From 0f2a9099056d0f7abf34ae215b014e3c3b0bbb13 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Sun, 10 Jul 2022 23:34:30 +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 186276 -> 188179 bytes advanced-math/exercise/1-limit/limit.tex | 12 +++ ...ntegal-of-functions-of-single-variable.pdf | Bin 250821 -> 260755 bytes ...ntegal-of-functions-of-single-variable.tex | 70 +++++++++++++++++- 4 files changed, 80 insertions(+), 2 deletions(-) diff --git a/advanced-math/exercise/1-limit/limit.pdf b/advanced-math/exercise/1-limit/limit.pdf index dac4619c3cf2478900eb6f72a2822e0d2bd0dde7..7533c5275dca4c7342796bdb5291d2f51844c749 100644 GIT binary patch delta 106389 zcmY(JQ+Osk)UIpW*3`yZ+wIi$)VAHXwr$(CZQHgvK#)ceL)#YcUCV!W{n^)Pr!d>@so2gbeg4>XZ7 zZWFq1yKAh+Ki=w^()S*P7uoI=XMHMjf_~fIz{aa}=%&xP*?qgng&o()8^2xdkgm^* 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$x_{n+1}-x_n=\sqrt{2+x_n}-x_n=\dfrac{2+x_n-x_n^2}{\sqrt{2+x_n}+x_n}=\dfrac{-(x_n \subsection{数列和} +使用放缩法进行夹逼定理失败时可以使用定积分定义。 + 使用定积分的精确定义$\displaystyle\int_a^bf(x)\,\textrm{d}x=\lim\limits_{n\to\infty}\sum\limits_{i=1}^nf\left(a+\dfrac{b-a}{n}\cdot i\right)\dfrac{b-a}{n}$。 +将$a$、$b$设为$0$和$1$可以得出普通形式$\displaystyle\int_0^1f(x)\,\textrm{d}x=\lim\limits_{n\to\infty}\sum\limits_{i=1}^nf\left(\dfrac{i}{n}\right)\dfrac{1}{n}$。 + +\begin{enumerate} + \item 提出$\dfrac{1}{n}$。 + \item 凑出$\dfrac{i}{n}$。 + \item 由于$\dfrac{i}{n}=0=\dfrac{1-0}{n}i$,所以$\dfrac{i}{n}$可以读作0到1上的$x$,且$\dfrac{1}{n}=\dfrac{1-0}{n}$读作0到1上的$\textrm{d}x$。 +\end{enumerate} + +\textbf{例题:}计算$\lim\limits_{n\to\infty}\left(\dfrac{n+1}{n^2+1}+\cdots+\dfrac{n+n}{n^2+n^2}\right)$。 + \end{document} 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 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zE3M0{i>eE(tE-EuE2_(>JE_~KJEj|`yQf>H|Ce+^4;EiS?YIH6a5HYhJlu)7xDOBFNj!o_u?SCy`Vx>~#QA@u(Ug!Jw1M5{H zPw4;Lv~pJD-$xB4e-KEta!vwXrk&4AAUMH)HFT+!3laz`_Dgu7m1+q@)Y-XK0#QHs zuSO5Ga!tJH@pi5eFD1#&Hyh4H4Oe=z#ma5*%7X2DQM}5p>|1CP@v6F7za-v;3OnD$ z`&=X9eNkuEdl-bl*c2l$3Y%dIj1lj1$时收敛,在$p\leqslant1$时发散。 + +无界函数的反常积分$\int_0^1\dfrac{\textrm{d}x}{x^p}$($p>1$,奇点$x=0$)在$00$,判断反常积分$\displaystyle{\int_1^{+\infty}\dfrac{\left(\arctan\dfrac{1}{x}\right)^a}{\left[\ln\left(1+\dfrac{1}{x}\right)\right]^{2\beta}}\textrm{d}x}$的敛散性。 + +解:积分存在唯一奇点$+\infty$,当$x\to+\infty$时,$\arctan\dfrac{1}{x}\sim\dfrac{1}{x}$,$\ln\left(1+\dfrac{1}{x}\right)\sim\dfrac{1}{x}$。 + +$\dfrac{\left(\arctan\dfrac{1}{x}\right)^a}{\left[\ln\left(1+\dfrac{1}{x}\right)\right]^{2\beta}}=\dfrac{1}{x^{\alpha-2\beta}}$。 + +当$\alpha-2\beta>1$时收敛,$\alpha-2\beta\leqslant1$时发散。 + +\subsubsection{计算} + +可以直接通过计算反常积分判断收敛或发散。 + \section{一元函数积分应用} \subsection{几何应用} @@ -1120,6 +1170,22 @@ $\rho gA(y)\,\textrm{d}y=\rho g\pi(16-y^2)\,\textrm{d}y$ 包括证明带有积分的等式、方程根、积分中值定理等。 +对于这种等式要求其值,往往是给出部分式子的条件,所以必须想办法进行转换。 + +一种是已知$f(x)$,式子中含有未知的$f'(x)$,一种是已知$f'(x)$(多以$f(x)=\int f'(x)\,\textrm{d}x$的形式给出),式子中含有未知的$f(x)$。 + +\subsubsection{分部积分} + +分部积分法用于消除未知的式子的导数或者积分。 + +\textbf{例题:}设$f''(u)$在区间$[0,2]$上连续,$f(2)=a$,$f'(2)=b$,$\int_0^2f(u)\,\textrm{d}u=c$,求$\int_0^1x^2f''(2x)\,\textrm{d}x$。 + +解:已知条件是多个导数和一个积分的值,要将这些值换成目标积分的样式。所以要考虑换元积分或分部积分。 + +这里目标积分的值是乘积形式且变量无法换元,且存在一个未知的$f''(2x)$,所以使用分部积分将$f''(2x)$换到$\textrm{d}x$中降低求导幂次。 + +$\int_0^1x^2f''(2x)\,\textrm{d}x=\dfrac{1}{2}\int_0^1x^2\,\textrm{d}[f'(2x)]=\dfrac{1}{2}[x^2f'(2x)]\bigg\vert_0^1-\int_0^1xf'(2x)\,\textrm{d}x=\dfrac{1}{2}f'(2)-\dfrac{1}{2}\int_0^1x\,\textrm{d}[f(2x)]=\dfrac{b}{2}-\dfrac{1}{2}[xf(2x)]\bigg\vert_0^1+\dfrac{1}{2}\int_0^1f(2x)\,\textrm{d}x=\dfrac{b}{2}-\dfrac{a}{2}+\dfrac{1}{4}\int_0^2f(u)\,\textrm{d}(u)=-\dfrac{a}{2}+\dfrac{b}{2}+\dfrac{c}{4}$。 + \subsubsection{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)