From 58b404b5ded785ed1016a1b384009221b6a797cb Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Tue, 29 Jun 2021 18:03:21 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=A7=AF=E5=88=86=E5=B9=B6?= =?UTF-8?q?=E5=A2=9E=E5=8A=A0=E8=A7=A3=E4=B8=8E=E8=AF=81=E6=98=8E=E6=A0=87?= =?UTF-8?q?=E8=AF=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../function-and-limit.pdf | Bin 188359 -> 271997 bytes .../function-and-limit.tex | 78 ++++---- .../continuity-and-discontinuity.pdf | Bin 66494 -> 91903 bytes .../continuity-and-discontinuity.tex | 8 +- .../derivative-and-differentiate.pdf | Bin 134181 -> 199030 bytes .../derivative-and-differentiate.tex | 32 ++-- ...efinite-integral-and-definite-integral.pdf | Bin 215795 -> 247848 bytes ...efinite-integral-and-definite-integral.tex | 166 +++++++++++++++--- advanced-math/knowledge/0-perpare/perpare.pdf | Bin 295981 -> 296016 bytes advanced-math/knowledge/0-perpare/perpare.tex | 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a/advanced-math/exercise/1-function-and-limit/function-and-limit.tex +++ b/advanced-math/exercise/1-function-and-limit/function-and-limit.tex @@ -55,7 +55,7 @@ \textbf{例题:}证明多项式$f(x)=x^3-3x+a$在$[0,1]$上不可能有两个零点。 -假设$f(x)=x^3-3x+a$在$[0,1]$有两个零点$x_1$和$x_2$,其中$x_1b>0$,证明:$\dfrac{a-b}{a}<\ln\dfrac{a}{b}<\dfrac{a-b}{b}$。 -因为$\ln\dfrac{a}{b}=\ln a-\ln b$,所以令$f(x)=\ln x$。 +证明:因为$\ln\dfrac{a}{b}=\ln a-\ln b$,所以令$f(x)=\ln x$。 所以根据拉格朗日中值定理:$\ln a-\ln b=f'(\xi)(a-b)$($\xi\in(b,a)$)。 @@ -103,7 +103,7 @@ \textbf{例题:}当$x>1$时,证明$e^x>ex$。 -题目中没有差式,所以需要选择一个函数作为基准函数,里面有一个指数函数和一个幂函数,所以选择$e^x$作为基准函数。 +证明:题目中没有差式,所以需要选择一个函数作为基准函数,里面有一个指数函数和一个幂函数,所以选择$e^x$作为基准函数。 然后选择一个常数作为$b$值,可以先选一个1作为$b$值:$f(x)-f(1)=f'(\xi)(x-1)$。 @@ -115,7 +115,7 @@ \textbf{例题:}设$00$时,$x\to 0^+$,两边都乘以10,$10-x0,b>0,c>0)$。\medskip -$=e^{\lim\limits_{x\to 0}\frac{\ln\left(\frac{a^x+b^x+c^x}{3}\right)}{x}}=e^{\lim\limits_{x\to 0}\frac{\ln(a^x+b^x+c^x)-\ln 3}{x}}=e^{\lim\limits_{x\to 0}\frac{a^x\ln a+b^x\ln b+c^x\ln c}{a^x+b^x+c^x}}$(洛必达法则) +解:$=e^{\lim\limits_{x\to 0}\frac{\ln\left(\frac{a^x+b^x+c^x}{3}\right)}{x}}=e^{\lim\limits_{x\to 0}\frac{\ln(a^x+b^x+c^x)-\ln 3}{x}}=e^{\lim\limits_{x\to 0}\frac{a^x\ln a+b^x\ln b+c^x\ln c}{a^x+b^x+c^x}}$(洛必达法则) $=e^{\lim\limits_{x\to 0}\frac{\ln a+\ln b+\ln c}{1+1+1}}=e^{\lim\limits_{x\to 0}\frac{\ln(abc)}{3}}=\sqrt[3]{abc}$。\medskip \textbf{例题:}求$\lim\limits_{n\to\infty}n\left[\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}-\sqrt{e}\right]$。 -首先对于幂指函数需要取指数,所以$\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}=e^{\frac{n}{2}\ln(1+\frac{1}{n})}$。\medskip +解:首先对于幂指函数需要取指数,所以$\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}=e^{\frac{n}{2}\ln(1+\frac{1}{n})}$。\medskip 而后面的多一个$\sqrt{e}$导致整个式子变为一个复杂的式子,而与$e^x$相关的是$e^x-1\sim x$。 @@ -276,7 +276,7 @@ $=\dfrac{e^{\frac{1}{2}}}{2}\lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}-\dfrac{1 \textbf{例题:}求极限$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)$。\medskip -$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)=\lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{\sin^2x\cdot x^2} (\sin x\sim x)$ \medskip +解:$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)=\lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{\sin^2x\cdot x^2} (\sin x\sim x)$ \medskip $=\lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{x^4} (\sin x\cos x\sim\dfrac{1}{2}\sin 2x)=\lim\limits_{x\to 0}\dfrac{x^2-\dfrac{1}{4}\sin^22x}{x^4}$ \medskip @@ -294,9 +294,7 @@ $=\lim\limits_{x\to 0}\dfrac{2-\dfrac{1}{2}\cos 4x\cdot 4}{12x^2}=\dfrac{1}{6}\l \textbf{例题:}求$\lim\limits_{x\to 4}\dfrac{x^2-6x+8}{x^5-5x+4}$。 -需要先提取公因子: - -$=\lim\limits_{x\to 4}\dfrac{(x-2)(x-4)}{(x-1)(x-4)}=\lim\limits_{x\to 4}\dfrac{x-2}{x-1}=\dfrac{2}{3}$。 +解:需要先提取公因子:$=\lim\limits_{x\to 4}\dfrac{(x-2)(x-4)}{(x-1)(x-4)}=\lim\limits_{x\to 4}\dfrac{x-2}{x-1}=\dfrac{2}{3}$。 (当然可以使用洛必达法则得到极限为$\lim\limits_{x\to 4}\dfrac{2x-6}{2x-5}=\lim\limits_{x\to 4}\dfrac{8-6}{8-5}$) @@ -306,7 +304,7 @@ $=\lim\limits_{x\to 4}\dfrac{(x-2)(x-4)}{(x-1)(x-4)}=\lim\limits_{x\to 4}\dfrac{ \textbf{例题:}求$\lim\limits_{x\to-\infty}\left[\sqrt{4x^2+x}\ln\left(2+\dfrac{1}{x}\right)+2\ln 2x\right]$。\medskip -题目的形式为$\infty-\infty$,所以必须使用后面的倒代换转换为商的形式。\medskip +解:题目的形式为$\infty-\infty$,所以必须使用后面的倒代换转换为商的形式。\medskip $=\lim\limits_{x\to-\infty}-x\left[\sqrt{4+\dfrac{1}{x}}\ln\left(2+\dfrac{1}{x}\right)-2\ln 2\right]$。 \medskip @@ -328,7 +326,7 @@ $=\lim\limits_{t\to 0^-}\left(\dfrac{1}{2}\cdot\dfrac{\ln(2+t)}{\sqrt{t+4}}+\dfr \textbf{例题:}求极限$\lim\limits_{x\to-\infty}x(\sqrt{x^2+100}+x)$。 -首先定性分析:$\lim\limits_{x\to-\infty}x\cdot(\sqrt{x^2+100}+x)$。 +解:首先定性分析:$\lim\limits_{x\to-\infty}x\cdot(\sqrt{x^2+100}+x)$。 在$x\to-\infty$趋向时,$x$就趋向无穷大。 @@ -342,7 +340,7 @@ $\xRightarrow{\text{令}x=-t}\lim\limits_{t\to+\infty}\dfrac{-100t}{\sqrt{t^2+10 \textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x\sqrt{1+\sin^2x}-x}$。 -$=\lim\limits_{x\to 0}\sqrt{1+\tan x}-\sqrt{1+\sin x}\cdot\lim\limits_{x\to 0}\dfrac{1}{x\sqrt{1+\sin^2x}-x}$\medskip +解:$=\lim\limits_{x\to 0}\sqrt{1+\tan x}-\sqrt{1+\sin x}\cdot\lim\limits_{x\to 0}\dfrac{1}{x\sqrt{1+\sin^2x}-x}$\medskip $=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{\sqrt{1+\tan x}+\sqrt{1+\sin x}}\cdot\lim\limits_{x\to 0}\dfrac{x\sqrt{1+\sin^2x}+x}{x^2(1+\sin^2x)-x^2}$\medskip @@ -358,7 +356,7 @@ $=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{x\sin^2x}=\lim\limits_{x\to 0}\dfrac \textbf{例题:}求极限$\lim\limits_{x\to 1^-}\ln x\ln(1-x)$。 -当$x\to 1^-$时,$\ln x$趋向0,$\ln(1-x)$趋向$-\infty$。 +解:当$x\to 1^-$时,$\ln x$趋向0,$\ln(1-x)$趋向$-\infty$。 又$x\to 0$,$\ln(1+x)\sim x$,所以$x\to 1$,$\ln x\sim x-1$: @@ -374,7 +372,7 @@ $=-\lim\limits_{t\to 0^+}\dfrac{\ln t}{\dfrac{1}{t}}=-\lim\limits_{t\to 0^+}\dfr \textbf{例题:}求极限$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}$ \medskip -$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}=\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}\cdot 2x^{-3}}{100x^{99}}=\lim\limits_{x\to 0}\dfrac{1}{50}\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{102}}$ +解:$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}=\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}\cdot 2x^{-3}}{100x^{99}}=\lim\limits_{x\to 0}\dfrac{1}{50}\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{102}}$ \medskip @@ -390,7 +388,7 @@ $=\lim\limits_{t\to+\infty}\dfrac{50!}{e^t}=0$ \textbf{例题:}求极限$\lim\limits_{x\to+\infty}[x^2(e^{\frac{1}{x}}-1)-x]$。 -该式子含有分数,所以尝试使用倒数代换:\medskip +解:该式子含有分数,所以尝试使用倒数代换:\medskip $\lim\limits_{x\to+\infty}[x^2(e^{\frac{1}{x}}-1)-x]\xRightarrow{\text{令}x=\frac{1}{t}}\lim\limits_{t\to 0^+}\left(\dfrac{e^t-1}{t^2}-\dfrac{1}{t}\right)=\lim\limits_{t\to 0^+}\dfrac{e^t-1-t}{t^2}$ @@ -408,7 +406,7 @@ $\xRightarrow{\text{泰勒展开}e^t}\lim\limits_{t\to 0^+}\dfrac{\dfrac{1}{2}t^ \textbf{例题:}求$\lim\limits_{n\to\infty}\dfrac{(n+1)(n+2)(n+3)\cdots(n+6)}{6n^6}$。 -需要将分子和分母都拆为6项: +解:需要将分子和分母都拆为6项: $=\dfrac{1}{6}\lim\limits_{n\to\infty}\dfrac{n+1}{n}\times\dfrac{n+2}{n}\times\cdots\dfrac{n+6}{n}=\dfrac{1}{6}\lim\limits_{n\to\infty}(1+\dfrac{1}{n})(1+\dfrac{2}{n})\cdots(1+\dfrac{6}{n})=\dfrac{1}{6}$。 @@ -420,7 +418,7 @@ $=\dfrac{1}{6}\lim\limits_{n\to\infty}\dfrac{n+1}{n}\times\dfrac{n+2}{n}\times\c \textbf{例题:}求极限$\lim\limits_{x\to 0}\left(\dfrac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)^{\frac{e}{x}}$。($n\in N^+$) -这里可以使用等价无穷小$e^x-1\sim x$。 +解:这里可以使用等价无穷小$e^x-1\sim x$。 $\lim\limits_{x\to 0}\left(\dfrac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)^{\frac{e}{x}}=e^{\lim\limits_{x\to 0}\frac{e}{x}\ln\left(\frac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)}$ @@ -430,7 +428,7 @@ $=e^{\frac{e}{n}\lim\limits_{x\to 0}\left(\frac{e^x-1}{x}+\frac{e^{2x}-1}{x}+\cd \textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{\ln\cos x}$\medskip -可以使用$\cos x-1\sim\dfrac{x^2}{2}$。\medskip +解:可以使用$\cos x-1\sim\dfrac{x^2}{2}$。\medskip $\lim\limits_{x\to 0}\dfrac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{\ln\cos x}$ @@ -500,7 +498,7 @@ $\therefore \lim u^v=e^{\lim v\cdot\ln u}=e^{\lim v(u-1)}$ \textbf{例题:}已知$\lim\limits_{x\to 0}\dfrac{\ln(1-x)+xf(x)}{x^2}=0$,求$\lim\limits_{x\to 0}\dfrac{f(x)-1}{x}$。\medskip -令目标$\dfrac{f(x)-1}{x}=t$,$\therefore f(x)=tx+1$。\medskip +解:令目标$\dfrac{f(x)-1}{x}=t$,$\therefore f(x)=tx+1$。\medskip $\lim\limits_{x\to 0}\dfrac{\ln(1-x)+xf(x)}{x^2}=\lim\limits_{x\to 0}\dfrac{\ln(1-x)+tx^2+x}{x^2} (\text{泰勒展开})$\medskip @@ -512,7 +510,7 @@ $\therefore\lim\limits_{x\to 0}t=\lim\limits_{x\to 0}\dfrac{f(x)-1}{x}=\dfrac{1} \textbf{例题:}如果$\lim\limits_{x\to 0}\dfrac{x-\sin x+f(x)}{x^4}$存在,则$\lim\limits_{x\to 0}\dfrac{x^3}{f(x)}$为常数多少? -由$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$,而目标是$x^3$,所以需要变形: +解:由$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$,而目标是$x^3$,所以需要变形: $\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$ @@ -532,7 +530,7 @@ $\lim\limits_{x\to x_0}f(x)\Leftrightarrow f(x)=A+\alpha(x),\lim\limits_{x\to x_ \textbf{例题:}如果$\lim\limits_{x\to 0}\dfrac{x-\sin x+f(x)}{x^4}$存在,则$\lim\limits_{x\to 0}\dfrac{x^3}{f(x)}$为常数多少? -由$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$脱帽:$\dfrac{x\sin x+f(x)}{x^4}=A+\alpha$。 +解:由$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$脱帽:$\dfrac{x\sin x+f(x)}{x^4}=A+\alpha$。 得到:$f(x)=Ax^4+\alpha\cdot x^4-(x-\sin x)$。 @@ -552,7 +550,7 @@ $\therefore \lim\limits_{x\to 0}\dfrac{x^3}{f(x)}=-6$。 \textbf{例题:}设$\lim\limits_{x\to 0}\dfrac{\ln(1+x)-(ax+bx^2)}{x^2}=2$,求常数a,b。 -根据泰勒展开式:$x\to 0,\ln(1+x)=x-\dfrac{x^2}{x}+o(x^2)$,$x-\ln(1+x)\sim\dfrac{1}{2}x^2\sim 1-\cos x$。 +解:根据泰勒展开式:$x\to 0,\ln(1+x)=x-\dfrac{x^2}{x}+o(x^2)$,$x-\ln(1+x)\sim\dfrac{1}{2}x^2\sim 1-\cos x$。 $\lim\limits_{x\to 0}\dfrac{\ln(1+x)-(ax+bx^2)}{x^2}=2$ @@ -568,7 +566,7 @@ $\therefore a=1;b=-\dfrac{5}{2}$。 \textbf{例题:}确定常数$a$和$b$,使得$f(x)=x-(a+b\cos x)\sin x$当$x\to 0$时关于$x$的5阶等价无穷小。 -使用泰勒展开展开到五阶: +解:使用泰勒展开展开到五阶: $f(x)=x-(a+b\cos x)\sin x$ @@ -618,7 +616,7 @@ $\therefore a=-1,b=\dfrac{4}{3}$。 \textbf{例题:}设$a$为常数,$\lim\limits_{x\to 0}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)$存在,求出极限值。 -因为求$x\to 0$,所以需要分两种情况讨论: +解:因为求$x\to 0$,所以需要分两种情况讨论: \medskip @@ -647,7 +645,7 @@ $\lim\limits_{x\to 0^-}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cd e^{-\frac{1}{2}}, & & x\leqslant 0 \end{array}\right.$在$x=0$处的连续性。 -因为$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^+}\left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]^{\frac{1}{x}}=e^{\lim\limits_{x\to 0^+}\frac{1}{x}\ln[\frac{(1+x)^{\frac{1}{x}}}{e}]}$。 +解:因为$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^+}\left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]^{\frac{1}{x}}=e^{\lim\limits_{x\to 0^+}\frac{1}{x}\ln[\frac{(1+x)^{\frac{1}{x}}}{e}]}$。 又$\lim\limits_{x\to 0^+}\dfrac{1}{x}\ln\left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]=\lim\limits_{x\to 0^+}\dfrac{1}{x}\left[\dfrac{1}{x}\ln(1+x)-1\right]=\lim\limits_{x\to 0^+}\dfrac{\ln(1+x)-x}{x^2}$ @@ -663,7 +661,7 @@ $\therefore\lim\limits_{x\to 0^+}f(x)=e^{-\frac{1}{2}}$。 \textbf{例题:}函数在$f(x)$在$x=1$处连续,且$f(1)=1$,求$\lim\limits_{x\to+\infty}\ln\left[2+f\left(x^{\frac{1}{x}}\right)\right]$。 -根据题目,所求的$\lim\limits_{x\to+\infty}\ln\left[2+f\left(x^{\frac{1}{x}}\right)\right]$中,唯一未知的且会随着$x\to+\infty$而变换就是$f\left(x^{\frac{1}{x}}\right)$。如果我们可以求出这个值就可以了。 +解:根据题目,所求的$\lim\limits_{x\to+\infty}\ln\left[2+f\left(x^{\frac{1}{x}}\right)\right]$中,唯一未知的且会随着$x\to+\infty$而变换就是$f\left(x^{\frac{1}{x}}\right)$。如果我们可以求出这个值就可以了。 而我们对于$f(x)$的具体的关系是未知的,只知道$f(1)=1$。那么先需要考察$\lim\limits_{x\to+\infty}x^{\frac{1}{x}}$的整数最大值。 @@ -679,7 +677,7 @@ $\therefore\lim\limits_{x\to+\infty}f(x^{\frac{1}{x}})=f(1)=1$。 \textbf{例题:}数列$\{a_n\}$满足$a_0=0,a_1=1,2a_{n+1}=a_n+a_{n-1},n=1,2,\cdots$。计算$\lim\limits_{n\to\infty}a_n$。 -首先看题目,给出的递推式设计到二阶递推,即存在三个数列变量,所以我们必须先求出对应的数列表达式。因为这个表达式涉及三个变量,所以尝试对其进行变型: +解:首先看题目,给出的递推式设计到二阶递推,即存在三个数列变量,所以我们必须先求出对应的数列表达式。因为这个表达式涉及三个变量,所以尝试对其进行变型: $a_{n+1}-a_n=\dfrac{a_{n-1}-a_n}{2}=\left(-\dfrac{1}{2}\right)(a_n-a_{n-1})=\left(-\dfrac{1}{2}\right)^2(a_{n-1}-a_{n-2})$ @@ -705,7 +703,7 @@ $\therefore\lim\limits_{n\to\infty}a_n=\dfrac{2}{3}$ \textbf{例题:}$x_0=0$,$x_n=\dfrac{1+2x_{n-1}}{1+x_{n-1}}(n\in N*)$,求$\lim\limits_{n\to\infty}x_n$。\medskip -首先应该知道数列的趋向都是趋向正无穷。 +解:首先应该知道数列的趋向都是趋向正无穷。 然后对关系式进行变形:$x_n=\dfrac{1+2x_{n-1}}{1+x_{n-1}}=1+\dfrac{x_{n-1}}{1+x_{n-1}}=2-\dfrac{1}{1+x_{n-1}}$。 @@ -733,7 +731,7 @@ $\therefore\lim\limits_{n\to\infty}x_n=\dfrac{1+\sqrt{5}}{2}$。 \textbf{例题:}求出数列$\sqrt{2}$,$\sqrt{2+\sqrt{2}}$,$\sqrt{2+\sqrt{2+\sqrt{2}}}$$\cdots$的极限。 -根据数列样式,无法通过普通的通项公式来表达,所以需要考虑使用递推式来表示:$x_{n+1}=\sqrt{2+x_n}$。 +解:根据数列样式,无法通过普通的通项公式来表达,所以需要考虑使用递推式来表示:$x_{n+1}=\sqrt{2+x_n}$。 首先证明有界性: diff --git a/advanced-math/exercise/2-continuity-and-discontinuity/continuity-and-discontinuity.pdf b/advanced-math/exercise/2-continuity-and-discontinuity/continuity-and-discontinuity.pdf index 880a9e30a47a08235319093714cd0b28192e668d..0fa381ccf7fde627895a5a1fc25fb1af3577ad9c 100644 GIT binary patch delta 88621 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1$,当$x\in(-\infty,-1)\cup(1,+\infty)$时,当$n\to\infty$时,$x^n\to\infty$,而$x\in(-1,1)$时,当$n\to\infty$时,$x^n\to 0$。 @@ -138,7 +138,7 @@ $\therefore\lim\limits_{n\to\infty}\dfrac{1+x}{1+x^{2n}}=\left\{\begin{array}{lc \textbf{例题:}$f(x)=\dfrac{e^x-b}{(x-a)(x-b)}$有无穷间断点$x=e$,可去间断点$x=1$,求$ab$的值。 -已知有两个间断点$x=a,x=b$,其中无穷间断点指极限值为无穷的点,可去间断点表示极限值存在且两侧相等,但是与函数值不相等的点。 +解:已知有两个间断点$x=a,x=b$,其中无穷间断点指极限值为无穷的点,可去间断点表示极限值存在且两侧相等,但是与函数值不相等的点。 已经给出两个间断点的值为$x=1$和$x=e$,所以$ab$必然对应其中一个,但是不清楚到底谁是谁。 diff --git a/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.pdf b/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.pdf index 2967248c63e9c1e44f097db7931ba00cf24c90c5..406fda30c8dcb0d5a024a8217521731e6c96f60f 100644 GIT binary patch delta 195708 zcmZs>Q*7Xm6ZYHP+S=OI*0yb%Tib5GwQburwl=r6ZQFf+@5#A1|K!}wOfs2^$xJ?Z z@)V~*Jte@!u>u^NjC_1BPR@=dhBh$nYnSPQwet*ce%BunU;Y>ugL1ggU7#q04qkJ% 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z!Ya_Xu2BNdUhomkz(X5_qCybV$gj=-*T5&%K)ZVJwz#2h+hu8|4X1{`reIRa*{}gy zrS*v1R;QZQwX%`@GPiXBUD0HoyfJmf)7hQ$-OsKX&Z-*g>?)7985W3tNs@;J TCq!UlW8>t8qo9ybl!W_V%O=Q> diff --git a/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex b/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex index 10dbc7d..2c9a7bd 100644 --- a/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex +++ b/advanced-math/exercise/3-derivative-and-differentiate/derivative-and-differentiate.tex @@ -41,7 +41,7 @@ \textbf{例题:}求$f(x)=x^{\sin x}(x>0)$的导数。 -取对数: +解:取对数: $\therefore\ln y=\sin x\ln x$ @@ -65,7 +65,7 @@ $e^{\sin x\cdot\ln x}(\sin x\cdot\ln x)'=x^{\sin x}\left(\cos\ln x+\dfrac{\sin x \textbf{例题:}设$f(x)=\left\{\begin{array}{lcl} \arctan x, & & x\leqslant 1 \\ - \dfrac{1}{2}(e^{x^2-1}-x)+\dfrac{\pi}{4}, & & x>1 +解: \dfrac{1}{2}(e^{x^2-1}-x)+\dfrac{\pi}{4}, & & x>1 \end{array} \right.$,求$f'(x)$。 @@ -97,7 +97,7 @@ $f(x)=\left\{\begin{array}{lcl} \end{array} \right.$,其中$b$为某常数,$f(x)$在定义域上处处可导,求$f'(x)$。 -首先需要求出参数$b$,而定义域上可导则在分段点$x=0$处也必然可导。 +解:首先需要求出参数$b$,而定义域上可导则在分段点$x=0$处也必然可导。 而可导必连续,所以当$x=0$时$f(x)$也是连续的,而连续的定义就是两边极限相等,且两边极限等于该点函数值。\medskip @@ -151,7 +151,7 @@ $\therefore f'(0)=\lim\limits_{x\to 0}f'(x)$。计算过程类似。 \end{array} \right.$,若$f'(x)$连续,则$\alpha$应该满足? -若导数连续,则两侧导数相等。 +解:若导数连续,则两侧导数相等。 $\lim\limits_{x\to 0^-}f'(x)=\lim\limits_{x\to 0^-}2x=0$。 @@ -175,7 +175,7 @@ $\therefore\alpha-2>0$,从而$\alpha>2$。 \textbf{例题:}已知$f(x)$是周期为5的连续函数,它在$x=0$的某个邻域内满足关系式:$f(1+\sin x)-3f(1-\sin x)= 8x+o(x)$,且$f(x)$在$x=1$处可导,求曲线$y=f(x)$在点$(6,f(6))$处的切线方程。 -因为这是个函数等式,而我们最后要求的是一个导数,所以先尝试对其直接求极限,令$x\to0$: +解:因为这是个函数等式,而我们最后要求的是一个导数,所以先尝试对其直接求极限,令$x\to0$: $f(1)-3f(1)=0$,从而得到了一个函数值$f(1)=0$。 @@ -217,7 +217,7 @@ L01 \textbf{例题:}设$f(x)$是以3为周期的可导函数,且是偶函数,$f'(-2)=-1$,求$\lim\limits_{h\to 0}\dfrac{h}{f(5-2\sin h)-f(5)}$。\medskip -根据导数与函数的基本性质,原函数为偶函数,则其导函数为奇函数,所以$f'(5)=f'(2)=-f'(-2)=1$。 +解:根据导数与函数的基本性质,原函数为偶函数,则其导函数为奇函数,所以$f'(5)=f'(2)=-f'(-2)=1$。 然后需要转换目标的极限式子,因为目标式子倒过来的式子类似于导数定义的$f'(x)=\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}$结构。所以我们可以先求其倒数式子:\medskip @@ -233,7 +233,7 @@ $\therefore\lim\limits_{h\to 0}\dfrac{h}{f(5-2\sin h)-f(5)}=-\dfrac{1}{2}$。 \textbf{例题:}设$f(x)$在$x=0$处可导且$f(0)=1$,$f'(0)=3$,则数列极限$I=\lim\limits_{n\to\infty}\left(f\left(\dfrac{1}{n}\right)\right)^{\frac{\frac{1}{n}}{1-\cos\frac{1}{n}}}$。\medskip -设$\dfrac{1}{n}=x$,则: +解:设$\dfrac{1}{n}=x$,则: $=\lim\limits_{x\to 0}(f(x))^{\frac{x}{1-\cos x}}=e^{\lim\limits_{x\to 0}\frac{x}{1-\cos x}\ln f(x)}=e^{2\lim\limits_{x\to 0}\frac{\ln f(x)}{x}}=e^{2\lim\limits_{x\to 0}\frac{f(x)-1}{x}}$ @@ -251,7 +251,7 @@ $=e^{2\lim\limits_{x\to 0}\frac{f(x)-f(0)}{x-0}}=e^{2f'(0)}=e^6$。 \textbf{例题:}已知$f(x)=x^2(x+1)^2(x+2)^2\cdots(x+n)^2$,求$f''(0)$。 -因为式子中带有未知数$n$,所以结果很可能会带有$n$。 +解:因为式子中带有未知数$n$,所以结果很可能会带有$n$。 而这个式子项数为$n+1$项,所以求导结果必然很大,所以一定会消去一部分。 @@ -289,13 +289,13 @@ $\quad\quad\quad+x^2(x+1)^2(x+2)^2\cdots 2(x+n)$ \textbf{例题:}已知$y=x+e^x$,求其反函数的二阶导数。 -$y=x+e^x$的反函数的一阶导数为$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{1}{1+e^x}$。\medskip +解:$y=x+e^x$的反函数的一阶导数为$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{1}{1+e^x}$。\medskip 所以二阶导数为$\dfrac{\textrm{d}^2x}{\textrm{d}y^2}=\dfrac{\textrm{d}\left(\dfrac{1}{1+e^{x}}\right)}{\textrm{d}y}=\dfrac{\dfrac{\textrm{d}\left(\dfrac{1}{1+e^{x}}\right)}{\textrm{d}x}}{\dfrac{\textrm{d}y}{\textrm{d}x}}=-\dfrac{e^x}{(1+e^x)^3}$。 \textbf{例题:}已知$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{1}{y'}$,求$\dfrac{\textrm{d}^2x}{\textrm{d}y^2}$和$\dfrac{\textrm{d}^3x}{\textrm{d}y^3}$。 -其实就是求$\dfrac{\textrm{d}^2x}{\textrm{d}y^2}$和$\dfrac{\textrm{d}^3x}{\textrm{d}y^3}$,$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{1}{y'}$这个条件只是让我们用$y'$来表示结果而已。 +解:其实就是求$\dfrac{\textrm{d}^2x}{\textrm{d}y^2}$和$\dfrac{\textrm{d}^3x}{\textrm{d}y^3}$,$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{1}{y'}$这个条件只是让我们用$y'$来表示结果而已。 $\dfrac{\textrm{d}^2x}{\textrm{d}y^2}=\dfrac{\textrm{d}\dfrac{\textrm{d}x}{\textrm{d}y}}{\textrm{d}y}=\dfrac{\dfrac{\textrm{d}\dfrac{\textrm{d}x}{\textrm{d}y}}{\textrm{d}x}}{\dfrac{\textrm{d}y}{\textrm{d}x}}=\dfrac{-\dfrac{y''}{(y')^2}}{y'}=-\dfrac{y''}{(y')^3}$。\medskip @@ -315,7 +315,7 @@ $\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{\dfrac{\textrm{d}y}{\textrm{d}t}}{\dfra \textbf{例题:}落在平静水面上的石头,产生同心波纹。若最外一圈波半径的增大速率总是6m/s,问在2s末扰动水面面积增大的速率为多少? -首先根据题目最后的要求的是面积,所以肯定要设一个面积变量,随时间变动而改变,所以也一定会设一个时间变量,同时还给出一个条件是半径增大速度,所以也会有一个半径的变量。同时要求的是面积增大速率,正好跟另外两个变量相关,时间跟半径和面积都相关,所以时间就是中间变量。 +解:首先根据题目最后的要求的是面积,所以肯定要设一个面积变量,随时间变动而改变,所以也一定会设一个时间变量,同时还给出一个条件是半径增大速度,所以也会有一个半径的变量。同时要求的是面积增大速率,正好跟另外两个变量相关,时间跟半径和面积都相关,所以时间就是中间变量。 从而设最外一圈波的半径为$r=r(t)$,圆的面积$S=S(t)$。根据$S$和$r$的公式$S=\pi r^2$,因为求的是随时间变化的速率,所以其两端分别对$t$求导,得: @@ -331,7 +331,7 @@ $\dfrac{\textrm{d}S}{\textrm{d}t}\bigg|_{t=2}=2\pi\cdot12\cdot6=144\pi$。 \end{array} \right.$确定,求其一阶导数与二阶导数。 -$\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{\dfrac{\textrm{d}y}{\textrm{d}t}}{\dfrac{\textrm{d}x}{\textrm{d}t}}=\dfrac{\dfrac{1}{2}\cdot\dfrac{2t}{1+t^2}}{\dfrac{1}{1+t^2}}=\dfrac{1}{t}$。 +解:$\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{\dfrac{\textrm{d}y}{\textrm{d}t}}{\dfrac{\textrm{d}x}{\textrm{d}t}}=\dfrac{\dfrac{1}{2}\cdot\dfrac{2t}{1+t^2}}{\dfrac{1}{1+t^2}}=\dfrac{1}{t}$。 $\dfrac{\textrm{d}^2y}{\textrm{d}x^2}=\dfrac{\textrm{d}\left(\dfrac{\textrm{d}y}{\textrm{d}x}\right)}{\textrm{d}x}=\dfrac{\dfrac{\textrm{d}\left(\dfrac{\textrm{d}y}{\textrm{d}x}\right)}{\textrm{d}t}}{\dfrac{\textrm{d}x}{\textrm{d}t}}=\dfrac{-\dfrac{1}{t^2}}{\dfrac{t}{1+t^2}}=-\dfrac{1+t^2}{t^3}$。 @@ -341,7 +341,7 @@ $\dfrac{\textrm{d}^2y}{\textrm{d}x^2}=\dfrac{\textrm{d}\left(\dfrac{\textrm{d}y} \textbf{例题:}求$y=x+\vert\sin 2x\vert$的单调区间。 -因为函数的定义域为$R$。 +解:因为函数的定义域为$R$。 又$y=\left\{\begin{array}{lcl} x+\sin 2x, & & n\pi\leqslant x\leqslant n\pi+\dfrac{\pi}{2} \\ @@ -375,7 +375,7 @@ $\left[x=n\pi+\dfrac{5\pi}{6},(n+1)\pi\right]$($n=0,\pm 1,\pm2,\cdots$)。 \textbf{例题:}决定曲线$y=ax^3+bx^2+cx+d$中参数,使得$x=-2$处曲线有水平切线,$(1,-10)$为拐点,且点$(-2,44)$在曲线上。 -$y'=3ax^2+2bx+c$,$y''=6ax+2b$。 +解:$y'=3ax^2+2bx+c$,$y''=6ax+2b$。 因为$x=-2$处曲线有水平切线,即$y'\vert_{x=-2}=12a-4b+c=0$。 @@ -401,7 +401,7 @@ $(1,-10)$为拐点,代入:$y''\vert_{x=1}=6a+2b=0$,$y\vert_{x=1}=a+b+c+d=- \textbf{例题:}求$y=\sin x$在$x=\dfrac{\pi}{4}$对应的曲率 -$y'=\cos x$,$y'(\dfrac{\pi}{4})=\dfrac{\sqrt{2}}{2}$。 +解:$y'=\cos x$,$y'(\dfrac{\pi}{4})=\dfrac{\sqrt{2}}{2}$。 $y''=-\sin x$,$y''(\dfrac{\pi}{4})=-\dfrac{\sqrt{2}}{2}$。 @@ -413,7 +413,7 @@ $\therefore k=\dfrac{\dfrac{\sqrt{2}}{2}}{\dfrac{3}{2}\cdot\sqrt{\dfrac{3}{2}}}= \textbf{例题:}求$y=x^2-4x+11$曲率最大值所在的点。 -简单得$y'=2x-4$,$y''=2$。 +解:简单得$y'=2x-4$,$y''=2$。 曲率为$\dfrac{2}{[1+(2x-4)^2]^{\frac{3}{2}}}$。 diff --git a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf 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a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex +++ b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -41,7 +41,7 @@ \textbf{例题:}汽车以20m/s的速度行驶,刹车后匀减速行驶了50m停止,求刹车加速度。 -已知题目含有两个变量:距离和时间,设距离为$s$,时间为$t$。 +解:已知题目含有两个变量:距离和时间,设距离为$s$,时间为$t$。 因为汽车首先按20m/s匀速运动,所以$\dfrac{\textrm{d}s}{\textrm{d}t}\bigg\vert_{t=0}=20$,最开始距离为0,所以$s\vert_{t=0}=0$。 @@ -69,11 +69,13 @@ \textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{x\ln x\ln\ln x}}$。 \medskip -$=\displaystyle{\int\dfrac{\textrm{d}(\ln x)}{\ln x\ln\ln x}}=\displaystyle{\int\dfrac{\textrm{d}(\ln\ln x)}{\ln\ln x}}=\ln\vert\ln\ln x\vert+C$。\medskip +解:$=\displaystyle{\int\dfrac{\textrm{d}(\ln x)}{\ln x\ln\ln x}}=\displaystyle{\int\dfrac{\textrm{d}(\ln\ln x)}{\ln\ln x}}=\ln\vert\ln\ln x\vert+C$。\medskip \textbf{例题:}求$\displaystyle{\int\dfrac{10^{2\arccos x}}{\sqrt{1-x^2}}\,\textrm{d}x}$。 -$=-\displaystyle{\int10^{2\arccos x}\,\textrm{d}(\arccos x)}=-\dfrac{1}{2}\displaystyle{\int10^{2\arccos x}\,\textrm{d}(2\arccos x)}=-\dfrac{10^{2\arccos x}}{2\ln10}+C$。 +解:$=-\displaystyle{\int10^{2\arccos x}\,\textrm{d}(\arccos x)}=-\dfrac{1}{2}\displaystyle{\int10^{2\arccos x}\,\textrm{d}(2\arccos x)}$ + +$=-\dfrac{10^{2\arccos x}}{2\ln10}+C$。 \paragraph{积化和差} \leavevmode \medskip @@ -81,7 +83,7 @@ $=-\displaystyle{\int10^{2\arccos x}\,\textrm{d}(\arccos x)}=-\dfrac{1}{2}\displ \textbf{例题:}求$\displaystyle{\int\sin2x\cos3x\,\textrm{d}x}$。 -$=\displaystyle{\int\cos3x\sin2x\,\textrm{d}x=\dfrac{1}{2}\int(\sin5x-\sin x)\,\textrm{d}x}$ +解:$=\displaystyle{\int\cos3x\sin2x\,\textrm{d}x=\dfrac{1}{2}\int(\sin5x-\sin x)\,\textrm{d}x}$ $=\dfrac{1}{2}\int\sin5x\,\textrm{d}x-\dfrac{1}{2}\int\sin x\,\textrm{d}x=-\dfrac{1}{10}\cos5x+\dfrac{1}{2}\cos x+C$。 @@ -91,7 +93,9 @@ $=\dfrac{1}{2}\int\sin5x\,\textrm{d}x-\dfrac{1}{2}\int\sin x\,\textrm{d}x=-\dfra \textbf{例题:}$\displaystyle{\int\tan^3x\sec x\,\textrm{d}x}$。 -$=\displaystyle{\int(\sec^2x-1)}\tan x\sec x\,\textrm{d}x=\displaystyle{\int(\sec^2x-1)}\,\textrm{d}(\sec x)=\dfrac{1}{3}\sec^3x-\sec x+C$。 +解:$=\displaystyle{\int(\sec^2x-1)}\tan x\sec x\,\textrm{d}x=\displaystyle{\int(\sec^2x-1)}\,\textrm{d}(\sec x)$ + +$=\dfrac{1}{3}\sec^3x-\sec x+C$。 需要利用到有理积分的高阶多项式分配与低阶多项式因式分解。 @@ -101,7 +105,7 @@ $=\displaystyle{\int(\sec^2x-1)}\tan x\sec x\,\textrm{d}x=\displaystyle{\int(\se \textbf{例题:}求$\int(\arcsin x)^2\,\textrm{d}x$。 -令$\arcsin x=t$,$\therefore x=\sin t$,$\textrm{d}x=\cos t\,\textrm{d}t$。 +解:令$\arcsin x=t$,$\therefore x=\sin t$,$\textrm{d}x=\cos t\,\textrm{d}t$。 $=\int t^2\cos t\,\textrm{d}t$。除了换元法还需要使用分部积分法,可以直接进行分部积分也可以使用下面讲到的分部积分拓展公式的表格法来计算,$\cos t$更难求导,所以对其积分,放在下面,$t^2$容易求导,所以对其求导,放在上面:\medskip @@ -130,7 +134,7 @@ $\therefore=t^2\sin t-2t\cos t-\sin t+C=(\arcsin x)^2x+2\arcsin x\sqrt{1-x^2}-2x \textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{1+\sqrt[3]{x+1}}}$。 -令$u=\sqrt[3]{x+1}$,从而$x=u^3-1$,$\textrm{d}x=3u^2\,\textrm{d}u$。 +解:令$u=\sqrt[3]{x+1}$,从而$x=u^3-1$,$\textrm{d}x=3u^2\,\textrm{d}u$。 $=\displaystyle{\int\dfrac{3u^2}{1+u}\textrm{d}u=\int\dfrac{3u^2+3u-3u-3+3}{1+u}\textrm{d}u=\int\left(3u-3+\dfrac{3}{1+u}\right)\textrm{d}u}$ @@ -138,7 +142,9 @@ $=\dfrac{3}{2}u^2-3u+3\ln\vert1+u\vert+C=\dfrac{3}{2}\sqrt[3]{(x+1)^2}-3\sqrt[3] \textbf{例题:}求$\displaystyle{\int\sqrt{\dfrac{1-x}{1+x}}\dfrac{\textrm{d}x}{x}}$。\medskip -令$u=\sqrt{\dfrac{1-x}{1+x}}$,$x=\dfrac{1-u^2}{1+u^2}$,$\textrm{d}x=\textrm{d}\left(\dfrac{1-u^2}{1+u^2}\right)=\dfrac{-2u(1+u^2)-2u(1-u^2)}{(1+u^2)^2}$ +解:令$u=\sqrt{\dfrac{1-x}{1+x}}$,$x=\dfrac{1-u^2}{1+u^2}$。 + +$\therefore\textrm{d}x=\textrm{d}\left(\dfrac{1-u^2}{1+u^2}\right)=\dfrac{-2u(1+u^2)-2u(1-u^2)}{(1+u^2)^2}$ $=\displaystyle{\int u\cdot\dfrac{1+u^2}{1-u^2}\cdot\dfrac{-4u}{(1+u^2)^2}\textrm{d}u=\int\dfrac{-4u^2}{(1-u)(1+u)(1+u^2)}\textrm{d}u}$ @@ -166,7 +172,7 @@ $=2\arctan\sqrt{\dfrac{1-x}{1+x}}+\ln\left\vert\dfrac{\sqrt{1+x}-\sqrt{1-x}}{\sq \textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{3+\cos x}}$。 -令$u=\tan\dfrac{x}{2}$,从而$\cos x=\dfrac{1-u^2}{1+u^2}$: +解:令$u=\tan\dfrac{x}{2}$,从而$\cos x=\dfrac{1-u^2}{1+u^2}$: $=\displaystyle{\int\dfrac{1}{3+\dfrac{1-u^2}{1+u^2}}\cdot\dfrac{2}{1+u^2}\textrm{d}u=\dfrac{1}{2+u^2}\textrm{d}u=\dfrac{1}{\sqrt{2}}\arctan\dfrac{u}{\sqrt{2}}+C}$ @@ -184,7 +190,7 @@ $=\dfrac{1}{\sqrt{2}}\arctan\dfrac{\tan\dfrac{x}{2}}{\sqrt{2}}+C$。 \textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{1+\sqrt{1-x^2}}}$。\medskip -令$x=\sin t$($t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$),所以$\sqrt{1-x^2}=\cos t$,$\textrm{d}x=\cos t\,\textrm{d}t$,$t=\arcsin x$。 +解:令$x=\sin t$($t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$),所以$\sqrt{1-x^2}=\cos t$,$\textrm{d}x=\cos t\,\textrm{d}t$,$t=\arcsin x$。 因为式子$\dfrac{1}{1+\sqrt{1-x^2}}>0$,单调递减,所以不用讨论正负号。 @@ -200,7 +206,7 @@ $=\arcsin x-\dfrac{x}{1+\sqrt{1-x^2}}+C$。 \textbf{例题:}求$\displaystyle{\int\dfrac{x^3+1}{(x^2+1)^2}\textrm{d}x}$。\medskip -虽然本题目看着可以从分母解开平方,然后低阶分配,但是这分母是平方的式子很难分配,所以需要使用换元法。 +解:虽然本题目看着可以从分母解开平方,然后低阶分配,但是这分母是平方的式子很难分配,所以需要使用换元法。 令$x=\tan t$,$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$,$x^2+1=\sec^2t$,$\textrm{d}x=\sec^2t\,\textrm{d}t$。 @@ -224,7 +230,7 @@ $=\dfrac{1+x}{2(1+x^2)}+\dfrac{1}{2}\ln(1+x^2)+\dfrac{1}{2}\textrm{arctan}\,x+C$ \textbf{例题:}求$\displaystyle{\int\dfrac{\sqrt{x^2-9}}{x}\textrm{d}x}$。 -令$x=3\sec t$。$\therefore\sqrt{x^2-9}=3\tan t$,$\textrm{d}x=3\sec t\tan t\,\textrm{d}t$。\medskip +解:令$x=3\sec t$。$\therefore\sqrt{x^2-9}=3\tan t$,$\textrm{d}x=3\sec t\tan t\,\textrm{d}t$。\medskip 因为式子$\dfrac{\sqrt{x^2-9}}{x}$的分子必然为为正,而对于分子在0两边的单调性不同,所以需要对$x$进行正负区分,又$x\in(-\infty,-3]\cup[3,+\infty)$,所以: @@ -252,7 +258,7 @@ $=\sqrt{x^2-9}-3\arccos\dfrac{3}{-x}+C$。 \textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{x+\sqrt{1-x^2}}}$。\medskip -令$x=\sin t$,所以$\sqrt{1-x^2}=\cos t$,$\textrm{d}x=\cos t\,\textrm{d}t$。 +解:令$x=\sin t$,所以$\sqrt{1-x^2}=\cos t$,$\textrm{d}x=\cos t\,\textrm{d}t$。 $\because x+\sqrt{1-x^2}$可能为正可能为负,正负时单调性不同,所以令$ x+\sqrt{1-x^2}=0$,即$\sin t=-\dfrac{\sqrt{2}}{2}$,从而$t\in(-\dfrac{\pi}{2},-\dfrac{\pi}{4})\cup(-\dfrac{\pi}{4},\dfrac{\pi}{2})$。 @@ -292,7 +298,7 @@ $I_1-I_2=\displaystyle{\int\dfrac{\cos t-\sin t}{\sin t+\cos t}\textrm{d}t=\int\ \textbf{例题:}求$\int x^2\arctan x\,\textrm{d}x$。 -$=\dfrac{1}{3}\int\arctan x\,\textrm{d}(x^3)=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{3}\int x^3\,\textrm{d}(\arctan x)$ +解:$=\dfrac{1}{3}\int\arctan x\,\textrm{d}(x^3)=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{3}\int x^3\,\textrm{d}(\arctan x)$ $=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{3}\displaystyle{\int\dfrac{x^3}{1+x^2}\textrm{d}x}=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{3}\displaystyle{\int\dfrac{x+x^3-x}{1+x^2}\textrm{d}x}$ @@ -308,7 +314,7 @@ $=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{6}x^2+\dfrac{1}{6}\ln(1+x^2)+C$。 \textbf{例题:}求$\int x\tan^2x\,\textrm{d}x$。 -$=\int x(\sec^2-1)\,\textrm{d}x=\int x\,\textrm{d}(\tan x)-\dfrac{x^2}{2}=x\tan x+\ln\vert\cos x\vert-\dfrac{x^2}{2}+C$。 +解:$=\int x(\sec^2-1)\,\textrm{d}x=\int x\,\textrm{d}(\tan x)-\dfrac{x^2}{2}=x\tan x+\ln\vert\cos x\vert-\dfrac{x^2}{2}+C$。 \subsubsection{多次分部} @@ -320,7 +326,7 @@ $=\int x(\sec^2-1)\,\textrm{d}x=\int x\,\textrm{d}(\tan x)-\dfrac{x^2}{2}=x\tan \textbf{例题:}求$\int e^x\sin^2x\,\textrm{d}x$。 -$=\sin^2x\cdot e^x-\int e^x\,\textrm{d}(\sin^2x)=\sin^2x\cdot e^x-\int e^x\cdot\sin 2x\,\textrm{d}x$ +解:$=\sin^2x\cdot e^x-\int e^x\,\textrm{d}(\sin^2x)=\sin^2x\cdot e^x-\int e^x\cdot\sin 2x\,\textrm{d}x$ $=\sin^2x\cdot e^x-\int\sin2x\,\textrm{d}(e^x)=\sin^2x\cdot e^x-\sin2x\cdot e^x+\int e^x\,\textrm{d}(\sin2x)$ @@ -346,7 +352,9 @@ $=\dfrac{e^x(5\sin^2x-\sin2x+2\cos2x)}{5}+C=e^x\left(\dfrac{1}{2}-\dfrac{1}{5}\s \subsubsection{分部积分推广公式} -分部积分法可以直接利用表格简便计算,以三次导数的积分为例: +分部积分法可以直接利用表格简便计算,特别是积分式子中含有$f(x)$这种不定的因子。 + +如何得到分部积分推广公式?以三次导数的积分为例: $\int uv'''\,\textrm{d}x=\int u\,\textrm{d}(v'')=uv''-\int v''u'\,\textrm{d}x$ @@ -370,7 +378,9 @@ $\therefore\int uv'''\,\textrm{d}x=uv''-u'v'+u''v-\int u'''v\,\textrm{d}x$。 \textbf{例题:}求$\int(x^3+2x+6)e^{2x}\,\textrm{d}x$。 -如果要一般求,则需要拆分:$\int(x^3+2x+6)e^{2x}\,\textrm{d}x=\int x^3e^{2x}\,\textrm{d}x+2\int xe^{2x}\,\textrm{d}x+6\int e^2x\,\textrm{d}x$。 +解:如果要一般求,则需要拆分: + +$\int(x^3+2x+6)e^{2x}\,\textrm{d}x=\int x^3e^{2x}\,\textrm{d}x+2\int xe^{2x}\,\textrm{d}x+6\int e^2x\,\textrm{d}x$。 而如果使用分部积分的推广公式,令$u=x^3+2x+6$,$v=e^{2x}$。\medskip @@ -385,6 +395,8 @@ $\therefore=(x^3+2x+6)\left(\dfrac{1}{2}e^{2x}\right)-(3x^2+2)\left(\dfrac{1}{4} \textbf{例题:}求$\int x^2\ln x\,\textrm{d}x$。\medskip +解: + \begin{tabular}{|c|c|} \hline $\ln x$ & $\dfrac{1}{x}$ \\ \hline @@ -400,7 +412,7 @@ $\therefore=\dfrac{1}{3}x^3\ln x-\displaystyle{\int\dfrac{1}{x}\dfrac{1}{3}x^3\, \textbf{例题:}求$\int e^{\sqrt[3]{x}}\,\textrm{d}x$。 -令$\sqrt[3]{x}=u$,从而$x=u^3$,$\textrm{d}x=3u^2\,\textrm{d}u$。 +解:令$\sqrt[3]{x}=u$,从而$x=u^3$,$\textrm{d}x=3u^2\,\textrm{d}u$。 $=3\int e^uu^2\,\textrm{d}u=3\int u^2\,\textrm{d}(e^u)=3u^2e^u-3\int e^u\,\textrm{d}(u^2)=3u^2e^u-6\int e^uu\,\textrm{d}u$ @@ -410,7 +422,7 @@ $=3e^u(u^2-2u+2)+C=3e^{\sqrt[3]{x}}(x^{\frac{2}{3}}-2x^{\frac{1}{3}}+2)+C$。 \textbf{例题:}求$\int e^{\sqrt{3x+9}}\,\textrm{d}x$。 -令$\sqrt{3x+9}=u$,从而$x=\dfrac{1}{3}(u^2-9)$,$\textrm{d}x=\dfrac{2}{3}u\,\textrm{d}u$: +解:令$\sqrt{3x+9}=u$,从而$x=\dfrac{1}{3}(u^2-9)$,$\textrm{d}x=\dfrac{2}{3}u\,\textrm{d}u$: $=\displaystyle{\dfrac{2}{3}\int ue^u\,\textrm{d}u=\dfrac{2}{3}\int u\,\textrm{d}(e^u)=\dfrac{2}{3}ue^u-\int\dfrac{2}{3}e^u\,\textrm{d}u=\dfrac{2}{3}ue^u-\dfrac{2}{3}e^u+C}$ @@ -426,7 +438,7 @@ $=\dfrac{2}{3}e^{\sqrt{3x+9}}(\sqrt{3x+9}-1)+C$。 \textbf{例题:}$\displaystyle{\int\dfrac{x^3}{x^2+9}\,\textrm{d}x}$。 \medskip -$=\displaystyle{\int\dfrac{x^3+9x-9x}{x^2+9}\,\textrm{d}x=\int\dfrac{x^3+9x}{x^2+9}\,\textrm{d}x-\int\dfrac{9x}{x^2+9}\,\textrm{d}x}$ \medskip +解:$=\displaystyle{\int\dfrac{x^3+9x-9x}{x^2+9}\,\textrm{d}x=\int\dfrac{x^3+9x}{x^2+9}\,\textrm{d}x-\int\dfrac{9x}{x^2+9}\,\textrm{d}x}$ \medskip $\displaystyle{=\int x\,\textrm{d}x-\dfrac{9}{2}\int\dfrac{\textrm{d}(x^2+9)}{x^2+9}}=\dfrac{x^2}{2}-\dfrac{9}{2}\ln(9+x^2)+C$。 @@ -447,7 +459,7 @@ $\displaystyle{=\int x\,\textrm{d}x-\dfrac{9}{2}\int\dfrac{\textrm{d}(x^2+9)}{x^ \textbf{例题:}求$\displaystyle{\int\dfrac{x^2+1}{(x+1)^2(x-1)}\textrm{d}x}$。\medskip -令$\dfrac{A}{x+1}+\dfrac{B}{(x+1)^2}+\dfrac{C}{x-1}=\dfrac{x^2+1}{(x+1)^2(x-1)}$。 \medskip +解:令$\dfrac{A}{x+1}+\dfrac{B}{(x+1)^2}+\dfrac{C}{x-1}=\dfrac{x^2+1}{(x+1)^2(x-1)}$。 \medskip 通分:$=A(x+1)(x-1)+B(x-1)+C(x+1)^2$ @@ -479,7 +491,7 @@ $=\dfrac{1}{2}\ln\vert x-1\vert+\dfrac{1}{2}\ln\vert x+1\vert+\dfrac{1}{x+1}+C=\ \textbf{例题:}求$\displaystyle{\int\dfrac{x-1}{x^2+2x+3}\textrm{d}x}$。 -因为$x^2+2x+3$不能因式分解,所以考虑将分子按照分母形式进行分配。优先对高阶的$x$进行分配。 +解:因为$x^2+2x+3$不能因式分解,所以考虑将分子按照分母形式进行分配。优先对高阶的$x$进行分配。 首先因为分子最高阶为$x$只比分母最高阶$x^2$低一阶,所以考虑将$x-1$分配到微分号内。 @@ -497,7 +509,7 @@ $=\dfrac{1}{2}\ln(x^2+2x+3)-\sqrt{2}\arctan\dfrac{x+1}{\sqrt{2}}+C$。 \textbf{例题:}求$\displaystyle{\int\dfrac{x^5+x^4-8}{x^3-x}\textrm{d}x}$。 -$=\displaystyle{\int\dfrac{x^5-x^3+x^4-x^2+x^3-x+x^2+x-8}{x^3-x}\textrm{d}x}$ +解:$=\displaystyle{\int\dfrac{x^5-x^3+x^4-x^2+x^3-x+x^2+x-8}{x^3-x}\textrm{d}x}$ $=\int x^2\,\textrm{d}x+\int x\,\textrm{d}x+\int\textrm{d}x+\displaystyle{\int\dfrac{x^x+x-8}{x^3-x}\textrm{d}x}=\dfrac{x^3}{3}+\dfrac{x^2}{2}+x+\displaystyle{\int\dfrac{x^2+x-8}{x^3-x}\textrm{d}x}$ @@ -515,7 +527,7 @@ $=\dfrac{x^3}{3}+\dfrac{x^2}{2}+x+8\ln\vert x\vert-4\ln\vert x+1\vert-3\ln\vert \textbf{例题:}求$\displaystyle{\int\dfrac{-x^2-2}{(x^2+x+1)^2}\textrm{d}x}$。 -首先根据因式分解:$\dfrac{-x^2-2}{(x^2+x+1)^2}=\dfrac{Ax+B}{x^2+x+1}+\dfrac{Cx+D}{(x^2+x+1)^2}$。 +解:首先根据因式分解:$\dfrac{-x^2-2}{(x^2+x+1)^2}=\dfrac{Ax+B}{x^2+x+1}+\dfrac{Cx+D}{(x^2+x+1)^2}$。 $\therefore Ax^3+(A+B)x^2+(A+B+C)x+(B+D)=-x^2-2$。 @@ -569,7 +581,7 @@ $=-\dfrac{4x+3}{2(x^2+x+1)}-\dfrac{6}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}+C$ \textbf{例题:}求$\lim\limits_{n\to\infty}\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}\cdots+\dfrac{1}{n+n}\right)$。 -即求$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}$。如果我们要传统求的话一般使用夹逼准则,找到放缩的两个函数。 +解:即求$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}$。如果我们要传统求的话一般使用夹逼准则,找到放缩的两个函数。 所以找到两个:$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+n}<\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}<\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+1}$。 @@ -581,7 +593,7 @@ $\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}=\lim\limits_{n\to\inf \textbf{例题:}求$\lim\limits_{n\to\infty}\left(\dfrac{n+1}{n^2+1}+\dfrac{n+2}{n^2+4}+\cdots+\dfrac{n+n}{n^2+n^2}\right)$。 -即求$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{n+i}{n^2+i^2}=\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{n^2+ni}{n^2+i^2}\cdot\dfrac{1}{n}=\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1+\dfrac{i}{n}}{1+\left(\dfrac{i}{n}\right)^2}\cdot\dfrac{1}{n}$ +解:即求$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{n+i}{n^2+i^2}=\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{n^2+ni}{n^2+i^2}\cdot\dfrac{1}{n}=\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1+\dfrac{i}{n}}{1+\left(\dfrac{i}{n}\right)^2}\cdot\dfrac{1}{n}$ $=\displaystyle{\int_0^1\dfrac{1+x}{1+x^2}\textrm{d}x}=\displaystyle{\int_0^1\dfrac{1}{1+x^2}\textrm{d}x+\int_0^1\dfrac{x}{1+x^2}\textrm{d}x}$ @@ -603,6 +615,106 @@ $=\left[\arctan x+\dfrac{1}{2}\ln(1+x^2)\right]_0^1=\dfrac{\pi}{4}+\dfrac{1}{2}\ \subsection{体积} +\subsection{平均值} + \subsection{弧长} +\section{积分等式} + +包括证明带有积分的等式、方程根、积分中值定理等。 + +\subsection{中值定理} + +\textbf{例题:}设$f(x)$、$g(x)$在$[a,b]$上连续且$g(x)$不变号,证明至少存在一点$\xi\in[a,b]$使得$\int_a^bf(x)g(x)\,\textrm{d}x=f(\xi)\int_a^bg(x)\,\textrm{d}x$。(推广中值定理) + +若$g(x)\equiv1$,则结论就是$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$积分中值定理。 + +证明:若$g(x)\equiv0$则等式显然成立。若$g(x)$不变号,则不妨设$g(x)>0$。 + +令$F(x)=\int_a^xf(t)g(t)\,\textrm{d}t$,$G(x)=\int_a^xg(t)\,\textrm{d}t$。 + +由柯西中值定理可得$\dfrac{F(b)-F(a)}{G(b)-G(a)}=\dfrac{F'(\xi)}{G'(\xi)}=\dfrac{\int_a^bf(x)g(x)\,\textrm{d}x}{\int_a^bg(x)\,\textrm{d}x}=\dfrac{f(\xi)g(\xi)}{g(\xi)}$。 + +$\therefore\int_a^bf(x)g(x)\,\textrm{d}x=f(\xi)\int_a^bg(x)\,\textrm{d}x$,$\xi\in(a,b)\subset[a,b]$。 + +同理$g(x)<0$也如此。 + +\subsection{夹逼准则} + +夹逼准则一般用于求数列极限,若极限号后面有积分号可以考虑使用夹逼准则。 + +\section{积分不等式} + +带积分号的不等问题。 + +\subsection{函数单调性} + +将某一限一般是上限变量化,然后移项构造辅助函数,由辅助函数的单调性来证明不等式,多用于所给条件为“$f(x)$在$[a,b]$上连续”的情况,因为函数连续必然可积分与求导。 + +\textbf{例题:}设$f(x)$在$[a,b]$上连续,且$f(x)>0$,证明$\displaystyle{\int_a^bf(x)\,\textrm{d}x\int_a^b\dfrac{1}{f(x)}\textrm{d}x}\geqslant(b-a)^2$。 + +证明:令$F(x)=\displaystyle{\int_a^xf(t)\,\textrm{d}t\int_a^x\dfrac{1}{f(t)}\textrm{d}t-(x-a)^2}$。目标是证明$F(b)\geqslant0$。 + +$F'(x)=f(x)\cdot\displaystyle{\int_a^x\dfrac{1}{f(t)}\textrm{d}t+\int_a^xf(t)\,\textrm{d}t\dfrac{1}{f(x)}}-2(x-a)$ + +$=\displaystyle{\int_a^x\dfrac{f(x)}{f(t)}\textrm{d}t+\int_a^x\dfrac{f(t)}{f(x)}\textrm{d}t-2(x-a)=\int_a^x\left[\dfrac{f(x)}{f(t)}+\dfrac{f(t)}{f(x)}\right]\textrm{d}t-2(x-a)}$ + +又$\dfrac{f(x)}{f(t)}+\dfrac{f(t)}{f(x)}\geqslant2\sqrt{\dfrac{f(x)}{f(t)}\cdot\dfrac{f(t)}{f(x)}}=2$。 + +$\therefore\geqslant\int_a^x2\,\textrm{d}t-2(x-a)=2(x-a)-2(x-a)=0$。 + +从而$F'(x)\geqslant0$,等于0不恒成立,所以$F(x)$在$[a,b]$上单调增,从而$F(b)>F(a)$,$\displaystyle{\int_a^xf(t)\,\textrm{d}t\int_a^x\dfrac{1}{f(t)}\textrm{d}t-(x-a)^2}>0$。 + +所以$\displaystyle{\int_a^bf(x)\,\textrm{d}x\int_a^b\dfrac{1}{f(x)}\textrm{d}x}\geqslant(b-a)^2$ + +\textbf{例题:}设$f(x)$在$[0,1]$上连续递减,证明$0<\lambda<1$时$\int_0^\lambda f(x)\,\textrm{d}x\geqslant\lambda\int_0^1f(x)\,\textrm{d}x$。 + +证明:看到$\lambda$和1,可以做出辅助函数$\dfrac{\int_0^xf(t)\,\textrm{d}t}{x}$,即要证明$F(\lambda)\geqslant F(1)$。 + +又$\lambda<1$,所以只用证明$F(x)$在$[0,1]$上递减。 + +则$F'(x)=\dfrac{f(x)x-\int_0^xf(t)\,\textrm{d}t}{x^2}$,又根据积分中值定理$\int_0^xf(t)\,\textrm{d}t=f(\xi)(x-0)$,$\xi\in(0,x)$。 + +$\therefore=\dfrac{f(x)x-f(\xi)x}{x^2}=\dfrac{f(x)-f(\xi)}{x}$,因为$f(x)$在$[0,1]$上递减,$\xif(x)$。 + +所以$F'(x)<0$,所以$F(x)$单调递减,从而$F(\lambda)\geqslant F(1)$,所以得证。 + + + +\subsection{拉格朗日中值定理} + +多用于所给条件为“$f(x)$一阶可导”且某一端点值较简单甚至为0的题目。 + +一阶导数不代表导数连续,导数可能存在震荡间断点。 + +\textbf{例题:}设$f(x)$在$[0,1]$上具有一阶连续导数,且$f(0)=f(1)=0$,证明:$\vert\int_0^1f(x)\,\textrm{d}x\vert\leqslant\dfrac{1}{4}\max\limits_{x\in[0,1]}\{\vert f'(x)\vert\}$。 + +证明:在$[0,1]$中任意取一点$x$,进行两次拉格朗日中值定理。 + +$f(x)-f(0)=f'(\xi_1)(x-0)$($\xi_1\in(0,x)$),$f(x)=f'(\xi_1)x$。 + +$f(1)-f(x)=f'(\xi_2)(1-x)$($\xi_2\in(x,1)$),$-f(x)=f'(\xi_2)(1-x)$。 + +根据不等式性质:$\vert\int_0^1f(x)\,\textrm{d}x\vert=\vert\int_0^xf(t)\,\textrm{d}t+\int_x^1f(t)\,\textrm{d}t\vert$ + +$\leqslant\vert\int_0^xf(t)\,\textrm{d}t\vert+\vert\int_x^1f(t)\,\textrm{d}t\vert\leqslant\int_0^x\vert f(t)\vert\,\textrm{d}t+\int_x^1\vert f(t)\vert\,\textrm{d}t$ + +$\xi_1\in(0,x)$上代入$f(x)=f'(\xi_1)x$,$\xi_2\in(x,1)$上代入$-f(x)=f'(\xi_2)(1-x)$。 + +$=\int_0^x\vert f'(\xi_1)\vert t\,\textrm{d}t+\int_x^1\vert f'(\xi_2)\vert(1-t)\,\textrm{d}t$,令$M=\max\limits_{x\in[0,1]}\{\vert f'(x)\vert\}$。 + +$\leqslant\int_0^xMt\,\textrm{d}t+\int_x^1M(1-t)\,\textrm{d}t=M\left(\int_0^xt\,\textrm{d}t+\int_x^1(1-t)\,\textrm{d}t\right)$ + +$=M\left(\dfrac{1}{2}x^2+\dfrac{1}{2}(1-x)^2\right)=M\left(\left(x-\dfrac{1}{2}\right)^2+\dfrac{1}{4}\right)$ + +即$\vert\int_0^1f(x)\,\textrm{d}x\vert\leqslant M\left(\left(x-\dfrac{1}{2}\right)^2+\dfrac{1}{4}\right)$,$\min\left(M\left(\left(x-\dfrac{1}{2}\right)^2+\dfrac{1}{4}\right)\right)=\dfrac{1}{4}M$ + +$\therefore\vert\int_0^1f(x)\,\textrm{d}x\vert\leqslant\dfrac{1}{4}M=\dfrac{1}{4}\max\limits_{x\in[0,1]}\{\vert f'(x)\vert\}$ + +\subsection{泰勒公式} + +多用于所给条件为“$f(x)$二阶可导”且某一端点值较简单甚至为0的题目。 + +\textbf{例题:}设$f(x)$在$[0,2]$上二阶导数连续,且$f(1)=0$,当$x\in[0,2]$时,记$M=\max\{\vert f''(x)\vert\}$,证明$\vert\int_0^2f(x)\,\textrm{d}x\vert\leqslant\dfrac{1}{3}M$。 + \end{document} diff --git a/advanced-math/knowledge/0-perpare/perpare.pdf b/advanced-math/knowledge/0-perpare/perpare.pdf index aff87838f6e18b69dc44e192fa3a0ddca0351dd5..02517d05667538f592b38ee62bb72fc884eba0ec 100644 GIT binary patch delta 210875 zcmYhCQ*bU?u&ra;wr!g$Ua`4i+sPl>wr$(CZQFM4KKHzws($RJ(N%Mf{;EfH7|L82 zieD3O0~vsen?*nX#>v^y#Lx!DePdf&CKj(1x#v{<$pLfZFI^9G?JE)yOk5`Nwh`e8 zHdt8D8`|%eH(95YE8-_}V`Bn3R3Lu+xKaH@C8bx#F3$zmkD{5lS+v=V+vqIT8?C_a z6aaAf^Hu(N7tcd-(u}cPDpLw}_|_sLx?Q>`s0RppNw{T{ly~-c{s_*UO4l!f_$Or15DyIPEov=xypDylchIif0C#p;eR%f zDNT#dT@zSIU0$(CW2C5zu-@HXU)xA7gjO?8noCi9?iO-!5GnrMC%3FCneG=`^W~7%K zQ)eImQcvXodg^k|e~#m4hH16DFc_9TUnsKp`d2b_i%(D?xe^B77BSp#u`#JnPyhsC zejW>SNrU;)oCcKd3fSUt$;!n)_N4t!i@v8hRCq!-^?D^99*2<^l$(*~l1(zC_VgA* 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又因为是证明该中间式在一个区间,所以很明显会想到拉格朗日中值定理:$f(b)-f(a)=f'(\xi)(b-a)$。 @@ -1059,7 +1059,7 @@ $a^\alpha\cdot a^\beta=a^{\alpha+\beta},\dfrac{a^\alpha}{a^\beta}=a^{\alpha-\bet \textbf{例题:}计算$\int_0^{\frac{\pi}{2}}\sin^{10}x\textrm{d}x$与$\int_0^{\frac{\pi}{2}}\cos^9x\textrm{d}x$。\medskip -原式1为偶数次幂,所以$=\dfrac{9}{10}\cdot\dfrac{7}{8}\cdot\dfrac{5}{6}\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2}=\dfrac{\pi}{2}\cdot\dfrac{9!!}{10!!}$。\medskip +解:原式1为偶数次幂,所以$=\dfrac{9}{10}\cdot\dfrac{7}{8}\cdot\dfrac{5}{6}\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2}=\dfrac{\pi}{2}\cdot\dfrac{9!!}{10!!}$。\medskip 原式2为奇数次幂,所以$=\dfrac{8}{9}\cdot\dfrac{6}{7}\cdot\dfrac{4}{5}\cdot\dfrac{2}{3}=\dfrac{8!!}{9!!}$ @@ -1093,7 +1093,7 @@ $a,b,c>0$: \textbf{例题:}证明函数$f(x)=\dfrac{x}{1+x^2}$在$(-\infty,+\infty)$内有界。 -可以使用极限,若极限存在则函数有界,这里使用有界性定义与不等式来完成。 +证明:可以使用极限,若极限存在则函数有界,这里使用有界性定义与不等式来完成。 \ding{172}当$x=0$时,$f(0)=\dfrac{0}{1}$,有界。 diff --git a/advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf 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f(x)=x^2$,$\therefore f[\psi(x)]=\psi^2(x)=-x^2+2x+3$ +解:广义化:$\because f(x)=x^2$,$\therefore f[\psi(x)]=\psi^2(x)=-x^2+2x+3$ 又$\because\psi(x)\geqslant 0$, $\therefore\sqrt{\psi^2(x)}=\sqrt{-x^2+2x+3}=\psi(x)\geqslant 0$ @@ -105,7 +105,7 @@ $\therefore\psi(x)\in[0,2]$ \textbf{例题:}求函数$y=f(x)=\ln(x+\sqrt{x^2+1})$的反函数$f^{-1}(x)$的表达式及其定义域 -首先研究$f(x)$本身,因为$\ln(x)$的定义域必然要求大于0,而任意实数x都有下面不等式成立: +解:首先研究$f(x)$本身,因为$\ln(x)$的定义域必然要求大于0,而任意实数x都有下面不等式成立: $x+\sqrt{x^2+1}>x+\vert x\vert \geqslant 0$,所以$x\in R$。 @@ -418,7 +418,7 @@ $\therefore\exists N>0$,当$n>N$时,$\vert a_n-A\vert<\dfrac{A}{2}\Rightarro \textbf{例题:}求$\lim\limits_{n\to\infty}\left(n\tan\dfrac{1}{n}\right)^{n^2}$($n\in N^+$)。 -$\because \lim\limits_{x\to 0}\left(\dfrac{\tan x}{x}\right)^{\frac{1}{x^2}}$ +解:$\because \lim\limits_{x\to 0}\left(\dfrac{\tan x}{x}\right)^{\frac{1}{x^2}}$ 又$u^v=e^{v\ln u}$ @@ -634,7 +634,7 @@ $\Rightarrow A-\varepsilon0$,证明$a_{n+1}=\dfrac{1}{2}\left(a_n+\dfrac{2}{a_n}\right)$的极限存在并求出。 -$\because a_1=a>0$,且递推式中没有负数与减的操作,所以$a_n>0$。 +解:$\because a_1=a>0$,且递推式中没有负数与减的操作,所以$a_n>0$。 由重要不等式$\dfrac{a+b}{2}\geqslant\sqrt{ab}$,所以$a_{n+1}=\dfrac{1}{2}\left(a_n+\dfrac{2}{a_n}\right)\geqslant\sqrt{a_n\cdot\dfrac{2}{a_n}})=\sqrt{2}$ @@ -777,7 +777,7 @@ $\therefore a_{n+1}\leqslant a_n$,得到数列单调减少。 \textbf{例题:}求证$x_{n+1}=\sin x_n$极限存在,$00 \end{array} \right. $在$(-\infty,+\infty)$内连续,求$a$。 -因为连续,所以$f(0)=\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^-}f(x)$。 +解:因为连续,所以$f(0)=\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^-}f(x)$。 $\therefore a=1$。 \textbf{例题:}若函数$f(x)=\dfrac{\ln\vert x\vert}{\vert x-1\vert}\sin x$,则x的间断点类型是?\medskip -由式子的分式部分可知有两个无定义的间断点:$x=0$,$x=1$。\medskip +解:由式子的分式部分可知有两个无定义的间断点:$x=0$,$x=1$。\medskip $\lim\limits_{x\to 1}f(x)=\lim\limits_{x\to 1}\dfrac{x-1}{\vert x-1\vert}\sin x=\left\{ \begin{array}{lcl} @@ -1090,7 +1090,7 @@ $\lim\limits_{x\to 0}f(x)=\lim\limits_{x\to 0}\ln\vert x\vert\cdot\sin x=\lim\li \textbf{例题:}证明方程$x=a\sin x+b(a>0,b>0)$中至少有一个正根,并且不超过$a+b$。 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b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex index 61c631d..38e668b 100644 --- a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex +++ b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex @@ -106,7 +106,7 @@ $=\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}=f'(x)$。 \textbf{例题:}设$f(x)$是二阶可导的以2为周期的奇函数,且$f(\dfrac{1}{2})>0$,$f'(\dfrac{1}{2})>0$,比较$f(-\dfrac{1}{2})$、$f'(\dfrac{3}{2})$、$f''(0)$的大小。\medskip -$\because f(x)$为二阶奇函数,$\therefore f(x)\text{奇函数}\Rightarrow f'(x)\text{偶函数}\Rightarrow f''(x)\text{奇函数}\Rightarrow f''(0)=0$。 +解:$\because f(x)$为二阶奇函数,$\therefore f(x)\text{奇函数}\Rightarrow f'(x)\text{偶函数}\Rightarrow f''(x)\text{奇函数}\Rightarrow f''(0)=0$。 $\therefore f(-\dfrac{1}{2})=-f(\dfrac{1}{2})<0$。 @@ -116,7 +116,7 @@ $\therefore f'(\dfrac{3}{2})>f''(0)>f(-\dfrac{1}{2})$。\medskip \textbf{例题:}$\left(x^\alpha\right)'=\alpha x^{\alpha-1}(x>0)$。\medskip -$\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}=\lim\limits_{\Delta x\to 0}\dfrac{\left(x+\Delta x\right)^\alpha-x^\alpha}{\Delta x}$\medskip +解:$\lim\limits_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}=\lim\limits_{\Delta x\to 0}\dfrac{\left(x+\Delta x\right)^\alpha-x^\alpha}{\Delta x}$\medskip $=\lim\limits_{\Delta x\to 0}\dfrac{x^\alpha\left[\left(1+\dfrac{\Delta x}{x}\right)^\alpha-1\right]}{\Delta x}=\lim\limits_{\Delta x\to 0}\dfrac{x^\alpha\cdot\alpha\cdot\dfrac{\Delta x}{x}}{\Delta x}=\alpha x^{\alpha-1}$ @@ -182,7 +182,7 @@ $y=f(x)$可导,且$f'(x)\neq 0$就是指严格单调,而严格单调必有 \textbf{例题:}求$y=\arcsin x,x\in(-1,1)$与$y=\arctan x$的导数。 -首先反三角函数就是三角函数的反函数。 +解:首先反三角函数就是三角函数的反函数。 求$y=\arcsin x$,即$x=\sin y$。\medskip @@ -202,7 +202,7 @@ $=-\dfrac{x_{yy}''}{(x_y')^2}\cdot\dfrac{1}{x_y'}=-\dfrac{x_{yy}''}{(x_y')^3}$\m \textbf{例题:}设$y=f(x)$的反函数是$x=\varphi(y)$,且$f(x)=\int_1^{2x}e^{t^2}\textrm{d}t+1$,求$\varphi''(1)$。 -$\because y=f(x)$,$\therefore x=\varphi(y)$,$x_{yy}''=\varphi''(y)=-\dfrac{y_{xx}''}{(y_x')^3}=-\dfrac{f''(x)}{[f'(x)]^3}$。\medskip +解:$\because y=f(x)$,$\therefore x=\varphi(y)$,$x_{yy}''=\varphi''(y)=-\dfrac{y_{xx}''}{(y_x')^3}=-\dfrac{f''(x)}{[f'(x)]^3}$。\medskip 其中根据变限积分求导公式:$f'(x)=2e^{4x^2}$,$f''(x)=2e^{4x^2}\cdot 8x=16xe^{4x^2}$。\medskip @@ -214,7 +214,7 @@ $\because y=f(x)$,$\therefore x=\varphi(y)$,$x_{yy}''=\varphi''(y)=-\dfrac{y \textbf{例题:}设$f(x)=\prod\limits_{n=1}^{100}\left(\tan\dfrac{\pi x^n}{4}-n\right)$,则$f'(1)$为? -原式=$\left(\tan\dfrac{\pi x}{4}-1\right)\left(\tan\dfrac{\pi x^2}{4}-2\right)\cdots\left(\tan\dfrac{\pi x^{100}}{4}-100\right)$。 +解:原式=$\left(\tan\dfrac{\pi x}{4}-1\right)\left(\tan\dfrac{\pi x^2}{4}-2\right)\cdots\left(\tan\dfrac{\pi x^{100}}{4}-100\right)$。 令$\left(\tan\dfrac{\pi x^2}{4}-2\right)\cdots\left(\tan\dfrac{\pi x^{100}}{4}-100\right)=g(x)$。\medskip @@ -249,7 +249,7 @@ $=f'(x)\vert_{x=1}=\dfrac{\pi}{2}\cdot g(1)+0\cdot g'(x)=\dfrac{\pi}{2}\cdot g(1 \textbf{例题:}求$y=\sqrt[3]{\dfrac{(x+1)(2x-1)^2}{(4-3x)^5}}$的导数。 -取对数:$\ln\vert y\vert=\dfrac{1}{3}[\ln\vert x+1\vert+2\ln\vert 2x-1\vert-5\ln\vert 4-3x\vert]$。 +解:取对数:$\ln\vert y\vert=\dfrac{1}{3}[\ln\vert x+1\vert+2\ln\vert 2x-1\vert-5\ln\vert 4-3x\vert]$。 $\because \ln\vert y\vert'=\ln y'$。 @@ -271,11 +271,11 @@ $=u(x)^{v(x)}\left[v'(x)\ln u(x)+v(x)\cdot\dfrac{u'(x)}{u(x)}\right]$。 \textbf{例题:}求$y=x^x(x>0)$的导数。 -$\because x^x=e^{x\ln x}$,$\therefore (x^x)'=(e^{x\ln x})'=x^x\cdot(\ln x+1)$。 +解:$\because x^x=e^{x\ln x}$,$\therefore (x^x)'=(e^{x\ln x})'=x^x\cdot(\ln x+1)$。 \textbf{例题:}求解$y=x^{\frac{1}{x}}(x>0)$的整数最大值。 -$\because y=x^{\frac{1}{x}}=e^{\frac{1}{x}\ln x}$。 +解:$\because y=x^{\frac{1}{x}}=e^{\frac{1}{x}\ln x}$。 $\therefore y'=\left(x^{\frac{1}{x}}\right)=\left(e^{\frac{1}{x}\ln x}\right)'=x^{\frac{1}{x}}\cdot\dfrac{1-\ln x}{x^2}$。 @@ -334,7 +334,7 @@ $(a^x)^n=a^x(\ln a)^{(n)}$,如$y=2^x$,则$y'=2^x\ln 2$,$y''=2^x(\ln 2)^2\c \textbf{例题:}求$\sin x$的$n$阶导数。 -$\because \sin x'=\cos x$而不断求导会发现正负号会++--++--地变化而难以归纳为公式,所以需要另想办法。 +解:$\because \sin x'=\cos x$而不断求导会发现正负号会++--++--地变化而难以归纳为公式,所以需要另想办法。 使用诱导公式: @@ -412,7 +412,7 @@ $y^{(n)}=\sin(x+\dfrac{\pi}{2}\cdot n)$ \textbf{例题:}已知函数$y=e^x\cos x$,求$y^{(4)}$。 -根据莱布尼兹公式: +解:根据莱布尼兹公式: $(e^x\cos x)^{(4)}$ @@ -430,7 +430,7 @@ $=-4e^x\cos x$ \textbf{例题:}设$y=y(x)$是由方程$\sin(xy)=\ln\dfrac{x+e}{y}+1$确定的隐函数,求$y'(0)$。 -两边求导: +解:两边求导: $ \begin{aligned} @@ -465,7 +465,7 @@ $ \right. $($t$为参数)确定,求$\dfrac{\textrm{d}^2y}{\textrm{d}x^2}\vert_{t=\frac{\pi}{4}}$。 -求参数方程的二阶导数首先就要求出其一阶导数:\medskip +解:求参数方程的二阶导数首先就要求出其一阶导数:\medskip $\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{y_t'}{x_t'}=\dfrac{t\cos t}{\cos t}=t$。\medskip @@ -541,7 +541,7 @@ $\therefore \textrm{d}y\vert_{x=x_0}=A\Delta x=y'(x_0)\cdot\Delta x=y'(x_0)\cdot \textbf{例题:}设$y=e^{\sin(\ln x)}$,求$\textrm{d}y$。 -$\because y=e^{\sin(\ln x)} \therefore$ +解:$\because y=e^{\sin(\ln x)} \therefore$ $ \begin{aligned} diff --git 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a/advanced-math/knowledge/3-differential-mean-value-theorem-and-applications-of-derivatives/differential-mean-value-theorem-and-applications-of-derivatives.tex b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex similarity index 96% rename from advanced-math/knowledge/3-differential-mean-value-theorem-and-applications-of-derivatives/differential-mean-value-theorem-and-applications-of-derivatives.tex rename to advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex index f824da3..d6c3086 100644 --- a/advanced-math/knowledge/3-differential-mean-value-theorem-and-applications-of-derivatives/differential-mean-value-theorem-and-applications-of-derivatives.tex +++ b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex @@ -38,7 +38,7 @@ \usepackage{yhmath} % 圆弧 \author{Didnelpsun} -\title{微分中值定理与导数的应用} +\title{导数应用} \date{} \begin{document} \maketitle @@ -108,7 +108,7 @@ $\text{罗尔定理}\xrightleftharpoons[\text{特例:}f(a)=f(b)]{\text{泛化 \textbf{例题:}证明$x>0$时,$\dfrac{x}{1+x}<\ln(1+x)0$时,$x-\dfrac{x^3}{6}<\sin xg(0)=0$,即$\sin x>x-\dfrac{x^3}{6}$。 \textbf{例题:}证明凹凸性与二阶导数的关系。 -不妨先证明凹函数与二阶导数的关系。已知$f''(x)>0$ +证明:不妨先证明凹函数与二阶导数的关系。已知$f''(x)>0$ 不妨设$x_1m2KO$ZQHhOW0&n+n$r{05%Wj)#ktG8%ZoRkSh?a! zGDT5wIwpEHXtI<0n|EjyHbO>1dt)nTUS0-qfQ^fp6N9*ok&BtAnTfrr8H22uorQ}f zArm7D2R}cwvx}3Nku9{x)=lPY+%_j{z@0BN9#6QERAdUJp^_nt4aSDfVrMJgJH>4& z-*ZcSzwFqYXjJlq9mHsxIHD#aZ4$XaGOKML7Hfe3>tZHmQmScJz1NZbw|}4Km-!xV zhKBtjX9E!@J3_FKpq{rgL9gql%6GVc;W}~1IZ`)pBiVso$O?=w!a!v>|3zmQk|=}C z$xC^-ziJ(eYsWa;g-N(0bv1zCgt1)nHFH z>??S&>qHC2wID5*#jVmpnUd2LF6*Gc4Xs@!?B(bT#Iho760~AuTBP{)Mr`|GvvskY z6~FF_Os=-mAo4qm*I8Tl@(sUcL`nDe9>#5l&aaa(U*4_bqHjQwPhWY;7}_73YU^rhjpVTDQ3w2^ryI3`22CVdL8u#BMP+2jZg}PCg;vE_Z+Lk` z92M^7oCMt(YJDLT7n?u^;K6?Al608+xAqBaQybOc&vr{jQ zm^_pe;*n`Vs6D3@XqodGpxi(wx?PXPxcGsV+;Aa-$t_Go?y$h^P(FRT;k9Em5=C%( zGLSr4F@NB|kDNpj#kQ|$#t7pCfbN 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