From 71a77dacebca81d1077c4681d5ce5a49b3654a1e Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Wed, 29 Jun 2022 22:57:27 +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 184003 -> 184894 bytes advanced-math/exercise/1-limit/limit.tex | 5 +- .../function-and-limit.pdf | Bin 235396 -> 243847 bytes .../function-and-limit.tex | 76 ++++++++++++------ 4 files changed, 55 insertions(+), 26 deletions(-) diff --git a/advanced-math/exercise/1-limit/limit.pdf b/advanced-math/exercise/1-limit/limit.pdf index 0486e43c3cd7d98fc021811f8d614e7dc4835306..41153097822a0768c06c596da906f6b939bb8f30 100644 GIT binary patch delta 105051 zcmZ6yV{j&1u(ms~HPOVjZQHhOn@?=peq!6UF|lnsnIzx4>zt~yzw>YX>bh5Vt*)#3 z?lqJP*INQtFVF{IW8~+Db#ZkzGqQvA+&I>eiO1Psy4n4&3z^kY!tMQ zerwGR&hv=`yP|r3e~;RxOQd@#x2V47hKUI@t(08p>NRfS=(DplxsPX031aETRKW 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zh!xS;Kr9GW6kAM;B~h>-7IZR+88gX@c}ONtDUTUUCS@{XIRDzde6!}i7ddxx&pumY zjCuFY%}BFZhRHa<%@{AUf_Wq6;TGJ4MSoa;+i)vCWke#!8pAB;hAwo+R@fTbU|Vd5 z9@ri`peK5vH~L^l^hH1Hgq_70Z+i{?T-zY#E*OA;*cH2BckF>Zu@?qmF!shi*cU@E z6#HR+3`1SXfvDFIjvrtI4#pu|=XTtQyKxWhK zgR)2@B+!=3mhzRocN%deB@SodY+9Gr{u=&P9HF#!|lt8oow5%UvVOIeIda49{T zIT@E>3Vj{(a!kcEdJg7N(wU#)dW!y34(8G~Fzc&%xRI{&W-P!WdOmKX=zjYQH&J%t zE-azzzfg+%a6f$u^8q}Fhkxh=ScpaR!_3cdE2Rv}v4URAd<;)uC4C$7Nvy(Z`gYtw zImO(Bdi`hd9G<7^C#t~K zdJmpr);m$}zHItw<^T*teMX;Q?u~s=pTB2Ozb(@-1~cgAnRTsn1&-78yX1lGQBNBB z?efIV=ug-0TP^B0t$z#ZIpH#MFAPFGEnH!qf>TkSn^&0!;}D!dzlI-CW@0SmImc1eTAoq-evk5 zbv<>>ui*{URn;}r71Jx%_0cua6}XLeuo3U0Ub$Xp9oAz5KEQ|g2p@B8Zp=;i1fSwF ze9n7xSJ9p7CB=xV=sKUVnVqVcJ*Sy{shM4?+1a&PY=geo9^0V@dSeIlL@)Hgj;Lp5 zKkS5^u@82^0DtU(f!GzhVR!6}J+T)CVKDZ?z8Hd`s3(?448sTz^W$`KMFXHQ2IT zB-g_!dF8gGisb+1l!9to(nN~Vol+TVOS;I(MmuMS)PMLpuU=VYOQy)x-<@*npe-vz zZvWtvyS(#WqGR1<`|3)O`hV=4CDIV>y!y1xmQ^Cp{&336SGKGc`NG>NU;gO$D$MpZ zB3~yt<#nkoYel}fJmi7Tkd&V}L%4t2iC z!Fy~e7EV`Iyq$Bzm6KrS@|O9gopZ&NH^|OMTjpXruNT+m^>#knGQa;V8(Qv^1$N$n zyRZaHaUbr-gLoKE;7P2;(^!KS@sha8TUPRa^C)zkmtMF79Sk!wH!(H}B_%~qMhf1j BY-|7k diff --git a/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex b/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex index 5c3eb58..c0fa7c6 100644 --- a/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex +++ b/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex @@ -645,28 +645,6 @@ $\Rightarrow A-\varepsilonX$,$X$为充分大的正数)存在,且$F'(x)\neq0$,$\lim\limits_{x\to a}\dfrac{f'(x)}{F'(x)}$或$\lim\limits_{x\to\infty}\dfrac{f'(x)}{F'(x)}$存在或无穷大时,则$\lim\limits_{x\to a}\dfrac{f(x)}{F(x)}=\lim\limits_{x\to a}\dfrac{f'(x)}{F'(x)}$或$\lim\limits_{x\to\infty}\dfrac{f(x)}{F(x)}=\lim\limits_{x\to\infty}\dfrac{f'(x)}{F'(x)}$。 + +\textcolor{aqua}{\textbf{定理:}}当$x\to a$或$x\to\infty$时,函数$f(x)$以及$F(x)$都趋于无穷,$f'(x)$、$F'(x)$在点$a$的某去心邻域内(或当$\vert x\vert>X$,$X$为充分大的正数)存在,且$F'(x)\neq0$,$\lim\limits_{x\to a}\dfrac{f'(x)}{F'(x)}$或$\lim\limits_{x\to\infty}\dfrac{f'(x)}{F'(x)}$存在或无穷大时,则$\lim\limits_{x\to a}\dfrac{f(x)}{F(x)}=\lim\limits_{x\to a}\dfrac{f'(x)}{F'(x)}$或$\lim\limits_{x\to\infty}\dfrac{f(x)}{F(x)}=\lim\limits_{x\to\infty}\dfrac{f'(x)}{F'(x)}$。 + +同理如果导数存在也可以不断求导:$\lim\limits_{x\to a}\dfrac{f(x)}{F(x)}=\lim\limits_{x\to a}\dfrac{f(x)'}{F(x)'}=\lim\limits_{x\to a}\dfrac{f(x)''}{F(x)''}$。 + +\textcolor{orange}{注意:}洛必达法则求不出值也不能说其左边的值不存在。如$\lim\limits_{x\to0}\dfrac{x^2\sin\dfrac{1}{x}}{x}=\lim\limits_{x\to0}\sin\dfrac{1}{x}=0$,通过洛必达就求不出值。 + +\subsection{泰勒公式} + +与洛必达法则不同,适用于$\dfrac{A}{B}$上下同阶型和$A-B$幂次最低型。如$\dfrac{x-\sin x}{x^3}$和$\cos x-e^{-\frac{x^2}{2}}$。 + +是一个用函数在某点的信息描述其附近取值的公式。如果函数满足一定的条件,泰勒公式可以用函数在某一点的各阶导数值做系数构建一个多项式来近似表达这个函数,即用多项式拟合不规则曲线。 + +\begin{enumerate} + \item $e^x=\sum\limits_{i=0}^n\dfrac{1}{i!}x^i$,$=1+\dfrac{1}{1!}x+\dfrac{1}{2!}x^2+\dfrac{1}{3!}x^3+o(x^3)$。 + \item $\ln(1+x)=\sum\limits_{i=1}^n(-1)^{i+1}\dfrac{1}{i}x^i$,$=x-\dfrac{1}{2}x^2+\dfrac{1}{3}x^3+o(x^3)$。 + \item $\sin x=\sum\limits_{i=1}^{2i-1}(-1)^{2i-1}\dfrac{1}{(2i-1)!}x^{2i-1}$,$=x-\dfrac{1}{3!}x^3+\dfrac{1}{5!}x^5+o(x^5)$。 + \item $\cos x=\sum\limits_{i=1}^{2i}(-1)^{2i-1}\dfrac{1}{(2i-2)!}x^{2i-2}$,$=x-\dfrac{1}{2!}x^2+\dfrac{1}{4!}x^4+o(x^4)$。 + \item $\arcsin x=\sum\limits_{i=1}^{2i-1}\dfrac{(2i-3)!!}{(2i-2)!!}\dfrac{x^{2i-1}}{2i-1}$,$=x+\dfrac{1}{2}\dfrac{x^3}{3}+\dfrac{1\times3}{2\times4}\dfrac{x^5}{5}+\dfrac{1\times3\time5}{2\times4\times6}\dfrac{x^7}{7}+o(x^7)$。(假定$-1!=0!$) + \item $\dfrac{1}{1-x}=\sum\limits_{i=0}^nx^i$,$=1+x+x^2+x^3+o(x^3)$。 + \item $(1+x)^a=1+\sum\limits_{i=1}^n\dfrac{\prod_{j=1}^i(a-j+1)}{i!}x^i$,$=1+\dfrac{a}{1!}x+\dfrac{a(a-1)}{2!}x^2\\+\dfrac{a(a-1)(a-2)}{3!}x^3+o(x^3)$。 +\end{enumerate} + \subsection{常用等价无穷小} \textcolor{aqua}{\textbf{定理:}}若$\alpha\sim\alpha_1$,$\beta\sim\beta_1$,则$\lim\dfrac{\alpha}{\beta}=\lim\dfrac{\alpha_1}{\beta}=\lim\dfrac{\alpha}{\beta_1}=\lim\dfrac{\alpha_1}{\beta_1}$。 -所以可以使用等价无穷小替换对应式子,这些等价无穷小都是使用泰勒展开得到的。 +所以可以使用等价无穷小替换对应式子,这些等价无穷小都是使用泰勒展开得到的。等价无穷小只是泰勒公式在某个固定阶数上(通常为一阶)的特例。 + +泰勒一般用于替换整体,等价无穷小一般用于替换部分式子。 \textcolor{red}{警告:}一般只有所替换的式子为乘除的整个因子才能替换,加减一般都不能替换,如$x-\sin x\sim\dfrac{1}{6}x^3$。