From f56b675dd906e43e32d1c9bd8430c1e6fcc8d9d9 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 22 Aug 2022 18:24:49 +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 --- ...ntegal-of-functions-of-single-variable.pdf | Bin 264771 -> 269847 bytes ...ntegal-of-functions-of-single-variable.tex | 40 ++++- ...ial-calculus-of-multivariate-functions.pdf | Bin 118610 -> 143081 bytes ...ial-calculus-of-multivariate-functions.tex | 142 ++++++++++++++++-- ...efinite-integral-and-definite-integral.pdf | Bin 204792 -> 208292 bytes ...efinite-integral-and-definite-integral.tex | 24 ++- ...ial-calculus-of-multivariate-functions.pdf | Bin 140099 -> 141879 bytes ...ial-calculus-of-multivariate-functions.tex | 6 + 8 files changed, 201 insertions(+), 11 deletions(-) diff --git a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.pdf 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$=-\dfrac{4x+3}{2(x^2+x+1)}-\dfrac{6}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}+C$ \begin{enumerate} \item 先提出$\dfrac{1}{n}$。 \item 凑出$\dfrac{i}{n}$。 - \item 写出$\int_0^1f(x)\,\textrm{d}x$。 + \item 写出$\int_0^1f(x)\,\textrm{d}x$,其中$\dfrac{1}{n}$没有了,将所有$\dfrac{i}{n}$换为在$x$。 \end{enumerate} \textbf{例题:}求$\lim\limits_{n\to\infty}\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}\cdots+\dfrac{1}{n+n}\right)$。 @@ -781,6 +781,10 @@ $\int_0^\frac{\pi}{2}e^{2x}\cos x\,\textrm{d}x=\dfrac{1}{5}[e^{2x}(\sin x+2\cos 与不定积分等式一样,存在一些问题给出带有不定积分的等式,需要求里面的包含的函数。其中不同的是定限积分的值都是常数,所以解题时可以令其为一个参数求。 +\subsubsection{函数} + +求函数时表达式中含有积分式子,这时可以令其为常数项,然后反代计算定积分。 + \textbf{例题:}设$f(x)=x^2-x\int_0^2f(x)\,\textrm{d}x+2\int_0^1f(x)\,\textrm{d}x$,求$f(x)$。 解:令$\int_0^2f(x)\,\textrm{d}x=a$,$\int_0^1f(x)\,\textrm{d}x=b$。 @@ -795,6 +799,40 @@ $\left[\dfrac{x^3}{3}-\dfrac{ax^2}{2}+2bx\right]_0^2=\dfrac{8}{3}-2a+4b=a$,$\l 则$f(x)=x^2-\dfrac{4}{3}x+\dfrac{2}{3}$。 +\subsubsection{积分值} + +基本上是给出一个一个已知表达式的积分并给出积分值,然后求另一个表达式的积分值。可以令这个积分表达式为$t$代入原积分表达式,也可以将积分式子初等运算。 + +\textbf{例题:}$f(x)$为连续函数,$\int_0^\frac{\pi}{2}f(x\cos x)\cos x\,\textrm{d}x=A$,求$\int_0^\frac{\pi}{2}f(x\cos x)x\sin x\,\textrm{d}x$。 + +解:$\int_0^\frac{\pi}{2}f(x\cos x)\cos x\,\textrm{d}x-\int_0^\frac{\pi}{2}f(x\cos x)x\sin x\,\textrm{d}x=\int_0^\frac{\pi}{2}f(x\cos x)(\cos x\,\textrm{d}x+x\,\textrm{d}\cos x)=\int_0^\frac{\pi}{2}f(x\cos x)\textrm{d}(x\cos x)=F(x\cos x)\vert_0^\frac{\pi}{2}=F(0)-F(0)=0$。 + +$\therefore\int_0^\frac{\pi}{2}f(x\cos x)\cos x\,\textrm{d}x=\int_0^\frac{\pi}{2}f(x\cos x)x\sin x\,\textrm{d}x=A$。 + +\subsection{定积分不等式} + +主要是为了对定积分值进行对比。注意这里不用求出定积分的值,很多不等式中的定积分很难求出。主要是利用保号性根据原函数的符号判断积分的符号。 + +对于判断$\int f(x)\,\textrm{d}0$。($x\in\left(0,\dfrac{\pi}{4}\right)$) + +所以原函数在$\left(0,\dfrac{pi}{4}\right)$上增,$\varphi\left(\dfrac{\pi}{4}\right)$最大,其小于0,则在该区间$\varphi(x)<0$,所以其定积分也小于0。 + +\subsubsection{划分区域} + +如果要比较积分大小,要么在同一个区间内比较,如果这个区间内两个积分有时大有时小就划分区域,让区域上的积分能明确大小。 + \subsection{中值定理} 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z2h)!vV}X*|<=u!_cVR2r@!j}Y*^bI(g6lZoz7f6dOl7%)v=P(g%yk7AQ1eE4-bdw+ zrqXb2)ZeZ-DE)CC5*rpZTEzQ~by%H3o~fE=FBc>`;-r>yZO>Dzdg$a;DEr9ern_J+ zmniFyLi5Pbic!t}mOXzEI3%aO^~G}dDWo@3<^NCBmWyUMR?m`Cczs(=yQgI^@ibC? zH8fTPL^5D&1WIP#(Jd*b+0oXHlr?wOJB$5uj{Y@L{Y)|U%CzgsQ+Ray&45|ukm_ml zFH+ID>7@T2vZP)GD6P) diff --git a/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex b/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex index bb95b14..b8f63aa 100644 --- a/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex +++ b/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex @@ -294,6 +294,12 @@ $C=2$,$f(x,y)=x^2-\cos y+2$。 \textcolor{violet}{\textbf{定义:}}设$(x_0,y_0)$为$f(x,y)$定义域内一点,若对于$f(x,y)$的定义域内任意一个异于$(x_0,y_0)$的点$(x,y)$均有$f(x,y)f(x_0,y_0)$成立,则称$(x_0,y_0)$为$f(x,y)$的\textbf{真正的最大值点/最小值点},$f(x_0,y_0)$为$f(x,y)$的\textbf{真正的最大值/最小值}。 +\textcolor{violet}{\textbf{定义:}}函数的一阶导数为0的点(驻点也称为稳定点,临界点)。对于多元函数,驻点是所有一阶偏导数都为零的点。 + +极值点不一定是驻点:只有可导的极值点才可能是驻点;驻点也不一定是极值点:只有驻点两端变号才能成为极值点。 + +\textcolor{aqua}{\textbf{定理:}}若$f(x,y)$在$(x_0,y_0)$处取得极大/极小值,则$f(x,y_0)$在$x=x_0$取得极大/极小值,$f(x_0,y)$在$y=y_0$取得极大/极小值。 + \subsection{无条件极值} \textcolor{aqua}{\textbf{定理:}}二元函数取极值的必要条件:设$z=f(x,y)$在点$(x_0,y_0)$一阶偏导数存在且取极值,则$f_x'(x_0,y_0)=0$,$f_y'(x_0,y_0)=0$。三元及以上可以类推。

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