0, \\2x_1 + b^2x_2 + (c+1)x_3 = 0\end{cases}$同解，则 + (A) $a = 1, b = 0, c = 1$; + (B) $a = 1, b = 1, c = 2$; + (C) $a = 2, b = 0, c = 1$; + (D) $a = 2, b = 1, c = 2$. + +>答案：**D** +>解析：$\text{(I)}:\begin{bmatrix}1&2&3\\2&3&5\\1&1&a\end{bmatrix}x=0$，$\text{(II)}: \begin{bmatrix}1&b&c\\2&b^2&c+1\end{bmatrix}x=0$，对方程做初等行变换： +>$\text{(I)}:\begin{bmatrix}1&0&1\\0&1&1\\0&0&a-2\end{bmatrix}x=0$，$\text{(II)}: \begin{bmatrix}1&b&c\\0&b^2-2b&1-c\end{bmatrix}x=0$，记系数矩阵分别为$A,B$ +>因为方程(I),(II)同解，所以$\text{rank}A=\text{rank}B$，而$\text{rank}A\ge 2,\text{rank}B\le 2$，故$\text{rank}A=\text{rank}B=2$，故$a-2=0 \to a=2$；所以方程组(I)的解为$x=k(1,1,-1)^T$； +>令$k=1,x=(1,1,-1)^T$代入方程组(II)得$\begin{bmatrix}1&b&c\\0&b^2-2b&1-c\end{bmatrix}\begin{bmatrix}1\\1\\-1\end{bmatrix}=\begin{bmatrix}1+b-c\\b^2-2b+c-1\end{bmatrix}=0$，解得$\begin{cases}b=0\\c=1\end{cases}$或$\begin{cases}b=1\\c=2\end{cases}$；然而，当$\begin{cases}b=0\\c=1\end{cases}$时，$\begin{bmatrix}1&b&c\\0&b^2-2b&1-c\end{bmatrix}=\begin{bmatrix}1&0&1\\0&0&0\end{bmatrix}$，不符合$\text{rank}B=2$的约束，故舍去； +>综上，$\begin{cases}a=2\\b=1\\c=2\end{cases}$ + +# 线性方程组的系数矩阵与解关系 + +在研究线性方程组的解的性质（例如维数）时，我们通常要与其系数矩阵本身的性质产生联系，下面是一个链接了方程组解空间与方程组系数秩的公式。 + +>[!note] 秩零化度定理： +>对于齐次方程组 $\boldsymbol{A}_{m \times n}\boldsymbol{x}=\boldsymbol{0}$，设$\mathrm{rank}\boldsymbol{A}=r$，则 +> $$\dim N(\boldsymbol{A})=n-r$$ + +> [!example] 例1 +> 已知三阶方阵 $A=\begin{bmatrix}\alpha_1&\alpha_2&\alpha_3\end{bmatrix}$ 有三个不同的特征值，其中$\alpha_3=2\alpha_1+\alpha_2$，若 $\beta=\alpha_1+3\alpha_2+4\alpha_3$ ，求线性方程组 $Ax=\beta$ 的通解. + +**答案：** + $$\begin{bmatrix}1\\3\\4\end{bmatrix}+k\begin{bmatrix}2\\1\\-1\end{bmatrix}, k\in\mathbb{R}$$ +**分析：** 在求解非齐次方程组通解的题目中，若是题目给出了特解与齐次方程组的解，那么大概率来说这个齐次方程组的解就可以拓展为齐次方程组通解（根据问题导向，不然写不出来了），那么如何由齐次方程组的解拓展为齐次方程组通解呢，那就要根据题目具体的条件进行分析了，这就要用到我们的秩零化度定理来求齐次方程组解空间的维数 +**解析：** 由 $\alpha_3=2\alpha_1+\alpha_2$ 可得 $A$ 的列向量组线性相关， $|A|=0$；又因为 $A$ 的三个特征值各不相同，故 $A$ 有两个不为零的特征值 $\lambda_1,\lambda_2$，且 $A$ 可相似对角化，即 $A=P^{-1}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}P$，$\mathrm{rank}A=\mathrm{rank}(P^{-1}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}P)=\mathrm{rank}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}=2$ +故 $Ax=0$ 的解空间维数是 $1$ （5分） +$\beta=\alpha_1+3\alpha_2+4\alpha_3$，所以 $(1,3,4)^\mathrm{T}$ 为特解；（5分） +$\alpha_3=2\alpha_1+\alpha_2$，所以$A\begin{bmatrix}2k\\k\\-k\end{bmatrix}=2\alpha_1+\alpha_2-\alpha_3=0$，所以 $(2,1,-1)^\mathrm{T}$ 为基础解系；（10分） +解空间维数是 $1$ ，方程的解 $\begin{bmatrix}1\\3\\4\end{bmatrix}+k\begin{bmatrix}2\\1\\-1\end{bmatrix}, k\in\mathbb{R}$ 维数是 $1$，该解完备 + +> [!example] 例2 +> 设 $$ +A = \begin{bmatrix} +1 & -1 & 0 & -1 \\ +1 & 1 & 0 & 3 \\ +2 & 1 & 2 & 6 +\end{bmatrix}, \quad +B = \begin{bmatrix} +1 & 0 & 1 & 2 \\ +1 & -1 & a & a-1 \\ +2 & -3 & 2 & -2 +\end{bmatrix},$$ +向量 $$ +\alpha = \begin{bmatrix} 0 \\ 2 \\ 3 \end{bmatrix}, \quad +\beta = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}.$$ +(1) 证明：方程组 $Ax = \alpha$ 的解均为方程组 $Bx = \beta$ 的解； +(2) 若方程组 $Ax = \alpha$ 与方程组 $Bx = \beta$ 不同解，求 $a$ 的值。 + + +**解：** +(1) 由于 +$$ +\begin{bmatrix} +A \quad \alpha \\ +B \quad \beta +\end{bmatrix} = +\begin{bmatrix} +1 & -1 & 0 & -1 & 0 \\ +1 & 1 & 0 & 3 & 2 \\ +2 & 1 & 2 & 6 & 3 \\ +1 & 0 & 1 & 2 & 1 \\ +1 & -1 & a & a-1 & 0 \\ +2 & -3 & 2 & -2 & -1 +\end{bmatrix} +\rightarrow +\begin{bmatrix} +1 & -1 & 0 & -1 & 0 \\ +0 & 1 & 0 & 2 & 1 \\ +0 & 0 & 2 & 2 & 0 \\ +0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 +\end{bmatrix}, +$$ +故 +$$ +\mathrm{rank} \begin{bmatrix}A&\alpha\\B&\beta\end{bmatrix} = \mathrm{rank}[A\ \alpha]$$ +故 $A\boldsymbol{x} = \alpha$ 的解均为 $B\boldsymbol{x} = \beta$ 的解。 + +(2) 分析：不同解，却要可以求出$a$的具体值，说明这是一个与秩相关的题，而与解相关的秩的问题我们就可以考虑秩零化度定理 +由于 $A\boldsymbol{x} = \alpha$ 的解均为 $Bx = \beta$ 的解，若 $A\boldsymbol{x} = \alpha$ 与 $B\boldsymbol{x} = \beta$ 同解，则与题意矛盾，故 $Ax = \alpha$ 的解是 $Bx = \beta$ 解的真子集。于是 $Ax = 0$ 的基础解系中解向量的个数小于 $B\boldsymbol{x} = 0$ 的基础解系中解向量的个数，即 +$$ +4 - r(A) < 4 - r(B), +$$ +故 $r(A) > r(B)$。又因 $r(A) = 3$，故 $r(B) < 3$，则 +$$ +\left| \begin{array}{ccc} +1 & 0 & 1 \\ +1 & -1 & a \\ +2 & -3 & 2 +\end{array} \right| = 0, +$$ +解得 $a = 1$。 +# 通过秩反过来得方程是否有解 + +>[!example] 例1 +>已知$\boldsymbol{A},\boldsymbol{B}$均为$m\times n$矩阵，$\beta_1,\beta_2$为$m$维列向量，则下列选项正确的有[ ] +(A)若$\mathrm{rank}\boldsymbol{A}=m$，则对于任意$m$维列向量$\boldsymbol{b},\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$总有解. +(B)若$\boldsymbol{A}$与$\boldsymbol{B}$等价，则齐次线性方程组$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$与$\boldsymbol{B}\boldsymbol{x}=\boldsymbol{0}$同解. +(C)矩阵方程$\boldsymbol{A}\boldsymbol{X}=\boldsymbol{B}$有解，但$\boldsymbol{B}\boldsymbol{Y}=\boldsymbol{A}$无解的充要条件是$$\mathrm{rank}\boldsymbol{B}<\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ B}].$$ +(D)线性方程组$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_1}$与$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_2}$同时有解当且仅当$$\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1\ \beta_2}].$$ + +**解：** +(A)一方面$\mathrm{rank}[\boldsymbol{A\ b}]\ge \mathrm{rank}\boldsymbol{A}=m$,另一方面矩阵$[\boldsymbol{A\ b}]$只有$m$行，所以它的秩必然不大于$m$，所以$\mathrm{rank}[\boldsymbol{A\ b}]=m=\mathrm{rank}\boldsymbol{A}$，即方程$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$总有解。 + +(B)等价的矩阵只需要是经过初等变换可以变成同一个矩阵就行了，但齐次线性方程组同解需要只经过初等行变换就能变成同一个矩阵才行，后一个条件明显更强，所以后一种更“难”达成，B就不对。 + +(C)方程$\boldsymbol{A}\boldsymbol{X}=\boldsymbol{B}$有解$\Leftrightarrow\mathrm{rank}[\boldsymbol{A\ B}]=\mathrm{rank}\boldsymbol{A}$,方程$\boldsymbol{B}\boldsymbol{Y}=\boldsymbol{A}$无解$\Leftrightarrow\mathrm{rank}\boldsymbol{B}<\mathrm{rank}[\boldsymbol{B\ A}]$,而$\mathrm{rank}[\boldsymbol{A\ B}]=\mathrm{rank}[\boldsymbol{B\ A}]$,故C正确。 +这是纯形式化的解答，不过当然是正确的。但是怎么理解这个结果呢？$\boldsymbol{A}\boldsymbol{X}=\boldsymbol{B}$有解，就是说我们可以用矩阵$\boldsymbol{A}$表示矩阵$\boldsymbol{B}$,也就是说，$\boldsymbol{A}$中包含了$\boldsymbol{B}$中的所有信息，也就是$\mathrm{rank}\boldsymbol{A}\ge\mathrm{rank}\boldsymbol{B}$；另一方面，$\boldsymbol{BY}=\boldsymbol{A}$无解说明我们无法用矩阵$\boldsymbol{B}$表示矩阵$\boldsymbol{A}$，也就是说，$\boldsymbol{B}$中没有包含$\boldsymbol{A}$中的所有信息，那么$\mathrm{rank}\boldsymbol{B}<\mathrm{rank}\boldsymbol{A}$；再加上有解的充要条件得出正向是正确的。 +反过来，如果$\mathrm{rank}{A}=\mathrm{rank}[{A\ B}]$，说明$A$中已经包含了矩阵$[A\ B]$的所有信息，所以$A$中就也包含了$B$中的所有信息，所以方程组$AX=B$有解；而如果又有$\mathrm{rank}B<\mathrm{rank}A$，则$B$没有完全包含$A$中的所有信息，或者说，$B$的信息真包含于$A$的信息，所以无法用$B$表示$A$，即$BY=A$无解。 + +(D)我们同样有两种方法去解这道题，一种是形式化的、严谨的，另一种是理解性的、直观的。 +1)线性方程组$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_1}$与$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_2}$同时有解$\Leftrightarrow\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1}]=\mathrm{rank}[\boldsymbol{A\ \beta_2}]$,故$\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1\ \beta_2}]$。 +2)也可以从初等变换的角度来解答，方程$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_1}$有解说明$\boldsymbol{\beta_1}$可以用$\boldsymbol{A}$的列向量线性表示，从而$[\boldsymbol{A\ \beta_1}]$可以通过初等列变换变成$[\boldsymbol{A\ O}]$，故$\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1}]$；同理可以得出关于$\boldsymbol{\beta_2}$的结论。 +3)同样，怎么直观地理解？我们一样用信息量的观点去看。方程$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_1}$有解，意味着$\boldsymbol{A}$中包含了$\boldsymbol{\beta_1}$中的所有信息，同理，$\boldsymbol{A}$中也包含了$\boldsymbol{\beta_2}$中的所有信息，这就意味着矩阵$[\boldsymbol{A\ \beta_1\ \beta_2}]$中所有的信息其实只需要用$\boldsymbol{A}$就可以表示，故$\mathrm{rank}\boldsymbol{A}=\mathrm{rank}[\boldsymbol{A\ \beta_1\ \beta_2}]$。反过来也一样，如果有$\mathrm{rank}{A}=\mathrm{rank}[{A\ \boldsymbol{\beta_1\ \beta_2}}]$，那么$A$中就包含了$\boldsymbol{\beta_1},\boldsymbol{\beta_2}$中的所有的信息，所以可以用$A$去表示向量$\boldsymbol{\beta_1}$和$\boldsymbol{\beta_2}$，也就是方程组$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_1}$与$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta_2}$同时有解。这就说明D是正确的。 + +# 矩阵秩与线性方程组解的关系图解说明 + +--- + +## 概念回顾 + +- __$\mathrm{rank}A$__ 代表矩阵$A$的秩。 +- 秩的定义：矩阵列向量组中**极大线性无关组所含列向量的个数**。 +- 可以用“圆”或“空间”来表示矩阵列向量组张成的向量空间。 + +--- + +## 图解说明 + +假设 +$$ +A = [\alpha_1, \alpha_2, \alpha_3] +$$ +是$m \times 3$矩阵， +$$ +\beta_1, \beta_2 +$$ +是$m$维列向量。 + +### 1. 方程组有解的条件 + +方程组 +$$ +Ax = \beta_1 \quad \text{和} \quad Ax = \beta_2 +$$ +有解 +$\Leftrightarrow$$\beta_1, \beta_2$可由$A$的列向量线性表示。 + +在几何上，这表示： + +- 设$A$的列向量张成的空间为$S_A$。 +-$\beta_1, \beta_2 \in S_A$。 +- 即$S_A$“包含”$\beta_1, \beta_2$。 + +因此，$S_A$这个“圆”应当能够**覆盖**$\beta_1$和$\beta_2$。 + +--- + +### 2. 秩等价条件 + +已知： +$$ +\operatorname{rank}[A] = \operatorname{rank}[A \quad \beta_1 \quad \beta_2] +$$ +表示： +- 矩阵$A$的秩与增广矩阵$[A \mid \beta_1 \mid \beta_2]$的秩相等。 +- 这意味着$\beta_1, \beta_2$并没有“扩大”$A$的列空间。 + +因此： +$$ +\operatorname{rank}[A] = \operatorname{rank}[A \quad \beta_1 \quad \beta_2] +\quad\Leftrightarrow\quad +\beta_1, \beta_2 \in S_A +$$ +即$Ax = \beta_1$和$Ax = \beta_2$有解。 + +--- + +### 3. 等价写法 + +把$\beta_1, \beta_2$放在$A$的右侧构成一个更大的矩阵： +$$ +[A \quad \beta_1 \quad \beta_2] +$$ +其秩与$A$相同，说明： +1. 空间$S_{[A \ \beta_1 \ \beta_2]}$与$S_A$相同。 +2. 从初等行变换角度看：在行阶梯形中，$\beta_1, \beta_2$对应的列会被$A$的列线性表示，从而可化为零列（在解方程时体现为消去）。 +3. 存在$x_1, x_2$使得： + $$ + A(-x_1) = \beta_1, \quad A(-x_2) = \beta_2 + $$ + 这样在增广矩阵中可以通过列操作消去$\beta_1, \beta_2$，使其变为零列。 + +--- + +## 总结 + +- 秩相等 ⇔ 列空间相同 ⇔ 方程组有解。 +- 图示法：把$A$的列空间画成一个圆，$\beta_1, \beta_2$若落在圆内，则方程有解。 +- 矩阵的秩是判断线性方程组解的存在性的核心工具。 + +--- + +**注**：这里的“圆”是比喻，实际为**线性子空间**。 + +# 秩的不等式 + +### 1. 和的秩不超过秩的和 + +设 $A, B$ 为同型矩阵，则 +$$ \operatorname{rank}(A+B) \leq \operatorname{rank} A + \operatorname{rank} B $$ + +### 2. 积的秩不超过任何因子的秩 + +设 $A_{m \times n}, B_{n \times k}$，则 +$$ \operatorname{rank}(AB) \leq \min\{\operatorname{rank} A, \operatorname{rank} B\} $$ + +### 3. Sylvester(西尔维斯特)不等式 + +设 $A_{m \times n}, B_{n \times k}$，则 +$$ \operatorname{rank}(AB) \geq \operatorname{rank} A + \operatorname{rank} B - n $$ +特别地，当 $AB = 0$ 时，有 $\operatorname{rank} A + \operatorname{rank} B \leq n$。 + +### 4. 分块式 + +设 $A_{n \times n}$， $B_{n \times n}$，则 + +$$(1)\ \mathrm{rank} + +\begin{bmatrix} +A \\ +B +\end{bmatrix} \geq \text{rank } A, \quad \text{rank } +\begin{bmatrix} +A \\ +B +\end{bmatrix} \geq \text{rank } B +$$ + +$$(2)\ \mathrm{rank} +\begin{bmatrix} +A & 0 \\ +0 & B +\end{bmatrix} = \text{rank } A + \text{rank } B +$$ + +$$(3)\ \mathrm{rank} +\begin{bmatrix} +A & E_n \\ +0 & B +\end{bmatrix} \geq \text{rank } A + \text{rank } B +$$ + +$$(4)\ \mathrm{rank} +\begin{bmatrix} +A & 0 \\ +0 & B +\end{bmatrix} = \text{rank } +\begin{bmatrix} +A & B \\ +0 & B +\end{bmatrix} = \text{rank } +\begin{bmatrix} +A + B & B \\ +B & B +\end{bmatrix} \geq \text{rank } (A + B) +$$ + +注：（2）与（4）结合即第一个不等式的证明方法 + +> [!note] 证明1： +$$\operatorname{rank}(AB) \leq \min\{\operatorname{rank}(A), \operatorname{rank}(B)\}$$ + +--- + +### 证明思路 +设 +$$ +A \in \mathbb{R}^{m \times n}, \quad B \in \mathbb{R}^{n \times p}, \quad C = AB \in \mathbb{R}^{m \times p}. +$$ + +--- + +#### 1. 先证$\operatorname{rank}(AB) \leq \operatorname{rank}(A)$ + +- 考虑$C$的列向量： + 设$B = [b_1, b_2, \dots, b_p]$，则 + $$ + C = [A b_1, A b_2, \dots, A b_p]. + $$ + 因此$C$的每一列都是$A$的列向量的线性组合。 +- 所以$C$的列空间是$A$的列空间的子空间，故 + $$ + \operatorname{rank}(C) \leq \operatorname{rank}(A)。 + $$ + +--- + +#### 2. 再证$\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ + +- 考虑$C$的行向量： + 设$A = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_m \end{bmatrix}$，则 + $$ + C = \begin{bmatrix} a_1 B \\ a_2 B \\ \vdots \\ a_m B \end{bmatrix}. + $$ + 因此$C$的每一行都是$B$的行向量的线性组合。 +- 所以$C$的行空间是$B$的行空间的子空间，故 + $$\operatorname{rank}(C) \leq \operatorname{rank}(B)$$ + +--- + +#### 3. 综合 +由 1 和 2 得 +$$ +\operatorname{rank}(AB) \leq \operatorname{rank}(A) \quad \text{且} \quad \operatorname{rank}(AB) \leq \operatorname{rank}(B), +$$ +即 +$$ +\operatorname{rank}(AB) \leq \min\{\operatorname{rank}(A), \operatorname{rank}(B)\}. +$$ + +--- + +**证毕。** + +> [!note] 证明2 +> 证明 Sylvester 秩不等式： +$$\operatorname{rank}(AB) \ge \operatorname{rank}(A) + \operatorname{rank}(B) - n$$ +其中 +$A \in \mathbb{R}^{m \times n}, \; B \in \mathbb{R}^{n \times p}, \; AB \in \mathbb{R}^{m \times p}$。 + +--- + +### 证明思路 +设： +-$\operatorname{rank}(A) = r$ +-$\operatorname{rank}(B) = s$ +-$n$是矩阵乘法的中间维度，即$A$的列数、$B$的行数。 + +--- + +#### 1. 利用分块矩阵构造 +构造如下分块矩阵： +$$ +M = \begin{bmatrix} +A & O \\ +I_n & B +\end{bmatrix} +\in \mathbb{R}^{(m+n) \times (n+p)} +$$ +其中$I_n$是$n \times n$单位矩阵，$O$是零矩阵。 + +--- + +#### 2. 对$M$进行初等变换 +从$M$的第二块行减去第一块行左乘某个矩阵（这里相当于对$M$做列初等变换），实际上我们可以对$M$做以下变换： +$$ +\begin{bmatrix} +A & O \\ +I_n & B +\end{bmatrix} +\xrightarrow{\text{右乘 } \begin{bmatrix} I_n & -B \\ O & I_p \end{bmatrix}} +\begin{bmatrix} +A & -AB \\ +I_n & O +\end{bmatrix} +$$ +初等变换不改变矩阵的秩，所以： +$$ +\operatorname{rank}(M) = \operatorname{rank}\begin{bmatrix} +A & -AB \\ +I_n & O +\end{bmatrix} +$$ + +--- + +#### 3. 估计$\operatorname{rank}(M)$ +另一方面，由分块矩阵的秩不等式： +$$ +\operatorname{rank}(M) \ge \operatorname{rank}(A) + \operatorname{rank}(B) +$$ +这是因为$M$左上块为$A$，右下块为$B$，中间有单位矩阵，所以$A$和$B$的秩可以同时取到。 + +利用： +$$ +\operatorname{rank}\begin{bmatrix} +A & O \\ +I_n & B +\end{bmatrix} \ge \operatorname{rank}(A) + \operatorname{rank}(B) +$$ +因为$I_n$的存在使得两个子块的秩可以同时保持。 + +--- + +#### 4. 从变换后的矩阵得到下界 + +实际上更直接的方法是注意到： +$$ +\operatorname{rank}\begin{bmatrix} +A & -AB \\ +I_n & O +\end{bmatrix} += \operatorname{rank}\begin{bmatrix} +O & -AB \\ +I_n & O +\end{bmatrix} \quad (\text{行变换}) +$$ +即： +$$ += \operatorname{rank}\begin{bmatrix} +I_n & O \\ +O & AB +\end{bmatrix} = \operatorname{rank}(I_n) + \operatorname{rank}(AB) = n + \operatorname{rank}(AB) +$$ + +--- + +#### 5. 联立 +由初等变换保秩，得： +$$ +n + \operatorname{rank}(AB) = \operatorname{rank}(M) \ge \operatorname{rank}(A) + \operatorname{rank}(B) +$$ +整理得： +$$ +\operatorname{rank}(AB) \ge \operatorname{rank}(A) + \operatorname{rank}(B) - n +$$ + +--- + +## 重点思路 + +>[!information] 思路1 +>通过矩阵的秩的不等式，最大限度限制所求的表达式的取值范围，或者将其**限制到一个具体的值**. +>在希望求一个矩阵的秩的确切值时，也可以考虑用不等式关系来“夹逼”，常见的不等式： +>1. $\mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}-n\le\mathrm{rank}(\boldsymbol{AB})\le\min{\{\mathrm{rank}\boldsymbol{A}, \mathrm{rank}\boldsymbol{B}\}}$ +>2. $\mathrm{rank}(\boldsymbol{A+B})<\mathrm{rank}\boldsymbol A+\mathrm{rank}\boldsymbol B$ +>3. 矩阵加边不会减小秩； +> + +> [!note] 思路2 +> 在遇到诸如 $AB=O$ 的情况，务必要想到$\mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}-n\le\mathrm{rank}(\boldsymbol{AB}) \Rightarrow \mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}\le n$ + +> [!example] 例1 +> 设 $A$ 是 $m\times n$ 实矩阵， $\beta \neq 0$ 是 $m$ 维实列向量，证明： +> (1) $\mathrm{rank}A=\mathrm{rank}(A^\mathrm{T}A)$ . +> (2) 线性方程组 $A^\mathrm{T}Ax = A^\mathrm{T}\beta$ 有解.（这一问用到这个方法） + +解析：证明如下： +>(1)（10分） +> 对于方程组$Ax=0$ (a)和 $A^\mathrm{T}Ax=0$ (b)，b的解空间一定包含a的解空间（5分）； +>而方程b两边同时乘以$x^\mathrm{T}$，得 $x^\mathrm{T}A^\mathrm{T}Ax=0$ ，即 $(x^\mathrm{T}A^\mathrm{T})(Ax)=0 \to Ax=0$， +>所以a的解空间包含b的解空间（5分）， +>所以a,b同解，所以$\mathrm{rank}A=\mathrm{rank}{A^\mathrm{T}A}$ +>(2) （10分） +>$A^\mathrm{T}Ax=A^\mathrm{T}\beta \iff \mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})$ +>而由(1)的结论得等式左边 $\mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}A$； +>等式右边 $\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})\ge \mathrm{rank}A^\mathrm{T}A=\mathrm{rank}A$ （5分） +>又$\mathrm{rank}(A^\mathrm{T}\begin{bmatrix}A&\beta\end{bmatrix})\le \min{(\mathrm{rank}A^\mathrm{T},\ \mathrm{rank}\begin{bmatrix}A&\beta\end{bmatrix})}=\mathrm{rank}A$ （5分） +>所以$\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})=\mathrm{rank}A$ +>故 $\mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})$ 得证. + +>[!example] 例2 +>已知$A, B, C, D$都是 4 阶非零矩阵，且$ABCD = O$，如果$|BC| \neq 0$，记$$r(A) + r(B) + r(C) + r(D) = r$$ +>则$r$的最大值是（ ）。 +>(A) 11 +>(B) 12 +>(C) 13 +>(D) 14 + +**解析思路**： +- 由$|BC| \neq 0$知$B, C$均可逆。 +- 由$ABCD = O$，故$r(AB) + r(CD) \le 4$，$r(A) + r(D) \le 4$ +- 又$B, C$满秩，即$r(B) = r(C) = 4$。 +- 于是$r = r(A) + r(B) + r(C) + r(D) \le 4 + 4 + 4 = 12$。 +- 存在构造使等号成立，故最大值为$12$。 +- 例如$$A = +\begin{pmatrix} +1 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{pmatrix}, +\quad +D = +\begin{pmatrix} +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +1 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 +\end{pmatrix}$$ +**答案**： (B) 12 diff --git a/编写小组/讲义/证明题方法：单调有界定理，介值定理.md b/编写小组/讲义/证明题方法：单调有界定理，介值定理.md index 98d80bd..8c4903b 100644 --- a/编写小组/讲义/证明题方法：单调有界定理，介值定理.md +++ b/编写小组/讲义/证明题方法：单调有界定理，介值定理.md @@ -16,6 +16,12 @@ tags: - **函数单调有界性质**： 若函数 $f(x)$ 在区间 $I$ 上单调且有界，则 $f(x)$ 在 $I$ 的端点或无穷远处存在单侧极限。 +- **这类题的基本思路**： + + 先看有界性：尝试用不等式或者使用数学归纳法，注意在上下界不好找的时候，可以先取极限，来确定一个大致范围，再对猜测的有界性尝试证明 + + 再看单调性：尝试作差、作商或者使用数学归纳法 + ## 适用情况 适用于证明数列或函数极限的存在性，尤其是： diff --git a/编写小组/讲义/证明题方法：单调有界定理，介值定理（解析版）.md b/编写小组/讲义/证明题方法：单调有界定理，介值定理（解析版）.md index 4bc4449..8c868c6 100644 --- a/编写小组/讲义/证明题方法：单调有界定理，介值定理（解析版）.md +++ b/编写小组/讲义/证明题方法：单调有界定理，介值定理（解析版）.md @@ -4,6 +4,7 @@ tags: --- **内部资料，禁止传播** **编委会（不分先后，姓氏首字母顺序）：程奕铭 韩魏 刘柯妤 卢吉辚 王轲楠 支宝宁 郑哲航 + # 单调有界准则 ## 原理 @@ -15,6 +16,12 @@ tags: - **函数单调有界性质**： 若函数 $f(x)$ 在区间 $I$ 上单调且有界，则 $f(x)$ 在 $I$ 的端点或无穷远处存在单侧极限。 +- **这类题的基本思路**： + + 先看有界性：尝试用不等式或者使用数学归纳法，注意在上下界不好找的时候，可以先取极限，来确定一个大致范围，再对猜测的有界性尝试证明 + + 再看单调性：尝试作差、作商或者使用数学归纳法 + ## 适用情况 适用于证明数列或函数极限的存在性，尤其是： diff --git a/编写小组/试卷/0103高数模拟试卷.md b/编写小组/试卷/0103高数模拟试卷.md new file mode 100644 index 0000000..33fabc5 --- /dev/null +++ b/编写小组/试卷/0103高数模拟试卷.md @@ -0,0 +1,388 @@ +## 一、单选题（共5小题，每小题2分，共10分） + +1.设函数 $f(x)$ 在 $x = 0$ 处可导，且 $f(0) = 0$。若 +$$ +\lim_{x \to 0} \frac{f(2x) - 3f(x) + f(-x)}{x} = 2, +$$ +则 $f'(0)$ 的值为（ ）。 + +A. $-2$ +B. $-1$ +C. $1$ +D. $2$ + +2.求极限 $\displaystyle \lim_{x \to 0} \frac{e^{x^2} \cos x - 1 - \frac{1}{2}x^2}{\ln(1 + x^2) - x^2}$ +A.$-\frac{1}{12}$ +B.$\frac{1}{12}$ +C.$-\frac{1}{6}$ +D$\frac{1}{6}$ + +3.已知函数 $f(x) = 2e^x \sin x - 2ax - bx^2$ 与 $g(x) = \int \arctan(x^2) \mathrm{d}x$（取满足 $g(0) = 0$ 的那个原函数）是 $x \to 0$ 过程的同阶无穷小量，则（ ）。 + +(A) $a = 1, \, b = 1$ +(B) $a = 1, \, b = 2$ +(C) $a = 2, \, b = 1$ +(D) $a = 2, \, b = 2$ + +4.下列级数中绝对收敛的是（ ）。 + +(A) $\displaystyle \sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n+(-1)^n}}$ + +(B) $\displaystyle \sum_{n=2}^{\infty} \frac{1}{(\ln n)^{\ln n}}$ + +(C) $\displaystyle \sum_{n=1}^{\infty} \frac{n! \cdot 3^n}{n^n}$ + +(D) $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{1+1/n}}$ + +## 二、填空题（共5小题，每小题2分，共10分） + +1.设 $y = \arctan x$，则 $y^{(n)}(0)=\_\_\_\_.$ + + +2.已知 $f(x)$ 是三次多项式，且有 $\lim\limits_{x \to 2a} \frac{f(x)}{x-2a} = \lim\limits_{x \to 4a} \frac{f(x)}{x-4a} = 1$，则$\lim\limits_{x \to 3a} \frac{f(x)}{x-3a}=\_\_\_.$ + + +3.若级数$\sum\limits_{n=1}^{\infty}\frac{n^p}{(-1)^n}\sin(\frac{1}{\sqrt{n}})$绝对收敛，则常数$p$的取值范围是$\underline{\quad\quad\quad}.$ + + +4.求不定积分$\int{x^3\sqrt{4-x^2}\mathrm{d}x}=\underline{\quad\quad\quad}.$ + + +5.设$y=f(x)$由$\begin{cases}x=t^2+2t\\t^2-y+a\sin y=1\end{cases}$确定，若$y(0)=b$，$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}|_{t=0}=\underline{\quad\quad\quad}.$ + +## 三、解答题（共11小题，共80分） +1.求下列不定积分(提示，换元），其中 $a > 0$ + +(1) $\displaystyle \int \frac{x^2}{\sqrt{a^2 - x^2}} \mathrm{d}x$; (2) $\displaystyle \int \frac{\sqrt{x^2 + a^2}}{x^2} \mathrm{d}x$; +(3) $\displaystyle \int \frac{\sqrt{x^2 - a^2}}{x} \mathrm{d}x$; (4) $\displaystyle \int \sqrt{1 + e^x} \mathrm{d}x$. +```text + + + + + + + + + + + + + + + +``` +2.求极限 $\displaystyle \lim_{x \to 0} \frac{e^{x^2} - 1 - \ln\left(x^2 + 1\right)}{x^3 \arcsin x}$。 +```text + + + + + + + + + + + + + + + + + + +``` + +3.设 $f(x)$ 在 $[0, \frac{1}{2}]$ 上二阶可导，$f(0) = f'(0)$，$f\left(\frac{1}{2}\right) = 0$。 +证明：存在 $\xi \in (0, \frac{1}{2})$，使得 $f''(\xi) = \frac{3f'(\xi)}{1-2\xi}$。 +```text + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +``` + +4.求 $y = e^{ax} \sin bx$ 的 $n$ 阶导数 $y^{(n)}\text{,其中a，b为非零常数}$。 +```text + + + + + + + + + + + + + + + + + + + + + + + + + + + +``` + +5.将函数 $f(x) = x^2 e^x + x^6$ 展开成六阶带佩亚诺余项的麦克劳林公式，并求 $f^{(6)}(0)$ 的值。 +```text + + + + + + + + + + + + + + + + + + + + + + + + + +``` + +6.已知对于一数列$\{\alpha_n\}$，若满足$$\forall N\in\mathbb{N}_+,\exists\delta>0,当n>N时 ，有,\lvert \alpha_{n+1}-\alpha_n\rvert<\delta,$$则数列$\{\alpha_n\}$收敛.满足以上条件的数列称为柯西数列.设函数 $f(x)$ 在 $(-\infty, +\infty)$ 内可导，且 $|f'(x)| \leq r$ ($0 < r < 1$)。取实数 $x_1$，记 +$$x_{n+1} = f(x_n), \quad n = 1, 2, \cdots.$$ 试证明： + +（1）数列 $\{x_n\}$ 收敛（记 $\lim\limits_{n \to \infty} x_n = a$）； +（2）方程 $f(x) = x$ 有唯一实根 $x = a$. +```text + + + + + + + + + + + + + + + + + + + + + + + + + + +``` + +7.设在$\mathbb{R}$上的连续函数$f(x)$ 满足 $\sin f(x) - \frac{1}{3} \sin f\left(\frac{1}{3}x\right) = x$，求 $f(x)$。 + +```text + + + + + + + + + + + + + + + + + + + + + + + + + +``` + +8.已知当$x \to 0$时，函数$f(x) = a + bx^2 - \cos x$与$x^2$是等价无穷小。 +(1) 求参数$a, b$的值;（5分） +(2) 计算极限$$\lim\limits_{x \to 0} \frac{f(x) - x^2}{x^4}$$的值.（5分） +```text + + + + + + + + + + + + + + + + + + + + + +``` + +9.设$f(x)$在$[0,1]$上连续，在$(0,1)$上可导,且 $f(0)=0, f(1)=1$。试证： +（1）在$(0,1)$内存在不同的$\xi, \eta$使$f'(\xi)f'(\eta)=1$; +（2）对任意给定的正数$a, b$，在$(0,1)$内存在不同的$\xi, \eta$使$\frac{a}{f'(\xi)}+\frac{b}{f'(\eta)}=a+b$. +```text + + + + + + + + + + + + + + + + + + + + + + + + + + + +``` + + +10.讨论级数 $$\sum_{n=2}^{\infty} \frac{(-1)^n}{[n+(-1)^n]^p}$$ ( $p>0$ )的敛散性。 +```text + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +``` + +11.(10分) + +(1)证明： 对任意的正整数 $n$，方程 $$ x^n + n^2 x - 1 = 0 $$ +有唯一正实根（记为 $x_n$）。 + (2) 证明： 级数$\sum\limits_{n=1}^{\infty} x_n$收敛，且其和不超过 2。 +```text + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +``` diff --git a/编写小组/试卷/0103高数模拟试卷（解析版）.md b/编写小组/试卷/0103高数模拟试卷（解析版）.md new file mode 100644 index 0000000..8846b2e --- /dev/null +++ b/编写小组/试卷/0103高数模拟试卷（解析版）.md @@ -0,0 +1,826 @@ +--- +tags: + - 编写小组 +--- +## 一、单选题（共5小题，每小题2分，共10分） + +1.设函数 $f(x)$ 在 $x = 0$ 处可导，且 $f(0) = 0$。若 +$$ +\lim_{x \to 0} \frac{f(2x) - 3f(x) + f(-x)}{x} = 2, +$$ +则 $f'(0)$ 的值为（ ）。 + +A. $-2$ +B. $-1$ +C. $1$ +D. $2$ + + **解析** +因为 $f(x)$ 在 $x=0$ 处可导且 $f(0)=0$，所以 $f'(0) = \lim\limits_{x \to 0} \frac{f(x)}{x}$ 存在。 + +将极限式分解： +$$ +\frac{f(2x) - 3f(x) + f(-x)}{x} = \frac{f(2x)}{x} - 3\frac{f(x)}{x} + \frac{f(-x)}{x}. +$$ + +分别计算各项极限： +$$ +\begin{aligned} +\lim_{x \to 0} \frac{f(2x)}{x} &= 2 \lim_{x \to 0} \frac{f(2x)}{2x} = 2f'(0), \\ +\lim_{x \to 0} \frac{f(x)}{x} &= f'(0), \\ +\lim_{x \to 0} \frac{f(-x)}{x} &= -\lim_{x \to 0} \frac{f(-x)}{-x} = -f'(0). +\end{aligned} +$$ + +因此， +$$ +\lim_{x \to 0} \frac{f(2x) - 3f(x) + f(-x)}{x} = 2f'(0) - 3f'(0) + (-f'(0)) = -2f'(0). +$$ + +由已知条件 $-2f'(0) = 2$，解得 $f'(0) = -1$。 + +**答案：B** + + +2.求极限 $\displaystyle \lim_{x \to 0} \frac{e^{x^2} \cos x - 1 - \frac{1}{2}x^2}{\ln(1 + x^2) - x^2}$ +A.$-\frac{1}{12}$ +B.$\frac{1}{12}$ +C.$-\frac{1}{6}$ +D$\frac{1}{6}$ + +**解：** + +分别对分子和分母进行泰勒展开： + +分子： +$$\begin{aligned} +e^{x^2} \cos x &= \left(1 + x^2 + \frac{x^4}{2} + o(x^4)\right) \left(1 - \frac{x^2}{2} + \frac{x^4}{24} + o(x^4)\right) \\&= 1 + \frac{1}{2}x^2 + \left(-\frac{1}{2} + \frac{1}{2} + \frac{1}{24}\right)x^4 + o(x^4) \\&= 1 + \frac{1}{2}x^2 + \frac{1}{24}x^4 + o(x^4) +\end{aligned}$$ +所以 $e^{x^2} \cos x - 1 - \frac{1}{2}x^2 = \frac{1}{24}x^4 + o(x^4)$ + +分母： +$$ +\ln(1+x^2) = x^2 - \frac{1}{2}x^4 + o(x^4) +$$ +所以 $\ln(1+x^2) - x^2 = -\frac{1}{2}x^4 + o(x^4)$ + +因此 +$$ +\lim_{x \to 0} \frac{e^{x^2} \cos x - 1 - \frac{1}{2}x^2}{\ln(1 + x^2) - x^2} = \lim_{x \to 0} \frac{\frac{1}{24}x^4 + o(x^4)}{-\frac{1}{2}x^4 + o(x^4)} = \frac{1/24}{-1/2} = -\frac{1}{12} +$$ + + +3.已知函数 $f(x) = 2e^x \sin x - 2ax - bx^2$ 与 $g(x) = \int \arctan(x^2) \mathrm{d}x$（取满足 $g(0) = 0$ 的那个原函数）是 $x \to 0$ 过程的同阶无穷小量，则（ ）。 + +(A) $a = 1, \, b = 1$ +(B) $a = 1, \, b = 2$ +(C) $a = 2, \, b = 1$ +(D) $a = 2, \, b = 2$ + +**解：** +**法一**：为使 $f(x)$ 与 $g(x)$ 在 $x \to 0$ 时为同阶无穷小，需使二者最低阶非零项的阶数相同。下面分别展开 $f(x)$ 和 $g(x)$ 的麦克劳林公式。 + +对于 $f(x)$，利用已知展开式： +$$ +e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + O(x^3), \quad \sin x = x - \frac{x^3}{3!} + O(x^5), +$$ +则（把大于等于$5$次的项全都放在高阶无穷小里边） +$$ +e^x \sin x = \left(1 + x + \frac{x^2}{2} + \frac{x^3}{6} + O(x^3)\right)\left(x - \frac{x^3}{6} + O(x^5)\right) = x + x^2 + \frac{x^3}{3} + O(x^5). +$$ +因此 +$$ +2e^x \sin x = 2x + 2x^2 + \frac{2}{3}x^3 + O(x^5). +$$ +代入 $f(x)$ 得 +$$ +f(x) = 2x + 2x^2 + \frac{2}{3}x^3 - 2ax - bx^2 + O(x^5) = (2 - 2a)x + (2 - b)x^2 + \frac{2}{3}x^3 + O(x^5). +$$ + +对于 $g(x)$，由 $g(0)=0$ 及 $\arctan(x^2)$ 的展开： +$$ +\arctan(x^2) = x^2 - \frac{x^6}{3} + O(x^{10}), +$$ +积分得 +$$ +g(x) = \int_0^x \arctan(t^2) \mathrm{d}t = \frac{x^3}{3} - \frac{x^7}{21} + O(x^{11}) = \frac{1}{3}x^3 + O(x^7). +$$ +可见 $g(x)$ 是 $x \to 0$ 时的三阶无穷小，其主项为 $\dfrac{1}{3}x^3$。 + +为使 $f(x)$ 也是三阶无穷小（即与 $g(x)$ 同阶），$f(x)$ 中 $x$ 和 $x^2$ 的系数必须为零： +$$ +2 - 2a = 0, \quad 2 - b = 0, +$$ +解得 $a = 1$, $b = 2$。此时 +$$ +f(x) = \frac{2}{3}x^3 + O(x^5), +$$ +与 $g(x)$ 同阶（但不等价，因为系数比值为 $2$）。 + +因此正确选项为 (B)。 + +**法二**：洛必达＋泰勒 +先考虑两者之比的极限$$\lim\limits_{x\to0}\frac{f(x)}{g(x)}=\lim\limits_{x\to0}\frac{f'(x)}{g'(x)}=\lim\limits_{x\to0}\frac{2\mathrm{e}^x(\sin x+\cos x)-2a-2bx}{\arctan x^2}$$由麦克劳林公式得$$\begin{aligned}\mathrm{e}^x&=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+O(x^3)\\\sin x&=x-\frac{x^3}{6}+O(x^3)\\\cos x&=1-\frac{x^2}{2}+O(x^3)\end{aligned}$$于是（把所有大于3次的项都放在高阶无穷小里面，这样可以简化计算）$$\mathrm{e}^x(\sin x+\cos x)=1+2x+x^2+O(x^3)$$从而极限式的分子等于$$2-2a+(4-2b)+2x^2+O(x^3).$$又由麦克劳林公式得$$\arctan x=x-\frac{x^3}{6}+O(x^3)$$故分母为$$x^2-\frac{x^6}{6}+O(x^6)=x^2+O(x^3).$$上面的分子分母带入极限式中得$$\lim\limits_{x\to0}\frac{2-2a+(4-2b)+2x^2+O(x^3)}{x^2+O(x^3)}$$要让上式为有限值且不为$0$,只有$$\begin{cases}2-2a&=0\\4-2b&=0\end{cases}\implies\begin{cases}a&=1\\b&=2\end{cases}$$故选(B) +**答案：** (B) + +4.下列级数中绝对收敛的是（ ）。 + +(A) $\displaystyle \sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n+(-1)^n}}$ + +(B) $\displaystyle \sum_{n=2}^{\infty} \frac{1}{(\ln n)^{\ln n}}$ + +(C) $\displaystyle \sum_{n=1}^{\infty} \frac{n! \cdot 3^n}{n^n}$ + +(D) $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{1+1/n}}$ + + **解析** +- **(A)** g根据莱布尼兹判别法，级数条件收敛，但其绝对值接近于$p-$级数（$p=\frac{1}{2}$）的情形，故不绝对收敛。 +- **(B)** 由于 $(\ln n)^{\ln n} = n^{\ln \ln n}$，当 $n$ 足够大时，$\ln \ln n > 2$，故 $\frac{1}{(\ln n)^{\ln n}} < \frac{1}{n^2}$，由 $p$-级数收敛知原级数绝对收敛。 +- **(C)** 用比值判别法：$\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n} = \lim\limits_{n\to\infty} \frac{3}{(1+1/n)^n} = \frac{3}{e} > 1$，发散。 +- **(D)** 由于 $n^{1/n} \to 1$，故 $\frac{1}{n^{1+1/n}} \sim \frac{1}{n}$，与调和级数比较，发散。 + +**答案：(B)** + +5.一个倒置的圆锥形容器（顶点在下，底面在上），高度为 10 米，底面半径为 5 米。容器内装有水，水从底部的一个小孔（面积为 $0.1\pi$ 平方米）流出，流速为 $v = 0.6\sqrt{2gh}$ 米/秒，其中 $g = 10$ 米/秒$^2$。同时，以恒定速率 $Q = 0.3\pi$ 立方米/秒从顶部注入水。当水面高度为 5 米时，水面高度的瞬时变化率是多少？ + +选项: +A. $-0.048$ 米/秒 +B. $-0.024$ 米/秒 +C. 0 +D. 0.048 米/秒 +**解析：** +设水面高度为 $h$（从圆锥顶点算起）。由相似关系，水面半径 +$$ +r = \frac{R}{H}h = \frac{5}{10}h = 0.5h, +$$ +水的体积为 +$$ +V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (0.5h)^2 h = \frac{1}{12}\pi h^3. +$$ +对时间 $t$ 求导，得 +$$ +\frac{\mathrm{d}V}{\mathrm{d}t} = \frac{1}{4}\pi h^2 \frac{\mathrm{d}h}{\mathrm{d}t}. +$$ +另一方面，体积变化率由注入速率 $Q$ 和流出速率 $A \cdot v$ 决定： +$$ +\frac{\mathrm{d}V}{\mathrm{d}t} = Q - A \cdot v = 0.3\pi - 0.1\pi \cdot 0.6\sqrt{2gh}. +$$ +代入 $h=5$，$g=10$，则 +$$ +\sqrt{2gh} = \sqrt{2 \cdot 10 \cdot 5} = \sqrt{100} = 10, +$$ +于是 +$$ +A \cdot v = 0.1\pi \cdot 0.6 \cdot 10 = 0.6\pi, +$$ +$$ +\frac{dV}{dt} = 0.3\pi - 0.6\pi = -0.3\pi. +$$ +代入微分式： +$$ +\frac{1}{4}\pi \cdot 5^2 \cdot \frac{\mathrm{d}h}{\mathrm{d}t} = -0.3\pi \quad \Rightarrow \quad \frac{25}{4}\pi \frac{\mathrm{d}h}{\mathrm{d}t} = -0.3\pi. +$$ +解得 +$$ +\frac{\mathrm{d}h}{\mathrm{d}t} = -0.3 \times \frac{4}{25} = -0.048 \ \text{米/秒}. +$$ +因此水面高度以每秒 $0.048$ 米的速度下降，故选 A。 +## 二、填空题（共5小题，每小题2分，共10分） + +1.设 $y = \arctan x$，则 $y^{(n)}(0)=\_\_\_\_.$ + +**分析** +逐次求导以找到 $n$ 阶导数的规律。由于 $y' = \frac{1}{1 + x^2}$，即 $(1 + x^2)y' = 1$，故想到用莱布尼茨公式。 + +**解** + +**方法1** +因为 $y' = \frac{1}{1 + x^2}$，所以 $(1 + x^2)y' = 1$。 +上述两端对 $x$ 求 $n$ 阶导数，并利用莱布尼茨公式，得 +$$\sum_{k=0}^{n} C_n^k (1 + x^2)^{(k)} (y')^{(n-k)} = (1 + x^2)y^{(n+1)} + 2nxy^{(n)} + n(n-1)y^{(n-1)} = 0.$$ +在上式中令 $x = 0$，得 +$$y^{(n+1)}(0) = -n(n-1)y^{(n-1)}(0).$$ +由此递推公式，再加上 $y'(0) = 1, y''(0) = 0$，可得： +$$y^{(n)}(0) = \begin{cases} 0, & n\text{为偶数} \\ (-1)^n (n-1)!, & n\text{为奇数} \end{cases}.$$ + +**方法2** +因为 $y' = \frac{1}{1 + x^2} = \sum\limits_{n=0}^{\infty} (-1)^n x^{2n}, (|x| < 1)$， +所以 （积分得） +$$y = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1}, (|x| < 1).$$ +又知 $y(x)$ 在点 $x = 0$ 处的麦克劳林展开式为 +$$y(x) = \sum_{n=0}^{\infty} \frac{y^{(n)}(0)}{n!} x^n.$$ +比较系数可得 +$$y^{(n)}(0) = \begin{cases} 0, & n\text{为偶数} \\ (-1)^n (n-1)!, & n\text{为奇数} \end{cases}.$$ + +**方法3** +由 $y = \arctan x$，得 $x = \tan y$，则 +$$y' = \frac{1}{1 + x^2} = \frac{1}{1 + \tan^2 y} = \cos^2 y.$$ +利用复合函数求导法则： +$$y'' = -2 \cos y \sin y \cdot y' = -\sin(2y) \cos^2 y = \cos^2 y \sin 2\left( y + \frac{\pi}{2} \right),$$ +$$\begin{aligned}y''' &= \left[ -2 \cos y \sin y \sin 2\left( y + \frac{\pi}{2} \right) + 2 \cos^2 y \cos 2\left( y + \frac{\pi}{2} \right) \right] y'\\& = 2 \cos^3 y \cos\left( y + \frac{\pi}{2} \right) y' + 2 \cos^3 y \sin 3\left( y + \frac{\pi}{2} \right),\end{aligned}$$ +$$y^{(4)} = 6 \cos^4 y \cos\left( y + \frac{\pi}{2} \right) y' + 3 \cos^4 y \sin 4\left( y + \frac{\pi}{2} \right).$$ +由此归纳出 +$$y^{(n)} = (n-1)! \cos^n y \sin n\left( y + \frac{\pi}{2} \right).$$ +下面用归纳法证明以上结论： +当 $n = 1$ 时结论成立； +假设 $n = k$ 时结论成立，则当 $n = k + 1$ 时， +$$\begin{aligned} y^{(k+1)} &= (y^{(k)})' \\ &= (k-1)! \left[ -k \cos^{k+1} y \sin y \sin k\left( y + \frac{\pi}{2} \right) + k \cos^k y \cos k\left( y + \frac{\pi}{2} \right) \right] y' \\ &= k \cos^{k+1} y \cos\left( y + \frac{\pi}{2} \right) y' + k \cos^{k+1} y \sin (k+1)\left( y + \frac{\pi}{2} \right). \end{aligned}$$ +由于 $y(0) = \arctan 0 = 0, \cos 0 = 1$，因此 +$$y^{(n)}(0) = (n-1)! \sin \frac{n\pi}{2}.$$ +根据 $\sin \frac{n\pi}{2}$ 的取值（$n$ 为偶数时为 $0$，$n=4k+1$ 时为 $1$，$n=4k+3$ 时为 $-1$），可得到与方法1、2相同的结果。 + + + +2.已知 $f(x)$ 是三次多项式，且有 $\lim\limits_{x \to 2a} \frac{f(x)}{x-2a} = \lim\limits_{x \to 4a} \frac{f(x)}{x-4a} = 1$，则$\lim\limits_{x \to 3a} \frac{f(x)}{x-3a}=\_\_\_.$ + +**分析** +由已知的两个极限式可确定 $f(x)$ 的两个一次因子以及两个待定系数，从而完全确定 $f(x)$。 + +**解** +由已知有 $\lim\limits_{x \to 2a} f(x) = \lim\limits_{x \to 4a} f(x) = 0$。 +由于 $f(x)$ 处处连续，故 $f(2a) = f(4a) = 0$。 +因此 $f(x)$ 含有因式 $(x-2a)(x-4a)$，可设 +$$f(x) = (Ax + B)(x - 2a)(x - 4a).$$ +由条件： +$$\lim_{x \to 2a} \frac{f(x)}{x-2a} = \lim_{x \to 2a} (Ax + B)(x - 4a) = (2aA + B)(-2a) = 1,$$ +$$\lim_{x \to 4a} \frac{f(x)}{x-4a} = \lim_{x \to 4a} (Ax + B)(x - 2a) = (4aA + B)(2a) = 1.$$ +解方程组： +$$\begin{cases} (2aA + B)(-2a) = 1 \\ (4aA + B)(2a) = 1 \end{cases} \Rightarrow \begin{cases} -4a^2A - 2aB = 1 \\ 8a^2A + 2aB = 1 \end{cases}.$$ +相加得 $4a^2A = 2$，故 $A = \frac{1}{2a^2}$；代入第一个方程得 $-4a^2 \cdot \frac{1}{2a^2} - 2aB = 1$，即 $-2 - 2aB = 1$，解得 $B = -\frac{3}{2a}$。 +于是 +$$f(x) = \left( \frac{1}{2a^2}x - \frac{3}{2a} \right)(x - 2a)(x - 4a) = \frac{1}{2a^2}(x - 3a)(x - 2a)(x - 4a).$$ +最后， +$$\begin{aligned}\lim_{x \to 3a} \frac{f(x)}{x-3a} &= \lim_{x \to 3a} \frac{\frac{1}{2a^2}(x-3a)(x-2a)(x-4a)}{x-3a} \\&= \frac{1}{2a^2} \cdot (3a-2a)(3a-4a) \\&= \frac{1}{2a^2} \cdot a \cdot (-a) \\&= -\frac{1}{2}.\end{aligned}$$ + +**答案**：$\displaystyle \lim_{x \to 3a} \frac{f(x)}{x-3a} = -\frac{1}{2}$. + +3.若级数$\sum\limits_{n=1}^{\infty}\frac{n^p}{(-1)^n}\sin(\frac{1}{\sqrt{n}})$绝对收敛，则常数$p$的取值范围是$\underline{\quad\quad\quad}.$ + +解：首先，考虑级数$\sum\limits_{n=1}^{\infty}|\frac{n^p}{(-1)^n}\sin(\frac{1}{\sqrt{n}})|=\sum\limits_{n=1}^{\infty}{n^p}|\sin(\frac{1}{\sqrt{n}})|$ + 当$n\to\infty$时，$\frac{1}{\sqrt{n}}\to 0$，此时有等价无穷小关系：$\sin(\frac{1}{\sqrt{n}}) \sim \frac{1}{\sqrt{n}}.$ + 因此，级数的通项可以近似为$\frac{1}{n^p}\frac{1}{\sqrt{n}}=\frac{1}{n^{p+\frac{1}{2}}}$ + 根据**p级数**的收敛性结论，级数$\sum\limits_{n=1}^{\infty}\frac{1}{n^{p+\frac{1}{2}}}$当且仅当$p+\frac{1}{2}>1$时收敛，即$p>\frac{1}{2}$， +所以，$p\in(\frac{1}{2},+\infty)$ + +4.求不定积分$\int{x^3\sqrt{4-x^2}\mathrm{d}x}=\underline{\quad\quad\quad}.$ + +方法1： + 令$x=2\sin t$，则$\mathrm{d}x=2\cos t \mathrm{d}t$， +$$\begin{align}\int{x^3\sqrt{4-x^2}\mathrm{d}x}&=\int{(2\sin t)^3\sqrt{4-4\sin^2 t} \cdot 2\cos t\mathrm{d}t}\\ +&=32\int{\sin^3t\cos^2t\mathrm{d}t}\\ +&=32\int{\sin t(1-\cos^2t)\cos^2t\mathrm{d}t}\\ +&=-32\int{(\cos^2t-cos^4t)\mathrm{d}\cos t}\\ +&=-32(\frac{\cos^3t}{3}-\frac{cos^5t}{5})+C\\ +&=-\frac{4}{3}(\sqrt{4-x^2})^3+\frac{1}{5}(\sqrt{4-x^2})^5+C +\end{align} +$$ + +方法2： + 令$\sqrt{4-x^2}=t$，$x^2=4-t^2$，$x\mathrm{d}x=-t\mathrm{d}t$， +$$ +\begin{align} +\int{x^3\sqrt{4-x^2}\mathrm{d}x}&=-\int{(4-t^2)t^2\mathrm{d}t}\\ +&=\frac{t^5}{5}-\frac{4t^3}{3}+C\\ +&=\frac{(\sqrt{4-x^2})^5}{5}-\frac{4(\sqrt{4-x^2})^3}{3}+C +\end{align} +$$ + +5.设$y=f(x)$由$\begin{cases}x=t^2+2t\\t^2-y+a\sin y=1\end{cases}$确定，若$y(0)=b$，$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}|_{t=0}=\underline{\quad\quad\quad}.$ + +**解**： +方程两边对$t$求导，得 +$$ +\begin{cases} +\frac{\mathrm{d}x}{\mathrm{d}t}=2t+2\\ +2t-\frac{\mathrm{d}y}{\mathrm{d}t}+a\frac{\mathrm{d}y}{\mathrm{d}t}\cos y=0 +\end{cases} +\Rightarrow +\begin{cases} +\frac{\mathrm{d}x}{\mathrm{d}t}=2(t+1)\\ +\frac{\mathrm{d}y}{\mathrm{d}t}=\frac{2t}{1-a\cos y} +\end{cases} +\Rightarrow +\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{t}{(t+1)(1-a\cos y)} +$$ +$$ +\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\frac{\mathrm{d}\frac{\mathrm{d}y}{\mathrm{d}x}}{\mathrm{d}x}=\frac{\frac{\mathrm{d}\frac{\mathrm{d}y}{\mathrm{d}x}}{\mathrm{d}t}}{\frac{\mathrm{d}x}{\mathrm{d}t}}=\frac{\frac{(1-a\cos y)-at(t+1)\frac{\mathrm{d}y}{\mathrm{d}t}\sin y}{(t+1)^2(1-a\cos y)^2}}{2(t+1)} +$$ +注意到$y|_{t=0}=b,\frac{\mathrm{d}y}{\mathrm{d}t}|_{t=0}=0$，得 +$$ +\frac{\mathrm{d}^2y}{\mathrm{d}x^2}|_{t=0}=\frac{1}{2(1-a\cos b)} +$$ + + +## 三、解答题（共11小题，共80分） +1.求下列不定积分(提示，换元），其中 $a > 0$ + +(1) $\displaystyle \int \frac{x^2}{\sqrt{a^2 - x^2}} \mathrm{d}x$ + +**解：** +令 $x = a \sin t$，有$\mathrm{d}x=a\cos t\mathrm{d}t$,则： + +$$ +\begin{aligned} +\int \frac{x^2}{\sqrt{a^2 - x^2}} \mathrm{d}x &= \int \frac{a^2 \sin^2 t}{a \cos t} \cdot a \cos t \, \mathrm{d}t \\ +&= a^2 \int \sin^2 t \, \mathrm{d}t \\ +&= \frac{a^2}{2} \int (1 - \cos 2t) \, \mathrm{d}t \\ +&= \frac{a^2}{2} \left( t - \frac{1}{2} \sin 2t \right) + C \\ +&= \frac{a^2}{2} \arcsin \frac{x}{a} - \frac{x}{2} \sqrt{a^2 - x^2} + C +\end{aligned} +$$ + +--- + + (2) $\displaystyle \int \frac{\sqrt{x^2 + a^2}}{x^2} \mathrm{d}x$ + +**解：** +**方法一：** 令 $x = a \tan t$，则： + +$$ +\begin{aligned} +\int \frac{\sqrt{x^2 + a^2}}{x^2} \mathrm{d}x &= \int \frac{a \sec t}{a^2 \tan^2 t} \cdot a \sec^2 t \, \mathrm{d}t \\ +&= \int \frac{\sec^3 t}{\tan^2 t} \, \mathrm{d}t \\ +&= \int \frac{1}{\sin^2 t \cos t} \, \mathrm{d}t \\ +&= \int \frac{\sin^2 t + \cos^2 t}{\sin^2 t \cos t} \, \mathrm{d}t \\ +&= \int \sec t \, \mathrm{d}t + \int \frac{\cos t}{\sin^2 t} \, \mathrm{d}t \\ +&= \ln |\sec t + \tan t| - \frac{1}{\sin t} + C \\ +&= \ln (x + \sqrt{x^2 + a^2}) - \frac{\sqrt{x^2 + a^2}}{x} + C +\end{aligned} +$$ + +**方法二：** + +$$ +\begin{aligned} +\int \frac{\sqrt{x^2 + a^2}}{x^2} \mathrm{d}x &= \int \frac{x^2 + a^2}{x^2 \sqrt{x^2 + a^2}} \mathrm{d}x \\ +&= \int \frac{1}{\sqrt{x^2 + a^2}} \mathrm{d}x + a^2 \int \frac{1}{x^2 \sqrt{x^2 + a^2}} \mathrm{d}x \\ +&= \ln (x + \sqrt{x^2 + a^2}) - \frac{\sqrt{x^2 + a^2}}{x} + C +\end{aligned} +$$ + +--- + +(3) $\displaystyle \int \frac{\sqrt{x^2 - a^2}}{x} \mathrm{d}x$ + +**解：** +令 $x = a \sec t$，则： + +$$ +\begin{aligned} +\int \frac{\sqrt{x^2 - a^2}}{x} \mathrm{d}x &= \int \frac{a \tan t}{a \sec t} \cdot a \sec t \tan t \, \mathrm{d}t \\ +&= a \int \tan^2 t \, \mathrm{d}t \\ +&= a \int (\sec^2 t - 1) \, \mathrm{d}t \\ +&= a (\tan t - t) + C \\ +&= \sqrt{x^2 - a^2} - a \arccos \frac{a}{x} + C +\end{aligned} +$$ + +--- + +(4) $\displaystyle \int \sqrt{1 + e^x} \mathrm{d}x$ + +**解：** +令 $t = \sqrt{1 + e^x}$，则 $e^x = t^2 - 1$，$x = \ln(t^2 - 1)$，$\mathrm{d}x = \frac{2t}{t^2 - 1} \mathrm{d}t$： + +$$ +\begin{aligned} +\int \sqrt{1 + e^x} \mathrm{d}x &= \int t \cdot \frac{2t}{t^2 - 1} \mathrm{d}t \\ +&= \int \frac{2t^2}{t^2 - 1} \mathrm{d}t \\ +&= 2 \int \left( 1 + \frac{1}{t^2 - 1} \right) \mathrm{d}t \\ +&= 2 \int 1 \, \mathrm{d}t + \int \left( \frac{1}{t-1} - \frac{1}{t+1} \right) \mathrm{d}t \\ +&= 2t + \ln \left| \frac{t-1}{t+1} \right| + C \\ +&= 2 \sqrt{1 + e^x} + \ln \frac{\sqrt{1 + e^x} - 1}{\sqrt{1 + e^x} + 1} + C +\end{aligned} +$$ + + +2.求极限 $\displaystyle \lim_{x \to 0} \frac{e^{x^2} - 1 - \ln\left(x^2 + 1\right)}{x^3 \arcsin x}$。 + +**解：** + +当 $x \to 0$ 时，利用等价无穷小替换和泰勒展开：9 + +分母：$x^3 \arcsin x \sim x^3 \cdot x = x^4$ + +分子：$e^{x^2} - 1 - \ln(1+x^2) = \left(1+x^2+\frac{x^4}{2}+o(x^4)\right) - 1 - \left(x^2-\frac{x^4}{2}+o(x^4)\right) = x^4 + o(x^4)$ + +所以 +$$ +\lim_{x \to 0} \frac{e^{x^2} - 1 - \ln\left(x^2 + 1\right)}{x^3 \arcsin x} = \lim_{x \to 0} \frac{x^4 + o(x^4)}{x^4} = 1 +$$ + + + +3.设 $f(x)$ 在 $[0, \frac{1}{2}]$ 上二阶可导，$f(0) = f'(0)$，$f\left(\frac{1}{2}\right) = 0$。 +证明：存在 $\xi \in (0, \frac{1}{2})$，使得 $f''(\xi) = \frac{3f'(\xi)}{1-2\xi}$。 + +**证明：** +构造辅助函数 $g(x) = (1-2x)^{3/2} f'(x)$，$x \in [0, \frac{1}{2}]$。 +由于 $f$ 二阶可导，故 $g$ 在 $[0, \frac{1}{2}]$ 上连续，在 $(0, \frac{1}{2})$ 内可导。 +计算 $g$ 的导数： + +$$ +\begin{aligned} +g'(x) &= (1-2x)^{3/2} f''(x) + \frac{3}{2}(1-2x)^{1/2} \cdot (-2) \cdot f'(x) \\ +&= (1-2x)^{1/2} \left[ (1-2x) f''(x) - 3f'(x) \right] +\end{aligned} +$$ + +当 $x \in (0, \frac{1}{2})$ 时，$(1-2x)^{1/2} > 0$，因此 $g'(x) = 0$ 当且仅当： + +$$ +(1-2x) f''(x) - 3f'(x) = 0 \quad \text{即} \quad f''(x) = \frac{3f'(x)}{1-2x} +$$ + +所以只需证明存在 $\xi \in (0, \frac{1}{2})$ 使得 $g'(\xi) = 0$。 + +由已知条件： + +$$ +g(0) = (1-0)^{3/2} f'(0) = f'(0) = f(0), \quad g\left(\frac{1}{2}\right) = (1-1)^{3/2} f'\left(\frac{1}{2}\right) = 0 +$$ + +分两种情况讨论： + +1. **若 $f'(0) = 0$** + 此时 $g(0) = 0$，所以 $g(0) = g\left(\frac{1}{2}\right) = 0$。由罗尔定理，存在 $\xi \in (0, \frac{1}{2})$ 使得 $g'(\xi) = 0$。 + +2. **若 $f'(0) \neq 0$** + 对 $f$ 在 $[0, \frac{1}{2}]$ 上应用拉格朗日中值定理，存在 $c \in (0, \frac{1}{2})$ 使得： + + $$ + f'(c) = \frac{f\left(\frac{1}{2}\right) - f(0)}{\frac{1}{2} - 0} = \frac{0 - f(0)}{\frac{1}{2}} = -2f(0) = -2f'(0) + $$ + + 由于 $f'(0) \neq 0$，故 $f'(c) \neq 0$ 且与 $f'(0)$ 异号。 + 计算 $g(c) = (1-2c)^{3/2} f'(c)$，因为 $(1-2c)^{3/2} > 0$，所以 $g(c)$ 与 $f'(c)$ 同号，从而 $g(c)$ 与 $g(0) = f'(0)$ 异号。 + 由连续函数的介值定理，存在 $d \in (0, c)$ 使得 $g(d) = 0$。 + 在区间 $[d, \frac{1}{2}]$ 上，$g(d) = g\left(\frac{1}{2}\right) = 0$，由罗尔定理，存在 $\xi \in (d, \frac{1}{2}) \subset (0, \frac{1}{2})$ 使得 $g'(\xi) = 0$。 + +综上所述，无论何种情况，均存在 $\xi \in (0, \frac{1}{2})$ 使得 $g'(\xi) = 0$，从而有： + +$$ +f''(\xi) = \frac{3f'(\xi)}{1-2\xi} +$$ + +原命题得证。 + + + +4.求 $y = e^{ax} \sin bx$ 的 $n$ 阶导数 $y^{(n)}\text{,其中a，b为非零常数}$。 + + + **方法一：逐阶求导归纳法** + +由 +$$ +y' = a e^{ax} \sin bx + b e^{ax} \cos bx += e^{ax} \bigl( a \sin bx + b \cos bx \bigr), +$$ +令 +$$ +\varphi = \arctan \frac{b}{a}, +$$ +则 +$$ +a \sin bx + b \cos bx = \sqrt{a^2 + b^2} \, \sin(bx + \varphi). +$$ +于是 +$$ +y' = e^{ax} \cdot \sqrt{a^2 + b^2} \, \sin(bx + \varphi). +$$ +同理， +$$ +y'' = \sqrt{a^2 + b^2} \cdot e^{ax} \bigl[ a \sin(bx + \varphi) + b \cos(bx + \varphi) \bigr] += (a^2 + b^2) e^{ax} \sin(bx + 2\varphi). +$$ +依此类推，由归纳法可得 +$$ +y^{(n)} = (a^2 + b^2)^{n/2} \, e^{ax} \sin(bx + n\varphi), +$$ +其中 +$$ +\varphi = \arctan \frac{b}{a}. +$$ +下用数学归纳法证明.设$y^{(n)}=(a^2+b^2)^{n/2}\mathrm{e}^{ax}\sin(bx+n\varphi)$,则$$\begin{aligned}y^{(n+1)}&=(a^2+b^2)^{n/2}(a\mathrm{e}^{ax}\sin(bx+n\varphi)+b\mathrm{e}^{ax}\cos(bx+n\varphi))\\&=(a^2+b^2)^{(n+1)/2}\mathrm{e}^{ax}\sin(bx+(n+1)\varphi)\end{aligned}$$得证. + +--- + + **方法二：欧拉公式法** + +设 +$$ +u = e^{ax} \cos bx, \quad v = e^{ax} \sin bx, +$$ +则 +$$ +u + \mathrm{i}v = e^{ax} (\cos bx + \mathrm{i} \sin bx) = e^{(a + b\mathrm{i})x}. +$$ +求 $n$ 阶导数： +$$ +u^{(n)} + \mathrm{i} v^{(n)} = \bigl[ e^{(a + b\mathrm{i})x} \bigr]^{(n)} += (a + b\mathrm{i})^n e^{(a + b\mathrm{i})x}. +$$ +记 +$$ +a + bi = \sqrt{a^2 + b^2} \, e^{\mathrm{i}\varphi}, +\quad \varphi = \arctan \frac{b}{a}, +$$ +则 +$$ +(a + b\mathrm{i})^n = (a^2 + b^2)^{n/2} e^{n\varphi\mathrm{i}}. +$$ +于是 +$$ +u^{(n)} + \mathrm{i} v^{(n)} += (a^2 + b^2)^{n/2} e^{ n\varphi\mathrm{i}} \cdot e^{ax} e^{\mathrm{i}bx} += (a^2 + b^2)^{n/2} e^{ax} e^{\mathrm{i}(bx + n\varphi)}. +$$ +取虚部，即得 +$$ +\bigl( e^{ax} \sin bx \bigr)^{(n)} += (a^2 + b^2)^{n/2} e^{ax} \sin(bx + n\varphi). +$$ +类似可得 +$$ +\bigl( e^{ax} \cos bx \bigr)^{(n)} += (a^2 + b^2)^{n/2} e^{ax} \cos(bx + n\varphi). +$$ + +5.将函数 $f(x) = x^2 e^x + x^6$ 展开成六阶带佩亚诺余项的麦克劳林公式，并求 $f^{(6)}(0)$ 的值。 + +**解：** +已知 $e^x$ 的麦克劳林展开为： +$$ +e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + o(x^6). +$$ +则 +$$ +\begin{aligned} +x^2 e^x &= x^2 \left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + o(x^6)\right)\\[1em] &= x^2 + x^3 + \frac{x^4}{2!} + \frac{x^5}{3!} + \frac{x^6}{4!} + \frac{x^7}{5!} + \frac{x^8}{6!} + o(x^8). +\end{aligned} +$$ +由于我们只关心六阶展开，保留到 $x^6$ 项，$x^7$ 及更高次项可并入 $o(x^6)$，故 +$$ +x^2 e^x = x^2 + x^3 + \frac{x^4}{2} + \frac{x^5}{6} + \frac{x^6}{24} + o(x^6). +$$ +于是 +$$ +\begin{aligned} +f(x) &= x^2 e^x + x^6\\[1em] &= x^2 + x^3 + \frac{x^4}{2} + \frac{x^5}{6} + \frac{x^6}{24} + x^6 + o(x^6)\\[1em] &= x^2 + x^3 + \frac{x^4}{2} + \frac{x^5}{6} + \frac{25}{24}x^6 + o(x^6). +\end{aligned} +$$ +这就是 $f(x)$ 的六阶带佩亚诺余项的麦克劳林公式。 + +由泰勒展开的唯一性，$x^6$ 项的系数为 $\frac{f^{(6)}(0)}{6!}$，即 +$$ +\frac{f^{(6)}(0)}{6!} = \frac{25}{24}. +$$ +所以 +$$ +f^{(6)}(0) = 6! \times \frac{25}{24} = 720 \times \frac{25}{24} = 30 \times 25 = 750. +$$ + +**答案：** +展开式为 $f(x) = x^2 + x^3 + \frac{1}{2}x^4 + \frac{1}{6}x^5 + \frac{25}{24}x^6 + o(x^6)$，$f^{(6)}(0) = 750$. + + +6.已知对于一数列$\{\alpha_n\}$，若满足$$\forall N\in\mathbb{N}_+,\exists\delta>0,当n>N时 ，有,\lvert \alpha_{n+1}-\alpha_n\rvert<\delta,$$则数列$\{\alpha_n\}$收敛.满足以上条件的数列称为柯西数列.设函数 $f(x)$ 在 $(-\infty, +\infty)$ 内可导，且 $|f'(x)| \leq r$ ($0 < r < 1$)。取实数 $x_1$，记 +$$x_{n+1} = f(x_n), \quad n = 1, 2, \cdots.$$ +证明： + +（1）数列 $\{x_n\}$ 收敛（记 $\lim\limits_{n \to \infty} x_n = a$）； +（2）方程 $f(x) = x$ 有唯一实根 $x = a$. + +**证明：** + +（1）首先证明 $\{x_n\}$ 是柯西数列.对任意 $n \geq 1$，由拉格朗日中值定理，存在 $\xi_n$ 介于 $x_n$ 和 $x_{n-1}$ 之间，使得 +$$|x_{n+1} - x_n| = |f(x_n) - f(x_{n-1})| = |f'(\xi_n)| \cdot |x_n - x_{n-1}| \leq r |x_n - x_{n-1}|.$$ +反复应用此不等式，得 +$$|x_{n+1} - x_n| \leq r |x_n - x_{n-1}| \leq r^2 |x_{n-1} - x_{n-2}| \leq \cdots \leq r^{n-1} |x_2 - x_1|.$$ +于是对任意正整数 $m > n$，有 +$$ +\begin{aligned} +|x_m - x_n| &\leq |x_m - x_{m-1}| + |x_{m-1} - x_{m-2}| + \cdots + |x_{n+1} - x_n| \\[1em] +&\leq (r^{m-2} + r^{m-3} + \cdots + r^{n-1}) |x_2 - x_1| \\[1em] +&= r^{n-1} \cdot \frac{1 - r^{m-n}}{1 - r} \cdot |x_2 - x_1| \\[1em] +&\leq \frac{r^{n-1}}{1 - r} |x_2 - x_1|. +\end{aligned} +$$ +因为 $0 < r < 1$，所以当 $n \to \infty$ 时，$r^{n-1} \to 0$，从而对任意 $\varepsilon > 0$，存在 $N$，当 $m > n > N$ 时 $|x_m - x_n| < \varepsilon$。故 $\{x_n\}$ 是柯西数列，因此在实数域中收敛，记 $\lim_{n \to \infty} x_n = a$。 + +（2）由于 $f$ 可导，故连续。在递推式 $x_{n+1} = f(x_n)$ 两边取极限 $n \to \infty$，得 +$$a = \lim_{n \to \infty} x_{n+1} = \lim_{n \to \infty} f(x_n) = f(\lim_{n \to \infty} x_n) = f(a),$$ +即 $a$ 是方程 $f(x) = x$ 的一个实根。 + +下证唯一性。假设另有 $b \neq a$ 满足 $f(b) = b$，则由拉格朗日中值定理，存在 $\eta$ 介于 $a$ 和 $b$ 之间，使得 +$$|a - b| = |f(a) - f(b)| = |f'(\eta)| \cdot |a - b| \leq r |a - b|.$$ +由于 $|a - b| > 0$，两边除以 $|a - b|$ 得 $1 \leq r$，与 $0 < r < 1$ 矛盾。故方程 $f(x) = x$ 有唯一实根 $x = a$ + +综上所述，数列 $\{x_n\}$ 收敛于 $a$，且 $a$ 是方程 $f(x) = x$ 的唯一实根。 + + +7.设在$\mathbb{R}$上的连续函数$f(x)$ 满足 $\sin f(x) - \frac{1}{3} \sin f\left(\frac{1}{3}x\right) = x$，求 $f(x)$。 + +**解** +令 $g(x) = \sin f(x)$，则 +$$ +g(x) - \frac{1}{3} g\left(\frac{1}{3}x\right) = x. +$$ + +依次将 $x$ 替换为 $\frac{x}{3}, \frac{x}{3^2}, \cdots, \frac{x}{3^{n-1}}$，并乘以相应系数，得 +$$ +\begin{aligned} +g(x) - \frac{1}{3} g\left(\frac{x}{3}\right) &= x, \\ +\frac{1}{3} g\left(\frac{x}{3}\right) - \frac{1}{3^2} g\left(\frac{x}{3^2}\right) &= \frac{1}{3^2} x, \\ +\frac{1}{3^2} g\left(\frac{x}{3^2}\right) - \frac{1}{3^3} g\left(\frac{x}{3^3}\right) &= \frac{1}{3^4} x, \\ +&\vdots \\ +\frac{1}{3^{n-1}} g\left(\frac{x}{3^{n-1}}\right) - \frac{1}{3^n} g\left(\frac{x}{3^n}\right) &= \frac{1}{3^{2(n-1)}} x. +\end{aligned} +$$ + +以上各式相加，得 +$$ +g(x) - \frac{1}{3^n} g\left(\frac{x}{3^n}\right) = x \left(1 + \frac{1}{9} + \frac{1}{9^2} + \cdots + \frac{1}{9^{n-1}}\right). +$$ + +因为 $|g(x)| \leq 1$，所以 $\lim\limits_{n \to \infty} \frac{1}{3^n} g\left(\frac{x}{3^n}\right) = 0$。 +而 $\lim\limits_{n \to \infty} \left(1 + \frac{1}{9} + \frac{1}{9^2} + \cdots + \frac{1}{9^{n-1}}\right) = \frac{1}{1 - \frac{1}{9}} = \frac{9}{8}$， +因此 由函数的连续性， +$$ +g(x) = \frac{9}{8} x. +$$ + +于是 $\sin f(x) = \frac{9}{8} x$，解得 +$$ +f(x) = 2k\pi + \arcsin \frac{9}{8} x \quad \text{或} \quad f(x) = (2k-1)\pi - \arcsin \frac{9}{8} x \quad (k \in \mathbb{Z}). +$$ + +8.已知当$x \to 0$时，函数$f(x) = a + bx^2 - \cos x$与$x^2$是等价无穷小。 + +(1) 求参数$a, b$的值；（5分） +(2) 计算极限$$\lim_{x \to 0} \frac{f(x) - x^2}{x^4}$$的值。（5分） + +**解** + +(1) +由于 +$$ +\lim_{x \to 0} f(x) = \lim_{x \to 0} \left( a + bx^2 - \cos x \right) = a - 1 = 0, +$$ +故$a = 1$。 +又因为$f(x)$与$x^2$等价无穷小，所以 +$$ +\lim_{x \to 0} \frac{f(x)}{x^2} = \lim_{x \to 0} \frac{1 + bx^2 - \cos x}{x^2} = b + \lim_{x \to 0} \frac{1 - \cos x}{x^2} = b + \frac{1}{2} = 1, +$$ +因此$b = \frac{1}{2}$。 + +(2) +由$\cos x$的麦克劳林展开： +$$ +\cos x = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4 + o(x^4). +$$ +代入$f(x)$： +$$ +f(x) = 1 + \frac{1}{2}x^2 - \left( 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4 + o(x^4) \right) = x^2 - \frac{1}{24}x^4 + o(x^4). +$$ +于是 +$$ +\lim_{x \to 0} \frac{f(x) - x^2}{x^4} = \lim_{x \to 0} \frac{-\frac{1}{24}x^4 + o(x^4)}{x^4} = -\frac{1}{24}. +$$ + +--- + + +9.设$f(x)$在$[0,1]$上连续，在$(0,1)$上可导,且 $f(0)=0, f(1)=1$。试证： +（1）在$(0,1)$内存在不同的$\xi, \eta$使$f'(\xi)f'(\eta)=1$； +（2）对任意给定的正数$a, b$，在$(0,1)$内存在不同的$\xi, \eta$使$\frac{a}{f'(\xi)}+\frac{b}{f'(\eta)}=a+b$。 + +**分析** +（1）只需将$[0,1]$分成两个区间，使$f(x)$在两个区间各用一次微分中值定理。设分点为$x_0 \in (0,1)$，由 +$$ +f(x_0)-f(0)=f'(\xi)(x_0-0),\quad f(1)-f(x_0)=f'(\eta)(1-x_0) \quad (0<\xi<x_0<\eta<1), +$$ +得$f'(\xi)=\frac{f(x_0)}{x_0}$，$f'(\eta)=\frac{1-f(x_0)}{1-x_0}$，则 +$$ +f'(\xi)f'(\eta)=1 \Leftrightarrow \frac{f(x_0)}{x_0} \cdot \frac{1-f(x_0)}{1-x_0} = 1. +$$ +等价于$x_0$是方程$f(x)[1-f(x)] = x(1-x)$的根。取$x_0$满足$f(x_0)=1-x_0$即可。 + +**证明**： +（1）令$F(x)=f(x)-1+x$，则$F(x)$在$[0,1]$上连续，且$F(0)=-1<0$，$F(1)=1>0$。由介值定理知，存在$x_0 \in (0,1)$使$F(x_0)=0$，即$f(x_0)=1-x_0$。 +在$[0,x_0]$和$[x_0,1]$上分别应用拉格朗日中值定理，存在$\xi \in (0,x_0)$，$\eta \in (x_0,1)$，使得 +$$f'(\xi)=\frac{f(x_0)-f(0)}{x_0-0},\quad f'(\eta)=\frac{f(1)-f(x_0)}{1-x_0}.$$ +于是 +$$f'(\xi)f'(\eta)=\frac{f(x_0)}{x_0} \cdot \frac{1-f(x_0)}{1-x_0} = \frac{1-x_0}{x_0} \cdot \frac{x_0}{1-x_0} = 1.$$ （2） +给定正数$a, b$，令$c = \frac{a}{a+b}$，则$0<c<1$。由于$f(x)$在$[0,1]$上连续，且$f(0)=0$, $f(1)=1$，由介值定理，存在$x_1 \in (0,1)$使得$f(x_1) = c = \frac{a}{a+b}$。 +在区间$[0, x_1]$和$[x_1, 1]$上分别应用拉格朗日中值定理，存在$\xi \in (0, x_1)$, $\eta \in (x_1, 1)$，使得 +$$f'(\xi) = \frac{f(x_1) - f(0)}{x_1 - 0} = \frac{f(x_1)}{x_1}, \quad +f'(\eta) = \frac{f(1) - f(x_1)}{1 - x_1} = \frac{1 - f(x_1)}{1 - x_1}.$$ +于是 +$$\frac{a}{f'(\xi)} + \frac{b}{f'(\eta)} = a \cdot \frac{x_1}{f(x_1)} + b \cdot \frac{1 - x_1}{1 - f(x_1)}.$$ +代入$f(x_1) = \frac{a}{a+b}$, $1 - f(x_1) = \frac{b}{a+b}$，得 +$$\frac{a}{f'(\xi)} + \frac{b}{f'(\eta)} = a \cdot \frac{x_1}{a/(a+b)} + b \cdot \frac{1 - x_1}{b/(a+b)} = (a+b)x_1 + (a+b)(1 - x_1) = a+b.$$ +故命题得证。 + + + +10.讨论级数 $$\sum_{n=2}^{\infty} \frac{(-1)^n}{[n+(-1)^n]^p}$$ ( $p>0$ )的敛散性。 + +**补充整理：** + +1. 若两级数均收敛，则其和也收敛；若两个都绝对收敛，则和也绝对收敛；若一个绝对收敛一个条件收敛，则和条件收敛。 + +2. 若两级数仅一个收敛，则其和是发散的。 + +**级数添加与去掉括号的敛散性有以下结论：** + +1. 收敛级数任意添加括号也收敛。 +2. 若收敛级数去掉括号后的通项仍以0为极限，则去掉括号后的级数也收敛，且和不变。 + +**去括号情况的证明** + +设 $(a_1 + \cdots + a_{n_1}) + (a_{n_1+1} + \cdots + a_{n_2}) + \cdots + (a_{n_{k-1}+1} + \cdots + a_{n_k}) + \cdots$ 收敛于 $S$，且 $\lim\limits_{n \to \infty} a_n = 0$。记该级数的部分和为 $T_k$，$\sum_{n=1}^{\infty} a_n$ 的部分和为 $S_n$，则 $T_k = S_{n_k}$，$\lim\limits_{k \to \infty} S_{n_k} = \lim\limits_{k \to \infty} T_k = S$。 + +由于 $\lim\limits_{n \to \infty} a_n = 0$，对 $\forall i \in \{1, 2, \cdots, n_{k+1} - n_k - 1\}$，有 + +$\lim\limits_{k \to \infty} S_{n_k+i} = \lim\limits_{k \to \infty} ( T_k + a_{n_k+1} + \cdots + a_{n_k+i} ) = S$ + +所以 $\lim\limits_{n \to \infty} S_n = S$，即 $\sum\limits_{n=1}^{\infty} a_n$ 也收敛于 $S$。 + +**分析** 这是交错级数，且通项趋于0，但通项不单调，不适用莱布尼茨准则。可考虑用添加括号的方式来证明。也可采用交换相邻两项顺序的方式使通项满足单调性。 + +**解**： +$$|a_n| = \left| \frac{(-1)^n}{[n+(-1)^n]^p} \right| = \frac{1}{n^p} \cdot \frac{1}{\left[ 1 + \frac{(-1)^n}{n} \right]^p} \sim \frac{1}{n^p}$$ ($n \to \infty$) + +当 $p>1$ 时，级数绝对收敛；当 $0<p<1$ 时，级数不绝对收敛。 +下面讨论 $0<p\leq 1$ 时，级数的收敛性。 +首先，级数的通项 $a_n \to 0$。 + +**方法1** 将原级数按如下方式添加括号 +$$\left( \frac{1}{3^p} - \frac{1}{2^p} \right) + \left( \frac{1}{5^p} - \frac{1}{4^p} \right) + \cdots + \left( \frac{1}{(2n+1)^p} - \frac{1}{(2n)^p} \right) + \cdots$$ + +记 $b_n = \frac{1}{(2n+1)^p} - \frac{1}{(2n)^p}$，则 $b_n < 0$，$\sum_{n=2}^{\infty} (-b_n)$是正项级数。由于 +$$-b_n = \frac{1}{(2n)^p} - \frac{1}{(2n+1)^p} = \frac{1}{(2n+1)^p} \left[ \left( 1 + \frac{1}{2n} \right)^p - 1 \right] \sim \frac{1}{(2n+1)^p} \cdot \frac{p}{2n} \sim \frac{p}{(2n)^{p+1}}$$ + +而 $p+1>1$，所以 $\sum_{n=2}^{\infty} (-b_n)$ 收敛，由上面补充中去括号的讨论知，原级数收敛。 + +**方法2** 同样考虑方法1中的级数 $\sum\limits_{n=2}^{\infty} (-b_n)$，其部分和为 +$$ +\begin{aligned} +S_n &= \left( \frac{1}{2^p} - \frac{1}{3^p} \right) + \left( \frac{1}{4^p} - \frac{1}{5^p} \right) + \cdots + \left( \frac{1}{(2n)^p} - \frac{1}{(2n+1)^p} \right) \\ +&= \frac{1}{2^p} \left( \frac{1}{3^p} - \frac{1}{4^p} \right) - \left( \frac{1}{5^p} - \frac{1}{6^p} \right) - \cdots - \left( \frac{1}{(2n-1)^p} - \frac{1}{(2n)^p} \right) - \frac{1}{(2n+1)^p} < \frac{1}{2^p} +\end{aligned} +$$ +正项级数部分和数列有界，级数收敛，从而原级数收敛。 + +**方法3** 原级数是 +$$\frac{1}{3^p} - \frac{1}{2^p} + \frac{1}{5^p} - \frac{1}{4^p} + \cdots + \frac{1}{(2n+1)^p} - \frac{1}{(2n)^p} + \cdots$$ +奇偶项互换后的新级数为 +$$\frac{1}{2^p} - \frac{1}{3^p} + \frac{1}{4^p} - \frac{1}{5^p} + \cdots + \frac{1}{(2n)^p} - \frac{1}{(2n+1)^p} + \cdots$$ + +记 $c_n = \frac{1}{n^p}$，该级数为 $\sum\limits_{n=2}^{\infty}(-1)^{n-1}c_n$，由于 $c_n$ 单减趋于0，由莱布尼茨判别法知，该交错级数收敛，从而原级数收敛。 + +**评注** “方法3” 用到了收敛级数的性质：收敛级数交换相邻两项的位置后的级数仍收敛，且和不变。 + +证明如下： +设 $a_1 + a_2 + a_3 + a_4 + \cdots + a_{2n-1} + a_{2n} + \cdots$ 收敛于 $S$，其部分和为 $S_n$。交换相邻两项的位置后的级数为 $a_2 + a_1 + a_4 + a_3 + \cdots + a_{2n} + a_{2n-1} + \cdots$，其部分和为 $T_n$，则 +$$T_{2n} = S_{2n} \Rightarrow \lim_{n \to \infty} T_{2n} = \lim_{n \to \infty} S_{2n} = S, \quad \lim_{n \to \infty} T_{2n+1} = \lim_{n \to \infty} T_{2n} + \lim_{n \to \infty} a_{2n+2} = S,$$ +所以 $\lim\limits_{n \to \infty} T_n = S$。 + +11.(10分) + +(1)证明： 对任意的正整数 $n$，方程 +$$ x^n + n^2 x - 1 = 0 $$ +有唯一正实根（记为 $x_n$）。 + +**证明：** +令 $f_n(x) = x^n + n^2x - 1$ 。显然 $f_n(x)$ 在 $[0,1]$ 上连续，且 +$$ f_n(0) = -1 < 0, \quad f_n(1) = n^2 > 0, $$ +由闭区间上连续函数的零值定理可知，至少存在一点 $\xi \in (0,1)$，使得 $f_n(\xi) = 0$，即 +$$ x^n + n^2x - 1 = 0 $$ +至少有一个正实根。 + +又 +$$ f'_n(x) = nx^{n-1} + n^2, $$ +易知当 $x > 0$ 时，$f'_n(x) > 0$，故函数 $f_n(x)$ 在 $(0,+\infty)$ 内严格单调增加，因此 $f_n(x)$ 在 $(0,+\infty)$ 内至多只有一个零点。综上，函数 $f_n(x)$ 在$(0,+\infty)$ 内有唯一零点，即方程 $x^n + n^2x - 1 = 0$ 有唯一正实根。 + +--- + + (2) 证明： 级数 +$$ \sum_{n=1}^\infty x_n $$ +收敛，且其和不超过 2。 + +**证明：** +记方程$x^n + n^2 x - 1 = 0$ 的唯一正实根为 $x_n$，则 $x_n^n + n^2 x_n - 1 = 0$，故 +$$ 0 < x_n = \frac{1}{n^2} - \frac{x_n^n}{n^2} < \frac{1}{n^2}. $$ +根据比较判别法，由于级数 $\sum\limits_{n=1}^{\infty} \frac{1}{n^2}$ 收敛，故级数$\sum\limits_{n=1}^{\infty} x_n$收敛。 + +记$\sum\limits_{n=1}^{\infty} \frac{1}{n^2}$的前$n$ 项部分和为 $S_n$,$\sum\limits_{n=1}^{\infty} x_n$的前$n$ 项部分和为$T_n$，显然$T_n < S_n$，且 +$$\begin{aligned} +S_n &= \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2} < \frac{1}{1} + \frac{1}{1 \cdot 2} + \cdots + \frac{1}{(n-1)n} + \\&= 1 + 1 - \frac{1}{2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{n-1} - \frac{1}{n} + \\&= 2 - \frac{1}{n}<2, + \end{aligned}$$ +根据数列极限的保号性（更准确地说是保序性）， +$$ \sum_{n=1}^{\infty} x_n = \lim_{n \to \infty} T_n \leq \lim_{n \to \infty} S_n \leq 2. $$ \ No newline at end of file diff --git a/编写小组/试卷/1231线性代数考试卷.md b/编写小组/试卷/1231线性代数考试卷.md new file mode 100644 index 0000000..ee4547a --- /dev/null +++ b/编写小组/试卷/1231线性代数考试卷.md @@ -0,0 +1,216 @@ +--- +tags: + - 编写小组 +--- +# 线性代数适应性调研 +## 一、选择题，共六道，每题3分，共18分 + +1. 设 $A$ 为 $n$ 阶对称矩阵，$B$ 为 $n$ 阶反对称矩阵，下列矩阵中为反对称矩阵的是 + (A) $AB - BA$; + (B) $AB + BA$; + (C) $BAB$; + (D) $(AB)^2$. + +2.  设 $e_1, e_2$ 和 $\varepsilon_1, \varepsilon_2$ 是线性空间 $\mathbb{R}^2$ 的两组基，并且已知关系式 $\varepsilon_1 = e_1 + 5e_2,\ \varepsilon_2 = e_2,$ 则由基 $\varepsilon_1, \varepsilon_2$ 到基 $e_1, e_2$ 到基的过渡矩阵是 + (A) $\begin{bmatrix}0 & -1 \\ -6 & 0\end{bmatrix}$ + (B) $\begin{bmatrix}-1 & 0 \\5 & -1\end{bmatrix}$ + (C) $\begin{bmatrix}1 & 0 \\-5 & -1\end{bmatrix}$ + (D) $\begin{bmatrix}1 & 0 \\-5 & 1\end{bmatrix}$ + +3. 设向量组 $\alpha_1 = (0, 0, c_1)^T,\quad \alpha_2 = (0, 1, c_2)^T,\quad \alpha_3 = (1, -1, c_3)^T,\quad \alpha_4 = (-1, 1, c_4)^T,$ + 其中 $c_1, c_2, c_3, c_4$ 为任意常数，则下列向量组线性相关的是 + (A) $\alpha_1, \alpha_2, \alpha_3$; + (B) $\alpha_1, \alpha_2, \alpha_4$; + (C) $\alpha_1, \alpha_3, \alpha_4$; + (D) $\alpha_2, \alpha_3, \alpha_4$. + +4. 设 $A, B$ 为 $n$ 阶矩阵，则 + (A) $\text{rank}[A \ AB] = \text{rank} A$; + (B) $\text{rank}[A \ BA] = \text{rank} A$; + (C) $\text{rank}[A \ B] = \max\{\text{rank} A, \text{rank} B\}$; + (D) $\text{rank}[A \ B] = \text{rank}[A^T \ B^T]$. + +5. 设 $A$ 可逆，将 $A$ 的第一列加上第二列的 2 倍得到 $B$，则 $A^*$ 与 $B^*$ 满足 + (A) 将 $A^*$ 的第一列加上第二列的 2 倍得到 $B^*$; + (B) 将 $A^*$ 的第一行加上第二行的 2 倍得到 $B^*$; + (C) 将 $A^*$ 的第二列加上第一列的 $(-2)$ 倍得到 $B^*$; + (D) 将 $A^*$ 的第二行加上第一行的 $(-2)$ 倍得到 $B^*$. + +6. 已知方程组$\quad\begin{cases}x_1 + 2x_2 + 3x_3 = 0, \\2x_1 + 3x_2 + 5x_3 = 0, \\x_1 + x_2 + ax_3 = 0,\end{cases}$与$\text{(II)} \quad\begin{cases}x_1 + bx_2 + cx_3 = 0, \\2x_1 + b^2x_2 + (c+1)x_3 = 0\end{cases}$同解，则 + (A) $a = 1, b = 0, c = 1$; + (B) $a = 1, b = 1, c = 2$; + (C) $a = 2, b = 0, c = 1$; + (D) $a = 2, b = 1, c = 2$. + +## 二、填空题，共六道，每题3分，共18分 + +7. 已知向量 $\alpha_1 = (1,0,-1,0)^T$，$\alpha_2 = (1,1,-1,-1)^T$，$\alpha_3 = (-1,0,1,1)^T$，则向量 $\alpha_1 + 2\alpha_2$ 与 $2\alpha_1 + \alpha_3$ 的内积$\langle \alpha_1 + 2\alpha_2,\, 2\alpha_1 + \alpha_3 \rangle = \underline{\qquad\qquad}.$ + +8. 设2阶矩阵A=$\begin{bmatrix}3&-1\\-9&3\end{bmatrix}$，n为正整数，则$A^n=\underline{\qquad\qquad\quad\quad}$。 + +9. 若向量组$\alpha_1 = (1,0,1)^T,\quad \alpha_2 = (0,1,1)^T,\quad \alpha_3 = (1,3,5)^T$不能由向量组$\beta_1 = (1,1,1)^T,\quad\beta_2 = (1,2,3)^T,\quad\beta_3 = (3,4,a)^T$线性表示，则$a = \underline{\qquad\qquad}.$ + +10. 设矩阵$A = \begin{bmatrix}1 & a_1 & a_1^2 & a_1^3 \\1 & a_2 & a_2^2 & a_2^3 \\1 & a_3 & a_3^2 & a_3^3 \\1 & a_4 & a_4^2 & a_4^3\end{bmatrix},x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix},b = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix},$其中常数 $a_1, a_2, a_3, a_4$ 互不相等，则线性方程组 $Ax = b$ 的解为$\underline{\qquad\qquad\qquad\qquad}.$ + +11. 矩阵(陈峰华原创题)$$A=\begin{bmatrix} +0 & 0 & 0 & \cdots & 0 & 1 & 1 & \cdots & 1 & 1 & 1 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 1 & \cdots & 1 & 1 & 1 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 1 & 1 & 1 \\ +\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 1 & 1 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 1 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ +\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 +\end{bmatrix}_{(mn) \times (mn)}$$其中第一行有$m$个$0$.若$A^k=O$，则 $k$ 的最小值为$\underline{\qquad\qquad\qquad\qquad}.$ + +12. 设 $A, B$ 均为 $n$ 阶方阵，满足$\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} = \text{rank}B,$ 且方程 $XA = B$ 有解。若 $\operatorname{rank} A = k$，则$\text{rank} \begin{bmatrix} B & O \\ A & E \end{bmatrix} =\underline{\hspace{3cm}}.$ + +## 三、解答题，共五道，共64分 + +13. （20 分）计算 下面的两个$n$阶行列式 +$$ + K_n = \begin{vmatrix} + 1 & 2 & 3 & \cdots & n-1 & n \\ + 2 & 1 & 2 & \cdots & n-2 & n-1 \\ + 3 & 2 & 1 & \cdots & n-3 & n-2 \\ + \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ + n-1 & n-2 & n-3 & \cdots & 1 & 2 \\ + n & n-1 & n-2 & \cdots & 2 & 1 + \end{vmatrix}. + $$ +$$ + M_n =\begin{vmatrix} +1+x_1 & 1+x_1^2 & \cdots & 1+x_1^n \\ +1+x_2 & 1+x_2^2 & \cdots & 1+x_2^n \\ +\vdots & \vdots & \ddots & \vdots \\ +1+x_n & 1+x_n^2 & \cdots & 1+x_n^n +\end{vmatrix} +$$ +```text + + + + + + + + + + + + + + + + + + + + + +``` +14. （10 分）设$A=\begin{bmatrix}1 & -1 & 0 & -1 \\ 1 & 1 & 0 & 3 \\ 2 & 1 & 2 & 6\end{bmatrix},B=\begin{bmatrix}1 & 0 & 1 & 2 \\ 1 & -1 & a & a-1 \\ 2 & -3 & 2 & -2\end{bmatrix}$,向量$\alpha=\begin{bmatrix}0\\2\\3\end{bmatrix},\beta=\begin{bmatrix}1\\0\\-1\end{bmatrix}$. + (1)证明：方程组 $Ax=\alpha$ 的解均为方程组 $Bx=\beta$ 的解； + (2)若方程组 $Ax=\alpha$ 与方程组 $Bx=\beta$ 不同解，求 $a$ 的值. +```text + + + + + + + + + + + + + + + + + + + + + +``` +15. （10 分）设 $\alpha_1 = (1,0,-1)^T,\quad \alpha_2 = (2,1,1)^T,\quad \alpha_3 = (1,1,1)^T$和$\beta_1 = (0,1,1)^T,\quad \beta_2 = (-1,1,0)^T,\quad \beta_3 = (0,2,1)^T$是 $\mathbb{R}^3$ 的两组基，求向量 $u = \alpha_1 + 2\alpha_2 - 3\alpha_3$在基 $\beta_1, \beta_2, \beta_3$ 下的坐标。 +```text + + + + + + + + + + + + + + + + + + + + + +``` +16. （12 分）设 $n$ 阶方阵 $A, B$ 满足 $AB = A + B$。 + （1）证明 $A - E$ 可逆； + （2）证明 $AB = BA$； + （3）证明 $\mathrm{rank}(A) = \mathrm{rank}(B)$； + （4）若矩阵$B = \begin{bmatrix}1 & -3 & 0 \\2 & 1 & 0 \\0 & 0 & 2\end{bmatrix}$，求矩阵 $A$。 +```text + + + + + + + + + + + + + + + + + + + + + +``` +17. （12 分）设矩阵$A=\begin{bmatrix}1&2&1&2\\0&1&t&t\\1&t&0&1\end{bmatrix}$，齐次线性方程组 $Ax=0$ 的基础解系中含有两个解向量，求 $Ax=0$ 的通解。 +```text + + + + + + + + + + + + + + + + + + + + + +``` \ No newline at end of file diff --git a/编写小组/试卷/1231线性代数考试卷（解析版）.md b/编写小组/试卷/1231线性代数考试卷（解析版）.md new file mode 100644 index 0000000..f9ef83c --- /dev/null +++ b/编写小组/试卷/1231线性代数考试卷（解析版）.md @@ -0,0 +1,617 @@ +--- +tags: + - 编写小组 +--- +# 线性代数适应性调研 +## 一、选择题，共六道，每题3分，共18分 + +1. 设 $A$ 为 $n$ 阶对称矩阵，$B$ 为 $n$ 阶反对称矩阵，下列矩阵中为反对称矩阵的是 + (A) $AB - BA$; + (B) $AB + BA$; + (C) $BAB$; + (D) $(AB)^2$. + +>答案：**B** +>解析：$A$ 为 $n$ 阶对称矩阵 $\Rightarrow$ $A^T=A$ ， $B$ 为 $n$ 阶反称矩阵 $\Rightarrow$ $B^T=-B$ ； +>逐个选项分析： +>(A) $(AB-BA)^T=(AB)^T-(BA)^T=B^TA^T-A^TB^T=-BA+AB$，故 $AB-BA$ 是对称矩阵 +>(B) $(AB+BA)^T=(AB)^T+(BA)^T=B^TA^T+A^TB^T=-BA-AB$，故 $(AB+BA)$ 是反称矩阵 +>(C) $(BAB)^T=B^TA^TB^T=BAB$，故 $BAB$ 是对称矩阵 +>(D) $((AB)^2)^T=(ABAB)^T=B^TA^TB^TA^T=(BA)^2$，故 $(AB)^2$ 不一定有对称性 + +2.  设 $e_1, e_2$ 和 $\varepsilon_1, \varepsilon_2$ 是线性空间 $\mathbb{R}^2$ 的两组基，并且已知关系式 $\varepsilon_1 = e_1 + 5e_2,\ \varepsilon_2 = e_2,$ 则由基 $\varepsilon_1, \varepsilon_2$ 到基 $e_1, e_2$ 到基的过渡矩阵是 + (A) $\begin{bmatrix}0 & -1 \\ -6 & 0\end{bmatrix}$ + (B) $\begin{bmatrix}-1 & 0 \\5 & -1\end{bmatrix}$ + (C) $\begin{bmatrix}1 & 0 \\-5 & -1\end{bmatrix}$ + (D) $\begin{bmatrix}1 & 0 \\-5 & 1\end{bmatrix}$ + +>答案：**D** +>解析：$\varepsilon_1 = e_1 + 5e_2,\quad \varepsilon_2 = 0e_1 + 1e_2.$ +>把它们按列排成矩阵形式：$[\varepsilon_1, \varepsilon_2] = [e_1, e_2] \begin{bmatrix}1 & 0 \\5 & 1\end{bmatrix}.$ +>而由基 $e_1, e_2$ 到基 $\varepsilon_1, \varepsilon_2$ 的过渡矩阵 $T$ 应该满足$\begin{bmatrix}e_1&e_2\end{bmatrix}=\begin{bmatrix}\varepsilon_1&\varepsilon_2\end{bmatrix}T$，即： +>$\begin{bmatrix}e_1&e_2\end{bmatrix}=\begin{bmatrix}e_1&e_2\end{bmatrix}\begin{bmatrix}1 & 0 \\ 5 & 1\end{bmatrix}T$，所以 $T=\begin{bmatrix}1 & 0 \\ 5 & 1\end{bmatrix}^{-1}=\begin{bmatrix}1 & 0 \\ -5 & 1\end{bmatrix}$ + +3. 设向量组 $\alpha_1 = (0, 0, c_1)^T,\quad \alpha_2 = (0, 1, c_2)^T,\quad \alpha_3 = (1, -1, c_3)^T,\quad \alpha_4 = (-1, 1, c_4)^T,$ + 其中 $c_1, c_2, c_3, c_4$ 为任意常数，则下列向量组线性相关的是 + (A) $\alpha_1, \alpha_2, \alpha_3$; + (B) $\alpha_1, \alpha_2, \alpha_4$; + (C) $\alpha_1, \alpha_3, \alpha_4$; + (D) $\alpha_2, \alpha_3, \alpha_4$. + +>答案：**C** +>解析：逐选项分析： +>A. $\begin{bmatrix}\alpha_1&\alpha_2&\alpha_3\end{bmatrix}=\begin{bmatrix}0&0&1\\0&1&-1\\c_1&c_2&c_3\end{bmatrix}$，当$c_1\ne 0$时，作初等列变换，$\text{rank}\begin{bmatrix}\alpha_1&\alpha_2&\alpha_3\end{bmatrix}=\text{rank}\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}=3$，故线性无关； +>B. $\begin{bmatrix}\alpha_1&\alpha_2&\alpha_4\end{bmatrix}=\begin{bmatrix}0&0&-1\\0&1&1\\c_1&c_2&c_4\end{bmatrix}$，当$c_1\ne 0$时，作初等列变换，$\text{rank}\begin{bmatrix}0&0&-1\\0&1&1\\c_1&c_2&c_4\end{bmatrix}=\text{rank}\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}=3$，故线性无关； +>C. $\begin{bmatrix}\alpha_1&\alpha_3&\alpha_4\end{bmatrix}=\begin{bmatrix}0&1&-1\\0&-1&1\\c_1&c_3&c_4\end{bmatrix}\cong\begin{bmatrix}0&0&-1\\0&0&1\\c_1&c_3+c_4&c_4\end{bmatrix}\cong\begin{bmatrix}0&0&0\\0&0&1\\c_1&c_3+c_4&c_4\end{bmatrix}$，故其必定线性相关； +>D. $\begin{bmatrix}\alpha_1&\alpha_2&\alpha_4\end{bmatrix}=\begin{bmatrix}0&1&-1\\1&-1&1\\c_2&c_3&c_4\end{bmatrix}\cong\begin{bmatrix}0&0&-1\\1&0&1\\c_2&c_3+c_4&c_4\end{bmatrix}$，当$c_3+c_4\ne 0$时，作初等列变换，$\text{rank}\begin{bmatrix}0&0&-1\\1&0&1\\c_2&c_3+c_4&c_4\end{bmatrix}=\text{rank}\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}=3$，故线性无关； + +4. 设 $A, B$ 为 $n$ 阶矩阵，则 + (A) $\text{rank}[A \ AB] = \text{rank} A$; + (B) $\text{rank}[A \ BA] = \text{rank} A$; + (C) $\text{rank}[A \ B] = \max\{\text{rank} A, \text{rank} B\}$; + (D) $\text{rank}[A \ B] = \text{rank}[A^T \ B^T]$. + +>答案：**A** +>重点：$AB$ 的每一列都是 $A$ 的列向量的线性组合，因此，$\text{rank}[A\quad AB]=\text{rank}A$ ，因为 $AB$ 的列都在 $A$ 的列空间中 + +5. 设 $A$ 可逆，将 $A$ 的第一列加上第二列的 2 倍得到 $B$，则 $A^*$ 与 $B^*$ 满足 + (A) 将 $A^*$ 的第一列加上第二列的 2 倍得到 $B^*$; + (B) 将 $A^*$ 的第一行加上第二行的 2 倍得到 $B^*$; + (C) 将 $A^*$ 的第二列加上第一列的 $(-2)$ 倍得到 $B^*$; + (D) 将 $A^*$ 的第二行加上第一行的 $(-2)$ 倍得到 $B^*$. + +>答案：**D** +>解析：$B=AP(2,1(2))$，故$B^{-1}=P(2,1(-2))A^{-1}$， +>又$|A|=|B|$，$A^*=|A|A^{-1}$， +>可得$B^*=|B|B^{-1}=|A|P(2,1(-2))A^{-1}=P(2,1(-2))A^*$，根据初等矩阵的性质，$B^*$应该是$A^*$的第一行乘以$(-2)$加到第二行的结果。 + +6. 已知方程组$\quad\begin{cases}x_1 + 2x_2 + 3x_3 = 0, \\2x_1 + 3x_2 + 5x_3 = 0, \\x_1 + x_2 + ax_3 = 0,\end{cases}$与$\text{(II)} \quad\begin{cases}x_1 + bx_2 + cx_3 = 0, \\2x_1 + b^2x_2 + (c+1)x_3 = 0\end{cases}$同解，则 + (A) $a = 1, b = 0, c = 1$; + (B) $a = 1, b = 1, c = 2$; + (C) $a = 2, b = 0, c = 1$; + (D) $a = 2, b = 1, c = 2$. + +>答案：**D** +>解析：$\text{(I)}:\begin{bmatrix}1&2&3\\2&3&5\\1&1&a\end{bmatrix}x=0$，$\text{(II)}: \begin{bmatrix}1&b&c\\2&b^2&c+1\end{bmatrix}x=0$，对方程做初等行变换： +>$\text{(I)}:\begin{bmatrix}1&0&1\\0&1&1\\0&0&a-2\end{bmatrix}x=0$，$\text{(II)}: \begin{bmatrix}1&b&c\\0&b^2-2b&1-c\end{bmatrix}x=0$，记系数矩阵分别为$A,B$ +>因为方程(I),(II)同解，所以$\text{rank}A=\text{rank}B$，而$\text{rank}A\ge 2,\text{rank}B\le 2$，故$\text{rank}A=\text{rank}B=2$，故$a-2=0 \to a=2$；所以方程组(I)的解为$x=k(1,1,-1)^T$； +>令$k=1,x=(1,1,-1)^T$代入方程组(II)得$\begin{bmatrix}1&b&c\\0&b^2-2b&1-c\end{bmatrix}\begin{bmatrix}1\\1\\-1\end{bmatrix}=\begin{bmatrix}1+b-c\\b^2-2b+c-1\end{bmatrix}=0$，解得$\begin{cases}b=0\\c=1\end{cases}$或$\begin{cases}b=1\\c=2\end{cases}$；然而，当$\begin{cases}b=0\\c=1\end{cases}$时，$\begin{bmatrix}1&b&c\\0&b^2-2b&1-c\end{bmatrix}=\begin{bmatrix}1&0&1\\0&0&0\end{bmatrix}$，不符合$\text{rank}B=2$的约束，故舍去； +>综上，$\begin{cases}a=2\\b=1\\c=2\end{cases}$ + +## 二、填空题，共六道，每题3分，共18分 + +7. 已知向量 $\alpha_1 = (1,0,-1,0)^T$，$\alpha_2 = (1,1,-1,-1)^T$，$\alpha_3 = (-1,0,1,1)^T$，则向量 $\alpha_1 + 2\alpha_2$ 与 $2\alpha_1 + \alpha_3$ 的内积$\langle \alpha_1 + 2\alpha_2,\, 2\alpha_1 + \alpha_3 \rangle = \underline{\qquad\qquad}.$ + +【答】4 + +【解析】由内积的性质可知 $$ \begin{aligned} \langle \alpha_1 + 2\alpha_2, 2\alpha_1 + \alpha_3 \rangle &= \langle \alpha_1, 2\alpha_1 + \alpha_3 \rangle + \langle 2\alpha_2, 2\alpha_1 + \alpha_3 \rangle \\[1em] &= \langle \alpha_1, 2\alpha_1 \rangle + \langle \alpha_1, \alpha_3 \rangle + \langle 2\alpha_2, 2\alpha_1 \rangle + \langle 2\alpha_2, \alpha_3 \rangle \\[1em] &= 2\langle \alpha_1, \alpha_1 \rangle + \langle \alpha_1, \alpha_3 \rangle + 4\langle \alpha_2, \alpha_1 \rangle + 2\langle \alpha_2, \alpha_3 \rangle, \end{aligned} $$ 再由题意，可知 $$ \langle \alpha_1, \alpha_1 \rangle = 2,\quad \langle \alpha_1, \alpha_3 \rangle = -2,\quad \langle \alpha_2, \alpha_1 \rangle = 2,\quad \langle \alpha_2, \alpha_3 \rangle = -3. $$ 从而 $$ \langle \alpha_1 + 2\alpha_2, 2\alpha_1 + \alpha_3 \rangle = 4. $$ + +--- + +8. 设2阶矩阵A=$\begin{bmatrix}3&-1\\-9&3\end{bmatrix}$，n为正整数，则$A^n=\underline{\quad\quad}$。 + +解析： +先计算$A^2$： + +$$\begin{align}A^2&=\begin{bmatrix}3&-1\\-9&3\end{bmatrix}\begin{bmatrix}3&-1\\-9&3\end{bmatrix}\\[1em] +&= \begin{bmatrix}3\times3 + (-1)\times(-9)&3\times(-1) + (-1)\times3\\-9\times3 + 3\times(-9)&-9\times(-1) + 3\times3\end{bmatrix}\\[1em] +&=\begin{bmatrix}18&-6\\-54&18\end{bmatrix}\\[1em] +&= 6\begin{bmatrix}3&-1\\-9&3\end{bmatrix} \\[1em] &= 6A +\end{align} +$$ + +由此递推：$A^3 = A^2 \cdot A = 6A \cdot A = 6A^2 = 6\times6A = 6^2A$，归纳可得当$n \geq 1$时，$A^n = 6^{n-1}A$ + + + +将A代入得： +$$A^n = 6^{n-1}\begin{bmatrix}3&-1\\-9&3\end{bmatrix} $$ + +9. 若向量组$\alpha_1 = (1,0,1)^T,\quad \alpha_2 = (0,1,1)^T,\quad \alpha_3 =(1,3,5)^T$不能由向量组$\beta_1 = (1,1,1)^T,\quad \beta_2 = (1,2,3)^T,\quad \beta_3 =(3,4,a)^T$线性表示，则$a = \underline{\qquad\qquad}.$ + +--- + +【答】5. + +【解析】（方法一）依题意知 $\beta_1, \beta_2, \beta_3$ 线性相关，否则，若 $\beta_1, \beta_2, \beta_3$ 线性无关，则 $\beta_1, \beta_2, \beta_3$ 为向量空间 $R^3$ 的一组基，$\alpha_1, \alpha_2, \alpha_3$ 能由 $\beta_1, \beta_2, \beta_3$ 线性表示，矛盾。记 $B = [\beta_1, \beta_2, \beta_3]$，则 $|B| = 0$，解得 $a = 5$。 + +（方法二）令 +$$A = [\beta_1, \beta_2, \beta_3, \alpha_1, \alpha_2, \alpha_3] = \begin{bmatrix} 1 & 1 & 3 & 1 & 0 & 1 \\ 1 & 2 & 4 & 0 & 1 & 3 \\ 1 & 3 & a & 1 & 1 & 5 \end{bmatrix}$$ +对其进行初等行变换 + +$$A \to \begin{bmatrix} 1 & 1 & 3 & 1 & 0 & 1 \\ 0 & 1 & 1 & -1 & 1 & 2 \\ 0 & 2 & a-3 & 0 & 1 & 4 \end{bmatrix} \to \begin{bmatrix} 1 & 1 & 3 & 1 & 0 & 1 \\ 0 & 1 & 1 & -1 & 1 & 2 \\ 0 & 0 & a-5 & 2 & -1 & 0 \end{bmatrix} = \begin{bmatrix} \gamma_1 & \gamma_2 & \gamma_3 & \gamma_4 & \gamma_5 & \gamma_6 \end{bmatrix}$$ + +由 $\alpha_1, \alpha_2, \alpha_3$ 不能由 $\beta_1, \beta_2, \beta_3$ 线性表示可知 $\gamma_4, \gamma_5, \gamma_6$ 不能由 $\gamma_1, \gamma_2, \gamma_3$ 线性表示，从而 $a = 5$。 + +--- + +10. 设矩阵 + $$ + A = \begin{bmatrix} + 1 & a_1 & a_1^2 & a_1^3 \\ + 1 & a_2 & a_2^2 & a_2^3 \\ + 1 & a_3 & a_3^2 & a_3^3 \\ + 1 & a_4 & a_4^2 & a_4^3 + \end{bmatrix},\quad + x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix},\quad + b = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, + $$ + 其中常数 $a_1, a_2, a_3, a_4$ 互不相等，则线性方程组 $Ax = b$ 的解为 + $$ + \underline{\qquad\qquad\qquad\qquad}. + $$ +--- + +**答**：$(1,0,0,0)^T$。 + +**解析**：由范德蒙行列式的性质可知 $|A| \neq 0$，从而线性方程组 $Ax = b$ 有唯一解。 + +又由 +$$ +\begin{bmatrix} +1 & a_1 & a_1^2 & a_1^3 \\ +1 & a_2 & a_2^2 & a_2^3 \\ +1 & a_3 & a_3^2 & a_3^3 \\ +1 & a_4 & a_4^2 & a_4^3 +\end{bmatrix} +\begin{bmatrix} +1 \\ +0 \\ +0 \\ +0 +\end{bmatrix} += +\begin{bmatrix} +1 \\ +1 \\ +1 \\ +1 +\end{bmatrix} +$$ +可知 $Ax = b$ 的解为 +$$ +\begin{bmatrix} +1 \\ +0 \\ +0 \\ +0 +\end{bmatrix} +$$ +--- + + +11. 矩阵(陈峰华原创题)$$A=\begin{bmatrix} +0 & 0 & 0 & \cdots & 0 & 1 & 1 & \cdots & 1 & 1 & 1 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 1 & \cdots & 1 & 1 & 1 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 1 & 1 & 1 \\ +\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 1 & 1 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 1 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ +\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ +0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 & 0 +\end{bmatrix}_{(mn) \times (mn)}$$其中第一行有$m$个$0$.若$A^k=O$，则 $k$ 的最小值为$\underline{\qquad\qquad\qquad\qquad}.$ + +--- + +解析：观察得每乘一次第一行少 $m$ 个0，故最少进行 $n$ 次即可 + +--- + +12. 设 $A, B$ 均为 $n$ 阶方阵，满足$\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} = \text{rank}B,$ 且方程 $XA = B$ 有解。若 $\operatorname{rank} A = k$，则$\text{rank} \begin{bmatrix} B & O \\ A & E \end{bmatrix} =\underline{\hspace{3cm}}.$ + +--- + +解析：类似于方程 $AX = B$ 有解的充要条件是$\text{rank} \begin{bmatrix} A & B \end{bmatrix} = \text{rank}A$，由方程 $XA = B$ 有解可知 $\text{rank} \begin{bmatrix} A \\ B \end{bmatrix} = \text{rank}B=\text{rank}A=k$，由初等变换不改变秩得$$\text{rank} \begin{bmatrix} B & O \\ A & E \end{bmatrix} =\text{rank} \begin{bmatrix} B & O \\ O & E \end{bmatrix}=n+k$$ + +--- +## 三、解答题，共五道，共64分 + +13. （20 分）计算 下面的两个$n$阶行列式 + + $$ + K_n = \begin{vmatrix} + 1 & 2 & 3 & \cdots & n-1 & n \\ + 2 & 1 & 2 & \cdots & n-2 & n-1 \\ + 3 & 2 & 1 & \cdots & n-3 & n-2 \\ + \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ + n-1 & n-2 & n-3 & \cdots & 1 & 2 \\ + n & n-1 & n-2 & \cdots & 2 & 1 + \end{vmatrix}. + $$ + + +$$ + M_n =\begin{vmatrix} +1+x_1 & 1+x_1^2 & \cdots & 1+x_1^n \\ +1+x_2 & 1+x_2^2 & \cdots & 1+x_2^n \\ +\vdots & \vdots & \ddots & \vdots \\ +1+x_n & 1+x_n^2 & \cdots & 1+x_n^n +\end{vmatrix} +$$ + +--- + +解析 +（1）$K_n$： +从第 $n-1$ 行开始，依次乘以 $(-1)$ 加到下一行，再把第 $n$ 列加到前面各列，得 + +$$ +\begin{aligned} +K_n &= +\begin{vmatrix} +1 & 2 & 3 & \cdots & n-1 & n \\ +2 & 1 & 2 & \cdots & n-2 & n-1 \\ +3 & 2 & 1 & \cdots & n-3 & n-2 \\ +\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ +n-1 & n-2 & n-3 & \cdots & 1 & 2 \\ +n & n-1 & n-2 & \cdots & 2 & 1 +\end{vmatrix} \\[4pt] +&= +\begin{vmatrix} +1 & 2 & 3 & \cdots & n-1 & n \\ +1 & -1 & -1 & \cdots & -1 & -1 \\ +1 & 1 & -1 & \cdots & -1 & -1 \\ +\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ +1 & 1 & 1 & \cdots & -1 & -1 \\ +1 & 1 & 1 & \cdots & 1 & -1 +\end{vmatrix} + +\end{aligned} +$$ + +继续化简： + +$$ +\begin{aligned} +&= +\begin{vmatrix} +n+1 & n+2 & n+3 & \cdots & 2n-1 & n \\ +0 & -2 & -2 & \cdots & -2 & -1 \\ +0 & 0 & -2 & \cdots & -2 & -1 \\ +\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ +0 & 0 & 0 & \cdots & -2 & -1 \\ +0 & 0 & 0 & \cdots & 0 & -1 +\end{vmatrix} + +\end{aligned} +$$ + +这是一个上三角行列式，因此 + +$$ +D_n = (-1)^{n-1} \cdot 2^{n-2} \cdot (n+1) + +$$ +--- +（2）加边 + + +$$ +\tilde{D} = +\begin{vmatrix} +1 & 0 & 0 & \cdots & 0 \\ +1 & 1 + x_1 & 1 + x_1^2 & \cdots & 1 + x_1^n \\ +1 & 1 + x_2 & 1 + x_2^2 & \cdots & 1 + x_2^n \\ +\vdots & \vdots & \vdots & \ddots & \vdots \\ +1 & 1 + x_n & 1 + x_n^2 & \cdots & 1 + x_n^n +\end{vmatrix} + +$$ + +$$ +\tilde{D} = +\begin{vmatrix} +1 & -1 & -1 & \cdots & -1 \\ +1 & x_1 & x_1^2 & \cdots & x_1^n \\ +1 & x_2 & x_2^2 & \cdots & x_2^n \\ +\vdots & \vdots & \vdots & \ddots & \vdots \\ +1 & x_n & x_n^2 & \cdots & x_n^n +\end{vmatrix} +$$ + +将第一行拆为 $(2,0,0,\dots,0)$ 与 $(-1,-1,\dots,-1)$ 之和： + +$$ +\tilde{D} = +\begin{vmatrix} +2 & 0 & 0 & \cdots & 0 \\ +1 & x_1 & x_1^2 & \cdots & x_1^n \\ +1 & x_2 & x_2^2 & \cdots & x_2^n \\ +\vdots & \vdots & \vdots & \ddots & \vdots \\ +1 & x_n & x_n^2 & \cdots & x_n^n +\end{vmatrix} ++ +\begin{vmatrix} +-1 & -1 & -1 & \cdots & -1 \\ +1 & x_1 & x_1^2 & \cdots & x_1^n \\ +1 & x_2 & x_2^2 & \cdots & x_2^n \\ +\vdots & \vdots & \vdots & \ddots & \vdots \\ +1 & x_n & x_n^2 & \cdots & x_n^n +\end{vmatrix} +$$ + +令左边为 $A$，右边为 $B$。 + +计算 $A$，按第一行展开： + +$$ +A = 2 \cdot +\begin{vmatrix} +x_1 & x_1^2 & \cdots & x_1^n \\ +x_2 & x_2^2 & \cdots & x_2^n \\ +\vdots & \vdots & \ddots & \vdots \\ +x_n & x_n^2 & \cdots & x_n^n +\end{vmatrix} += 2 \cdot \left( \prod_{i=1}^{n} x_i \right) \cdot +\begin{vmatrix} +1 & x_1 & \cdots & x_1^{n-1} \\ +1 & x_2 & \cdots & x_2^{n-1} \\ +\vdots & \vdots & \ddots & \vdots \\ +1 & x_n & \cdots & x_n^{n-1} +\end{vmatrix} +$$ + +右边为范德蒙德行列式： + +$$ +A = 2 \prod_{i=1}^{n} x_i \cdot \prod_{1 \leq i < j \leq n} (x_j - x_i) +$$ + +计算 $B$，提出第一行的因子 $-1$： + +$$ +B = (-1) \cdot +\begin{vmatrix} +1 & 1 & 1 & \cdots & 1 \\ +1 & x_1 & x_1^2 & \cdots & x_1^n \\ +1 & x_2 & x_2^2 & \cdots & x_2^n \\ +\vdots & \vdots & \vdots & \ddots & \vdots \\ +1 & x_n & x_n^2 & \cdots & x_n^n +\end{vmatrix} +$$ + +该行列式为 $n+1$ 阶范德蒙德行列式，变量为 $1, x_1, x_2, \dots, x_n$： + +$$ +B = (-1) \cdot \prod_{i=1}^{n} (x_i - 1) \cdot \prod_{1 \leq i < j \leq n} (x_j - x_i) +$$ + +因此： + +$$ +\tilde{D} = A + B = \left( 2 \prod_{i=1}^{n} x_i - \prod_{i=1}^{n} (x_i - 1) \right) \cdot \prod_{1 \leq i < j \leq n} (x_j - x_i) +$$ + +$$ +\boxed{\tilde{D} = \left(2\prod\limits_{i=1}^{n}x_i - \prod\limits_{i=1}^{n}(x_i-1)\right) \prod\limits_{1\leq i<j\leq n}(x_j-x_i)} +$$ + + +--- +14. 设$A=\begin{bmatrix}1 & -1 & 0 & -1 \\ 1 & 1 & 0 & 3 \\ 2 & 1 & 2 & 6\end{bmatrix},B=\begin{bmatrix}1 & 0 & 1 & 2 \\ 1 & -1 & a & a-1 \\ 2 & -3 & 2 & -2\end{bmatrix}$,向量$\alpha=\begin{bmatrix}0\\2\\3\end{bmatrix},\beta=\begin{bmatrix}1\\0\\-1\end{bmatrix}$. + + (1)证明：方程组$Ax=\alpha$的解均为方程组$Bx=\beta$的解； + (2)若方程组$Ax=\alpha$与方程组$Bx=\beta$不同解，求$a$的值. + +--- + +解析： +(1)证明：$[A\ \ \alpha] \rightarrow \begin{bmatrix}1 & 0 & 0 & 1 & 1\\0 & 1 & 0 & 2 & 1\\0 & 0 & 1 & 1 & 0\end{bmatrix}$,于是$Ax=\alpha$的通解为$$x=k\begin{bmatrix}-1\\-2\\-1\\1\end{bmatrix}+\begin{bmatrix}1\\1\\0\\0\end{bmatrix},$$把方程$Bx=\beta$还原成方程组得$$\begin{cases}x_1&+0x_2&+x_3&+2x_4&=1\\x_1&-x_2&+ax_3&+(a-1)x_4&=1\\2x_1&-3x_2&+2x_3&-2x_4&=-1\end{cases}$$把$Ax=\alpha$的解带入上方程组，显然符合，故方程组$Ax=\alpha$的解均为方程组$Bx=\beta$的解. + +(2)方程组$Bx=\beta$与方程组$Ax=\alpha$不同解，而由上一题，方程组$Ax=\alpha$的解是$Bx=\beta$的解的真子集，于是$\dim N(A)<\dim N(B),r(A)=3>r(B),r(B)\le2$.对$B$进行初等行变换得$$B\rightarrow\begin{bmatrix}1&0&1&2\\0&1&0&2\\0&0&a-1&a-1\end{bmatrix},$$于是$a=1$. + +--- + +15. （10 分）设 + $$ + \alpha_1 = (1,0,-1)^T,\quad \alpha_2 = (2,1,1)^T,\quad \alpha_3 = (1,1,1)^T + $$ + 和 + $$ + \beta_1 = (0,1,1)^T,\quad \beta_2 = (-1,1,0)^T,\quad \beta_3 = (0,2,1)^T + $$ + 是 $\mathbb{R}^3$ 的两组基，求向量 + $$ + u = \alpha_1 + 2\alpha_2 - 3\alpha_3 + $$ + 在基 $\beta_1, \beta_2, \beta_3$ 下的坐标。 + +--- + +解析： + +已知： + +$$ +A = (\alpha_1, \alpha_2, \alpha_3) = +\begin{bmatrix} +1 & 2 & 1 \\ +0 & 1 & 1 \\ +-1 & 1 & 1 +\end{bmatrix}, +$$ + +$$ +B = (\beta_1, \beta_2, \beta_3) = +\begin{bmatrix} +0 & -1 & 0 \\ +1 & 1 & 2 \\ +1 & 0 & 1 +\end{bmatrix}. +$$ + +设 $u$ 在基 $\alpha_1, \alpha_2, \alpha_3$ 下的坐标为 $x = (1, 2, -3)^T$，在基 $\beta_1, \beta_2, \beta_3$ 下的坐标为 $y$，则 + +$$ +u = (\alpha_1, \alpha_2, \alpha_3) x = (\beta_1, \beta_2, \beta_3) y, +$$ + +即 + +$$ +Ax = By. +$$ + +因为 $B$ 可逆，所以 + +$$ +y = B^{-1} A x. +$$ + +用增广矩阵求解 $y$： + +$$ +(B, Ax) = +\begin{bmatrix} +0 & -1 & 0 & \vert & 2 \\ +1 & 1 & 2 & \vert & -1 \\ +1 & 0 & 1 & \vert & -2 +\end{bmatrix} +$$ + +作行初等变换： + +$$ +\begin{aligned} +&\rightarrow +\begin{bmatrix} +1 & 0 & 1 & \vert & -2 \\ +0 & 1 & 1 & \vert & 1 \\ +0 & -1 & 0 & \vert & 2 +\end{bmatrix} \\[1em] +&\rightarrow +\begin{bmatrix} +1 & 0 & 1 & \vert & -2 \\ +0 & 1 & 1 & \vert & 1 \\ +0 & 0 & 1 & \vert & 3 +\end{bmatrix} \\[1em] +&\rightarrow +\begin{bmatrix} +1 & 0 & 0 & \vert & -5 \\ +0 & 1 & 0 & \vert & -2 \\ +0 & 0 & 1 & \vert & 3 +\end{bmatrix}. +\end{aligned} +$$ + +因此向量 + +$$ +u = \alpha_1 + 2\alpha_2 - 3\alpha_3 +$$ + +在基 $\beta_1, \beta_2, \beta_3$ 下的坐标为 + +$$ +y = (-5, -2, 3)^T. +$$ + +--- + + +16. （12 分）设 $n$ 阶方阵 $A, B$ 满足 $AB = A + B$。 + （1）证明 $A - E$ 可逆； + （2）证明 $AB = BA$； + （3）证明 $\mathrm{rank}(A) = \mathrm{rank}(B)$； + + （4）若矩阵 + $$ + B = \begin{bmatrix} + 1 & -3 & 0 \\ + 2 & 1 & 0 \\ + 0 & 0 & 2 + \end{bmatrix}, + $$ + 求矩阵 $A$。 + + +--- + + +**【解】** + +**(1)** +由 $AB = A + B$ 得 $(A - E)(B - E) = E$，因此 $A - E$ 可逆。 +$$\text{……3 分}$$ + +**(2)** +由 $(A - E)(B - E) = E$ 得 $(B - E)(A - E) = E$，因此 $AB = BA$。 +$$\text{……6 分}$$ + +**(3)** +由 $AB = A + B$ 得 $A = (A - E)B$，而 $A - E$ 可逆，故 +$$ +\mathrm{rank}(A) = \mathrm{rank}(B). +$$ +$$\text{……9 分}$$ + +**(4)** +由 $AB = A + B$ 得 $A(B - E) = B$，而 $B - E$ 可逆，故 +$$ +A = B(B - E)^{-1}. +$$ +已知 +$$ +B = \begin{bmatrix} +1 & -3 & 0 \\ +2 & 1 & 0 \\ +0 & 0 & 2 +\end{bmatrix}, +$$ +则 +$$ +B - E = \begin{bmatrix} +0 & -3 & 0 \\ +2 & 0 & 0 \\ +0 & 0 & 1 +\end{bmatrix}. +$$ +求逆得 +$$ +(B - E)^{-1} = \begin{bmatrix} +0 & \frac12 & 0 \\[2pt] +-\frac13 & 0 & 0 \\[2pt] +0 & 0 & 1 +\end{bmatrix}. +$$ +于是 +$$ +A = B(B - E)^{-1} = \begin{bmatrix} +1 & -3 & 0 \\ +2 & 1 & 0 \\ +0 & 0 & 2 +\end{bmatrix} +\begin{bmatrix} +0 & \frac12 & 0 \\[2pt] +-\frac13 & 0 & 0 \\[2pt] +0 & 0 & 1 +\end{bmatrix} += \begin{bmatrix} +1 & \frac12 & 0 \\[2pt] +-\frac13 & 1 & 0 \\[2pt] +0 & 0 & 2 +\end{bmatrix}. +$$ +$$\text{……12 分}$$ + +--- + +17. 设矩阵$A=\begin{bmatrix}1&2&1&2\\0&1&t&t\\1&t&0&1\end{bmatrix}$，齐次线性方程组Ax=0的基础解系中含有两个解向量，求Ax=0的通解。 + +--- + +解析： +因为n=4，$n-\text{rank}A=2$，所以$\text{rank}A=2$。 +对A施行初等行变换，得 +$$A=\begin{bmatrix}1&2&1&2\\0&1&t&t\\1&t&0&1\end{bmatrix}\to\begin{bmatrix}1&2&1&2\\0&1&t&t\\0&t-2&-1&-1\end{bmatrix}$$ + +$$\to\begin{bmatrix}1&2&1&2\\0&1&t&t\\0&0&-(1-t)^2&-(1-t)^2\end{bmatrix}\to\begin{bmatrix}1&0&1-2t&2-2t\\0&1&t&t\\0&0&-(1-t)^2&-(1-t)^2\end{bmatrix}$$ + + +要使$\text{rank}A=2$，则必有t=1。 +此时，与Ax=0同解的方程组为$\begin{cases}x_1=x_3\\x_2=-x_3-x_4\end{cases}$，得基础解系为 +$$\boldsymbol{\xi}_1=\begin{bmatrix}1\\-1\\1\\0\end{bmatrix},\ \boldsymbol{\xi}_2=\begin{bmatrix}0\\-1\\0\\1\end{bmatrix}$$ +方程组的通解为$$\boldsymbol{x}=k_1\boldsymbol{\xi}_1+k_2\boldsymbol{\xi}_2，（k_1,k_2为任意常数）$$ \ No newline at end of file diff --git a/编写小组/试卷/2018-19线性代数期末考试卷.md b/编写小组/试卷/2018-19线性代数期末考试卷.md new file mode 100644 index 0000000..11b3487 --- /dev/null +++ b/编写小组/试卷/2018-19线性代数期末考试卷.md @@ -0,0 +1,122 @@ +1. 设 $A$ 为 $n$ 阶对称矩阵，$B$ 为 $n$ 阶反对称矩阵，下列矩阵中为反对称矩阵的是【 】 + + (A) $AB - BA$; + (B) $AB + BA$; + (C) $BAB$; + (D) $(AB)^2$. + +2. 设 $A, B$ 是可逆矩阵，且 $A$ 与 $B$ 相似，则下列结论错误的是【 】 + + (A) $A^T$ 与 $B^T$ 相似； + (B) $A^{-1}$ 与 $B^{-1}$ 相似； + (C) $A + A^T$ 与 $B + B^T$ 相似； + (D) $A + A^{-1}$ 与 $B + B^{-1}$ 相似。 + +3. 设向量组 + $$ + \alpha_1 = (0, 0, c_1)^T,\quad + \alpha_2 = (0, 1, c_2)^T,\quad + \alpha_3 = (1, -1, c_3)^T,\quad + \alpha_4 = (-1, 1, c_4)^T, + $$ + 其中 $c_1, c_2, c_3, c_4$ 为任意常数，则下列向量组线性相关的是【 】 + + (A) $\alpha_1, \alpha_2, \alpha_3$; + (B) $\alpha_1, \alpha_2, \alpha_4$; + (C) $\alpha_1, \alpha_3, \alpha_4$; + (D) $\alpha_2, \alpha_3, \alpha_4$. + +4. 设 $A, B$ 为 $n$ 阶矩阵，则【 】 + + (A) $\text{rank}[A \ AB] = \text{rank} A$; + (B) $\text{rank}[A \ BA] = \text{rank} A$; + (C) $\text{rank}[A \ B] = \max\{\text{rank} A, \text{rank} B\}$; + (D) $\text{rank}[A \ B] = \text{rank}[A^T \ B^T]$. + +5. 设 $A$ 可逆，将 $A$ 的第一列加上第二列的 2 倍得到 $B$，则 $A^*$ 与 $B^*$ 满足【 】 + + (A) 将 $A^*$ 的第一列加上第二列的 2 倍得到 $B^*$; + (B) 将 $A^*$ 的第一行加上第二行的 2 倍得到 $B^*$; + (C) 将 $A^*$ 的第二列加上第一列的 $(-2)$ 倍得到 $B^*$; + (D) 将 $A^*$ 的第二行加上第一行的 $(-2)$ 倍得到 $B^*$. + +6. 已知方程组 + $$ + \text{(I)} \quad + \begin{cases} + x_1 + 2x_2 + 3x_3 = 0, \\ + 2x_1 + 3x_2 + 5x_3 = 0, \\ + x_1 + x_2 + ax_3 = 0, + \end{cases} + $$ + 与 + $$ + \text{(II)} \quad + \begin{cases} + x_1 + bx_2 + cx_3 = 0, \\ + 2x_1 + b^2x_2 + (c+1)x_3 = 0 + \end{cases} + $$ + 同解，则【 】 + + (A) $a = 1, b = 0, c = 1$; + (B) $a = 1, b = 1, c = 2$; + (C) $a = 2, b = 0, c = 1$; + (D) $a = 2, b = 1, c = 2$. + + +7. 已知向量 $\alpha_1 = (1,0,-1,0)^T$，$\alpha_2 = (1,1,-1,-1)^T$，$\alpha_3 = (-1,0,1,1)^T$，则向量 $\alpha_1 + 2\alpha_2$ 与 $2\alpha_1 + \alpha_3$ 的内积 + $$ + \langle \alpha_1 + 2\alpha_2,\, 2\alpha_1 + \alpha_3 \rangle = \underline{\qquad\qquad}. + $$ + +8. 设二阶矩阵 $A$ 有两个相异特征值，$\alpha_1, \alpha_2$ 是 $A$ 的线性无关的特征向量，且 $A^2 (\alpha_1 + \alpha_2) = \alpha_1 + \alpha_2$，则 + $$ + |A| = \underline{\qquad\qquad}. + $$ + +9. 若向量组 + $$ + \alpha_1 = (1,0,1)^T,\quad \alpha_2 = (0,1,1)^T,\quad \alpha_3 = (1,3,5)^T + $$ + 不能由向量组 + $$ + \beta_1 = (1,1,1)^T,\quad \beta_2 = (1,2,3)^T,\quad \beta_3 = (3,4,a)^T + $$ + 线性表示，则 + $$ + a = \underline{\qquad\qquad}. + $$ + +10. 设矩阵 + $$ + A = \begin{bmatrix} + 1 & a_1 & a_1^2 & a_1^3 \\ + 1 & a_2 & a_2^2 & a_2^3 \\ + 1 & a_3 & a_3^2 & a_3^3 \\ + 1 & a_4 & a_4^2 & a_4^3 + \end{bmatrix},\quad + x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix},\quad + b = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, + $$ + 其中常数 $a_1, a_2, a_3, a_4$ 互不相等，则线性方程组 $Ax = b$ 的解为 + $$ + \underline{\qquad\qquad\qquad\qquad}. + $$ + +11. 若 $n$ 阶实对称矩阵 $A$ 的特征值为 + $$ + \lambda_i = (-1)^i \quad (i=1,2,\dots,n), + $$ + 则 + $$ + A^{100} = \underline{\qquad\qquad\qquad\qquad}. + $$ + +12. 设 $n$ 阶矩阵 $A = [a_{ij}]_{n \times n}$，则二次型 + $f(x_1, x_2, \dots, x_n) = \sum_{i=1}^n (a_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n)^2$ + 的矩阵为 + $$ + \underline{\qquad\qquad\qquad\qquad}. + $$ + diff --git a/编写小组/试卷/2025秋期中考前押题卷.md b/编写小组/试卷/2025秋期中考前押题卷.md new file mode 100644 index 0000000..bdcbf09 --- /dev/null +++ b/编写小组/试卷/2025秋期中考前押题卷.md @@ -0,0 +1,71 @@ +时量：60分钟 ____ +**内部资料，禁止传播** +**编委会（不分先后，姓氏首字母顺序）：程奕铭 韩魏 刘柯妤 卢吉辚 王轲楠 支宝宁 郑哲航** + +1.设周期函数 $f(x)$ 在 $(-\infty,+\infty)$ 内可导，**周期为$4$**，又 $\lim\limits_{x\to0}\dfrac{f(1)-f(1-x)}{2x}=-1$，则曲线 $y=f(x)$ 在点 $(5,f(5))$ 处切线的斜率为（ ）。 +（A）$\dfrac{1}{2}$　　　（B）$0$　　　（C）$-1$　　　（D）$-2$ + + + +2.设 $f(x) = \dfrac{2 + e^{\frac{1}{x}}}{1 + e^{\frac{1}{x}}} + \dfrac{\sin x}{|x|}$，则 $x = 0$ 是 $f(x)$ 的（ ）。 + +(A) 可去间断点 +(B) 跳跃间断点 +(C) 无穷间断点 +(D) 振荡间断点 + + + +3.（多选）下列级数中收敛的有______。 + +A $\sin \frac{\pi}{2} + \sin \frac{\pi}{2^2} + \sin \frac{\pi}{2^3} + \cdots$ + +B $\sum_{n=1}^{\infty} \frac{1}{5^n} \cdot \frac{3n^3+2n^2}{4n^3+1}$ + +C $\sum_{n=1}^{\infty} \frac{1}{(a+n-1)(a+n)(a+n+1)} \quad (a > 0)$ + +D $\sum_{n=1}^{\infty} \frac{1}{n\sqrt[n]{n}}$ + +E $\sum_{n=1}^{\infty} \frac{1}{n} \arctan \frac{n}{n+1}$ + +F $\sum_{n=1}^{\infty} \frac{\sqrt{n+\sqrt{n}}}{n^2+1}$ + + + +4.设 $y=y(x)$ 由方程 $x^y + 2x^2 - y = 1$ 确定，求 $dy|_{x=1}$______。 + + + + +5.$\lim_{n \to \infty} \left( \frac{1}{n^2 + \sin 1} + \frac{2}{n^2 + 2\sin 2} + \cdots + \frac{n}{n^2 + n \sin n} \right)$=______。 + + + + +6.计算 $\lim_{x \to \infty} \left( \tan^2 \frac{2}{x} + \cos \frac{1}{x} \right)^{x^2}$______。 + + + +7.求曲线 $y = \ln(1+e^x) + \frac{2+x}{2-x} \arctan \frac{x}{2}$ 的渐近线方程。 + + +8.设 $a > 0, \sigma > 0$，定义 $a_1 = \dfrac{1}{2} \left( a + \dfrac{\sigma}{a} \right)$，$a_{n+1} = \dfrac{1}{2} \left( a_n + \dfrac{\sigma}{a_n} \right)$，$n = 1, 2, \ldots$ +证明：数列 ${a_n}$ 收敛，且极限为 $\sqrt{\sigma}$。 + + + + +9.一飞机在离地面$2 km$的高度，以$200 km/h$的速度水平飞行到某目标上空，以便进行航空摄影。试求飞机飞至该目标正上方时，摄影机转动的角速率。 + + + + +10.若函数 $f(x)$ 在 $(a,b)$ 内连续，任取 $x_i \in (a,b)$ $(i=1,2,\cdots,n)$ 使 +x1≤x2≤⋯≤xn +证明：存在 $\xi \in [x_1,x_n]$，使得 +$$f(\xi) = \frac{1}{n} \sum_{i=1}^n f(x_i)$$ + + + + + diff --git a/编写小组/试卷/2025秋期中考前押题卷解析版.md b/编写小组/试卷/2025秋期中考前押题卷解析版.md new file mode 100644 index 0000000..1786e05 --- /dev/null +++ b/编写小组/试卷/2025秋期中考前押题卷解析版.md @@ -0,0 +1,256 @@ +时量：60分钟 ____ +内部资料，禁止传播 +编委会（不分先后，姓氏首字母顺序）：程奕铭 韩魏 刘柯妤 卢吉辚 王轲楠 支宝宁 郑哲航 + + +1.设周期函数 $f(x)$ 在 $(-\infty,+\infty)$ 内可导，**周期为$4$**，又 $\lim\limits_{x\to0}\dfrac{f(1)-f(1-x)}{2x}=-1$，则曲线 $y=f(x)$ 在点 $(5,f(5))$ 处切线的斜率为[　]。 +（A）$\dfrac{1}{2}$　　　（B）$0$　　　（C）$-1$　　　（D）$-2$ + +**解析：** +由极限表达式变形：令 $h=-x$，则当 $x\to0$ 时 $h\to0$，于是 + +$$\begin{aligned} +\lim_{x \to 0} \frac{f(1) - f(1 - x)}{2x} &= \lim_{h \to 0} \frac{f(1) - f(1 + h)}{-2h} \\ &= \lim_{h \to 0} \frac{f(1 + h) - f(1)}{2h} \\ &= \frac{1}{2} \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h} \\ &= \frac{1}{2} f'(1). +\end{aligned}$$ + +已知该极限值为 $-1$，故 $\dfrac{1}{2}f'(1) = -1$，解得 $f'(1) = -2$。 +由于 $f(x)$ 是周期函数且周期为$4$，则有$$f(x)=f(x+4),$$两边对$x$求导得：$$f'(x)=f'(x+4),$$于是$$f'(5)=f'(1)=-2$$ +因此，曲线在点 $(5,f(5))$ 处的切线斜率为 $-2$，选项（D）正确。 + +**答案：** (D) + +--- + + + +2.设 $f(x) = \dfrac{2 + e^{\frac{1}{x}}}{1 + e^{\frac{1}{x}}} + \dfrac{\sin x}{|x|}$，则 $x = 0$ 是 $f(x)$ 的（ ）。 + +(A) 可去间断点 +(B) 跳跃间断点 +(C) 无穷间断点 +(D) 振荡间断点 + +**解析：** +分析函数在 $x=0$ 处的左右极限。 + +1. **第一部分**：$\dfrac{2 + e^{\frac{1}{x}}}{1 + e^{\frac{1}{x}}}$ + + - 当 $x \to 0^+$ 时，$\frac{1}{x} \to +\infty$，$e^{\frac{1}{x}} \to +\infty$，所以 + + $$ +\lim\limits_{x\to0^+}\frac{2 + e^{\frac{1}{x}}}{1 + e^{\frac{1}{x}}} = \lim\limits_{x\to0^+}\frac{\frac{2}{e^{\frac{1}{x}}} + 1}{\frac{1}{e^{\frac{1}{x}}} + 1} = \frac{0 + 1}{0 + 1} = 1. +$$ + - 当 $x \to 0^-$ 时，$\frac{1}{x} \to -\infty$，$e^{\frac{1}{x}} \to 0$，所以 + + $$ +\frac{2 + e^{\frac{1}{x}}}{1 + e^{\frac{1}{x}}} \rightarrow \frac{2 + 0}{1 + 0} = 2\ \ (x\to0^-). +$$ +1. **第二部分**：$\dfrac{\sin x}{|x|}$ + + - 当 $x \to 0^+$ 时，$|x| = x$，$\dfrac{\sin x}{|x|} = \dfrac{\sin x}{x} \to 1$。 + + - 当 $x \to 0^-$ 时，$|x| = -x$，$\dfrac{\sin x}{|x|} = \dfrac{\sin x}{-x} \to -1$。 + +2. **整体极限**： + + - 右极限：$\lim\limits_{x \to 0^+} f(x) = 1 + 1 = 2$。 + + - 左极限：$\lim\limits_{x \to 0^-} f(x) = 2 + (-1) = 1$。 + + +由于左右极限存在但不相等，故 $x=0$ 处为**跳跃间断点**。 + +**答案：** (B) + +--- +3.（多选）下列级数中收敛的有______。 + +A $\sin \frac{\pi}{2} + \sin \frac{\pi}{2^2} + \sin \frac{\pi}{2^3} + \cdots$ + +B $\sum_{n=1}^{\infty} \frac{1}{5^n} \cdot \frac{3n^3+2n^2}{4n^3+1}$ + +C $\sum_{n=1}^{\infty} \frac{1}{(a+n-1)(a+n)(a+n+1)} \quad (a > 0)$ + +D $\sum_{n=1}^{\infty} \frac{1}{n\sqrt[n]{n}}$ + +E $\sum_{n=1}^{\infty} \frac{1}{n} \arctan \frac{n}{n+1}$ + +F $\sum_{n=1}^{\infty} \frac{\sqrt{n+\sqrt{n}}}{n^2+1}$ + +**解析：** + +- **A**：由于 $0 < \sin \frac{\pi}{2^n} \leq \frac{\pi}{2^n}$，且几何级数 $\sum_{n=1}^{\infty} \frac{\pi}{2^n}$ 收敛，故由比较判别法知原级数收敛。 + +- **B**：由于 $\lim\limits_{n \to \infty} \left( \frac{1}{5^n} \cdot \frac{3n^3 + 2n^2}{4n^3 + 1} \right) / \left( \frac{1}{5^n} \right) = \lim\limits_{n \to \infty} \frac{3n^3 + 2n^2}{4n^3 + 1} = \frac{3}{4}$，又几何级数 $\sum_{n=1}^{\infty} \frac{1}{5^n}$ 收敛，故由极限形式的比较判别法知原级数收敛。 + +- **C**：由于 $\lim\limits_{n \to \infty} \left( \frac{1}{(a+n-1)(a+n)(a+n+1)} \right) / \left( \frac{1}{n^3} \right) = 1$，而 $p$ 级数 $\sum_{n=1}^{\infty} \frac{1}{n^3}$ 收敛（$p=3>1$），故原级数收敛。也可直接放缩：$\frac{1}{(a+n-1)(a+n)(a+n+1)} \leq \frac{1}{(a+n-1)^3} < \frac{1}{(n-1)^3}$（当 $n>1$），由比较判别法知收敛。 + +- **D**：由于 $\lim\limits_{n \to \infty} \frac{1}{n\sqrt[n]{n}} / \frac{1}{n} = \lim\limits_{n \to \infty} \frac{1}{\sqrt[n]{n}} = 1$，而调和级数 $\sum_{n=1}^{\infty} \frac{1}{n}$ 发散，故原级数发散。 + +- **E**：由于 $\lim\limits_{n \to \infty} \left( \frac{1}{n} \cdot \arctan \frac{n}{n+1} \right) / \frac{1}{n} = \lim\limits_{n \to \infty} \arctan \frac{n}{n+1} = \arctan 1 = \frac{\pi}{4}$，而调和级数 $\sum_{n=1}^{\infty} \frac{1}{n}$ 发散，故原级数发散。 + +- **F**：由于 $\lim\limits_{n \to \infty} \frac{\sqrt{n+\sqrt{n}}}{n^2+1} / \frac{1}{n^{3/2}} = \lim\limits_{n \to \infty} \sqrt{1 + \frac{1}{\sqrt{n}}} / \left( 1 + \frac{1}{n^2} \right) = 1$，而 $p$ 级数 $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$ 收敛（$p=3/2>1$），故原级数收敛。 + + +**答案：** ABCF + + + +4.设 $y=y(x)$ 由方程 $x^y + 2x^2 - y = 1$ 确定，求 $dy|_{x=1}$。 + +**解析：** + +1. 先求 $x=1$ 时的 $y$ 值：代入方程：$1^y + 2 \times 1^2 - y = 1 \implies 1 + 2 - y = 1 \implies y = 2$。 + +2. 隐函数求导：由原式得$$x^y=1+y-2x^2,$$取对数得$$y\ln x=\ln(1+y-2x^2)$$两边对$x$求导得$$y'\ln x+\frac{y}{x}=\frac{y'-4x}{1+y-2x^2}$$把$x=1,y=2$带入得$$2=\frac{y'-4}{1+2-2},y'=6$$于是$$dy|_{x=1}=6dx$$ +**答案：** $dy|_{x=1} = 6dx$ + + + +5.求 $\lim\limits_{n \to \infty} \left( \frac{1}{n^2 + \sin 1} + \frac{2}{n^2 + 2\sin 2} + \cdots + \frac{n}{n^2 + n \sin n} \right)$。 + +**解析：** +用夹逼准则： + +- 下界：$S_n \geq \sum_{k=1}^n \frac{k}{n^2 + n} = \frac{\frac{1}{2}n(n+1)}{n^2+n}=\frac{1}{2}$； + +- 上界：$S_n \leq \sum_{k=1}^n \frac{k}{n^2 - n} = \frac{n(n+1)}{2(n^2-n)} \to \frac{1}{2}$（$n \to \infty$）。 + 故极限为 $\frac{1}{2}$。 + + +**答案：** $\frac{1}{2}$ + + +6.计算 $\lim\limits_{x \to \infty} \left( \tan^2 \frac{2}{x} + \cos \frac{1}{x} \right)^{x^2}$。 + +**解析：** $$\begin{aligned} +\lim\limits_{x \to \infty} \left( \tan^2 \frac{2}{x} + \cos \frac{1}{x} \right)^{x^2} +&=\lim\limits_{x\to\infty}(1+\tan^2\frac{2}{x}+\cos \frac{1}{x}-1)^{\frac{1}{\tan^2\frac{2}{x}+\cos \frac{1}{x}-1}\cdot x^2\cdot(\tan^2\frac{2}{x}+\cos \frac{1}{x}-1)} +\\\\&=e^{\lim\limits_{x\to\infty}\frac{\tan^2\frac{2}{x}+\cos \frac{1}{x}-1}{\frac{1}{x^2}}} +\\\\&\overset{t=\frac{1}{x}}{=}e^{\lim\limits_{t\to0}\frac{\tan^2(2t)+\cos t-1}{t^2}} +\\\\&=e^{\lim\limits_{t\to0}\frac{4t^2}{t^2}-\frac{\frac{1}{2}t^2}{t^2}}(四则运算和等价无穷小) +\\\\&=e^{\frac{7}{2}} +\end{aligned}$$ + +**答案：** $e^{\frac{7}{2}}$ + +--- + + +7.求曲线 $y = \ln(1+e^x) + \frac{2+x}{2-x} \arctan \frac{x}{2}$ 的渐近线方程。 + +**解析：** +渐近线分垂直、水平、斜渐近线分析： + +1. **垂直渐近线**： + 分母 $2-x=0 \implies x=2$，计算 $\lim\limits_{x \to 2} y = \infty$，故垂直渐近线为 $x=2$。 + +2. **水平渐近线**（$x \to -\infty$）： + 当 $x \to -\infty$ 时，$\ln(1+e^x) \to 0$，$\frac{2+x}{2-x} \to -1$，$\arctan \frac{x}{2} \to -\frac{\pi}{2}$， + 所以 $\lim\limits_{x \to -\infty} y = (-1) \times (-\frac{\pi}{2}) = \frac{\pi}{2}$，水平渐近线为 $y = \frac{\pi}{2}$。 + +3. **斜渐近线**（$x \to +\infty$）： + + - 斜率 $k = \lim\limits_{x \to +\infty} \frac{y}{x} = \lim\limits_{x \to +\infty} \frac{\ln(1+e^x)}{x} + \lim\limits_{x \to +\infty} \frac{\frac{2+x}{2-x} \arctan \frac{x}{2}}{x}$。 + 由于 $\ln(1+e^x) =\ln e^x(1+e^{-x})= x + \ln(1+e^{-x})$，所以 $\frac{\ln(1+e^x)}{x} = 1 + \frac{\ln(1+e^{-x})}{x} \to 1$； + 又 $\frac{2+x}{2-x} \to -1$，$\frac{\arctan \frac{x}{2}}{x} \to 0$，所以第二项趋于0，因此 $k = 1$。 + + - 截距 $b = \lim\limits_{x \to +\infty} (y - x) = \lim\limits_{x \to +\infty} \left[ \ln(1+e^x) - x + \frac{2+x}{2-x} \arctan \frac{x}{2} \right]$。 + 由于 $\ln(1+e^x) - x = \ln(1+e^{-x}) \to 0$，且 $\frac{2+x}{2-x} \to -1$，$\arctan \frac{x}{2} \to \frac{\pi}{2}$，所以 $b = -\frac{\pi}{2}$。 + 故斜渐近线为 $y = x - \frac{\pi}{2}$。 + + +**答案：** +渐近线为：$x=2$，$y=\frac{\pi}{2}$，$y=x-\frac{\pi}{2}$。 + + +--- + + +8.设 $a > 0, \sigma > 0$，定义 $a_1 = \dfrac{1}{2} \left( a + \dfrac{\sigma}{a} \right)$，$a_{n+1} = \dfrac{1}{2} \left( a_n + \dfrac{\sigma}{a_n} \right)$，$n = 1, 2, \ldots$ +证明：数列 ${a_n}$ 收敛，且极限为 $\sqrt{\sigma}$。 + +**解析：** +**第一步：证明数列有下界。** +由算术-几何平均不等式，对任意正数 $x$，有 + +$$ +\frac{1}{2} \left( x + \frac{\sigma}{x} \right) \geq \sqrt{x \cdot \frac{\sigma}{x}} = \sqrt{\sigma}. +$$ + +因此，对 $n \geq 1$，有 $a_n \geq \sqrt{\sigma}$，即数列有下界 $\sqrt{\sigma}$。 + +**第二步：证明数列单调性。** +考虑差值： +考虑数列的递推式：$a_{n+1} = \frac{1}{2} \left( a_n + \frac{\sigma}{a_n} \right)$，则 +$$ +a_{n+1} - a_n = \frac{1}{2} \left( a_n + \frac{\sigma}{a_n} \right) - a_n = \frac{1}{2} \left( \frac{\sigma}{a_n} - a_n \right) = \frac{\sigma - a_n^2}{2a_n}. +$$ + +由于 $a_n > 0$，差值的符号由 $\sigma - a_n^2$ 决定。 + +- 若 $a_n > \sqrt{\sigma}$，则 $a_{n+1} - a_n < 0$，数列单调递减； + +- 若 $a_n < \sqrt{\sigma}$，则 $a_{n+1} - a_n > 0$，数列单调递增； + +- 若 $a_n = \sqrt{\sigma}$，则 $a_{n+1} = a_n$，数列为常数列。 + + +易证 $a_1 \geq \sqrt{\sigma}$，等号仅当 $a = \sqrt{\sigma}$ 时成立。 +若 $a = \sqrt{\sigma}$，则数列恒为 $\sqrt{\sigma}$，极限为$\sqrt{\sigma}$。 +若 $a > \sqrt{\sigma}$，则 $a_1 > \sqrt{\sigma}$，由归纳法所有 $a_n > \sqrt{\sigma}$，且数列单调递减。 +若 $0 < a < \sqrt{\sigma}$，则 $a_1 > \sqrt{\sigma}$（因为 $a_1 = \frac{1}{2}(a + \sigma/a) \geq \sqrt{\sigma}$ 且等号不成立），此时从 $n=1$ 起 $a_n > \sqrt{\sigma}$，且 $a_2 < a_1$（因 $a_1 > \sqrt{\sigma}$），之后单调递减。 + +因此，无论哪种情况，数列从某项起单调且有界（下界 $\sqrt{\sigma}$，上界为 $a_1$ ），故数列收敛。 + +**第三步：求极限。** +设 $\lim\limits_{n \to \infty} a_n = L$，则 $L \geq \sqrt{\sigma} > 0$。在递推式两边取极限： +$$ +L = \frac{1}{2} \left( L + \frac{\sigma}{L} \right). +$$ +整理得 $2L = L + \frac{\sigma}{L}$，即 $L = \frac{\sigma}{L}$，从而 $L^2 = \sigma$，故 $L = \sqrt{\sigma}$（正根）。 + +因此，数列 ${a_n}$ 收敛，且极限为 $\sqrt{\sigma}$。 + + +--- + + +9.一飞机在离地面$2 km$的高度，以$200 km/h$的速度水平飞行到某目标上空，以便进行航空摄影。试求飞机飞至该目标正上方时，摄影机转动的角速率。 + +**解析：** +几何关系：$\tan \theta = \frac{x}{2}$（$x$ 为水平距离，$\theta$ 为竖直与摄影机连线的夹角）。 + +求导：$\sec^2 \theta \cdot \frac{d\theta}{dt} = \frac{1}{2} \cdot \frac{dx}{dt}$。 + +代入 $x=0$（正上方）：$\theta = 0 \implies \cos \theta = 1$，$\frac{dx}{dt} = -200$ km/h，得 +$$ +\frac{d\theta}{dt} = \frac{1}{2} \times (-200) = -100 \text{ rad/h}. +$$ +角速率为 $100$ $rad/h$（速率取绝对值）。 + +**答案：** $100$ $rad/h$。 + + +--- + + +10.若函数 $f(x)$ 在 $(a,b)$ 内连续，任取 $x_i \in (a,b)$ $(i=1,2,\cdots,n)$ 使 $$x_1≤x_2≤⋯≤x_n$$ +证明：存在 $\xi \in [x_1,x_n]$，使得 +$$f(\xi) = \frac{1}{n} \sum_{i=1}^n f(x_i)$$ + +**解析：** + +由于 $f(x)$ 在 $(a,b)$ 内连续，而 $[x_1, x_n] \subset (a,b)$，所以 $f(x)$ 在闭区间 $[x_1, x_n]$ 上连续。根据闭区间上连续函数的最值定理，$f(x)$ 在 $[x_1, x_n]$ 上能取到最大值 $M$ 和最小值 $m$。 + +对于任意 $x_i \in [x_1, x_n]$，有 $m \leq f(x_i) \leq M$，因此 +$m≤$$\frac{1}{n} \sum_{i=1}^n f(x_i$)$≤M.$ +对于任意 $x_i \in [x_1, x_n]$，有 $m \leq f(x_i) \leq M$，因此可将所有不等式加起来， +从而得到 $m≤$$\frac{1}{n} \sum_{i=1}^n f(x_i$)$≤M.$ + +由连续函数的介值定理，存在 $\xi \in [x_1, x_n]$，使得 +$$f(\xi) = \frac{1}{n} \sum_{i=1}^n f(x_i)$$ +证毕。 + +--- + diff --git a/编写小组/试卷/图片/期中试卷-18.png b/编写小组/试卷/图片/期中试卷-18.png new file mode 100644 index 0000000000000000000000000000000000000000..388210cf40349fb02ccd61820c6a40eb6e7be8a9 GIT binary patch literal 27337 zcmeFZX;{<O_BI^rsnp}Z@mQr+gtnlF2vm@H(mG&3r80<s1TB-uJccmZ+DZilWt3TE zDpR7&Lm*Db3^EI05(r@mfdmr5ki2V0ZU5KvT<`nw{q*O$sy(MacJ^L-t$W?;Uh9{; zKVQ_}@$YZ{jlp1cV9%X4#$di|#bCae{AvsQKMy3yT=;8~x3T_D7-F0FH2jZ$Ii9?5 z5`%djwQc?Sm+-&0dYrTH#$a~7M*nQ8cmK-{gSm;ro<3>nkDD6`OzHJa6$t1m$|(+a zVjeg2OFh2z=Z{i;3Vm(;{Y5Iya^-Blfu^Fa5Nl_~gk`c-|I3z}TK)XFXVmM3y#v?o z-2Q8H>wo^Gv9-bJPFKrSir%DGbkwcKx3+szM%)xwAru9&S;0<G-CEr&uga@CVLPzt zys=Hfe_1kpsv-Or(?2JLfB*b_(q{B8C;qtc@zFj1`+(3e|7(Z;^@snB3(WuG!~fzW 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s}$，过渡矩阵的行列式不为$0$ +>按照第一行展开： +>$\begin{vmatrix}1&&&\cdots&1\\1&1&&&\\&1&1&&\\&&\ddots&\ddots&\\&&&1&1\end{vmatrix}_{s\times s}=(-1)^2\begin{vmatrix}1&&&&\\1&1&&&\\&1&1&&\\&&\ddots&\ddots&\\&&&1&1\end{vmatrix}_{(s-1)\times (s-1)}+(-1)^{s+1}\begin{vmatrix}1&1&&&\\&1&1&&\\&&1&\ddots&\\&&&\ddots&1\\&&&&1\end{vmatrix}_{(s-1)\times (s-1)}$ +>即$\begin{vmatrix}1&&&\cdots&1\\1&1&&&\\&1&1&&\\&&\ddots&\ddots&\\&&&1&1\end{vmatrix}_{s\times s}=1+(-1)^{s+1}=\begin{cases}0,s\text{为偶数}\\2,s\text{为奇数}\end{cases}$ +>所以 $s$ 是奇数 +4. （6分） 0 +>解析：因为$A,B$是正交矩阵且$|A|=-|B|$，所以$A^\mathrm{T}A=E, B^\mathrm{T}B=E, |A|=\pm 1, |B|=\mp 1$， +>而$A^{-1}(A+B)B^{-1}=B^{-1}+A^{-1}=A^\mathrm{T}+B^\mathrm{T}=(A+B)^\mathrm{T}$， +>所以$|A^{-1}||A+B||B^{-1}|=|(A+B)^\mathrm{T}|=|A+B|$，$|A^{-1}||B^{-1}|=-1$ +>所以$-|A+B|=|A+B|$，即$|A+B|=0$ +5. （7分） 0 +>$A\sim B\sim\begin{bmatrix}1&&&\\&2&&\\&&3&\\&&&4\end{bmatrix}$， +>所以$B^{-1}-E\sim\begin{bmatrix}1&&&\\&2&&\\&&3&\\&&&4\end{bmatrix}^{-1}-E=\begin{bmatrix}0&&&\\&-\frac{1}{2}&&\\&&-\frac{2}{3}&\\&&&-\frac{3}{4}\end{bmatrix}$， +>所以$|B^{-1}-E|=\begin{vmatrix}0&&&\\&-\frac{1}{2}&&\\&&-\frac{2}{3}&\\&&&-\frac{3}{4}\end{vmatrix}=0$ +6. （7分） 0 +>解析：设 $\alpha,\beta,\gamma$ 为 $x^3+px+q=0$ 的三个根，则 $x^3+px+q=$$(x-\alpha)(x-\beta)(x-\gamma)=x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\alpha\gamma+\beta\gamma)x+\alpha\beta\gamma$ ，所以$\alpha+\beta+\gamma=0$，将行列式第二、三行加在第一行，第一行全为$\alpha+\beta+\gamma=0$，故行列式为0 +7. （20分） $D_n=2^{n-1}(2+\sum\limits_{j=1}^n a_j^2)$ +>解析 +>方法1： +>行列式加边法：$D_n=\begin{vmatrix}1&a_1&a_2&\cdots&a_n\\0&2+a_1^2&a_1a_2&\cdots&a_1a_n\\0&a_2a_1&2+a_2^2&\cdots&a_2a_n\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&a_na_1&a_na_2&\cdots&2+a_n^2\end{vmatrix}$ +>将第一行乘以 $(-a_i)$ 加到第 $i+1$ 行： +>$D_n=\begin{vmatrix}1&a_1&a_2&\cdots&a_n\\-a_1&2&&&\\-a_2&&2&&\\\vdots&&&\ddots&\\-a_n&&&&2\end{vmatrix}$（模板：箭头行列式） +>再将第 $j+1$ 列乘以 $\frac{a_j}{2}$ 加到第一列： +>$D_n=\begin{vmatrix}1+\sum\limits_{j=1}^n\frac{a_j^2}{2}&a_1&a_2&\cdots&a_n\\&2&&&\\&&2&&\\&&&\ddots&\\&&&&2\end{vmatrix}$ +>所以$D_n=2^{n-1}(2+\sum\limits_{j=1}^n a_j^2)$ +>方法2： +>核心结论：$|E+AB|=|E+BA|$ +>$D_n=|2E_n+xx^\mathrm{T}|=2^n|E_n+\frac{1}{2}xx^\mathrm{T}|=2^n|E_1+\frac{1}{2}x^\mathrm{T}x|=2^n(1+x^\mathrm{T}x)=2^{n-1}(2+\sum\limits_{j=1}^{n}a_j^2)$ +8. （20分） $\begin{bmatrix}1\\3\\4\end{bmatrix}+k\begin{bmatrix}2\\1\\-1\end{bmatrix}, k\in\mathbb{R}$ +>解析：由 $\alpha_3=2\alpha_1+\alpha_2$ 可得 $A$ 的列向量组线性相关， $|A|=0$；又因为 $A$ 的三个特征值各不相同，故 $A$ 有两个不为零的特征值 $\lambda_1,\lambda_2$，且 $A$ 可相似对角化，即 $A=P^{-1}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}P$，$\mathrm{rank}A=\mathrm{rank}(P^{-1}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}P)=\mathrm{rank}\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&0\end{bmatrix}=2$ +>故 $Ax=0$ 的解空间维数是 $1$ （5分） +>$\beta=\alpha_1+3\alpha_2+4\alpha_3$，所以 $(1,3,4)^\mathrm{T}$ 为特解；（5分） +>$\alpha_3=2\alpha_1+\alpha_2$，所以$A\begin{bmatrix}2k\\k\\-k\end{bmatrix}=2\alpha_1+\alpha_2-\alpha_3=0$，所以 $(2,1,-1)^\mathrm{T}$ 为基础解系；（10分） +>解空间维数是 $1$ ，方程的解 $\begin{bmatrix}1\\3\\4\end{bmatrix}+k\begin{bmatrix}2\\1\\-1\end{bmatrix}, k\in\mathbb{R}$ 维数是 $1$，该解完备 +9. 证明如下： +>(1)（10分） +> 对于方程组$Ax=0$ (a)和 $A^\mathrm{T}Ax=0$ (b)，b的解空间一定包含a的解空间（5分）； +>而方程b两边同时乘以$x^\mathrm{T}$，得 $x^\mathrm{T}A^\mathrm{T}Ax=0$ ，即 $(x^\mathrm{T}A^\mathrm{T})(Ax)=0 \to Ax=0$， +>所以a的解空间包含b的解空间（5分）， +>所以a,b同解，所以$\mathrm{rank}A=\mathrm{rank}{A^\mathrm{T}A}$ +>(2) （10分） +>$A^\mathrm{T}Ax=A^\mathrm{T}\beta \iff \mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})$ +>而由(1)的结论得等式左边 $\mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}A$； +>等式右边 $\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})\ge \mathrm{rank}A^\mathrm{T}A=\mathrm{rank}A$ （5分） +>又$\mathrm{rank}(A^\mathrm{T}\begin{bmatrix}A&\beta\end{bmatrix})\le \min{(\mathrm{rank}A^\mathrm{T},\ \mathrm{rank}\begin{bmatrix}A&\beta\end{bmatrix})}=\mathrm{rank}A$ （5分） +>所以$\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})=\mathrm{rank}A$ +>故 $\mathrm{rank}(A^\mathrm{T}A)=\mathrm{rank}(\begin{bmatrix}A^\mathrm{T}A&A^\mathrm{T}\beta\end{bmatrix})$ 得证. \ No newline at end of file diff --git a/编写小组/试卷/线代期末复习模拟/1.10线代限时练.md b/编写小组/试卷/线代期末复习模拟/1.10线代限时练.md new file mode 100644 index 0000000..c76c6ad --- /dev/null +++ b/编写小组/试卷/线代期末复习模拟/1.10线代限时练.md @@ -0,0 +1,13 @@ +一、填空题 +1. （7分）设 $A=\begin{bmatrix}1&2&0\\0&2&0\\-2&-1&-1\end{bmatrix}$，则$A^{100}=$\_\_\_\_\_\_\_\_\_\_\_ +2. （6分）线性空间 $V = \{A\in\mathbb{R}^{n\times n}|A = -A^\mathrm{T}\}$ 的维数是\_\_\_\_\_\_\_\_\_\_\_ +3. （7分）设向量组 $\alpha_1,\alpha_2,\cdots,\alpha_s$ 线性无关，$\beta_1=\alpha_1+\alpha_2,\beta_2=\alpha_2+\alpha_3,\cdots,\beta_{s-1}=\alpha_{s-1}+\alpha_s, \beta_{s}=\alpha_{s}+\alpha_1$，则向量组 $\beta_1,\beta_2,\cdots,\beta_s$ 线性无关的充要条件是\_\_\_\_\_\_\_\_\_\_\_ +4. （6分）知 $A,B$ 均为 $n$ 阶正交矩阵，且 $|A|=-|B|$，则 $|A+B|$ 的值为\_\_\_\_\_\_\_\_\_\_\_ +5. （7分）已知 $4$ 阶矩阵 $A$ 与 $B$ 相似，$A$ 的全部特征值为 $1,2,3,4$，则行列式 $|B^{-1}-E|$ 为\_\_\_\_\_\_\_\_\_\_\_ +6. （7分）设 $\alpha,\beta,\gamma$ 为 $x^3+px+q=0$ 的三个根，则行列式 $\begin{vmatrix}\alpha&\beta&\gamma\\\gamma&\alpha&\beta\\\beta&\gamma&\alpha\end{vmatrix}$ 的值为\_\_\_\_\_\_\_\_\_\_\_ +二、解答题 +7. （20分）计算$n$阶行列式 $D_n=\begin{vmatrix}2+a_1^2 & a_1a_2 & \cdots&a_1a_n \\ a_2a_1& 2+a_2^2 & \cdots & a_2a_n \\ \vdots & \vdots & \ddots & \vdots \\ a_na_1&a_na_2&\cdots& 2+a_n^2\end{vmatrix}$. +8. （20分）已知三阶方阵 $A=\begin{bmatrix}\alpha_1&\alpha_2&\alpha_3\end{bmatrix}$ 有三个不同的特征值，其中$\alpha_3=2\alpha_1+\alpha_2$，若 $\beta=\alpha_1+3\alpha_2+4\alpha_3$ ，求线性方程组 $Ax=\beta$ 的通解. +9. （20分）设 $A$ 是 $m\times n$ 实矩阵， $\beta \neq 0$ 是 $m$ 维实列向量，证明： + (1) $\mathrm{rank}A=\mathrm{rank}(A^\mathrm{T}A)$ . + (2) 线性方程组 $A^\mathrm{T}Ax = A^\mathrm{T}\beta$ 有解. \ No newline at end of file diff --git a/编写小组/试卷/线代期末复习模拟/1.13线代测试答案.md b/编写小组/试卷/线代期末复习模拟/1.13线代测试答案.md new file mode 100644 index 0000000..4f0ed9d --- /dev/null +++ b/编写小组/试卷/线代期末复习模拟/1.13线代测试答案.md @@ -0,0 +1,223 @@ +# **1.13 线性代数限时练（题目 + 答案与解析）** + +## **第一部分：题目** + +### **1.** + +已知三阶行列式 $\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}=c$，代数余子式之和 $\sum_{i=1}^{3}\sum_{j=1}^{3}A_{ij}=3c$，则行列式 $\begin{vmatrix}a_{11}+1&a_{12}+1&a_{13}+1\\a_{21}+1&a_{22}+1&a_{23}+1\\a_{31}+1&a_{32}+1&a_{33}+1\end{vmatrix}=$？ + +### **2.（2013 秋 A）** + +$已知向量空间 V=\{(2a,2b,3b,3a)\mid a,b\in\mathbb{R}\}，则 V 的维数是\underline{\qquad}。$ + +### **3.（2018 秋 A）** + +$设 E 为 3 阶单位矩阵，\alpha 为一个 3 维单位列向量，则矩阵 E-\alpha\alpha^T 的全部 3 个特征值为\underline{\qquad}。$ + +### **4.（2018 秋 A）** + +$设 n 阶矩阵 A=[a_{ij}]_{n\times n}，则二次型 f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in}x_n)^2 的矩阵为\underline{\qquad}。$ +### **5.（2018 秋 A）** + +$若 n 阶实对称矩阵 A 的特征值为 \lambda_i=(-1)^i（i=1,2,\cdots,n），则 A^{100}=\underline{\qquad}。$ + +### **6.（2022 秋 A・5）** + +$已知 n（n\geq2）维列向量 \alpha,\beta 满足 \beta^T\alpha=-3，则方阵 (\beta\alpha^T)^2 的非零特征值为\underline{\qquad}。$ + +### **7.（2022 秋 A）** + +$已知向量组 \alpha_1,\alpha_2,\alpha_3 线性无关（\alpha_1,\alpha_2,\alpha_3\in\mathbb{R}^3），A 为 3 阶方阵，且满足：$ + +$$\begin{aligned} +A\alpha_1&=2\alpha_1-\alpha_2-\alpha_3,\\A\alpha_2&=\alpha_1+2\alpha_2+3\alpha_3,\\A\alpha_3&=2\alpha_1+4\alpha_2+\alpha_3 +\end{aligned} +$$ +(1) $证明 A\alpha_1,A\alpha_2,A\alpha_3 线性无关$； + +(2) $计算行列式 |E-A|（E 是 3 阶单位矩阵）$。 + +### **8.（2013 秋 A）** + +$求 n 阶方阵 A=\begin{bmatrix}1&1&1&\cdots&1\\1&0&1&\cdots&1\\1&1&0&\cdots&1\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&1&1&\cdots&0\end{bmatrix} 的逆矩阵。$ + +### **9.（2013 秋 A・三）** + +设 n 阶行列式： + +$$D_n=\begin{vmatrix}1&1&0&\cdots&0&0\\-1&1&1&\cdots&0&0\\0&-1&1&\cdots&0&0\\\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&0&\cdots&1&1\\0&0&0&\cdots&-1&1\end{vmatrix}$$ + +证明：$D_n=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]$。 + +## **第二部分：答案与解析** + +### 1. 答案：$\boxed{4c}$ + +#### **解析：** + +$设原矩阵为 A（即 |A|=c），全 1 矩阵 J=\boldsymbol{e}\boldsymbol{e}^T（其中 \boldsymbol{e}=(1,1,1)^T），需求解的行列式为 |A+J|。$ +由代数余子式性质：$\sum_{i,j}A_{ij}=\boldsymbol{e}^T A^*\boldsymbol{e}=3c$，且可逆矩阵的伴随矩阵满足$A^*=|A|A^{-1}=cA^{-1}$（因 $|A|=c\neq0，A$ 可逆），代入得 $\boldsymbol{e}^T A^{-1}\boldsymbol{e}=3$。 + +利用**Sherman-Morrison 行列式公式**：对可逆矩阵$A$ 和向量 $\boldsymbol{u},\boldsymbol{v}$，有 $|A+\boldsymbol{u}\boldsymbol{v}^T|=|A|(1+\boldsymbol{v}^T A^{-1}\boldsymbol{u})$。 + +此处 $\boldsymbol{u}=\boldsymbol{v}=\boldsymbol{e}$，代入得 $|A+J|=c(1+\boldsymbol{e}^T A^{-1}\boldsymbol{e})=c(1+3)=4c$。 + +### **2. 答案：$\boxed{2}$ + +#### **解析：** + +将向量空间 V 中的元素拆分为线性组合形式： + +$$(2a,2b,3b,3a)=a(2,0,0,3)+b(0,2,3,0)$$ + +设 $\boldsymbol{\alpha}=(2,0,0,3)，\boldsymbol{\beta}=(0,2,3,0)$，需验证 $\boldsymbol{\alpha},\boldsymbol{\beta}$ 线性无关： + +若 $k_1\boldsymbol{\alpha}+k_2\boldsymbol{\beta}=\boldsymbol{0}$（零向量），则 $\begin{cases}2k_1=0\\2k_2=0\\3k_2=0\\3k_1=0\end{cases}$，解得 $k_1=k_2=0$，故 $\boldsymbol{\alpha},\boldsymbol{\beta}$是 V 的一组基。 + +向量空间的维数等于基的个数，因此 $V$ 的维数为 $2$。 + +### **3. 答案：$\boxed{1,1,0}$ + +#### **解析：** + +设 $B=\alpha\alpha^T$，该矩阵为**秩 $1$ 矩阵**（因 $\alpha$ 是单位列向量，$\alpha^T\alpha=1$）。 + +• 秩 $1$ 矩阵的特征值性质：非零特征值为矩阵的迹 $\text{tr}(B)=\alpha^T\alpha=1$，其余 $n-1=2$ 个特征值为 $0$（秩 $1$ 矩阵的非零特征值个数等于秩）。 + +• 若 $B\boldsymbol{x}=\lambda\boldsymbol{x}$（$\boldsymbol{x}$为特征向量），则 $(E-B)\boldsymbol{x}=(1-\lambda)\boldsymbol{x}$，即$E-B$ 的特征值为 $1-\lambda$。 + +$代入 B 的特征值 \lambda=1,0,0，得 E-B 的特征值为 1-1=0，1-0=1，1-0=1，即 1,1,0$。 + +### **4. 答案：$\boxed{A^T A}$** + +#### **解析：** + +记$A_i$为$A$中除了第$i$行全都改为$0$的矩阵，$\boldsymbol{x}=\begin{bmatrix}x_1&x_2&\cdots&x_n\end{bmatrix}^T$。那么$$A_i\boldsymbol{x}=\begin{bmatrix}0 & 0 & \cdots & 0\\\vdots & \vdots & &\vdots\\a_{i1} & a_{i2} & \cdots &a_{in}\\\vdots & \vdots & &\vdots\\0 & 0 & \cdots & 0\end{bmatrix}\begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix}=\begin{bmatrix}0\\0\\\vdots\\\sum\limits_{j=1}^{n}a_{ij}x_j\\\vdots\\0\end{bmatrix}$$则$$(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in}x_n)^2=(\sum\limits_{j=1}^{n}a_{ij}x_j)^2=(A_i\boldsymbol{x})^TA_i\boldsymbol{x}=\boldsymbol{x}^TA_i^TA_i\boldsymbol{x}$$将二次型用上式展开得： + +$$\begin{aligned} +f(\boldsymbol{x})&=\sum_{i=1}^{n}(\boldsymbol{x}^TA_i^TA_i\boldsymbol{x})\\ +&=\boldsymbol{x}^T(\sum\limits_{i=1}^nA_i^TA_i)\boldsymbol{x}\\ +&=\boldsymbol{x}^T(\sum\limits_{i=1}^nA_i^T\sum\limits_{i=1}^nA_i)\boldsymbol{x}\\ +&=\boldsymbol{x}^T(A^T A)\boldsymbol{x} +\end{aligned}$$ +其中因为$A_i^TA_j=\boldsymbol{0},$如果$i\neq j$，所以 +$$\sum\limits_{i=1}^nA_i^T\sum\limits_{i=1}^nA_i=\sum\limits_{i=1}^nA_i^TA_i$$ + +• 二次型的矩阵需满足**对称性质**（即矩阵等于其转置），验证：$(A^T A)^T=A^T(A^T)^T=A^T A$，故 $A^T A$ 是对称矩阵，即为二次型的矩阵。 + +### **5. 答案：$\boxed{E}$（单位矩阵）** + +#### **解析：** + +实对称矩阵可对角化，即存在可逆矩阵 P，使得 $P^{-1}AP=\Lambda$（$\Lambda$ 为对角矩阵，对角元为 A 的特征值 $\lambda_i=(-1)^i$）。 + +• 矩阵幂运算性质：$A^{100}=P\Lambda^{100}P^{-1}$。 + +• 计算 $\Lambda^{100}$：对角元为 $\lambda_i^{100}=[(-1)^i]^{100}=1$，故 $\Lambda^{100}=E$（单位矩阵）。 + +• 因此 $A^{100}=P E P^{-1}=P P^{-1}=E$。 + +### **6. 答案：$\boxed{9}$** + +#### **解析：** + +设 $A=\beta\alpha^T$（秩 1 矩阵），计算 $A^2$： + +$A^2=(\beta\alpha^T)(\beta\alpha^T)=\beta(\alpha^T\beta)\alpha^T=(\alpha^T\beta)A$ + +• 注意：$\alpha^T\beta=(\beta^T\alpha)^T$（矩阵转置性质），而 $\beta^T\alpha=-3$（数，转置等于自身），故 $\alpha^T\beta=-3$，因此 $A^2=-3A$。 + +• 秩 $1$ 矩阵 $A$ 的非零特征值为 $\text{tr}(A)=\alpha^T\beta=-3$，设 $A\boldsymbol{x}=-3\boldsymbol{x}$，则 $A^2\boldsymbol{x}=(-3)A\boldsymbol{x}=(-3)^2\boldsymbol{x}=9\boldsymbol{x}$，即 $A^2$ 的非零特征值为 $9$。 + +### **7. 答案：(1) 证明见解析；(2) $\boxed{20}$** + +#### **(1) 证明 $A\alpha_1,A\alpha_2,A\alpha_3$ 线性无关** + +因 $\alpha_1,\alpha_2,\alpha_3$ 线性无关，故矩阵 $P=(\alpha_1,\alpha_2,\alpha_3)$ 可逆（列向量线性无关的矩阵可逆）。 + +由题设条件，将 $A$ 对 $\alpha_1,\alpha_2,\alpha_3$ 的作用表示为矩阵乘法： + +$$A(\alpha_1,\alpha_2,\alpha_3)=(\alpha_1,\alpha_2,\alpha_3)\begin{bmatrix}2&1&2\\-1&2&4\\-1&3&1\end{bmatrix}=P C$$ + +其中 $C=\begin{bmatrix}2&1&2\\-1&2&4\\-1&3&1\end{bmatrix}$，计算 $|C|$： + +$$\begin{aligned} +|C|&=2\times(2\times1-4\times3)-1\times(-1\times1-4\times(-1))+2\times(-1\times3-2\times(-1))\\ +&=2\times(-10)-1\times3+2\times(-1)\\ +&=-25\neq0 +\end{aligned} +$$ + +• 因 $|C|\neq0$，故 $C$ 可逆，$\text{rank}(C)=3$。 + +• 又 $P$ 可逆，故 $\text{rank}(A\alpha_1,A\alpha_2,A\alpha_3)=\text{rank}(P C)=\text{rank}(C)=3$，即 $A\alpha_1,A\alpha_2,A\alpha_3$ 线性无关。 + +#### **(2) 计算 $|E-A|$** + +由 (1) 知$P^{-1}AP=C（A 与 C 相似）$，则 $E-A$ 与 $E-C$ 相似（相似矩阵的 “单位矩阵减矩阵” 仍相似），而**相似矩阵的行列式相等**，故 $|E-A|=|E-C|$。 + +计算 $E-C$： + +$$E-C=\begin{bmatrix}1-2&0-1&0-2\\0-(-1)&1-2&0-4\\0-(-1)&0-3&1-1\end{bmatrix}=\begin{bmatrix}-1&-1&-2\\1&-1&-4\\1&-3&0\end{bmatrix}$$ + +按第三行展开计算行列式： + +$$ +\begin{aligned} +|E-C|&=1\times\begin{vmatrix}-1&-2\\-1&-4\end{vmatrix}-(-3)\times\begin{vmatrix}-1&-2\\1&-4\end{vmatrix}+0\times(\text{余子式})\\ +&=1\times(4-2)+3\times(4+2)\\ +&=2+18=20 +\end{aligned} +$$ + +故 $|E-A|=20$。 + +### **8. 答案：** + +$$A^{-1}=\begin{bmatrix}-(n-2)&1&1&\cdots&1\\1&-1&0&\cdots&0\\1&0&-1&\cdots&0\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&0&0&\cdots&-1\end{bmatrix} +$$ +#### **解析：** + +通过 “行变换法” 或 “规律归纳” 推导： + +• 观察矩阵 A 的结构：第一行全为 1，其余行的对角元为 0，非对角元为 1。可先计算 n=2,3 时的逆矩阵，归纳规律： + +◦ $当 n=2 时，A=\begin{bmatrix}1&1\\1&0\end{bmatrix}，逆矩阵为 \begin{bmatrix}0&1\\1&-1\end{bmatrix}（符合上述形式，-(2-2)=0）$； + +◦ 当 $n=3$ 时，$A=\begin{bmatrix}1&1&1\\1&0&1\\1&1&0\end{bmatrix}$，逆矩阵为 $\begin{bmatrix}-1&1&1\\1&-1&0\\1&0&-1\end{bmatrix}$（符合上述形式，$-(3-2)=-1$）。 + +• 验证规律：对 n 阶矩阵，逆矩阵的第一行第一列元素为 -(n-2)，第一行其余元素为 1，第一列其余元素为 1，对角元（除第一行第一列）为 -1，非对角元（除第一行、第一列）为 0，即为上述形式。 + +### **9. 证明：$D_n=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]$** + + **步骤 1：建立递推公式** + +对 $D_n$ 按**第一行展开**（第一行元素为 $a_{11}=1,a_{12}=1$，其余 $a_{1j}=0$）： + +$D_n=a_{11}\times(-1)^{1+1}M_{11}+a_{12}\times(-1)^{1+2}M_{12}$ + +• $M_{11}$：去掉第一行第一列后的子式，即 $n-1$ 阶行列式 $D_{n-1}$（结构与 $D_n$ 一致）； + +• $M_{12}$：去掉第一行第二列后的子式，按第一列展开（第一列仅首元素为 $-1$），得 $-D_{n-2}$（符号需结合 $(-1)^{1+2}=-1$）。 + +因此递推公式为： + +$$D_n=D_{n-1}+D_{n-2}$$ + + **步骤 2：确定初始条件** + +• 当 $n=1$ 时，$D_1=\begin{vmatrix}1\end{vmatrix}=1$； + +• 当 n=2 时，$D_2=\begin{vmatrix}1&1\\-1&1\end{vmatrix}=1\times1-1\times(-1)=2$。 +容易证明$n=1,n=2$时满足要证的式子 + +**步骤3：运用数学归纳法** +假设当$n\le k$时，有$D_n=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]$。当$n=k+1$时，有$$ +\begin{aligned} +D_{k+1}&=D_k+D_{k-1}\\ +&=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{k+1}-(\frac{1-\sqrt{5}}{2})^{k+1}+(\frac{1+\sqrt{5}}{2})^k-(\frac{1-\sqrt{5}}{2})^k)\\ +&=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{k}(\frac{3+\sqrt{5}}{2})-(\frac{1-\sqrt{5}}{2})^{k}(\frac{1-\sqrt{5}}{2}))\\ +&=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{k}(\frac{1+\sqrt{5}}{2})^2-(\frac{1-\sqrt{5}}{2})^{k}(\frac{1-\sqrt{5}}{2})^2)\\ +&=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{k+2}-(\frac{1-\sqrt{5}}{2})^{k+2}) +\end{aligned} +$$满足条件，故由数学归纳法知，$\forall{n}\in\mathbb{N}_+,D_n=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]$. \ No newline at end of file diff --git a/编写小组/试卷/线代期末复习模拟/1.13线代限时练.md b/编写小组/试卷/线代期末复习模拟/1.13线代限时练.md new file mode 100644 index 0000000..111f58e --- /dev/null +++ b/编写小组/试卷/线代期末复习模拟/1.13线代限时练.md @@ -0,0 +1,66 @@ +1. （[[线代2022秋B]]·1）已知 $\begin{vmatrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{vmatrix} = c$ ，代数余子式之和 $\sum\limits_{i=1}^{3}\sum\limits_{j=1}^{3} A_{ij} = 3c$ ，则 $\begin{vmatrix}a_{11}+1 & a_{12}+1 & a_{13}+1 \\a_{21}+1 & a_{22}+1 & a_{23}+1 \\a_{31}+1 & a_{32}+1 & a_{33}+1\end{vmatrix} =$ \_\_\_\_\_\_\_\_\_\_. + +2. （[[线代2013秋A]]·4）已知向量空间 $V=\{(2a,2b,3b,3a)|a,b\in\mathbb{R}\}$，则 $V$ 的维数是\_\_\_\_\_. + +3. （[[线代2019秋A]]·9）设 $\boldsymbol{E}$ 为 $3$ 阶单位矩阵，$\boldsymbol\alpha$ 为一个 $3$ 维单位列向量，则矩阵 $\boldsymbol E - \boldsymbol\alpha\boldsymbol\alpha^\mathrm{T}$ 的全部 $3$ 个特征值为\_\_\_\_\_\_\_. + +4. （[[线代2018秋A]]·12）设 $n$ 阶矩阵 $A=[a_{ij}]_{n\times n}$ ，则二次型 $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^n(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in}x_n)^2$ 的矩阵为\_\_\_\_\_\_\_. + +5. （[[线代2018秋A]]·11）若 $n$ 阶实对称矩阵 $\boldsymbol A$ 的特征值为 $\lambda_i=(-1)^i\ (i=1,2,\cdots,n)$，则 $\boldsymbol A^{100}=$ \_\_\_\_\_\_\_. + +6. ([[线代2022秋A]]·5）已知 $n(n\ge2)$ 维列向量 $\boldsymbol\alpha,\boldsymbol\beta$ 满足 $\boldsymbol\beta^\mathrm{T}\boldsymbol\alpha=-3$，则方阵 $(\boldsymbol{\beta\alpha}^\mathrm{T})^2$ 的非零特征值为\_\_\_\_\_\_\_\_\_\_. + +7. （[[线代2022秋A]]·七）已知向量组 $\boldsymbol\alpha_1,\boldsymbol\alpha_2,\boldsymbol\alpha_3$ 线性无关，其中 $\boldsymbol\alpha_1,\boldsymbol\alpha_2,\boldsymbol\alpha_3\in\mathbb{R}^3$，$\boldsymbol A$ 为 $3$ 阶方阵，且 +$$\boldsymbol{A\alpha_1}=2\boldsymbol\alpha_1-\boldsymbol\alpha_2-\boldsymbol\alpha_3, \boldsymbol A\boldsymbol\alpha_2=\boldsymbol\alpha_1+2\boldsymbol\alpha_2+3\boldsymbol\alpha_3, \boldsymbol A\boldsymbol\alpha_3=2\boldsymbol\alpha_1+4\boldsymbol\alpha_2+\boldsymbol\alpha_3.$$ +(1) 证明 $A\boldsymbol\alpha_1, A\boldsymbol\alpha_2, A\boldsymbol\alpha_3$ 线性无关； +(2) 计算行列式 $\boldsymbol E-\boldsymbol A$，其中 $\boldsymbol E$ 是 $3$ 阶单位矩阵. +```text + + + + + + + + + + + + + + + + + + + + + +``` + +8. （[[线代2013秋A]]）求 $n$ 阶方阵 $\boldsymbol{A} = \begin{bmatrix}1 & 1 & 1 & \dots & 1 \\1 & 0 & 1 & \dots & 1 \\1 & 1 & 0 & \dots & 1 \\\vdots & \vdots & \vdots & \ddots & \vdots \\1 & 1 & 1 & \dots & 0\end{bmatrix}$ 的逆。 +```text + + + + + + + + + + + + + + + + + + + + +``` + +9. （[[线代2013秋A]]·三）设 + $$D_n=\begin{vmatrix}1&1&0&\cdots&0&0\\-1&1&1&\cdots&0&0\\0&-1&1&\cdots&0&0\\\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&0&\cdots&1&1\\0&0&0&\cdots&-1&1\end{vmatrix}$$证明：$D_n=\frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^{n+1}-(\frac{1-\sqrt{5}}{2})^{n+1}]$ . \ No newline at end of file diff --git a/编写小组/试卷/高数期中试卷.md b/编写小组/试卷/高数期中试卷.md new file mode 100644 index 0000000..ffed476 --- /dev/null +++ b/编写小组/试卷/高数期中试卷.md @@ -0,0 +1,46 @@ +#官方试卷 +时量：120分钟 满分：100分 +#### 一、单选题（共5小题，每小题3分，共15分） +1. 若$\lim\limits_{n\rightarrow\infty}a_n=4$，则当$n$充分大时，恒有 + A. $|a_n|\le 1$ + B. $|a_n|>2$ + C. $|a_n|<2$ + D. $|a_n|>4$ +2. 已知$g(x)=\frac{1}{x^2}$，复合函数$y=f(g(x))$对$x$的导数为$-\frac{1}{2x}$，则$f'(\frac{1}{2})$的值为 + A. $1$ + B. $2$ + C. $\frac{\sqrt 2}{4}$ + D. $\frac{1}{2}$ +3. 设$f(x)$可导且$f'(x_0)=\frac{1}{3}$，则当$\Delta x \rightarrow 0$时，$f(x)$在$x_0$处的微分$\text{d}y$是$\Delta x$的 + A. 同阶无穷小 + B. 低阶无穷小 + C. 等价无穷小 + D. 高阶无穷小 +4. 设数列通项为$x_n=\begin{cases}\frac{n^2-\sqrt{n}}{n},n=2k, \\ \frac{1}{n},n=2k+1 \end{cases}(k\in \mathbb{N}^+)$，则当$n\rightarrow\infty$时，$x_n$是 + A. 无穷大量 + B. 无穷小量 + C. 有界变量 + D. 无界变量 +5. 下列四个级数，**发散**的是 + A. $\sum\limits_{n=1}^{\infty}{(a^{\frac{1}{n}}+a^{-\frac{1}{n}}-2)}(a>0)$ + B. $\sum\limits_{n=1}^{\infty}\frac{1}{n^{2n\sin{\frac{1}{n}}}}$ + C. $\sum\limits_{n=1}^{\infty}{(\frac{\cos{n!}}{n^2+1}+\frac{1}{\sqrt{n+12}})}$ + D. $\sum\limits_{n=1}^{\infty}\frac{[\sqrt{2}+(-1)^n]^n}{3^n}$ +#### 二、填空题（共5小题，每小题3分，共15分） +6. 函数$f(x)=\frac{\mathrm{e}^\frac{1}{x-1} \ln{|1+x|}}{(\mathrm{e}^x-1)(x-2)}$的第二类间断点的个数为____. +7. 设函数$y=x-\frac{1}{\sqrt{2}}\arctan{(\sqrt{2}\tan x)}$，则$\text{d}y|_{x=\frac{\pi}{4}}=$\_\_\_\_\_. +8. 已知级数$\sum\limits_{n=2}^{\infty}{(-1)^n\frac{1}{n^a}\ln\frac{n+1}{n-1}}$条件收敛，则常数$a$的取值范围是_____. +9. 已知函数$f(x)$在$x=1$处可导，且$f(1)=0,f'(1)=2$，则$\lim\limits_{x\rightarrow 0}{\frac{f(e^{x^2})}{\sin^2 x}}$=\_\_\_\_\_\_. +10. 曲线$y=\frac{1}{2}x^2$与曲线$y=c\ln x$相切，则常数$c$的值为\_\_\_\_\_\_. +#### 三、解答与证明题（11~19小题，共70分） +11. （6分）求极限$\lim\limits_{x\rightarrow 0}(\cos 2x + x\arcsin x)^\frac{1}{x^2}$. +12. （6分）求极限$\lim\limits_{n\rightarrow\infty}\sqrt[n+2]{2\sin^2 n+\cos^2 n}$. +13. （6分）已知函数$f(x)=\begin{cases}\sin 2x+1, x\le 0\\a^{2x}+b, x>0 \end{cases}$在$x=0$处可导，求常数$a,b$的值. +14. （6分）设$y=y(x)$是由方程$y=x\mathrm{e}^y=1$所确定的函数，求曲线$y=y(x)$在$x=0$对应点处的切线方程. +15. （8分）设函数$y=f(x)$的极坐标式为$\rho=\mathrm{e}^\theta$，求$\frac{\mathrm{d}y}{\mathrm{d}x}|_{\theta=\frac{\pi}{2}}$，$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}|_{\theta=\frac{\pi}{2}}$. +16. （8分）设$f(x)$在$(-\infty,+\infty)$内是以$2T$为周期的连续函数，证明对任一实数$x_0$，方程$f(x)=f(x+T)$在区间$[x_0-\frac{T}{2},x_0+\frac{T}{2}]$上至少有一个根. +17. （10分）求曲线$y=\frac{x^2-3x+2}{x-1}+2\ln|x-2|,(x>0)$的所有渐近线方程. +18. （10分）百米跑道上正在举行百米赛跑，摄影师站在50米处为4号跑道夺冠种子选手A录像，摄像头始终对准选手A，摄影师距离4号跑道5米，若选手A以10m/s的速度从摄影师正前方经过，问此时摄影镜头的角速度是多少？ ![[期中试卷-18.png]] +19. （10分）设$a_1=3,a_{n+1}=\frac{a_n}{2}+\frac{1}{n},n=1,2,\dots,$ + (1) 证明$\lim\limits_{n\rightarrow\infty}a_n$存在，并求其极限值；（5分） + (2) 证明：对于任意实数$p$，级数$\sum\limits_{n=1}{\infty}{n^p(\frac{a_n}{a_{n+1}}-1)}$收敛. （5分） \ No newline at end of file diff --git a/编写小组/试卷/高数期中试卷解析.md b/编写小组/试卷/高数期中试卷解析.md new file mode 100644 index 0000000..4874412 --- /dev/null +++ b/编写小组/试卷/高数期中试卷解析.md @@ -0,0 +1,62 @@ +#官方试卷 +#民间答案 +#### 一、单选题（共5小题，每小题3分，共15分） +1. 若$\lim\limits_{n\rightarrow\infty}a_n=4$，则当$n$充分大时，恒有 + A. $|a_n|\le 1$ + B. $|a_n|>2$ + C. $|a_n|<2$ + D. $|a_n|>4$ +> **答案：B** +> 解析：保号性 +> 若$\lim\limits_{n\rightarrow\infty}a_n=4$，按照定义，$\forall \epsilon >0,\exists N, n>N\text{时},|a_n-4|<\epsilon$， +> 取$\epsilon=2$，则$|a_n-4|<\epsilon$，则$2<a_n<6$，B选项符合题意 +2. 已知$g(x)=\frac{1}{x^2}$，复合函数$y=f(g(x))$对$x$的导数为$-\frac{1}{2x}$，则$f'(\frac{1}{2})$的值为 + A. $1$ + B. $2$ + C. $\frac{\sqrt 2}{4}$ + D. $\frac{1}{2}$ +>**答案：D** +>解析：复合函数求导法则 +>$y'=f'(g(x))g'(x)=\frac{-2}{x^3}f'(\frac{1}{x^2})=-\frac{1}{2x}\Rightarrow f'(\frac{1}{x^2})=\frac{x^2}{4}$，即$f'(\frac{1}{x})=\frac{x}{4}(x> 0)$ +>代入$x=2$得：$f'(\frac{1}{2})=\frac{1}{2}$ +3. 设$f(x)$可导且$f'(x_0)=\frac{1}{3}$，则当$\Delta x \rightarrow 0$时，$f(x)$在$x_0$处的微分$\text{d}y$是$\Delta x$的 + A. 同阶无穷小 + B. 低阶无穷小 + C. 等价无穷小 + D. 高阶无穷小 +> **答案：A** +> 解析：同阶无穷小；微分的定义 +> $\Delta x \rightarrow 0$时，$\Delta x \sim \mathrm{d}x$，又$\mathrm{d}y|_{x=x_0}=\frac{1}{3}\mathrm{d}x$，故$\mathrm{d}y$与$\Delta{x}$是同阶不等价的无穷小 +4. 设数列通项为$x_n=\begin{cases}\frac{n^2-\sqrt{n}}{n},n=2k, \\ \frac{1}{n},n=2k+1 \end{cases}(k\in \mathbb{N}^+)$，则当$n\rightarrow\infty$时，$x_n$是 + A. 无穷大量 + B. 无穷小量 + C. 有界变量 + D. 无界变量 +>**答案：D** +>解析：易错点10-无穷大与无界的辨析 +>$\lim\limits_{k\to\infty}a_{2k}=+\infty$，因此$a_n$无界；然而，$\lim\limits_{k\to\infty}a_{2k+1}=0$，因此$a_n(n\to\infty)$不是无穷大 +5. 下列四个级数，**发散**的是 + A. $\sum\limits_{n=1}^{\infty}{(a^{\frac{1}{n}}+a^{-\frac{1}{n}}-2)}(a>0)$ + B. $\sum\limits_{n=1}^{\infty}\frac{1}{n^{2n\sin{\frac{1}{n}}}}$ + C. $\sum\limits_{n=1}^{\infty}{(\frac{\cos{n!}}{n^2+1}+\frac{1}{\sqrt{n+12}})}$ + D. $\sum\limits_{n=1}^{\infty}\frac{[\sqrt{2}+(-1)^n]^n}{3^n}$ +>**答案：C** +> +#### 二、填空题（共5小题，每小题3分，共15分） +6. 函数$f(x)=\frac{\mathrm{e}^\frac{1}{x-1} \ln{|1+x|}}{(\mathrm{e}^x-1)(x-2)}$的第二类间断点的个数为____. +7. 设函数$y=x-\frac{1}{\sqrt{2}}\arctan{(\sqrt{2}\tan x)}$，则$\text{d}y|_{x=\frac{\pi}{4}}=$\_\_\_\_\_. +8. 已知级数$\sum\limits_{n=2}^{\infty}{(-1)^n\frac{1}{n^a}\ln\frac{n+1}{n-1}}$条件收敛，则常数$a$的取值范围是_____. +9. 已知函数$f(x)$在$x=1$处可导，且$f(1)=0,f'(1)=2$，则$\lim\limits_{x\rightarrow 0}{\frac{f(e^{x^2})}{\sin^2 x}}$=\_\_\_\_\_\_. +10. 曲线$y=\frac{1}{2}x^2$与曲线$y=c\ln x$相切，则常数$c$的值为\_\_\_\_\_\_. +#### 三、解答与证明题（11~19小题，共70分） +11. （6分）求极限$\lim\limits_{x\rightarrow 0}(\cos 2x + x\arcsin x)^\frac{1}{x^2}$. +12. （6分）求极限$\lim\limits_{n\rightarrow\infty}\sqrt[n+2]{2\sin^2 n+\cos^2 n}$. +13. （6分）已知函数$f(x)=\begin{cases}\sin 2x+1, x\le 0\\a^{2x}+b, x>0 \end{cases}$在$x=0$处可导，求常数$a,b$的值. +14. （6分）设$y=y(x)$是由方程$y=x\mathrm{e}^y=1$所确定的函数，求曲线$y=y(x)$在$x=0$对应点处的切线方程. +15. （8分）设函数$y=f(x)$的极坐标式为$\rho=\mathrm{e}^\theta$，求$\frac{\mathrm{d}y}{\mathrm{d}x}|_{\theta=\frac{\pi}{2}}$，$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}|_{\theta=\frac{\pi}{2}}$. +16. （8分）设$f(x)$在$(-\infty,+\infty)$内是以$2T$为周期的连续函数，证明对任一实数$x_0$，方程$f(x)=f(x+T)$在区间$[x_0-\frac{T}{2},x_0+\frac{T}{2}]$上至少有一个根. +17. （10分）求曲线$y=\frac{x^2-3x+2}{x-1}+2\ln|x-2|,(x>0)$的所有渐近线方程. +18. （10分）百米跑道上正在举行百米赛跑，摄影师站在50米处为4号跑道夺冠种子选手A录像，摄像头始终对准选手A，摄影师距离4号跑道5米，若选手A以10m/s的速度从摄影师正前方经过，问此时摄影镜头的角速度是多少？ ![[期中试卷-18.png]] +19. （10分）设$a_1=3,a_{n+1}=\frac{a_n}{2}+\frac{1}{n},n=1,2,\dots,$ + (1) 证明$\lim\limits_{n\rightarrow\infty}a_n$存在，并求其极限值；（5分） + (2) 证明：对于任意实数$p$，级数$\sum\limits_{n=1}{\infty}{n^p(\frac{a_n}{a_{n+1}}-1)}$收敛. （5分） \ No newline at end of file diff --git a/编写小组/试卷/高数期末复习模拟/1.10高数限时练.md b/编写小组/试卷/高数期末复习模拟/1.10高数限时练.md new file mode 100644 index 0000000..536e7fe --- /dev/null +++ b/编写小组/试卷/高数期末复习模拟/1.10高数限时练.md @@ -0,0 +1,133 @@ +--- +tags: + - 高数复习模拟 +--- +### 一、单选题（共3小题，每小题6分，共18分） + +1. 函数$f(x) = \frac{|x+1|}{x(x-1)(x+1)} \arctan x + \sin \frac{1}{x-2}$的可去间断点为（ ）。 + (A)$x = 0$ + (B)$x = 1$ + (C)$x = 2$ + (D)$x = -1$ + +2. “级数$\sum_{n=1}^\infty a_n$收敛”是“级数$a_1 - a_1 + a_2 - a_2 + \cdots + a_n - a_n + \cdots$收敛”的（ ）。 + (A) 充分但非必要条件 + (B) 必要但非充分条件 + (C) 充要条件 + (D) 既非充分也非必要条件 + +3. 已知函数$f(x) = x^2 \sin \frac{1}{x}$，则下列四个判断： + ①$f(x)$在$(0, +\infty)$上有界 + ②$f(x)$是$x \to +\infty$过程的无穷大量 + ③$f'(x)$在$(0, +\infty)$上有界 + ④$f'(x)$是$x \to +\infty$过程的无穷大量 + 正确的是（ ）。 + (A) ①③ + (B) ①④ + (C) ②③ + (D) ②④ + +--- + +### 二、填空题（共3小题，每小题6分，共18分） + +4. 设$y = y(x)$是由方程$xy + e^{2y} = \sin 3x + 1$确定的隐函数，则$dy|_{x=0} = \underline{\qquad}$。 + +5. 极限 + $$ + \lim_{n \to \infty} \left( \frac{1}{\sqrt{3n^4 + 1}} + \frac{2}{\sqrt{3n^4 + 2}} + \cdots + \frac{n}{\sqrt{3n^4 + n}} \right) + $$ + 的值为 $\underline{\qquad}$。 + +6. 已知 +$$\int x f(x) \, dx = \ln \left( 1 + x^2 \right) + C$$ + 则曲线$y = f(x) (x > 0)$的拐点为 $\underline{\qquad}$。 + +--- + +### 三、解答题（共4小题，共64分） + +7. （16分）求极限 + $$ + \lim_{x \to 0} \frac{e^{x^2} - 1 - \ln \left( x^2 + 1 \right)}{x^3 \arcsin x}. + $$ +```text + + + + + + + + + + + + + + + +``` +8. （16分）设函数 + $$ + f(x) = + \begin{cases} + |x|^x, & x \neq 0, \\ + a, & x = 0, + \end{cases} + $$ + 试求常数$a$的值，使得$f(x)$在$x = 0$处连续，并讨论此时$f(x)$在$x = 0$处的可导性。 +```text + + + + + + + + + + + + + + + +``` +9. （16分）将函数$f(x) = x^2 e^x + x^6$展开成六阶带佩亚诺余项的麦克劳林公式，并求$f^{(6)} (0)$的值。 +```text + + + + + + + + + + + + + + + +``` +10. （16分）已知函数$f(x)$在$[0,1]$上可导，$f(0)=0$。证明：至少存在一点$\xi \in (0,1)$，使得 + $$f'(1-\xi)=\frac{2}{\xi}f(1-\xi).$$ +```text + + + + + + + + + + + + + + + +``` \ No newline at end of file diff --git a/编写小组/试卷/高数期末复习模拟/1.17高数限时练.md b/编写小组/试卷/高数期末复习模拟/1.17高数限时练.md new file mode 100644 index 0000000..24e7993 --- /dev/null +++ b/编写小组/试卷/高数期末复习模拟/1.17高数限时练.md @@ -0,0 +1,137 @@ +--- +tags: + - 高数复习模拟 +--- +### 一、选择题（共3小题，每小题6分，共18分） + +1. 下列级数发散的是（　）。 + + (A) $\sum\limits_{n=1}^\infty \frac{4^n + 1}{n^4 + 1}$ + (B) $\sum\limits_{n=1}^\infty \frac{n^4 + 1}{4^n + 1}$ + (C) $\sum\limits_{n=1}^\infty (-1)^n \frac{1}{\sqrt{n} + 1}$ + (D) $\sum\limits_{n=1}^\infty (-1)^n \frac{\sqrt{n}}{n + 1}$ + +2. 当 $x \to 0$时，与 $x - \sin x$同阶的无穷小是（　）。 + + (A) $x + \tan x$ + (B) $x \tan x$ + (C) $x^2 + \tan x$ + (D) $x^2 \tan x$ + +3. 已知函数 $f(x)$在 $x = 1$的某邻域内三阶可导，且 + $$ + \lim_{x \to 1} \frac{f(x) - 2}{(x - 1)^2 \ln x} = \frac{1}{3}, + $$ + 则（　）。 + (A) $x = 1$为函数 $f(x)$的极大值点 + (B) 点 $(1, 2)$为曲线 $y = f(x)$的拐点 + (C) $x = 1$为函数 $f(x)$的极小值点 + (D) 点 $(1, 0)$为曲线 $y = f(x)$的拐点 + +--- + +### 二、填空题（共3小题，每小题6分，共18分） + +4. 极限 $$\lim_{n \to \infty} \frac{1}{n} \left( \tan \frac{\pi}{4n} + \tan \frac{2\pi}{4n} + \cdots + \tan \frac{n\pi}{4n} \right)$$ 的值为______。 + +5. 已知 $\int f(x^2) dx = x \ln x + C，$则 $f'(2) = \underline{\quad }。$ + +6. 极坐标曲线 $\rho = \theta$ 在点 $(\rho, \theta) = (\pi, \pi)$ 处的切线的直角坐标方程为______ + +--- + +### 三、解答题（共4小题，共64分） + +7. 计算数列极限 + $$ + \lim_{x \to +\infty} (x^2 + 2^x)^{\frac{1}{x}}。 + $$ + （12分） + +```text + + + + + + + + + + + + + + + +``` + +7. (16分)已知函数 $f(x)$ 在 $[0,1]$上连续，在 $(0,1)$内可导，且 $f(0)=0$ ，$f(1)=1$ 。试证明： + （1）（6分）存在 $\xi \in (0,1)$，使得 + $$ + f'(\xi) = 2\xi。 + $$ + （2）（10分）对任意正数 $a,b$，在 $(0,1)$内存在相异的两点 $x_1, x_2$，使得 + $$ + \frac{a}{f'(x_1)} + \frac{b}{f'(x_2)} = a+b。 + $$ + +```text + + + + + + + + + + + + + + + +``` + +7. （16分）已知当 $x \to 0$ 时，函数 $f(x) = \sqrt{a + bx^2} - \cos x$ 与 $x^2$ 是等价无穷小。 + （1）求参数 $a, b$ 的值；（6分） + （2）计算极限 $\lim\limits_{x \to 0} \frac{f(x) - x^2}{x^4}$的值。（10分） + +```text + + + + + + + + + + + + + + + +``` + +8. （20分）设函数 $f(x)$在 $[0,1]$上可导，$\large{\int_{0}^{\frac{1}{2}}} f(x) dx = 0$。证明存在 $\xi \in (0,1)$，使得 +$$f(\xi) = (1 - \xi) f'(\xi)。$$ +```text + + + + + + + + + + + + + + + +``` \ No newline at end of file diff --git a/编写小组/试卷/高数期末复习模拟/1.17高数限时练（解析版）.md b/编写小组/试卷/高数期末复习模拟/1.17高数限时练（解析版）.md new file mode 100644 index 0000000..9d6c9e1 --- /dev/null +++ b/编写小组/试卷/高数期末复习模拟/1.17高数限时练（解析版）.md @@ -0,0 +1,288 @@ +--- +tags: + - 高数复习模拟 +--- +### 一、选择题（共3小题，每小题6分，共18分） + +1. 下列级数发散的是（　）。 + + (A) $\sum\limits_{n=1}^\infty \frac{4^n + 1}{n^4 + 1}$ + (B) $\sum\limits_{n=1}^\infty \frac{n^4 + 1}{4^n + 1}$ + (C) $\sum\limits_{n=1}^\infty (-1)^n \frac{1}{\sqrt{n} + 1}$ + (D) $\sum\limits_{n=1}^\infty (-1)^n \frac{\sqrt{n}}{n + 1}$ + +2. 当 $x \to 0$时，与 $x - \sin x$同阶的无穷小是（　）。 + + (A) $x + \tan x$ + (B) $x \tan x$ + (C) $x^2 + \tan x$ + (D) $x^2 \tan x$ + +3. 已知函数 $f(x)$在 $x = 1$的某邻域内三阶可导，且 + $$ + \lim_{x \to 1} \frac{f(x) - 2}{(x - 1)^2 \ln x} = \frac{1}{3}, + $$ + 则（　）。 + (A) $x = 1$为函数 $f(x)$的极大值点 + (B) 点 $(1, 2)$为曲线 $y = f(x)$的拐点 + (C) $x = 1$为函数 $f(x)$的极小值点 + (D) 点 $(1, 0)$为曲线 $y = f(x)$的拐点 + +--- + +### 二、填空题（共3小题，每小题6分，共18分） + +4. 极限 $$\lim_{n \to \infty} \frac{1}{n} \left( \tan \frac{\pi}{4n} + \tan \frac{2\pi}{4n} + \cdots + \tan \frac{n\pi}{4n} \right)$$ 的值为______。 + +5. 已知 $\int f(x^2) dx = x \ln x + C，$则 $f'(2) = \underline{\quad }。$ + +6. 极坐标曲线 $\rho = \theta$ 在点 $(\rho, \theta) = (\pi, \pi)$ 处的切线的直角坐标方程为______ + +--- + +### 三、解答题（共4小题，共64分） + +7. 计算数列极限 + $$ + \lim_{x \to +\infty} (x^2 + 2^x)^{\frac{1}{x}}。 + $$ + （12分） + +```text + + + + + + + + + + + + + + + +``` + +7. (16分)已知函数 $f(x)$ 在 $[0,1]$上连续，在 $(0,1)$内可导，且 $f(0)=0$ ，$f(1)=1$ 。试证明： + （1）（6分）存在 $\xi \in (0,1)$，使得 + $$ + f'(\xi) = 2\xi。 + $$ + （2）（10分）对任意正数 $a,b$，在 $(0,1)$内存在相异的两点 $x_1, x_2$，使得 + $$ + \frac{a}{f'(x_1)} + \frac{b}{f'(x_2)} = a+b。 + $$ + +```text + + + + + + + + + + + + + + + +``` + +7. （16分）已知当 $x \to 0$ 时，函数 $f(x) = \sqrt{a + bx^2} - \cos x$ 与 $x^2$ 是等价无穷小。 + （1）求参数 $a, b$ 的值；（6分） + （2）计算极限 $\lim\limits_{x \to 0} \frac{f(x) - x^2}{x^4}$的值。（10分） + +```text + + + + + + + + + + + + + + + +``` + +8. （20分）设函数 $f(x)$在 $[0,1]$上可导，$\int_{0}^{\frac{1}{2}} f(x) dx = 0$。证明存在 $\xi \in (0,1)$，使得 +$$f(\xi) = (1 - \xi) f'(\xi)。$$ +```text + + + + + + + + + + + + + + + +``` + +# 高等数学（Ⅰ）考试试卷（A）卷 大题解答 + +## 7. 计算极限 +$$ +\lim_{x \to +\infty} (x^2 + 2^x)^{\frac{1}{x}} +$$ + +**解：** +令$L = \lim\limits_{x \to +\infty} (x^2 + 2^x)^{\frac{1}{x}}$，取对数得： +$$ +\ln L = \lim_{x \to +\infty} \frac{\ln(x^2 + 2^x)}{x} +$$ + +当$x \to +\infty$时，$2^x \gg x^2$，故： +$$ +x^2 + 2^x \sim 2^x +$$ +$$ +\ln(x^2 + 2^x) = x\ln 2 + \ln\left(1 + \frac{x^2}{2^x}\right) \sim x\ln 2 +$$ +因此： +$$ +\ln L = \lim_{x \to +\infty} \frac{x\ln 2}{x} = \ln 2 +$$ +$$ +L = e^{\ln 2} = 2 +$$ + +**答案：**$\boxed{2}$ + +--- + +## 8. 中值定理证明题 + +已知$f(x)$在$[0,1]$上连续，$(0,1)$内可导，$f(0)=0$，$f(1)=1$。 + +**(1)** 证明存在$\xi \in (0,1)$使$f'(\xi) = 2\xi$ + +**证：** +构造$g(x) = f(x) - x^2$ +-$g(0) = f(0) - 0 = 0$ +-$g(1) = f(1) - 1 = 0$ + +由罗尔定理，存在$\xi \in (0,1)$使$g'(\xi) = 0$，即： +$$ +f'(\xi) - 2\xi = 0 \quad \Rightarrow \quad f'(\xi) = 2\xi +$$ + +--- + +**(2)** 对任意$a,b > 0$，存在相异$x_1, x_2 \in (0,1)$使： +$$ +\frac{a}{f'(x_1)} + \frac{b}{f'(x_2)} = a+b +$$ + +**证：** +由介值定理，存在$c \in (0,1)$使$f(c) = \dfrac{a}{a+b}$（因$0 < \frac{a}{a+b} < 1$） + +1. 在$[0,c]$上用拉格朗日中值定理： +$$ +f'(x_1) = \frac{f(c)-f(0)}{c-0} = \frac{\frac{a}{a+b}}{c}, \quad x_1 \in (0,c) +$$ +$$ +\frac{a}{f'(x_1)} = a \cdot \frac{c}{\frac{a}{a+b}} = c(a+b) +$$ + +2. 在$[c,1]$上用拉格朗日中值定理： +$$ +f'(x_2) = \frac{f(1)-f(c)}{1-c} = \frac{1 - \frac{a}{a+b}}{1-c} = \frac{\frac{b}{a+b}}{1-c}, \quad x_2 \in (c,1) +$$ +$$ +\frac{b}{f'(x_2)} = b \cdot \frac{1-c}{\frac{b}{a+b}} = (1-c)(a+b) +$$ + +相加得： +$$ +\frac{a}{f'(x_1)} + \frac{b}{f'(x_2)} = c(a+b) + (1-c)(a+b) = a+b +$$ +且$x_1 < c < x_2$，故$x_1 \neq x_2$。 + +--- + +## 9. 等价无穷小与参数确定 + +已知$f(x) = \sqrt{a + bx^2} - \cos x \sim x^2 \ (x\to 0)$ + +**(1)** 求$a, b$ + +**解：** +展开至$x^4$： +$$ +\sqrt{a+bx^2} = \sqrt{a} + \frac{b\sqrt{a}}{2a}x^2 - \frac{b^2\sqrt{a}}{8a^2}x^4 + o(x^4) +$$ +$$ +\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} + o(x^4) +$$ +代入$f(x)$： +$$ +f(x) = (\sqrt{a}-1) + \left(\frac{b}{2\sqrt{a}} + \frac{1}{2}\right)x^2 + \left(-\frac{b^2}{8a^{3/2}} - \frac{1}{24}\right)x^4 + o(x^4) +$$ + +由$f(x) \sim x^2$得： +1.$\sqrt{a}-1 = 0 \Rightarrow a = 1$ +2.$\frac{b}{2} + \frac{1}{2} = 1 \Rightarrow b = 1$ + +**(2)** 计算$\lim\limits_{x\to 0} \dfrac{f(x) - x^2}{x^4}$ + +**解：** +$a=1, b=1$时： +$$ +f(x) = \sqrt{1+x^2} - \cos x = x^2 - \frac{1}{6}x^4 + o(x^4) +$$ +$$ +f(x) - x^2 = -\frac{1}{6}x^4 + o(x^4) +$$ +$$ +\lim_{x\to 0} \frac{f(x) - x^2}{x^4} = -\frac{1}{6} +$$ + +**答案：** +(1)$a=1, b=1$ +(2)$-\dfrac{1}{6}$ + +--- + +## 10. 积分条件与微分方程形式的中值定理 + +已知$f(x)$在$[0,1]$上可导，$\large{\int}_0^{1/2} f(x) dx = 0$ +证明存在$\xi \in (0,1)$使$f(\xi) = (1-\xi)f'(\xi)$ + +**证：** +令$g(x) = (1-x)f(x)$，则： +$$ +g'(x) = -f(x) + (1-x)f'(x) +$$ +要证等式等价于$g'(\xi) = 0$ + +设$h(x) = \large{\int_0^x }f(t)dt$，由已知$h(1/2) = 0$，且$h(0)=0$ +由罗尔定理，存在$c \in (0,1/2)$使$h'(c) = f(c) = 0$ + +于是： +$$ +g(c) = (1-c)f(c) = 0, \quad g(1) = (1-1)f(1) = 0 +$$ +在$[c,1]$上对$g(x)$应用罗尔定理，存在$\xi \in (c,1) \subset (0,1)$使： +$$ +g'(\xi) = 0 \quad \Rightarrow \quad f(\xi) = (1-\xi)f'(\xi) +$$ + +证毕。 \ No newline at end of file diff --git a/编写小组/黑马试卷/1.16黑马试卷.md b/编写小组/黑马试卷/1.16黑马试卷.md new file mode 100644 index 0000000..bfb7100 --- /dev/null +++ b/编写小组/黑马试卷/1.16黑马试卷.md @@ -0,0 +1,126 @@ +### 一、选择题（共3小题，每小题2分，共10分） + +1. 函数 +$$ + f(x) = \begin{cases} + \sqrt{x} \sin \frac{1}{x}, & x > 0, \\ + 0, & x \leq 0 + \end{cases} +$$ + 在点$x = 0$处（ ）。 + (A) 不连续 + (B) 连续但不可导 + (C) 可导且$f'(0) = 0$ + (D) 可导且$f'(0) \neq 0$ + +2. 数列极限$\lim_{n \to \infty} (e^{-n} + \pi^{-n})^{\frac{1}{n}}$的值为（ ）。 + (A)$e$ + (B)$\pi$ + (C)$\frac{1}{e}$ + (D)$\frac{1}{\pi}$ + +3. 曲线$y = \frac{x^3 - x^2}{2 + x^2}$的渐近线为（ ）。 + (A)$y = x - 1$ + (B)$y = x + 1$ + (C)$y = x$ + (D)$y = \frac{1}{2}x$ + +--- + +### 二、填空题（共3小题，每小题2分，共10分） + +4. 函数$f(x) = xe^{-x^2}$在$(-\infty,+\infty)$上的最大值为 $\underline{\qquad}$。 + +5. 曲线$C: x = \frac{1}{2} \cos t, y = \sin t, t \in [0,2\pi]$在点$(0,-1)$处的曲率为 $\underline{\qquad}$。 + +6. 不定积分$\int \frac{1}{x(1+2\ln x)} dx = \underline{\qquad}$。 + +--- + +### 三、解答题（共4小题，共80分） + +7. 设$y(x)$是由曲线方程$\sin x + y + e^x = 2$确定的隐函数，试计算$\frac{dy}{dx} \bigg|_{x=0}$的值，并求该曲线在点$P(0,1)$处的切线方程。（6分） + +```text + + + + + + + + + + + + + + + + + + + + + +``` + +8. 计算不定积分 +$$ + \int \frac{x}{1+\sqrt{1-x^2}} dx。 +$$ + （6分） + +```text + + + + + + + + + + + + + + + + + + + + + +``` + +9. 设曲线$f(x) = x^3 + ax^2 + 18x$（$a$为大于零的常数）的拐点正好位于$x$轴上，试求$a$的值及曲线$y = f(x)$的拐点坐标。（6分） + +```text + + + + + + + + + + + + + + + + + + + + + +``` + +10. 计算极限 + $$\lim_{x \to +\infty} \left[ x + x^2 \ln \left( 1 - \frac{1}{x} \right) \right]。$$ + （6分） diff --git a/试卷库/线性代数/线代2013秋A.md b/试卷库/线性代数/线代2013秋A.md new file mode 100644 index 0000000..83209e6 --- /dev/null +++ b/试卷库/线性代数/线代2013秋A.md @@ -0,0 +1,65 @@ +## 一、填空题（共6小题，每小题3分，共18分） +1. 设行列式 $D=\begin{vmatrix}-1&2&-3\\1&2&0\\-1&3&2\end{vmatrix}$ ，则 $M_{12}+A_{21}-M_{32}=$ ______ +2. 设矩阵 $\boldsymbol{A}=\begin{bmatrix}\boldsymbol{B}&\boldsymbol{C}\\\boldsymbol{0}&\boldsymbol{D}\end{bmatrix}$ ，其中 $\boldsymbol{B}$ 、 $\boldsymbol{D}$ 皆为可逆矩阵，则 $\boldsymbol{A}^{-1}=$ ______ +3. 设矩阵 $\boldsymbol{A}=\begin{bmatrix}2&1&0&0\\0&2&0&0\\0&0&-1&2\\0&0&-2&4\end{bmatrix}$ ， $n$ 为正整数，则 $\boldsymbol{A}^{n}=$ ______ +4. 已知向量空间 $V = \{(2a,2b,3b,3a)\mid a,b\in\mathbb{R}\}$ ，则 $V$ 的维数是______ +5. 已知矩阵 $\boldsymbol{A}=\begin{bmatrix}-2&1&1\\0&2&0\\-4&1&3\end{bmatrix}$ ，则 $\boldsymbol{A}$ 的特征值 $2$ 的几何重数是______ +6. 实二次型 $f(x_1,x_2,x_3)=2x_1x_2 - 2x_1x_3 + 2x_2x_3$ 的秩为______ + + +## 二、单选题（共6小题，每小题3分，共18分） +1. 在4阶行列式 $\det[a_{ij}]$ 的展开式中含有因子 $a_{31}$ 的项共有【】 + - (A) 4项 + - (B) 6项 + - (C) 8项 + - (D) 10项 +2. 设 $\boldsymbol{A}$ 、 $\boldsymbol{B}$ 是 $n$ 阶方阵，且 $\boldsymbol{B}$ 的第 $j$ 列元素全为零，则下列结论正确的是【】 + - (A) $\boldsymbol{AB}$ 的第 $j$ 列元素全等于零 + - (B) $\boldsymbol{AB}$ 的第 $j$ 行元素全等于零 + - (C) $\boldsymbol{BA}$ 的第 $j$ 列元素全等于零 + - (D) $\boldsymbol{BA}$ 的第 $j$ 行元素全等于零 +3. 设 $n$ 维向量组 $\boldsymbol{\alpha}_1$ 、 $\boldsymbol{\alpha}_2$ 、 $\boldsymbol{\alpha}_3$ 、 $\boldsymbol{\alpha}_4$ 、 $\boldsymbol{\alpha}_5$ 的秩为3，且满足 $\boldsymbol{\alpha}_1 + 2\boldsymbol{\alpha}_3 - 3\boldsymbol{\alpha}_5=\boldsymbol{0}$ ， $\boldsymbol{\alpha}_2 = 2\boldsymbol{\alpha}_4$ ，则该向量组的一个极大线性无关组为【】 + - (A) $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_5$ + - (B) $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_4$ + - (C) $\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_3,\boldsymbol{\alpha}_5$ + - (D) $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_3,\boldsymbol{\alpha}_5$ +4. 设 $\boldsymbol{A}$ 、 $\boldsymbol{B}$ 为 $n$ 阶方阵，给定以下命题：(1) $\boldsymbol{A}$ 与 $\boldsymbol{B}$ 等价；(2) $\boldsymbol{A}$ 与 $\boldsymbol{B}$ 相似；(3) $\boldsymbol{A}$ 、 $\boldsymbol{B}$ 的行向量组等价。下列命题正确的是【】 + - (A) (1) $\Rightarrow$ (2) $\Rightarrow$ (3) + - (B) (2) $\Rightarrow$ (1) $\Rightarrow$ (3) + - (C) (3) $\Rightarrow$ (2) $\Rightarrow$ (1) + - (D) 以上结论均不对 +1. 设矩阵 $\boldsymbol{A}\sim\boldsymbol{B}$ 、 $\boldsymbol{C}\sim\boldsymbol{D}$ ，则下列命题正确的是【】 + - (A) $\boldsymbol{A+B}\sim\boldsymbol{C+D}$ + - (B) $\boldsymbol{A-B}\sim\boldsymbol{C-D}$ + - (C) $\boldsymbol{A}^2\sim\boldsymbol{B}^2$ + - (D) $\boldsymbol{AB}\sim\boldsymbol{CD}$ +1. 如果 $\boldsymbol{A}$ 为反对称矩阵，那么 $\boldsymbol{B}=(\boldsymbol{E}-\boldsymbol{A})(\boldsymbol{E}+\boldsymbol{A})^{-1}$ 一定为【】 + - (A) 反对称矩阵 + - (B) 正交矩阵 + - (C) 对称矩阵 + - (D) 对角矩阵 + + +## 三、（10分） +设 $n$ 阶行列式 $D_n=\begin{vmatrix}1&1&0&\cdots&0&0\\-1&1&1&\cdots&0&0\\0&-1&1&\cdots&0&0\\\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&0&\cdots&1&1\\0&0&0&\cdots&-1&1\end{vmatrix}$ ，证明： $D_n=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]$ + + +## 四、（10分） +求 $n$ 阶方阵 $\boldsymbol{A}$ （原题 $\boldsymbol{A}$ 具体元素排版混乱，推测可能为 $\boldsymbol{A}=\begin{bmatrix}1&1&0&\cdots&1&1\\1&1&1&\cdots&1&0\\\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\1&0&0&\cdots&1&1\end{bmatrix}$ ，具体以标准题型为准）的逆矩阵 $\boldsymbol{A}^{-1}$ + + +## 五、（10分） +求解非齐次线性方程组（原题方程排版混乱，整理后推测为）： + $\begin{cases}x_1 + 2x_2 + 3x_3 = 4\\2x_1 + 3x_2 + x_3 = -4\\3x_1 + 8x_2 - 2x_3 = 13\\4x_1 - x_2 + 9x_3 = -6\\x_1 - x_2 + 2x_3 = -5\end{cases}$ （具体方程以标准题型为准） + + +## 六、（10分） +设 $\boldsymbol{A}$ 、 $\boldsymbol{B}$ 为3阶矩阵， $\boldsymbol{A}$ 相似于 $\boldsymbol{B}$ ， $\lambda_1=-1$ 、 $\lambda_2=1$ 为 $\boldsymbol{A}$ 的两个特征值， $|\boldsymbol{B}^{-1}|=\frac{1}{3}$ ，求行列式 $\begin{vmatrix}-(\boldsymbol{A}-3\boldsymbol{E})^{-1}&\boldsymbol{0}\\\boldsymbol{0}&\boldsymbol{B}^*+\left(-\frac{1}{4}\boldsymbol{B}\right)^{-1}\end{vmatrix}$ （其中 $\boldsymbol{B}^*$ 为 $\boldsymbol{B}$ 的伴随矩阵） + + +## 七、（12分） +求一可逆线性变换 $\boldsymbol{x}=\boldsymbol{P}\boldsymbol{y}$ ，将二次型 $f=2x_1^2 + 9x_2^2 + 3x_3^2 + 8x_1x_2 - 4x_1x_3 - 10x_2x_3$ 化成二次型 $g=2y_1^2 + 3y_2^2 + 6y_3^2 - 4y_1y_2 - 4y_1y_3 + 8y_2y_3$ （或按“将 $f$ 化为标准形”的常规题型修正，具体以题目意图为准） + + +## 八、（12分） +设 $\boldsymbol{A}$ 是 $n$ 阶方阵，证明： $\boldsymbol{A}^2=\boldsymbol{E}$ 的充分必要条件是 $\mathrm{rank}(\boldsymbol{E}-\boldsymbol{A})+\mathrm{rank}(\boldsymbol{E}+\boldsymbol{A})=n$ （其中 $\mathrm{rank}$ 表示矩阵的秩） \ No newline at end of file diff --git a/试卷库/线性代数/线代2018秋A.md b/试卷库/线性代数/线代2018秋A.md new file mode 100644 index 0000000..19fbf41 --- /dev/null +++ b/试卷库/线性代数/线代2018秋A.md @@ -0,0 +1,77 @@ + +## 一、单选题(共6小题，每小题3分，共18分) +1. 设 $\boldsymbol{A}$ 为 $n$ 阶对称矩阵， $\boldsymbol{B}$ 为 $n$ 阶反对称矩阵，下列矩阵中为反对称矩阵的是【】 + (A) $\boldsymbol{AB}-\boldsymbol{BA}$ ； + (B) $\boldsymbol{AB}+\boldsymbol{BA}$ ； + (C) $\boldsymbol{BAB}$ ； + (D) $(\boldsymbol{AB})^2$ 。 + +2. 设 $\boldsymbol{A}$ ， $\boldsymbol{B}$ 是可逆矩阵，且 $\boldsymbol{A}$ 与 $\boldsymbol{B}$ 相似，则下列结论错误的是【】 + (A) $\boldsymbol{A}^\mathrm{T}$ 与 $\boldsymbol{B}^\mathrm{T}$ 相似； + (B) $\boldsymbol{A}^{-1}$ 与 $\boldsymbol{B}^{-1}$ 相似； + (C) $\boldsymbol{A}+\boldsymbol{A}^\mathrm{T}$ 与 $\boldsymbol{B}+\boldsymbol{B}^\mathrm{T}$ 相似； + (D) $\boldsymbol{A}+\boldsymbol{A}^{-1}$ 与 $\boldsymbol{B}+\boldsymbol{B}^{-1}$ 相似。 + +3. 设向量组 $\boldsymbol{\alpha}_1=(0,0,c_1)^\mathrm{T}$ ， $\boldsymbol{\alpha}_2=(0,1,c_2)^\mathrm{T}$ ， $\boldsymbol{\alpha}_3=(1,-1,c_3)^\mathrm{T}$ ， $\boldsymbol{\alpha}_4=(-1,1,c_4)^\mathrm{T}$ ，其中 $c_1$ ， $c_2$ ， $c_3$ ， $c_4$ 为任意常数，则下列向量组线性相关的是【】 + (A) $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_3$ ； + (B) $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_4$ ； + (C) $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_3,\boldsymbol{\alpha}_4$ ； + (D) $\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_3,\boldsymbol{\alpha}_4$ 。 + +4. 设 $\boldsymbol{A}$ ， $\boldsymbol{B}$ 为 $n$ 阶矩阵，则【】 + (A) $\mathrm{rank}\begin{bmatrix}\boldsymbol{A}&\boldsymbol{AB}\end{bmatrix}=\mathrm{rank}\boldsymbol{A}$ ； + (B) $\mathrm{rank}\begin{bmatrix}\boldsymbol{A}&\boldsymbol{BA}\end{bmatrix}=\mathrm{rank}\boldsymbol{A}$ ； + (C) $\mathrm{rank}\begin{bmatrix}\boldsymbol{A}&\boldsymbol{B}\end{bmatrix}=max\{\mathrm{rank}\boldsymbol{A},\mathrm{rank}\boldsymbol{B}\}$ ； + (D) $\mathrm{rank}\begin{bmatrix}\boldsymbol{A}&\boldsymbol{B}\end{bmatrix}=\mathrm{rank}\begin{bmatrix}\boldsymbol{A}^\mathrm{T}&\boldsymbol{B}^\mathrm{T}\end{bmatrix}$ 。 + +5. 设 $\boldsymbol{A}$ 可逆，将 $\boldsymbol{A}$ 的第一列加上第二列的2倍得到 $\boldsymbol{B}$ ，则 $\boldsymbol{A}^*$ 与 $\boldsymbol{B}^*$ 满足【】 + (A) 将 $\boldsymbol{A}^*$ 的第一列加上第二列的2倍得到 $\boldsymbol{B}^*$ ； + (B) 将 $\boldsymbol{A}^*$ 的第一行加上第二行的2倍得到 $\boldsymbol{B}^*$ ； + (C) 将 $\boldsymbol{A}^*$ 的第二列加上第一列的 $(-2)$ 倍得到 $\boldsymbol{B}^*$ ； + (D) 将 $\boldsymbol{A}^*$ 的第二行加上第一行的 $(-2)$ 倍得到 $\boldsymbol{B}^*$ 。 + +6. 设齐次线性方程组 + (I) $\begin{cases}x_1 + 2x_2 + 3x_3 = 0\\2x_1 + 3x_2 + 5x_3 = 0\\x_1 + x_2 + ax_3 = 0\end{cases}$ + (II) $\begin{cases}x_1 + bx_2 + cx_3 = 0\\2x_1 + b^2x_2 + (c + 1)x_3 = 0\end{cases}$ + 同解，则 $a,b,c$ 的值为【】 + (A) $a=1,b=0,c=1$ ； + (B) $a=1,b=1,c=2$ ； + (C) $a=2,b=0,c=1$ ； + (D) $a=2,b=1,c=2$ 。 + +## 二、填空题(共6小题，每小题3分，共18分) +7. 已知向量 $\boldsymbol{\alpha}_1=(1,0,-1,0)^\mathrm{T}$ ， $\boldsymbol{\alpha}_2=(1,1,-1,-1)^\mathrm{T}$ ， $\boldsymbol{\alpha}_3=(-1,0,1,1)^\mathrm{T}$ ，则向量 $\boldsymbol{\alpha}_1 + 2\boldsymbol{\alpha}_2$ 与 $2\boldsymbol{\alpha}_1 + \boldsymbol{\alpha}_3$ 的内积 $<\boldsymbol{\alpha}_1 + 2\boldsymbol{\alpha}_2,2\boldsymbol{\alpha}_1 + \boldsymbol{\alpha}_3>=$ ________。 + +8. 设二阶矩阵 $\boldsymbol{A}$ 有两个相异特征值， $\boldsymbol{\alpha}_1$ ， $\boldsymbol{\alpha}_2$ 是 $\boldsymbol{A}$ 的线性无关的特征向量，且 $\boldsymbol{A}^2(\boldsymbol{\alpha}_1 + \boldsymbol{\alpha}_2)=\boldsymbol{\alpha}_1 + \boldsymbol{\alpha}_2$ ，则 $|\boldsymbol{A}|=$ ________。 + +9. 若向量组 $\boldsymbol{\alpha}_1=(1,0,1)^\mathrm{T}$ ， $\boldsymbol{\alpha}_2=(0,1,1)^\mathrm{T}$ ， $\boldsymbol{\alpha}_3=(1,3,5)^\mathrm{T}$ 不能由向量组 $\boldsymbol{\beta}_1=(1,1,1)^\mathrm{T}$ ， $\boldsymbol{\beta}_2=(1,2,3)^\mathrm{T}$ ， $\boldsymbol{\beta}_3=(3,4,a)^\mathrm{T}$ 线性表示，则 $a=$ ________。 + +10. 设矩阵 $\boldsymbol{A}=\begin{bmatrix}1&a_1&a_1^2&a_1^3\\1&a_2&a_2^2&a_2^3\\1&a_3&a_3^2&a_3^3\\1&a_4&a_4^2&a_4^3\end{bmatrix}$ ， $\boldsymbol{x}=\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}$ ， $\boldsymbol{b}=\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ ，其中 $a_1,a_2,a_3,a_4$ 互不相同，则线性方程组 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ 的解为________。 + +11. 若 $n$ 阶实对称矩阵 $\boldsymbol{A}$ 的特征值为 $\lambda_i=(-1)^i(i=1,2,\cdots,n)$ ，则 $\boldsymbol{A}^{100}=$ ________。 + +12. 设 $n$ 阶矩阵 $\boldsymbol{A}=[a_{ij}]_{n\times n}$ ，则二次型 $f(x_1,x_2,\cdots,x_n)=\sum_{i=1}^n(a_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n)^2$ 的矩阵为________。 + +## 三、计算与证明(共6小题，共64分) +13. (10分)计算 $n$ 阶行列式 $\begin{vmatrix}1&2&3&\cdots&n-1&n\\2&1&2&\cdots&n-2&n-1\\3&2&1&\cdots&n-3&n-2\\\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\n-1&n-2&n-3&\cdots&1&2\\n&n-1&n-2&\cdots&2&1\end{vmatrix}$ 。 + +14. (10分)设 $\boldsymbol{\alpha}_1=(1,0,-1)^\mathrm{T}$ ， $\boldsymbol{\alpha}_2=(2,1,1)^\mathrm{T}$ ， $\boldsymbol{\alpha}_3=(1,1,1)^\mathrm{T}$ 和 $\boldsymbol{\beta}_1=(0,1,1)^\mathrm{T}$ ， $\boldsymbol{\beta}_2=(-1,1,0)^\mathrm{T}$ ， $\boldsymbol{\beta}_3=(0,2,1)^\mathrm{T}$ 是 $\mathbb{R}^3$ 的两组基，求向量 $\boldsymbol{u}=\boldsymbol{\alpha}_1 + 2\boldsymbol{\alpha}_2 - 3\boldsymbol{\alpha}_3$ 在基 $\boldsymbol{\beta}_1$ ， $\boldsymbol{\beta}_2$ ， $\boldsymbol{\beta}_3$ 下的坐标。 + +15. (10分)设实二次型 $f(x_1,x_2,x_3)=x_1^2 + x_2^2 + x_3^2 - 2x_1x_2 - 2x_1x_3 + 2ax_2x_3$ 通过正交变换可化为标准型 $f=2y_1^2 + 2y_2^2 + by_3^2$ 。 + (1)求 $a$ ， $b$ 及所用正交变换矩阵 $\boldsymbol{Q}$ ； + (2)证明 $\boldsymbol{A} + 2\boldsymbol{E}$ 为正定矩阵。 + +16. (10分)设三阶矩阵 $\boldsymbol{A}=[\boldsymbol{\alpha}_1\ \boldsymbol{\alpha}_2\ \boldsymbol{\alpha}_3]$ 有三个不同的特征值，且满足 $\boldsymbol{\alpha}_3=\boldsymbol{\alpha}_1 + 2\boldsymbol{\alpha}_2$ ， $\boldsymbol{\beta}=\boldsymbol{\alpha}_1 + \boldsymbol{\alpha}_2 + \boldsymbol{\alpha}_3$ 。 + (1)证明 $\mathrm{rank}\boldsymbol{A}=2$ ； + (2)求方程组 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{\beta}$ 的通解。 + +17. (12分)设 $n$ 阶方阵 $\boldsymbol{A}$ ， $\boldsymbol{B}$ 满足 $\boldsymbol{AB}=\boldsymbol{A} + \boldsymbol{B}$ 。 + (1)证明 $\boldsymbol{A} - \boldsymbol{E}$ 可逆； + (2)证明 $\boldsymbol{AB}=\boldsymbol{BA}$ ； + (3)证明 $\mathrm{rank}\boldsymbol{A}=\mathrm{rank}\boldsymbol{B}$ ； + (4)若 $\boldsymbol{B}=\begin{bmatrix}1&-3&0\\2&1&0\\0&0&2\end{bmatrix}$ ，求矩阵 $\boldsymbol{A}$ 。 + +18. (12分)已知实矩阵 $\boldsymbol{A}=\begin{bmatrix}2&2\\2&a\end{bmatrix}$ ， $\boldsymbol{B}=\begin{bmatrix}4&b\\3&1\end{bmatrix}$ 。 + (1)证明矩阵方程 $\boldsymbol{AX}=\boldsymbol{B}$ 有解但 $\boldsymbol{BY}=\boldsymbol{A}$ 无解的充要条件是 $a\neq2$ ， $b=\frac{4}{3}$ ； + (2)证明矩阵 $\boldsymbol{A}$ 相似于 $\boldsymbol{B}$ 的充要条件是 $a=3$ ， $b=\frac{2}{3}$ ； + (3)证明矩阵 $\boldsymbol{A}$ 合同于 $\boldsymbol{B}$ 的充要条件是 $a<2$ ， $b=3$ 。 \ No newline at end of file diff --git a/试卷库/线性代数/线代2019秋A.md b/试卷库/线性代数/线代2019秋A.md new file mode 100644 index 0000000..ec00351 --- /dev/null +++ b/试卷库/线性代数/线代2019秋A.md @@ -0,0 +1,65 @@ + +## 一、单选题(共6小题,每小题3分,共18分) +1. n维向量组 $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, ..., \boldsymbol{\alpha}_{r}(3 \le r \le n)$ 线性无关的充要条件是【】 +A. 存在一组不全为零的数 $k_{1}, k_{2}, ..., k_{r}$ 使得 $\sum_{i=1}^{r} k_{i} \boldsymbol{\alpha}_{i} ≠0$ +B. $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, ..., \boldsymbol{\alpha}_{r}$ 中任意两个向量都线性无关 +C. $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, ..., \boldsymbol{\alpha}_{r}$ 中存在一个向量不能用其余向量线性表示 +D. $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, ..., \boldsymbol{\alpha}_{r}$ 中任意一个向量都不能用其余向量线性表示 + +2. 已知 $\boldsymbol{Q}$ 为n阶可逆矩阵， $\boldsymbol{P}$ 为n阶方阵，满足 $\boldsymbol{Q}\boldsymbol{P}=0$ ，则下列命题中正确的是【】 +A. $\mathrm{rank}(\boldsymbol{P})=0$ +B. $\boldsymbol{P}$ 可逆 +C. $\boldsymbol{P}$ 不可对角化 +D. 不存在这样的方阵 $\boldsymbol{P}$ + +3. 已知向量组 $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{4}$ 线性无关，则以下向量组中线性相关的是【】 +A. $\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{4}$ +B. $\boldsymbol{\alpha}_{2}-\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{3}-\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}-\boldsymbol{\alpha}_{4}, \boldsymbol{\alpha}_{4}-\boldsymbol{\alpha}_{1}$ +C. $\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}+\boldsymbol{\alpha}_{4}, \boldsymbol{\alpha}_{4}-\boldsymbol{\alpha}_{1}$ +D. $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}-\boldsymbol{\alpha}_{4}, \boldsymbol{\alpha}_{4}+\boldsymbol{\alpha}_{3}$ + +4. 设n阶非零方阵 $\boldsymbol{A}$ 满足 $\boldsymbol{A}^{2}=\boldsymbol{A}$ ，则下列命题中正确的是【】 +A. $\boldsymbol{A}$ 只有特征值1 +B. $\boldsymbol{A}$ 只有特征值0 +C. $\boldsymbol{A}$ 可对角化 +D. $\boldsymbol{A}$ 一定不可逆 + +5. 下列命题中正确的是【】 +A. 等价的矩阵必相似 +B. 合同的矩阵必相似 +C. 合同的矩阵必等价 +D. 等价的矩阵必合同 + +6. 设 $\boldsymbol{A}$ 是二次型 $f(x_{1}, x_{2}, ..., x_{n})$ 所对应的矩阵，则下列命题中正确的是【】 +A. 若 $f(x_{1}, x_{2}, ..., x_{n})$ 负定，则 $\boldsymbol{A}$ 的任意阶顺序主子式都大于零 +B. 若 $f(x_{1}, x_{2}, ..., x_{n})$ 正定，则 $\boldsymbol{A}$ 的任意阶顺序主子式都大于零 +C. 若 $f(x_{1}, x_{2}, ..., x_{n})$ 半负定，则 $\boldsymbol{A}$ 的任意阶顺序主子式都大于零 +D. 若 $f(x_{1}, x_{2}, ..., x_{n})$ 半正定，则 $\boldsymbol{A}$ 的任意阶顺序主子式都大于零 + +## 二、填空题(共6小题,每小题3分,共18分) +7. 如果 $\begin{vmatrix}1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 0 & 0 & 9 & x \\ 0 & 0 & 11 & 12\end{vmatrix}=0$ ，则 $x=$ _______ + +8. n阶方阵 $\begin{bmatrix}1 & a & \cdots & a \\ a & 1 & \cdots & a \\ \vdots & \vdots & \ddots & \vdots \\ a & a & \cdots & 1\end{bmatrix}$ 的秩为 $n - 1$ ，且 $n>2$ ，则 $a=$ _______ + +9. 设 $\boldsymbol{E}$ 为3阶单位矩阵， $\boldsymbol{\alpha}$ 为一个3维单位列向量，则矩阵 $\boldsymbol{E}-\boldsymbol{\alpha}\boldsymbol{\alpha}^{\mathrm{T}}$ 的全部3个特征值为_______ + +10. 设二次型 $f(x_{1}, x_{2}, x_{3})=x_{1}^{2}+4x_{2}^{2}+2x_{3}^{2}+2ax_{1}x_{2}+2x_{1}x_{3}$ 正定，则参数 $a$ 的取值范围是_______ + +11. $\begin{bmatrix}1 & a & 0 \\ 0 & 1 & a \\ 0 & 0 & 1\end{bmatrix}^{-1}=$ _______ + +12. 设 $\boldsymbol{A}=\begin{bmatrix}1 & 2 & 0 \\ 0 & 2 & 0 \\ -2 & -1 & -1\end{bmatrix}$ ，则 $\boldsymbol{A}^{100}=$ _______ + +## 三、计算与证明题(共6小题,共64分) +13. 计算行列式并解方程 $\begin{vmatrix}1 & 2 & 3 & 4 + x \\ 1 & 2 & 3 + x & 4 \\ 1 & 2 + x & 3 & 4 \\ 1 + x & 2 & 3 & 4\end{vmatrix}=0$ 。(10分) + +14. 设 $\boldsymbol{A}=\begin{bmatrix}1 & a & b \\ 0 & 1 & a \\ 0 & 0 & 1\end{bmatrix}$ ，其中 $a, b$ 都不等于0 +(1) 对任意自然数 $n \in \mathbb{N}$ ，计算 $\boldsymbol{A}^{n}$ 。(6分) +(2) 计算 $\boldsymbol{A}^{-1}$ 。(4分) + +15. 设 $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}$ 是线性方程组 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ 的解，其中 $\boldsymbol{x}=\begin{bmatrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{bmatrix}$ ， $\boldsymbol{b}=\begin{bmatrix}b_{1} \\ b_{2} \\ b_{3} \\ b_{4}\end{bmatrix}$ ，现已知 $\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}=\begin{bmatrix}2 \\ 2 \\ 4 \\ 6\end{bmatrix}$ ， $\boldsymbol{\alpha}_{1}+2\boldsymbol{\alpha}_{3}=\begin{bmatrix}0 \\ 3 \\ 0 \\ 6\end{bmatrix}$ ，求该方程组的通解。(10分) + +16. 设方阵 $\boldsymbol{A}$ 与 $\boldsymbol{B}$ 相似，其中 $\boldsymbol{A}=\begin{bmatrix}-2 & 0 & 0 \\ 2 & x & 2 \\ 3 & 1 & 1\end{bmatrix}$ ， $\boldsymbol{B}=\begin{bmatrix}-1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & y\end{bmatrix}$ ，求 $x, y$ 的值及可逆矩阵 $\boldsymbol{P}$ ，使得 $\boldsymbol{P}^{-1}\boldsymbol{A}\boldsymbol{P}=\boldsymbol{B}$ 。(10分) + +17. 已知二次型 $f=x_{1}^{2}+ax_{2}^{2}+x_{3}^{2}+2bx_{1}x_{2}+2x_{1}x_{3}+2x_{2}x_{3}$ 可经过正交变换 $\begin{bmatrix}x_{1} \\ x_{2} \\ x_{3}\end{bmatrix}=\boldsymbol{P}\begin{bmatrix}y_{1} \\ y_{2} \\ y_{3}\end{bmatrix}$ 化为 $y_{2}^{2}+4y_{3}^{2}$ ，求 $a, b$ 的值和正交矩阵 $\boldsymbol{P}$ 。(12分) + +18. 设 $\boldsymbol{A}$ 为n阶可逆矩阵，证明 $\boldsymbol{A}$ 可以相似对角化当且仅当 $\boldsymbol{A}^{2}$ 可以相似对角化。(12分) \ No newline at end of file diff --git a/试卷库/线性代数/线代2021秋A.md b/试卷库/线性代数/线代2021秋A.md new file mode 100644 index 0000000..108577c --- /dev/null +++ b/试卷库/线性代数/线代2021秋A.md @@ -0,0 +1,3 @@ +1. 设三阶方阵 \(A=\begin{bmatrix}a&b&b\\b&a&b\\b&b&a\end{bmatrix}\)，其伴随矩阵 \(A^*\) 的秩等于1，则（ ） + A. \(a=b\) 且 \(a+2b\neq0\) + B. \(a=b\) 或 \(a+2b\neq0\) C. \(a\neq b\) 且 \(a+2b=0\) D. \(a\neq b\) 且 \(a+2b\neq0\) 2. 设 \(A, B, A+B, A^{-1}+B^{-1}\) 均为 \(n(n\geq2)\) 阶可逆矩阵，则 \((A^{-1}+B^{-1})^{-1}\) 等于（ ） A. \(A^{-1}+B^{-1}\) B. \(A+B\) C. \(A(A+B)^{-1}B\) D. \((A+B)^{-1}\) 3. 设 \(n\) 维向量 \(\alpha, \beta, \gamma\) 与数 \(k, l, m\) 满足 \(k\alpha+l\beta+m\gamma=0\)，且 \(km\neq0\)，则（ ） A. \(\alpha, \beta\) 与 \(\alpha, \gamma\) 等价 B. \(\alpha, \beta\) 与 \(\beta, \gamma\) 等价 C. \(\alpha, \gamma\) 与 \(\beta, \gamma\) 等价 D. \(\alpha\) 与 \(\gamma\) 等价 4. 下列矩阵中，与 \(\begin{bmatrix}4&2&0\\2&4&0\\0&0&-8\end{bmatrix}\) 合同的是（ ） A. \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}\) B. \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}\) C. \(\begin{bmatrix}1&0&0\\0&-1&0\\0&0&0\end{bmatrix}\) D. \(\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}\) 5. 设 \(\lambda_1, \lambda_2\) 是矩阵 \(A\) 的两个相异特征值，对应的特征向量分别为 \(\alpha_1, \alpha_2\)，则 \(A(\alpha_1+\alpha_2), \alpha_1\) 线性无关的充要条件为（ ） A. \(\lambda_1\neq0\) B. \(\lambda_2\neq0\) C. \(\lambda_1=0\) D. \(\lambda_2=0\) 6. 设 \(A, B\) 均为 \(n\) 阶正定矩阵，则下列矩阵中必为正定矩阵的是（ ） A. \(kAB\)，其中 \(k\) 为常数 B. \(kA^*+lB^*\)，其中 \(kl>0\) C. \(A^{-1}+B^{-1}\) D. \(A^{-1}-B^{-1}\) ## 二、填空题(共6小题,每小题3分,共18分) 7. 设矩阵 \(A=\begin{bmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{bmatrix}\)，则矩阵 \(A^3\) 的秩为________。 8. 已知 \(\alpha_1=\begin{bmatrix}1\\0\end{bmatrix}, \alpha_2=\begin{bmatrix}0\\1\end{bmatrix}\) 和 \(\beta_1=\begin{bmatrix}1\\2\end{bmatrix}, \beta_2=\begin{bmatrix}2\\3\end{bmatrix}\) 为向量空间的两组基，则从 \(\beta_1, \beta_2\) 到 \(\alpha_1, \alpha_2\) 的过渡矩阵为________。 9. 在实数域中，二次型 \(f(x_1, x_2, x_3)=2x_1x_2-2x_1x_3+2x_2x_3\) 的规范形为________。 10. 设5阶实对称矩阵 \(A\) 满足 \(A^2-2A=O\)，\(\text{rank}(A)=3\)，则 \(|A+E|\) 等于________。 11. 设 \(A=[a_{ij}]_{3\times3}\) 是正交矩阵，且 \(a_{33}=1\)，\(b=\begin{bmatrix}0\\0\\3\end{bmatrix}\)，则线性方程组 \(Ax=b\) 的解为________。 12. 设 \(A\) 为三阶方阵，将 \(A\) 的第2列加到第1列得到 \(B\)，再交换 \(B\) 的第2行与第3行得到单位矩阵，则 \(A\) 等于________。 ## 三、计算与证明题(共6小题,共64分) 13. 求多项式 \(f(x)=\begin{vmatrix}x&x&1&2x\\1&x&2&-1\\2&1&x&1\\2&-1&1&x\end{vmatrix}\) 中 \(x^3\) 的系数。(10分) 14. 设三阶方阵 \(A, B\) 满足 \(A^*BA=2BA-8E\)，其中 \(A=\begin{bmatrix}1&0&0\\0&-2&0\\0&0&1\end{bmatrix}\)，求 \(B\)。(10分) 15. 已知线性方程组 \(\begin{cases}x_1+x_2=0\\-2x_1+2x_2+(2-\lambda)x_3=1\\-4x_1+(5-\lambda)x_2+2x_3=2\end{cases}\)，讨论当 \(\lambda\) 取何值时，方程组有唯一解、无解、有无穷多解，并求出有无穷多解时的通解。(10分) 16. 设 \(\beta_1, \beta_2, \cdots, \beta_m\) 均为实数域上的 \(n\) 维列向量，其中 \(m<n\)，证明 \(\beta_1, \beta_2, \cdots, \beta_m\) 线性无关的充要条件为 \(B^TB\) 可逆（其中 \(B=[\beta_1, \beta_2, \cdots, \beta_m]\)）。(10分) 17. 已知 \(\alpha=\begin{bmatrix}a_1\\a_2\\\vdots\\a_n\end{bmatrix}, \beta=\begin{bmatrix}b_1\\b_2\\\vdots\\b_n\end{bmatrix}\) 为 \(n\) 维非零列向量，且 \(\beta^T\alpha\neq0\)，\(A=\alpha\beta^T\)，证明 \(A\) 可相似对角化。(12分) 18. 已知实二次型 \(f(x_1, x_2, x_3)=3x_1^2+2x_2^2+ax_3^2+bx_1x_3\)，经过正交变换 \(x=Py\) 得标准形 \(y_1^2+2y_2^2+5y_3^2\)，其中 \(a, b\) 都是非负实数，求正交变换矩阵 \(P\)。(12分) \ No newline at end of file diff --git a/试卷库/线性代数/线代2022秋A.md b/试卷库/线性代数/线代2022秋A.md new file mode 100644 index 0000000..e5f0236 --- /dev/null +++ b/试卷库/线性代数/线代2022秋A.md @@ -0,0 +1,64 @@ +## 一、单选题(共6小题,每小题3分,共18分) +1. 设 $\boldsymbol{A}$ 为 $n$ 阶方阵,且 $|\boldsymbol{A}| = 5$ ,则 $||\boldsymbol{A}|\boldsymbol{A}^{T}|$ =【】 +A. $5n$ +B. $5^{n + 1}$ +C. $5^{n - 1}$ +D. 25 + +2. 设 $\boldsymbol{A}$ , $\boldsymbol{B}$ 均为 $n$ 阶方阵,则下列命题中正确的是【】 +A. $|\boldsymbol{A}+\boldsymbol{B}|=|\boldsymbol{A}| + |\boldsymbol{B}|$ +B. $\boldsymbol{AB}=\boldsymbol{BA}$ +C. $|\boldsymbol{AB}|=|\boldsymbol{BA}|$ +D. $(\boldsymbol{A}+\boldsymbol{B})^{-1}=\boldsymbol{A}^{-1}+\boldsymbol{B}^{-1}$ + +3. 下列关于矩阵秩的结论中,错误的是【】 +A. 若矩阵 $\boldsymbol{A}$ , $\boldsymbol{B}$ , $\boldsymbol{C}$ 满足 $\boldsymbol{A}=\boldsymbol{BC}$ 且 $\boldsymbol{B}$ 是列满秩的,则 $\boldsymbol{A}$ 列满秩当且仅当 $\boldsymbol{C}$ 列满秩 +B. 行阶梯形矩阵中1的个数等于矩阵的秩 +C. 设 $\boldsymbol{A}$ , $\boldsymbol{B}$ 均为 $n$ 阶方阵,则 $R(\boldsymbol{AB})=R(\boldsymbol{B})$ 的充要条件是线性方程组 $(\boldsymbol{AB})\boldsymbol{x}=\boldsymbol{0}$ 与 $\boldsymbol{B}\boldsymbol{x}=\boldsymbol{0}$ 同解 +D. 设 $\boldsymbol{A}$ 为 $n$ 阶方阵,则 $R(\boldsymbol{A}^{n})=R(\boldsymbol{A}^{n + 1})$ + +4. 设 $\boldsymbol{\eta}_{1}$ , $\boldsymbol{\eta}_{2}$ 是非齐次线性方程组 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ 的两个不同解, $\boldsymbol{\xi}_{1}$ , $\boldsymbol{\xi}_{2}$ 是导出方程组 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ 的两个不同解,则:① $\boldsymbol{\eta}_{1}-\boldsymbol{\eta}_{2}$ 是 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ 的解,② $3\boldsymbol{\xi}_{1}+\boldsymbol{\eta}_{1}$ 是 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ 的解,③ $2\boldsymbol{\xi}_{1}+2\boldsymbol{\xi}_{2}$ 是 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ 的解,④ $2\boldsymbol{\eta}_{1}-\boldsymbol{\eta}_{2}$ 是 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ 的解,⑤ $\boldsymbol{\eta}_{1}-\boldsymbol{\eta}_{2}+\boldsymbol{\xi}_{1}$ 是 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ 的解。上述结论正确的有多少个?【】 +A. 2 +B. 3 +C. 4 +D. 5 + +5. 已知 $n$ 阶方阵 $\boldsymbol{A}$ 与 $\boldsymbol{B}$ 相似, $\boldsymbol{C}$ 与 $\boldsymbol{D}$ 相似,则下列命题中正确的是【】 +A. $\boldsymbol{A}+\boldsymbol{C}$ 与 $\boldsymbol{B}+\boldsymbol{D}$ 相似 +B. $\boldsymbol{AC}$ 与 $\boldsymbol{BD}$ 相似 +C. $\boldsymbol{A}^{2}+\boldsymbol{I}$ 与 $\boldsymbol{B}^{2}+\boldsymbol{I}$ 相似 +D. $\boldsymbol{A}^{T}+\boldsymbol{A}$ 与 $\boldsymbol{B}^{T}+\boldsymbol{B}$ 相似 + +6. 设 $f(x_{1},x_{2},x_{3})=2x_{1}^{2}+6x_{2}^{2}+x_{3}^{2}-4x_{1}x_{2}-2x_{1}x_{3}=C$ ,则此二次曲面是【】 +A. $C = 0$ 为锥面 +B. $C>0$ 为椭球面 +C. $C<0$ 为柱面 +D. $C = 1$ 为单叶双曲面 + +## 二、填空题(共6小题,每小题3分,共18分) +1. 设 $\alpha,\beta,\gamma$ 为 $x^{3}+px + q = 0$ 的三个根,则行列式 $\begin{vmatrix}\boldsymbol{\alpha}&\boldsymbol{\beta}&\boldsymbol{\gamma}\\\boldsymbol{\gamma}&\boldsymbol{\alpha}&\boldsymbol{\beta}\\\boldsymbol{\beta}&\boldsymbol{\gamma}&\boldsymbol{\alpha}\end{vmatrix}$ =________ +2. 设 $\boldsymbol{A}=\begin{bmatrix}1&1&1\\a_{1}&a_{2}&a_{3}\\a_{1}^{2}&a_{2}^{2}&a_{3}^{2}\end{bmatrix}$ , $\boldsymbol{b}=\begin{bmatrix}1\\1\\1\end{bmatrix}$ ,其中 $a_{1},a_{2},a_{3}$ 互不相同,线性方程组 $\boldsymbol{A}^{T}\boldsymbol{x}=\boldsymbol{b}$ 的解为________ +3. 设3阶方阵 $\boldsymbol{A}=\begin{bmatrix}2&-1&2\\1&0&1\\0&0&2\end{bmatrix}$ ,3维向量 $\boldsymbol{\alpha}=\begin{bmatrix}t\\2\\1\end{bmatrix}$ ,若 $\boldsymbol{A}\boldsymbol{\alpha}$ 与 $\boldsymbol{\alpha}$ 线性相关,则 $t=$ ________ +4. 已知 $\|\boldsymbol{a}\| = 2$ , $\|\boldsymbol{b}\| = 5$ ,且 $\boldsymbol{a}$ 和 $\boldsymbol{b}$ 的夹角为 $\frac{2}{3}\pi$ ,若向量 $\boldsymbol{A}=\lambda\boldsymbol{a}+17\boldsymbol{b}$ 与 $\boldsymbol{B}=3\boldsymbol{a}-\boldsymbol{b}$ 垂直,则 $\lambda=$ ________ +5. 已知 $n(n\geq2)$ 维列向量 $\boldsymbol{\alpha}$ , $\boldsymbol{\beta}$ 满足 $\boldsymbol{\beta}^{T}\boldsymbol{\alpha}=-3$ ,其中 $\boldsymbol{\beta}^{T}$ 为 $\boldsymbol{\beta}$ 的转置,则方阵 $(\boldsymbol{\beta}\boldsymbol{\alpha}^{T})^{2}$ 的非零特征值为________ +6. 已知 $\boldsymbol{A}$ 为 $m\times n$ 实矩阵(其中 $m < n$ ), $\boldsymbol{I}$ 为 $n$ 阶单位矩阵, $t$ 为实数,则 $\boldsymbol{A}^{T}\boldsymbol{A}+t\boldsymbol{I}$ 为正定矩阵的充分必要条件是实数 $t$ 满足关系式________ + +## 三、计算题(10分) +计算 $n$ 阶行列式 $\begin{vmatrix}a&a_{2}&a_{2}&\cdots&a_{2}\\a_{2}&a&a_{2}&\cdots&a_{2}\\a_{2}&a_{2}&a&\cdots&a_{2}\\\vdots&\vdots&\vdots&\ddots&\vdots\\a_{2}&a_{2}&a_{2}&\cdots&a\end{vmatrix}$ + +## 四、计算题(10分) +设直线 $L:\frac{x - 1}{2}=y=\frac{z - 2}{3}$ 与平面 $\pi:2x - y+z - 10 = 0$ 的交点为 $P$ ,求过点 $P$ 且与平面 $\pi$ 垂直的直线方程。 + +## 五、计算题(10分) +已知线性方程组 $\begin{cases}\lambda x_{1}-x_{2}-x_{3}=1\\-x_{1}+\lambda x_{2}-x_{3}=-\lambda\\-x_{1}-x_{2}+\lambda x_{3}=\lambda^{2}\end{cases}$ 至少存在两个不同的解,求该线性方程组的通解。 + +## 六、计算题(10分) +判定向量组 $\boldsymbol{\alpha}_{1}=\begin{bmatrix}1\\1\\1\\-1\end{bmatrix}$ , $\boldsymbol{\alpha}_{2}=\begin{bmatrix}1\\4\\1\\0\end{bmatrix}$ , $\boldsymbol{\alpha}_{3}=\begin{bmatrix}-1\\2\\-1\\2\end{bmatrix}$ , $\boldsymbol{\alpha}_{4}=\begin{bmatrix}1\\1\\2\\3\end{bmatrix}$ , $\boldsymbol{\alpha}_{5}=\begin{bmatrix}2\\-1\\4\\5\end{bmatrix}$ 的线性相关性,求其一个极大线性无关组,并将其余向量用该极大线性无关组线性表示。 + +## 七、证明与计算题(共2小题,第1小题4分,第2小题8分,共12分) +已知向量组 $\boldsymbol{\alpha}_{1}$ , $\boldsymbol{\alpha}_{2}$ , $\boldsymbol{\alpha}_{3}$ 线性无关,其中 $\boldsymbol{\alpha}_{1}$ , $\boldsymbol{\alpha}_{2}$ , $\boldsymbol{\alpha}_{3}\in\mathbb{R}^{3}$ , $\boldsymbol{A}$ 为3阶方阵,且 $\boldsymbol{A}\boldsymbol{\alpha}_{1}=2\boldsymbol{\alpha}_{1}-\boldsymbol{\alpha}_{2}-\boldsymbol{\alpha}_{3}$ , $\boldsymbol{A}\boldsymbol{\alpha}_{2}=\boldsymbol{\alpha}_{1}+2\boldsymbol{\alpha}_{2}+3\boldsymbol{\alpha}_{3}$ , $\boldsymbol{A}\boldsymbol{\alpha}_{3}=2\boldsymbol{\alpha}_{1}+4\boldsymbol{\alpha}_{2}+3\boldsymbol{\alpha}_{3}$ 。 +1. 证明 $\boldsymbol{A}\boldsymbol{\alpha}_{1}$ , $\boldsymbol{A}\boldsymbol{\alpha}_{2}$ , $\boldsymbol{A}\boldsymbol{\alpha}_{3}$ 线性无关； +2. 计算行列式 $|\boldsymbol{E}-\boldsymbol{A}|$ ,其中 $\boldsymbol{E}$ 是3阶单位矩阵。 + +## 八、计算题(12分) +已知实二次型 $f(x_{1},x_{2},x_{3})=\boldsymbol{x}^{T}\boldsymbol{A}\boldsymbol{x}$ 在正交变换 $\boldsymbol{x}=\boldsymbol{P}\boldsymbol{y}$ 下的标准形为 $3y_{1}^{2}+3y_{2}^{2}$ ,且 $\boldsymbol{P}$ 的第3列为 $\frac{1}{\sqrt{3}}\begin{bmatrix}1\\1\\1\end{bmatrix}$ ,求 $\boldsymbol{A}$ 。 \ No newline at end of file diff --git a/试卷库/高数期末真题/2017高数期末考试卷.md b/试卷库/高数期末真题/2017高数期末考试卷.md new file mode 100644 index 0000000..082cd86 --- /dev/null +++ b/试卷库/高数期末真题/2017高数期末考试卷.md @@ -0,0 +1,146 @@ +--- +tags: + - 官方试卷 +--- +# 国防科技大学 2017—2018 学年秋季学期 +## 《高等数学》考试试卷（A）卷 + +**(2018 年 1 月 26 日)** + +**考试形式：闭卷** +**考试时间：150 分钟** +**满分：100 分** + +| 题号 | 一 | 二 | 三 | 四 | 五 | 六 | 七 | 八 | 九 | 十 | 十一 | 十二 | 总分 | 核分 | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +| 满分 | 15 | 15 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 100 | | +| 得分 | | | | | | | | | | | | | | | +| 评阅人 | | | | | | | | | | | | | | | + +**注意：** +1. 所有答题都须写在此试卷纸密封线右边，写在其它纸上一律无效。 +2. 密封线外不得有姓名及相关标记。 +3. 当题目留空不够时，可写在试卷反面，但密封线内请勿答题。 + +--- + +### 一、填空题（共 5 小题，每小题 3 分，共 15 分） + +1. 函数$y = \ln(2 - x^2)$在$x = 1$处的微分为 ______。 + +2. 曲线$y = 1 + xe^x$在点 (0,1) 处的曲率为 ______。 + +3. 函数$y = \frac{x^3 + 4}{x^2}$的单调递减区间为 ______。 + +4. 已知$\int f(x) dx = \arctan x + C$，则$f'(x) =$______。 + +5. 已知级数$\sum_{n=1}^{\infty} \frac{a^{2n}}{a^{2n} + 1}$收敛，则常数$a$的最大取值范围为 ______。 + +--- + +### 二、选择题（共 5 小题，每小题 3 分，共 15 分） + +1. 点$x = 0$为函数$f(x) = \frac{2e^x + 3}{3e^x + 2}$的（ ）。 + (A) 可去间断点 + (B) 跳跃间断点 + (C) 无穷间断点 + (D) 振荡间断点 + +2. 设函数$f(x), g(x)$在$[-a, a]$上均具有连续导数，且$f(x)$为奇函数，$g(x)$为偶函数，则定积分 + $$ + \int_{-a}^{a} [f'(x) + g'(x)] dx = \text{（ ）}。 + $$ + (A)$f(a) + g(a)$ + (B)$f(a) - g(a)$ + (C)$2g(a)$ + (D)$2f(a)$ + +3. 设$f(x)$是定义在$(-\infty, +\infty)$内的连续的奇函数，下列表格中给出了它的二阶导数$f''(x)$在$(0, +\infty)$内的符号信息，则曲线$y = f(x) (-\infty < x < +\infty)$的拐点个数为（ ）。 + + |$x$| (0,1) | 1 | (1,2) | 2 | (2,+\infty) | + |--------|--------|---|--------|---|-------------| + |$f''(x)$| + | 0 | - | 不存在 | - | + + (A) 1 + (B) 2 + (C) 3 + (D) 4 + +4. 下列级数中条件收敛的是（ ）。 + (A)$\sum_{n=1}^{\infty} (-1)^n \frac{n^2}{2^n + 1}$ + (B)$\sum_{n=1}^{\infty} (-1)^n \frac{2^n}{n! + 1}$ + (C)$\sum_{n=1}^{\infty} (-1)^n \frac{n}{n^2 + 1}$ + (D)$\sum_{n=1}^{\infty} (-1)^n \frac{2^n}{n^2 + 1}$ + +5. 极限 + $$ + \lim_{n \to \infty} \left( \frac{1}{n^2 + 1^2} + \frac{2}{n^2 + 2^2} + \cdots + \frac{n}{n^2 + n^2} \right) + $$ + 的值为（ ）。 + (A)$\frac{\ln 2}{2}$ + (B)$\frac{\pi}{4}$ + (C)$\frac{1}{2}$ + (D) 0 + +--- + +### 三、（6 分）求曲线$y = \frac{\ln(1-x)}{x}$的所有渐近线方程。 + +--- + +### 四、（6 分）已知函数$y = y(x)$由方程$x \cos y + e^y = 1$所确定，求曲线$y = y(x)$在点 (0,0) 处的切线方程。 + +--- + +### 五、（6 分）计算极限$\lim_{x \to +\infty} \left( 1 + 2^x + 3^x \right)^{\frac{1}{x}}$。 + +--- + +### 六、（6 分）已知 +$$ +\begin{cases} +x = \int_{0}^{t} \frac{\sin u}{u} du, \\ +y = \int_{0}^{t} \sin u^2 du, +\end{cases} +$$ +求$\frac{dy}{dx}$和$\frac{d^2 y}{dx^2}$。 + +--- + +### 七、（6 分）计算反常积分 +$$ +\int_{0}^{\infty} \frac{x^2}{\left(1 + x^2\right)^{\frac{5}{2}}} dx。 +$$ + +--- + +### 八、（8 分） +（I）写出函数$f(x) = \ln \sqrt{\frac{1 - x^2}{1 + x^2}}$带佩亚诺余项的 6 阶麦克劳林公式，并求$f''(0)$和$f'''(0)$的值； + +（II）已知当$x \to 0$时，$f(x)$与$g(x) = ae^{-x^2} + b\cos x$是等价无穷小，求常数$a, b$的值。 + +--- + +### 九、（8 分）求曲线$y = \sqrt{x}\sin x \ (0 \leq x \leq \pi)$与$x$轴所围的平面图形绕$x$轴旋转一周所得的旋转体体积。 + +--- + +### 十、（8 分）设$x_1 = 1, x_n = 1 + \frac{x_{n-1}}{1 + x_{n-1}}, n = 2, 3, \cdots$，试用单调有界原理证明极限$\lim_{n \to \infty} x_n$存在，并求出其值。 + +--- + +### 十一、（8 分）要造一个壁和底厚为$a(m)$、容积为$V(m^3)$、上端开口的圆柱形容器，问：当容器端口内半径尺寸$r(m)$为何值时，所用材料最省？ + +--- + +### 十二、（8 分）已知函数$f(x)$在$[0,+\infty)$内二阶可导，且 +$$ +\lim_{x \to 0^+} \frac{f(x)}{x} = \lambda < 0, \quad \lim_{x \to +\infty} f(x) = +\infty。 +$$ +证明： +（I）方程$f(x) = 0$在$(0,+\infty)$内至少有一个实根； +（II）方程$f(x)f''(x)+[f'(x)]^2=0$在$(0,+\infty)$内至少有两个实根。 + +--- + +**（试卷结束）** \ No newline at end of file diff --git a/试卷库/高数期末真题/2018高数期末考试卷.md b/试卷库/高数期末真题/2018高数期末考试卷.md new file mode 100644 index 0000000..a31ce35 --- /dev/null +++ b/试卷库/高数期末真题/2018高数期末考试卷.md @@ -0,0 +1,149 @@ +--- +tags: + - 官方试卷 +--- +# 国防科技大学 2018—2019 学年秋季学期 +## 《高等数学》(I) 考试试卷 (A) 卷 + +**(2019 年 1 月 25 日 8:00—10:30)** + +**考试形式：闭卷** +**考试时间：150 分钟** +**满分：100 分** + +**注意：** +1. 所有答题都须写在答题卡指定区域内，写在其它纸上一律无效。 +2. 答题区域内不得有姓名及相关标记。 + +--- + +### 一、单选题（共 5 小题，每小题 3 分，共 15 分） + +1. 点$x = 0$是函数$f(x) = \frac{e^x - 1}{|x|}$的（ ）。 + (A) 可去间断点 + (B) 跳跃间断点 + (C) 无穷间断点 + (D) 振荡间断点 + +2. 下列级数发散的是（ ）。 + (A)$\sum_{n=1}^{\infty} \frac{4^n + 1}{n^4 + 1}$ + (B)$\sum_{n=1}^{\infty} \frac{n^4 + 1}{4^n + 1}$ + (C)$\sum_{n=1}^{\infty} (-1)^n \frac{1}{\sqrt{n+1}}$ + (D)$\sum_{n=1}^{\infty} (-1)^n \frac{\sqrt{n}}{n+1}$ + +3. 当$x \to 0$时，与$x - \sin x$同阶的无穷小是（ ）。 + (A)$x + \tan x$ + (B)$x \tan x$ + (C)$x^2 + \tan x$ + (D)$x^2 \tan x$ + +4. 已知函数$f(x)$在$x = 1$的某邻域内三阶可导，且$\lim_{x \to 1} \frac{f(x) - 2}{(x-1)^2 \ln x} = \frac{1}{3}$，则（ ）。 + (A)$x = 1$为函数$f(x)$的极大值点 + (B) 点 (1, 2) 为曲线$y = f(x)$的拐点 + (C)$x = 1$为函数$f(x)$的极小值点 + (D) 点 (1, 0) 为曲线$y = f(x)$的拐点 + +5. 设$f(x)$与$g(x)$互为反函数，已知$f(x)$二阶可导，则依据下表知$g'(2)$与$g''(2)$的值分别为（ ）。 + + |$x$| 1 | 2 | + |--------|---|---| + |$f(x)$| 2 | 1 | + |$f'(x)$| -3 | -1 | + |$f''(x)$| 3 | 2 | + + (A) -1, 2 + (B) -1, -2 + (C)$\frac{1}{3}, \frac{1}{9}$ + (D)$\frac{1}{3}, -\frac{1}{3}$ + +--- + +### 二、填空题（共 5 小题，每小题 3 分，共 15 分） + +6. 函数$y = \frac{1 - x + x^2}{1 + x + x^2}$在$x = 0$处的微分是 ______。 + +7. 极限 + $$ + \lim_{n \to \infty} \frac{1}{n} \left( \tan \frac{\pi}{4n} + \tan \frac{2\pi}{4n} + \cdots + \tan \frac{n\pi}{4n} \right) + $$ + 的值为 ______。 + +8. 定积分 + $$ + \int_{-1}^{1} \frac{x(\cos x + x)}{1 + x^2} dx + $$ + 的值为 ______。 + +9. 已知$\int f(x^2) dx = x \ln x + C$，则$f'(2) =$______。 + +10. 极坐标曲线$\rho = \theta$在点$(\rho, \theta) = (\pi, \pi)$处的切线的直角坐标方程为 ______。 + +--- + +### 三、解答题（共 8 小题，共 54 分） + +11. （6分）计算极限 + $$ + \lim_{x \to 0} \frac{x \int_{0}^{x} \sqrt{1 + t^4} dt}{x - \ln (1 + x)}。 + $$ + +12. （6分）试确定常数$a, b$的值，使函数 + $$ + f(x) = + \begin{cases} + ae^x + b, & x > 0, \\ + \cos 3x, & x \leq 0 + \end{cases} + $$ + 在$x = 0$处可导。 + +13. （6分）计算不定积分 + $$ + \int \frac{2 - \sqrt{2x + 1}}{2 + \sqrt{2x + 1}} dx。 + $$ + +14. （6分）计算定积分 + $$ + \int_{0}^{\pi/2} x \sin x \cos^3 x dx。 + $$ + +15. （6分）试写出函数$f(x) = x \ln (2 + x) + \cos x$的 4 阶带佩亚诺余项的麦克劳林公式，并求函数图形在点$(0, 1)$处的曲率。 + +16. （8分）已知函数$f(x) = (2x + 3) e^{\frac{2}{x}}$。 + （1）求函数$f(x)$的单调区间与极值。 + （2）求曲线$y = f(x)$的渐近线方程。 + +17. （8分） + （1）证明：存在$\theta \in (0, 1)$使得 + $$ + \ln (1 + x) - \ln \left( 1 + \frac{x}{2} \right) = \frac{x}{2 + (1 + \theta)x}, \quad x > 0； + $$ + （2）证明不等式 + $$ + \left( 1 + \frac{1}{n} \right)^{n+1} < e \left( 1 + \frac{1}{2n} \right)，\quad n \text{为正整数}。 + $$ + +18. （8分）已知$y = y(x)$是由方程$x \cos y + \sin x + e^y = 1$所确定的隐函数。 + （1）求$\frac{dy}{dx}$； + （2）计算极限 + $$ + \lim_{x \to 0} \left( \frac{1 - y(x)}{1 + y(x)} \right)^{\frac{1}{x}} 的值。 + $$ + +--- + +### 四、应用题（8 分） + +19. 如图所示，某人在离水面 3m 高的岸上，用缆绳拉船靠岸。已知初始时刻船距离岸壁 20m，假设船的运动方向始终与岸壁垂直。问：当船离岸壁 4m，收缆速度为 2m/s 时，船靠岸的速度大小为多少？此时，缆绳倾斜角$\theta$关于时间的变化率是多少？ + +--- + +### 五、证明题（8 分） + +20. 设函数$f(x)$在$(-\infty, +\infty)$内可导，且$|f'(x)| \leq \lambda$，其中$\lambda (0 < \lambda < 1)$为常数。给定实数$a_1$，令$a_{n+1} = f(a_n) \quad (n=1,2,\cdots)$，证明： + （1）级数$\sum_{n=1}^{\infty} (a_{n+1} - a_n)$是绝对收敛的； + （2）数列$\{a_n\}$收敛，且其极限值为方程$f(x) = x$的唯一实数根。 + +--- + +**（试卷结束）** \ No newline at end of file diff --git a/试卷库/高数期末真题/2019高数期末考试卷.md b/试卷库/高数期末真题/2019高数期末考试卷.md new file mode 100644 index 0000000..2db1129 --- /dev/null +++ b/试卷库/高数期末真题/2019高数期末考试卷.md @@ -0,0 +1,144 @@ +--- +tags: + - 官方试卷 +--- +# 2019—2020学年秋季学期 +## 《高等数学》（I）考试试卷（A）卷 + +**考试形式：闭卷** +**考试时间：150分钟** +**满分：100分** + +| 题号 | 一 | 二 | 三 | 总分 | +|------|----|----|----|------| +| 得分 | | | | | +| 评阅人 | | | | | + +**注意：** +1. 所有答题都须写在答题卡对应位置，写在其它纸上一律无效。 +2. 答题卡密封线外不得有姓名及相关标记。 +3. 所有教学班做1-19题，空医、陆医三个教学班做20*题和21*题，其他教学班做20题和21题。 + +--- + +### 一、选择题（共5小题，每小题2分，共10分） + +1. 函数 + $$ + f(x) = \begin{cases} + \sqrt{x} \sin \frac{1}{x}, & x > 0, \\ + 0, & x \leq 0 + \end{cases} + $$ + 在点$x = 0$处（ ）。 + (A) 不连续 + (B) 连续但不可导 + (C) 可导且$f'(0) = 0$ + (D) 可导且$f'(0) \neq 0$ + +2. 数列极限$\lim_{n \to \infty} (e^{-n} + \pi^{-n})^{\frac{1}{n}}$的值为（ ）。 + (A)$e$ + (B)$\pi$ + (C)$\frac{1}{e}$ + (D)$\frac{1}{\pi}$ + +3. 曲线$y = \frac{x^3 - x^2}{2 + x^2}$的渐近线为（ ）。 + (A)$y = x - 1$ + (B)$y = x + 1$ + (C)$y = x$ + (D)$y = \frac{1}{2}x$ + +4. 图1中曲线分别为函数$y = f(x)$及其一、二阶导函数的图形，则编号为①、②、③的曲线依次对应的函数是（ ）。 + (A)$y = f(x)$、$y = f'(x)$、$y = f''(x)$ + (B)$y = f'(x)$、$y = f(x)$、$y = f''(x)$ + (C)$y = f'(x)$、$y = f''(x)$、$y = f(x)$ + (D)$y = f(x)$、$y = f''(x)$、$y = f'(x)$ + +5. 设$y = f(x)$为区间$[0,1]$上单调增加的连续函数，且$f(0) = 0$，$f(1) = 2$，$x = g(y)$为$y = f(x)$的反函数。若$\int_{0}^{1} f(x) dx = \frac{1}{3}$，则$\int_{0}^{2} g(y) dy$的值为（ ）。 + (A)$\frac{1}{3}$ + (B)$\frac{2}{3}$ + (C)$\frac{4}{3}$ + (D)$\frac{5}{3}$ + +--- + +### 二、填空题（共5小题，每小题2分，共10分） + +6. 已知函数$y = e^{\sin(2x)}$，则在$x = 0$处的微分$dy|_{x=0} = \underline{\qquad}$。 + +7. 函数$f(x) = xe^{-x^2}$在$(-\infty,+\infty)$上的最大值为 \underline{\qquad}。 + +8. 曲线$C: x = \frac{1}{2} \cos t, y = \sin t, t \in [0,2\pi]$在点$(0,-1)$处的曲率为 \underline{\qquad}。 + +9. 已知函数$f(x) = x^2 \sin x$，则$f^{(7)}(0) = \underline{\qquad}$。 + +10. 不定积分$\int \frac{1}{x(1+2\ln x)} dx = \underline{\qquad}$。 + +--- + +### 三、解答题（共11小题，共80分） + +11. 计算数列极限 + $$ + \lim_{n \to \infty} \left( \frac{1}{2n+1} + \frac{1}{2n+2} + \cdots + \frac{1}{2n+n} \right)。 + $$ + （6分） + +12. 设$y(x)$是由曲线方程$\sin x + y + e^x = 2$确定的隐函数，试计算$\frac{dy}{dx} \bigg|_{x=0}$的值，并求该曲线在点$P(0,1)$处的切线方程。（6分） + +13. 计算不定积分 + $$ + \int \frac{x}{1+\sqrt{1-x^2}} dx。 + $$ + （6分） + +14. 设曲线$f(x) = x^3 + ax^2 + 18x$（$a$为大于零的常数）的拐点正好位于$x$轴上，试求$a$的值及曲线$y = f(x)$的拐点坐标。（6分） + +15. 计算极限 + $$ + \lim_{x \to +\infty} \left[ x + x^2 \ln \left( 1 - \frac{1}{x} \right) \right]。 + $$ + （6分） + +16. 设函数$f(x)$在$[0,1]$上连续，且$f(x) = \arcsin x + x \int_0^1 f(x) \, dx$，试求$f(x)$的表达式。（6分） + +17. 无人机在海面执勤时，发现可疑目标后将实时拍照传回指挥部。已知无人机按直线飞行，高度为150米，飞行过程中摄像头始终对准可疑目标（如图2所示）。当无人机飞临目标正上方时，速度为3米/秒。问：此时无人机摄像头转动角速度为多少？（8分） + +18. 证明不等式 + $$ + 1 + x \int_0^x \frac{dt}{\sqrt{1 + t^2}} > \sqrt{1 + x^2} \quad (x > 0)。 + $$ + （8分） + +19. 记$I_n = \int_{0}^{\frac{\pi}{4}} \sec^n x \, dx, (n=0,1,2,\cdots)$。 + （1）证明：当$n \geq 2$时， + $$ + I_n = \frac{2^{\frac{n-2}{2}}}{n-1} + \frac{n-2}{n-1} I_{n-2}； + $$ + （2）计算$I_3$的值。（8分） + +--- + +**（以下为选做题，普通班做20、21题；空医、陆医班做20*、21*题）** + +20. （10分）已知当$x \to 0$时，函数$f(x) = \sqrt{a + bx^2} - \cos x$与$x^2$是等价无穷小。 + （1）求参数$a, b$的值；（5分） + （2）计算极限$\lim_{x \to 0} \frac{f(x) - x^2}{x^4}$的值。（5分） + +20*. （10分）已知当$x \to 0$时，函数$f(x) = a + bx^2 - \cos x$与$x^2$是等价无穷小。 + （1）求参数$a, b$的值；（5分） + （2）计算极限$\lim_{x \to 0} \frac{f(x) - x^2}{x^4}$的值。（5分） + +21. （10分）设函数$f(x)$在$[0,1]$上可导，$f(1) = 1$且$2 \int_{0}^{1} x^2 f(x) dx = 1$。证明存在$\xi \in (0,1)$，使得 + $$ + f(\xi) = -\frac{\xi}{2} f'(\xi)。 + $$ + +21*. （10分）设函数$f(x)$在$[0,1]$上可导，$\int_{0}^{\frac{1}{2}} f(x) dx = 0$。证明存在$\xi \in (0,1)$，使得 + $$ + f(\xi) = (1 - \xi) f'(\xi)。 + $$ + +--- + +**（试卷结束）** \ No newline at end of file diff --git a/试卷库/高数期末真题/2020高数期末考试卷.md b/试卷库/高数期末真题/2020高数期末考试卷.md new file mode 100644 index 0000000..869962d --- /dev/null +++ b/试卷库/高数期末真题/2020高数期末考试卷.md @@ -0,0 +1,147 @@ +--- +tags: + - 官方试卷 +--- +# 2020—2021 学年秋季学期 +## 《高等数学》（I）考试试卷（A）卷 + +**考试形式：闭卷** +**考试时间：150 分钟** +**满分：100 分** + +| 题号 | 一 | 二 | 三 | 总分 | +|------|----|----|----|------| +| 得分 | | | | | +| 评阅人 | | | | | + +**注意：** +1. 所有答题都须写在答题卡对应位置，写在其它纸上一律无效。 +2. 答题卡密封线外不得有姓名及相关标记。 + +--- + +### 一、单选题（共5小题，每小题2分，共10分） + +1. 函数$f(x) = \frac{|x+1|}{x(x-1)(x+1)} \arctan x + \sin \frac{1}{x-2}$的可去间断点为（ ）。 + (A)$x = 0$ + (B)$x = 1$ + (C)$x = 2$ + (D)$x = -1$ + +2. 已知函数$f(x)$可导，$y = \int_0^x f(t^2) dt$，则$y''(x)$等于（ ）。 + (A)$f(x^2)$ + (B)$2xf'(x^2)$ + (C)$2xf(x^2)$ + (D)$2f(x^2) + 4x^2 f'(x^2)$ + +3. “级数$\sum_{n=1}^\infty a_n$收敛”是“级数$a_1 - a_1 + a_2 - a_2 + \cdots + a_n - a_n + \cdots$收敛”的（ ）。 + (A) 充分但非必要条件 + (B) 必要但非充分条件 + (C) 充要条件 + (D) 既非充分也非必要条件 + +4. 已知函数$f(x), g(x)$在$(-\infty, +\infty)$内可导，且$f'(x) > 0, g'(x) < 0$，则（ ）。 + (A)$\int_0^1 f(x) dx > \int_1^2 f(x) dx$ + (B)$\int_0^1 [f(x)] dx > \int_1^2 [f(x)] dx$ + (C)$\int_0^1 f(x) g(x) dx > \int_1^2 f(x) g(x) dx$ + (D)$\int_0^1 [g(x)] dx > \int_1^2 [g(x)] dx$ + +5. 已知函数$f(x) = x^2 \sin \frac{1}{x}$，则下列四个判断： + ①$f(x)$在$(0, +\infty)$上有界 + ②$f(x)$是$x \to +\infty$过程的无穷大量 + ③$f'(x)$在$(0, +\infty)$上有界 + ④$f'(x)$是$x \to +\infty$过程的无穷大量 + 正确的是（ ）。 + (A) ①③ + (B) ①④ + (C) ②③ + (D) ②④ + +--- + +### 二、填空题（共5小题，每小题2分，共10分） + +6. 设$y = y(x)$是由方程$xy + e^{2y} = \sin 3x + 1$确定的隐函数，则$dy|_{x=0} = \underline{\qquad}$。 + +7. 极限 + $$ + \lim_{n \to \infty} \left( \frac{1}{\sqrt{3n^4 + 1}} + \frac{2}{\sqrt{3n^4 + 2}} + \cdots + \frac{n}{\sqrt{3n^4 + n}} \right) + $$ + 的值为 $\underline{\qquad}$。 + +8. 定积分 + $$ + \int \frac{\pi}{2} \left( |\sin x| + x \right) \cos^2 x \, dx + $$ + 的值为 $\underline{\qquad}$。 + +9. 不定积分 + $$ + \int x \tan^2 x \, dx = \underline{\qquad}。 + $$ + +10. 已知 + $$ + \int x f(x) \, dx = \ln \left( 1 + x^2 \right) + C, + $$ + 则曲线$y = f(x) (x > 0)$的拐点为 \underline{\qquad}。 + +--- + +### 三、解答题（共11小题，共80分） + +11. （6分）求极限 + $$ + \lim_{x \to 0} \frac{e^{x^2} - 1 - \ln \left( x^2 + 1 \right)}{x^3 \arcsin x}. + $$ + +12. （6分）设函数 + $$ + f(x) = + \begin{cases} + |x|^x, & x \neq 0, \\ + a, & x = 0, + \end{cases} + $$ + 试求常数$a$的值，使得$f(x)$在$x = 0$处连续，并讨论此时$f(x)$在$x = 0$处的可导性。 + +13. （6分）将函数$f(x) = x^2 e^x + x^6$展开成六阶带佩亚诺余项的麦克劳林公式，并求$f^{(6)} (0)$的值。 + +14. （6分）已知$f(x)$是周期为 2 的可导函数，且曲线$y = f(x)$与曲线$y = \int_0^x e^{-t^2} dt$在点$(0,0)$处相切。求曲线$y = f(x)$在$x = 4$处的切线方程。 + +15. （6分）已知函数$f(x)$在$[0,1]$上可导，$f(0)=0$。证明：至少存在一点$\xi \in (0,1)$，使得 + $$ + f'(1-\xi)=\frac{2}{\xi}f(1-\xi). + $$ + +16. （6分）在某直线公路上有 A, B, C, D 四个加油站，依次相距 20 km。现计划在该公路上建一个加油总站 M，并配备一台供油车给各加油站供油。已知 A, B, C, D 四个加油站每天所需的油量依次为 2 车、3 车、5 车及 4 车。问：加油总站 M 建在何处可使供油车每天行驶的总路程最少？并说明理由。 + + （图略：A、B、C、D 依次排列，间距均为 20 km） + +17. （8分）已知函数$f(x) = \frac{\ln(1+x)}{1+x}$。 + (1) 求函数$f(x)$的单调区间与极值；（4分） + (2) 求曲线$y = f(x)$的渐近线。（4分） + +18. （8分）求极坐标曲线$C: \rho = 1 + \cos \theta$在$\theta = \frac{\pi}{2}$对应点处的曲率与曲率半径。 + +19. （8分）已知函数$f(x)$在$(-\infty, +\infty)$内连续，且 + $$ + \int_{0}^{x} tf(x-t) dt = x^3 - \int_{0}^{x} f(t) dt. + $$ + (1) 验证：$f'(x) + f(x) = 6x$且$f(0) = 0$；（5分） + (2) 计算定积分$\int_{0}^{1} e^{x} f(x) dx$。（3分） + +20. （10分）设函数$f(x)$在$(-\infty, +\infty)$内可导，$f'(0) = 2$，且对任意$x, y$，恒有 + $$ + f(x+y) = f(x) + f(y) + 2xy. + $$ + (1) 求函数$f(x)$的表达式；（6分） + (2) 求曲线$y = f(x)$与$x$轴所围成的平面图形的面积。（4分） + +21. （10分） + (1) 证明：对任意的正整数$n$，方程 + $$ + x^n + n^2 x - 1 = 0 + $$ + 有唯一正实根（记为$x_n$）；（6分） + (2) 证明级数$\sum_{n=1}^{\infty} x_n$收敛，且其和不超过 2。（4分） \ No newline at end of file

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