From 8fa344170522c517e834f2d0cfd701bbeec1008c Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Wed, 9 Nov 2022 21:46:21 +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 191030 -> 194280 bytes advanced-math/exercise/1-limit/limit.tex | 28 ++++++++++++++++++ .../applications-of-derivatives.pdf | Bin 158818 -> 158815 bytes .../applications-of-derivatives.tex | 2 +- 4 files changed, 29 insertions(+), 1 deletion(-) diff --git a/advanced-math/exercise/1-limit/limit.pdf b/advanced-math/exercise/1-limit/limit.pdf index 7dc57655278d839cffda5f277164a475d30825f6..9984b6a2e2b4ce989f6b08bdfd68c52334b5b807 100644 GIT binary patch delta 116624 zcmY(JLy#^E52bJ0wr$(CZQFL=?ta_0ZQizR+qP|czM0iOt1OaAWtBR4a$1&P&aPqV zc7_3LtX%y3&@QgdW=3|8u&_2l8U@OcanG@*B{RWg9E+0(qdvF0zIWQxsZJgdvR-|zQZ!9%J{ 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z?wY^V{yV%0=Xu3{L3%js>6Mar{-!X5O9_pd^BwhR3lL~YHB4dkL`KUMZ&~fGeMDa)Y zGF)ukZs)IYC)ryIpRWr|Jx;m_ea+CKW<2~##N0$),$x_{n+1}=\sqrt{a+x_n}$,求极限。 + +解:$x_1=\sqrt{a}$,$x_2=\sqrt{a+\sqrt{a}}$,$a_n=\sqrt{a+\cdots\sqrt{a+\sqrt{a}}}$所以可得单调递增。 + +$x_{n+1}=\sqrt{a+x_n}$,$x_{n+1}^2=a+x_n$,$x_{n+1}=\dfrac{a}{x_{n+1}}+\dfrac{x_n}{x_{n+1}}$。 + +由单调性$x_{n+1}\geqslant x_n\geqslant x_1=\sqrt{a}$,$x_{n+1}\leqslant\dfrac{a}{x_{n+1}}+1\leqslant\sqrt{a}+1$,即有上界。 + +单调有界准则,$a_n$单调增且有上界,令极限为$A$,$A=\sqrt{a+A}$,解得$A=\dfrac{1\pm\sqrt{1+4a}}{2}$,负根舍去。 + +\textbf{例题:}$x_1=a\geqslant0$,$y_1=b\geqslant0$,$a\leqslant b$,$x_{n+1}=\sqrt{x_ny_n}$,$y_{n+1}=\dfrac{x_n+y_n}{2}$($n=1,2,\cdots$),证明$\lim\limits_{n\to\infty}x_n=\lim\limits_{n\to\infty}y_n$。 + +解:$x_1=a$,$x_2=\sqrt{a\cdot a}\geqslant\sqrt{a^2}=a$,$\therefore x_n$单调递增。 + +$y_1=b$,$y_2=\dfrac{a+b}{2}\leqslant\dfrac{2b}{2}=b$,$\therefore y_n$单调递减。 + +$x_{n+1}^2=x_ny_n$,$x_{n+1}=\dfrac{x_n}{x_{n+1}}y_n\leqslant y_n=b$,$\therefore x_n$有上界。 + +$y_{n+1}=\dfrac{x_n+y_n}{2}$,$x_n+y_n=2y_{n+1}$,$y_n=x_n-2y_{n+1}$,又$y_n$单调递减,所以$y_{n+1}\leqslant y_n$,$\therefore y_n\geqslant x_n-2y_n$,$3y_n\geqslant x_n$,$\therefore y_n$有下界。 + +所以根据单调有界准则,都有极限,令$\lim\limits_{n\to\infty}x_n=A$,$\lim\limits_{n\to\infty}y_n=B$。 + +代入第二个式子,$B=\dfrac{A+B}{2}$,解得$A=B$。 + \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。 diff --git a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.pdf b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.pdf index 0efeed86925867057f2c5c81f91d56717e029faf..b85fcc3082e48b943377ab96844038218676c268 100644 GIT binary patch delta 33864 zcmZ6RLy#^^uwZZ7wr$(iZQHhOez$Ggwr$(CZQJwDi&?ywRcw+SK;$vlO6+@tK zNA~ko{`+`*bo$<_-(~Oj=xy0#m*`yL39uUiG>vDNnY&aDv)#(?@>09P-u8I^hMw&1 z&f;8Hir{FQE>z`{UhiCs@?|8}zYkKI-sKZ4Q&s2`@^T0%%R zGjTq_FXVoI%`MMaX@TV$Nhj~}sf1Zwil>Gs;_olSbZy(8f?V} z5b#BZDw#?a*4~#*F}<*)47=QEQd2qC)DO4HB}08z&!8=XlfcxRv1u1ERV)*^7NXnY 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zR(W>XEiYDC$x(TE+byq}tqf6l{hwRjthO>#<)3Rd=c@es-2bV{UMqR3`p_-DIxE9e z15oO19i}@l&Dkmz~+4auVs(T wWjFyRVg**>6s*SSI1_7eHrC-hb!z-i>3>%xt;v@*Yyuz&H!%t&B}Gq03M?`xH~;_u diff --git a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex index 9cdbc92..05b510f 100644 --- a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex +++ b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex @@ -616,7 +616,7 @@ $\therefore\dfrac{\textrm{d}\alpha}{\textrm{d}x}=\dfrac{y''}{1+y'^2}\Rightarrow\ $\therefore \textcolor{aqua}{\textbf{定理:}}k=\left\lvert\dfrac{\textrm{d}\alpha}{\textrm{d}s}\right\rvert=\dfrac{\vert y''\vert}{(1+y'^2)^{\frac{3}{2}}}$。 -对于参数方程,$k=\dfrac{\vert y''x'-y'x''}{\left(x'^2+y'^2\right)^{\frac{3}{2}}}$。 +对于参数方程,$k=\dfrac{\vert y''x'-y'x''\vert}{\left(x'^2+y'^2\right)^{\frac{3}{2}}}$。 \subsection{曲率半径}