From 5cb2dba89648e550abe15b75ca8602559e04cafb Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Sat, 9 Jul 2022 23:23:50 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- advanced-math/exercise/1-limit/limit.pdf | Bin 184894 -> 186276 bytes advanced-math/exercise/1-limit/limit.tex | 8 +- .../exercise/2-function/function.pdf | Bin 137575 -> 137407 bytes .../exercise/2-function/function.tex | 18 +- ...iation-of-functions-of-single-variable.pdf | Bin 146156 -> 147402 bytes ...iation-of-functions-of-single-variable.tex | 2 + ...ntegal-of-functions-of-single-variable.pdf | Bin 249617 -> 250821 bytes ...ntegal-of-functions-of-single-variable.tex | 108 +++---- .../function-and-limit.pdf | Bin 243847 -> 243666 bytes .../function-and-limit.tex | 88 +++--- .../applications-of-derivatives.pdf | Bin 155638 -> 155457 bytes .../applications-of-derivatives.tex | 20 +- 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z`q8G;sQw8z^K;dIEqnREE#$bc{Md_QwxKh+qC0w`H?~5%+5$pE{sR@Nk68+3Ze(+G Va%GpE7y=gxH#9Q}B_%~qMhd)y6+Hj| diff --git a/advanced-math/exercise/2-function/function.tex b/advanced-math/exercise/2-function/function.tex index 5c4ec06..7c2db6d 100644 --- a/advanced-math/exercise/2-function/function.tex +++ b/advanced-math/exercise/2-function/function.tex @@ -180,9 +180,9 @@ $\therefore a=1,b=e$。 \paragraph{介值定理} \leavevmode \medskip -\textbf{例题:}设$f(x)$在$[0,3]$上连续,$(0,3)$上可导,且$f(0)+f(1)+f(2)=3$,$f(3)=1$,证明存在$\varepsilon\in(0,3)$,使得$f'(\varepsilon)=0$。 +\textbf{例题:}设$f(x)$在$[0,3]$上连续,$(0,3)$上可导,且$f(0)+f(1)+f(2)=3$,$f(3)=1$,证明存在$\xi\in(0,3)$,使得$f'(\xi)=0$。 -解:对于一个导数$f'(\varepsilon)=0$,只会想到罗尔定理和拉格朗日中值定理两种方式。由于这里没有差式,所以很可能是罗尔定理。所以必须找到$f(a)=f(b)$。 +解:对于一个导数$f'(\xi)=0$,只会想到罗尔定理和拉格朗日中值定理两种方式。由于这里没有差式,所以很可能是罗尔定理。所以必须找到$f(a)=f(b)$。 由于题目只给出特定条件而没有给出$f(x)$的表达形式,所以无法使用求原函数的方式证明。$f(3)=1$是唯一已知固定常数值的,所以想在$(0,3)$上找到一点$c$,使得$f(c)=f(3)=1$。 @@ -252,11 +252,11 @@ $f(a)\geqslant f(a+b)-f(b)$,所以$f(a+b)\leqslant f(a)+f(b)$。 这种题目就是证明某个式子成立,式子一边是常数一边是导数式子,要证明,就要将导数式子转换为原函数,方法跟罗尔定理使用的转换原函数的技巧一样。 -\textbf{例题:}设$f(x)$在$[0,1]$上连续且可导,证明存在一点$\varepsilon\in(0,1)$,使得$f(1)=3\varepsilon^2f(\varepsilon)+\varepsilon^3f'(\epsilon)$。 +\textbf{例题:}设$f(x)$在$[0,1]$上连续且可导,证明存在一点$\xi\in(0,1)$,使得$f(1)=3\xi^2f(\xi)+\xi^3f'(\epsilon)$。 -证明:由$3\varepsilon^2f(\varepsilon)+\varepsilon^3f'(\epsilon)$,可推出原函数为$x^3f(x)$,令$F(x)=x^3f(x)$,则其在$(0,1)$也可导。 +证明:由$3\xi^2f(\xi)+\xi^3f'(\epsilon)$,可推出原函数为$x^3f(x)$,令$F(x)=x^3f(x)$,则其在$(0,1)$也可导。 -即使用拉格朗日中值定理,$F(1)-F(0)=F'(\varepsilon)$,$\varepsilon\in(0,1)$。即$f(1)=3\varepsilon^2f(\varepsilon)+\varepsilon^3f'(\epsilon)$。 +即使用拉格朗日中值定理,$F(1)-F(0)=F'(\xi)$,$\xi\in(0,1)$。即$f(1)=3\xi^2f(\xi)+\xi^3f'(\epsilon)$。 \subsubsection{对数函数特性} @@ -278,13 +278,13 @@ $f(a)\geqslant f(a+b)-f(b)$,所以$f(a+b)\leqslant f(a)+f(b)$。 证明存在两个不同的点在同一个区间满足一个不等式。如果两个点彼此存在一定关系,如上面式子转换的例子$a+b$,$a$,$b$,那么我们可以使用转换,如果两个完全独立的变量,则这种方式没用,我们可以考虑划分区间,假定这两个点在不同的区间,中间以一个区间变量分隔,由于拉格朗日中值定理中两个变量只会出现一次,而间隔变量会出现多次,所以对其分别拉格朗日中值定理,就可以把两个变量换成以间隔变量表示的形式,将两个无关变量的式子变成一个变量的式子。 -\textbf{例题:}设函数$f(x)$在$[0,1]$上连续,在$(0,1)$内可导,且$f(0)=0$,$f(1)=1$,证明存在不同的$\varepsilon_1$、$\varepsilon_2$,使得$\dfrac{1}{f'(\varepsilon_1)}+\dfrac{1}{f'(\varepsilon_2)}=2$。 +\textbf{例题:}设函数$f(x)$在$[0,1]$上连续,在$(0,1)$内可导,且$f(0)=0$,$f(1)=1$,证明存在不同的$\xi_1$、$\xi_2$,使得$\dfrac{1}{f'(\xi_1)}+\dfrac{1}{f'(\xi_2)}=2$。 -证明:使用$\varepsilon$将$[0,1]$划分为$[0,\varepsilon]$和$[\varepsilon,1]$两个区间,假定$\varepsilon_1$、$\varepsilon_2$分别在这两个区间上。 +证明:使用$\xi$将$[0,1]$划分为$[0,\xi]$和$[\xi,1]$两个区间,假定$\xi_1$、$\xi_2$分别在这两个区间上。 -分别对其进行拉格朗日:$f(\varepsilon)-f(0)=f'(\varepsilon_1)(\varepsilon-0)$,即$\dfrac{1}{f'(\varepsilon_1)}=\dfrac{\varepsilon}{f(\varepsilon)}$,$f(1)-f(\varepsilon)=f'(\varepsilon_2)(1-\varepsilon)$,即$\dfrac{1}{f'(\varepsilon_2)}=\dfrac{1-\varepsilon}{1-f(\varepsilon)}$。 +分别对其进行拉格朗日:$f(\xi)-f(0)=f'(\xi_1)(\xi-0)$,即$\dfrac{1}{f'(\xi_1)}=\dfrac{\xi}{f(\xi)}$,$f(1)-f(\xi)=f'(\xi_2)(1-\xi)$,即$\dfrac{1}{f'(\xi_2)}=\dfrac{1-\xi}{1-f(\xi)}$。 -即$\dfrac{1}{f'(\varepsilon_1)}+\dfrac{1}{f'(\varepsilon_2)}=\dfrac{\varepsilon}{f(\varepsilon)}+\dfrac{1-\varepsilon}{1-f(\varepsilon)}$,任取$f(\varepsilon)=\dfrac{1}{2}$,原式等于2,得证。 +即$\dfrac{1}{f'(\xi_1)}+\dfrac{1}{f'(\xi_2)}=\dfrac{\xi}{f(\xi)}+\dfrac{1-\xi}{1-f(\xi)}$,任取$f(\xi)=\dfrac{1}{2}$,原式等于2,得证。 \subsubsection{查找特定值} diff --git a/advanced-math/exercise/3-differentiation-of-functions-of-single-variable/differentiation-of-functions-of-single-variable.pdf b/advanced-math/exercise/3-differentiation-of-functions-of-single-variable/differentiation-of-functions-of-single-variable.pdf index 23cf49bfaf1c0e76b6db6a9f19ab1bf6208fe3a3..576b252356119f27b616d27e8b0f862491cfbb05 100644 GIT binary patch delta 96044 zcmY)UQ*bU!*R>6=*tV?|+qP{xS+Sinwr$(CZQHi(eAj>1zwx|Pz1zKU)I`rQ=G8Og 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b/advanced-math/exercise/3-differentiation-of-functions-of-single-variable/differentiation-of-functions-of-single-variable.tex @@ -431,6 +431,8 @@ $F(t)$递增,所以$F(t)>F(0)=0$。 如果不等式中都是常数,可以将其中的一个或几个常数变量化再利用导数去证明。 +此外注意对式子进行化简,不要直接就计算不等式与计算导数。遇到$e^xf(x)$乘积形式就取对数让乘积变为加减法,遇到两边为平方项就两边开根号并加上绝对值。 + \textbf{例题:}设$02\dfrac{b-a}{a+b}$。 因为左边含有$\dfrac{b}{a}$不好处理,所以右边分子分母同时除以$a$全部变成统一变量:$\ln\dfrac{b}{a}>2\dfrac{\dfrac{b}{a}-1}{1+\dfrac{b}{a}}$。然后令$x=\dfrac{b}{a}$,所以即需要证明$\ln x>2\dfrac{x-1}{1+x}$,$x>1$。 diff --git a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.pdf b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.pdf index 7c743bcbe61de6b52a4cef3794cea05b7d9b382d..2d58ee48aa20fdbee54fdc789266cc8f3a33639f 100644 GIT binary patch delta 234100 zcmZU)V~{RP&?GvxZJe=f+xDEXZR;7^wryKyY}>Z&x$n2TKX&8(tBCHX=;)5ju8hh` z$VD73LyVUI$i!{3BlW&&+?|OvTy?*JEa1U31AD=Cwvj`wH+oz8Lr~8Be8t*~vp;J} 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