請標明題號,依序作答。
計算證明題- 設 f(x)=∫x1e3t√9t4+1dt,g(x)=xne3x,且 lim。求 n 之值。
- 求
- \displaystyle\lim_{x\to0^+}\left(\cot x-\frac1x\right)。
- 再用級數求 \displaystyle\lim_{x\to0^+}\left(\cot x-\frac1x\right)\left(\cot x+\frac1x\right)。
【方法一】 通分後使用羅必達法則計算如下
\displaystyle\lim_{x\to0^+}\left(\cot x-\frac1x\right)=\lim_{x\to0^+}\frac{x\cos x-\sin x}{x\sin x}=\lim_{x\to0^+}\frac{-x\sin x}{\sin x+x\cos x}=\lim_{x\to0^+}\frac{-\sin x-x\cos x}{2\cos x-x\sin x}=0
【方法二】 留意到 \displaystyle\cot x=\frac1{\tan x},根據正切函數的泰勒展開式有
\displaystyle\cot x=\frac1{\displaystyle x+\frac{x^3}3+\cdots}=\frac1x\cdot\frac1{\displaystyle1+\frac{x^2}3+\cdots}=\frac1x\cdot\left(1-\frac{x^2}3+\cdots\right)=\frac1x-\frac{x}3+\cdots
故可知\displaystyle\lim_{x\to0^+}\left(\cot x-\frac1x\right)=\lim_{x\to0^+}\left(-\frac{x}3+\cdots\right)=0
- 沿用前述的辦法已知
\displaystyle\cot x=\frac1x-\frac{x}3+\cdots
如此所給定的極限可表達為\displaystyle\lim_{x\to0^+}\left(\cot x-\frac1x\right)\left(\cot x+\frac1x\right)=\lim_{x\to0^+}\left[\left(-\frac{x}3+\cdots\right)\left(\frac1x+\cdots\right)\right]=\lim_{x\to0^+}\left(-\frac13+\cdots\right)=-\frac13
- 設 f\left(x\right)=\ln x+\tan^{-1}x,g\left(x\right) 為其反函數,求 \displaystyle g'\left(\frac\pi4\right) 和 \displaystyle g''\left(\frac\pi4\right) 之值。
- \displaystyle f\left(x\right)=\frac{x-\sin x}{x^3},求 f^{\left(6\right)}\left(0\right) 之值。
- 求
- \displaystyle\int_0^{\infty}\frac{\ln x}{1+x^2}dx。
- \displaystyle\int_0^{\infty}e^{-x}\sin xdx。
我們首先說明此瑕積分收斂,由於兩處瑕疵,故我們分為兩塊分別處理:
\displaystyle\int_0^{\infty}\frac{\ln x}{1+x^2}dx=\int_0^1\frac{\ln x}{1+x^2}dx+\int_1^{\infty}\frac{\ln x}{1+x^2}dx
針對前者,我們留意到被積分函數在 \left(0,1\right) 上恆負,故\displaystyle\int_0^1\frac{\ln x}{1+x^2}dx\geq\int_0^1\ln x dx=\lim_{t\to0^+}\int_t^1\ln xdx=\lim_{t\to0^+}\left(x\ln x-x\right)\Big|_t^1=\lim_{t\to0^+}\left(-1-t\ln t+t\right)=-1
其瑕積分收斂;而後者在 \left(1,\infty\right) 上恆正,故有\displaystyle\int_1^{\infty}\frac{\ln x}{1+x^2}dx\leq\int_1^{\infty}\frac{\ln x}{x^2}dx=-\lim_{t\to\infty}\int_1^t\ln xd\frac1x=\lim_{t\to\infty}\left.-\left(\frac{\ln x}x+\frac1x\right)\right|_1^t=\lim_{t\to\infty}\left(1-\frac{\ln t}t-\frac1t\right)=1
綜上可知兩個瑕積分皆收斂。現在特別去觀察前者的積分,我們令 \displaystyle u=\frac1x,那麼
- 當 x\to0^+ 時有 u\to\infty;
- 當 x=1 時有 u=1;
- 移項有 \displaystyle x=\frac1u,求導則得 \displaystyle dx=-\frac1{u^2}du。
\displaystyle\int_0^1\frac{\ln x}{1+x^2}dx=\int_\infty^1\frac{-\ln u}{u^2+1}\cdot-du=-\int_1^{\infty}\frac{\ln u}{1+u^2}du
從而與第二個瑕積分相加可恰好消去,從而所求為零。- 先考慮其對應的不定積分並使用分部積分法計算如下
\displaystyle\begin{aligned}\int e^{-x}\sin xdx&=-\int\sin xde^{-x}=-e^{-x}\sin x+\int e^{-x}\cos xdx\\&=-e^{-x}\sin x-\int\cos xde^{-x}=-e^{-x}\sin x-e^{-x}\cos x-\int e^{-x}\sin xdx\end{aligned}
因此得到不定積分為\displaystyle\int e^{-x}\sin xdx=-e^{-x}\frac{\sin x+\cos x}2+C
故所求為\displaystyle\int_0^{\infty}e^{-x}\sin xdx=-\frac12\lim_{t\to\infty}\left(e^{-x}\sin x+e^{-x}\cos x\right)\Big|_0^t=-\frac12\lim_{t\to\infty}\left(e^{-t}\sin t+e^{-t}\cos t-1\right)=\frac12
- 設 z=f\left(x,y\right) 有連續的各二偏導函數,且 x=r\cos\theta,y=r\sin\theta。試證
\displaystyle\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}=\frac{\partial^2z}{\partial r^2}+\frac1{r^2}\frac{\partial^2z}{\partial\theta^2}+\frac1r\frac{\partial z}{\partial r}。
- 用 Lagrane method 求 f\left(x,y,z\right)=y+xz-2x^2-y^2-z^2 之極大極小值,而要求點在 x+y+z=35 之上。
- 用變數變換求 \displaystyle\iint_RxdA 之值。此處 R 為第一象限內,由 y=x,4x^2-y^2=4,x 軸,4x^2-y^2=16 所圍成之區域。
- y=x 可表達為 \displaystyle v=\frac{5u}3;
- 4x^2-y^2=4 可表達為 u=4;
- x 軸(y=0)可表達為 v=u;
- 4x^2-y^2=16 則表達為 u=16
- 用 Stokes 定理求 \displaystyle\iint_S\left(\mbox{curl}~F\cdot n\right)dS 之值。此處 F=zi+xj+yk;S 是曲面 z=\sqrt{5-x^2-y^2}-1,z>0;n 朝上。
訣竅
使用微積分基本定理與微分的乘法公式求導後計算極限即可。解法
使用微積分基本定理可得f'\left(x\right)=e^{3x}\sqrt{9x^2+1}
而使用乘法的微分公式則有g'\left(x\right)=nx^{n-1}e^{3x}+3x^ne^{3x}
因此給定的極限式可表達為\displaystyle1=\lim_{x\to\infty}\left[\frac{f'\left(x\right)}{g'\left(x\right)}\right]=\lim_{x\to\infty}\frac{e^{3x}\sqrt{9x^2+1}}{nx^{n-1}e^{3x}+3x^ne^{3x}}=\lim_{x\to\infty}\frac{\sqrt{9x^2+1}}{nx^{n-1}+3x^n}=\lim_{x\to\infty}\frac{\displaystyle\sqrt{9+\frac1{x^2}}}{3x^{n-1}+nx^{n-2}}
因此根據極限的四則運算結果可知\displaystyle\lim_{x\to\infty}\left(3x^{n-1}+nx^{n-2}\right)=\lim_{x\to\infty}\frac{3x^{n-1}+nx^{n-2}}{\displaystyle\sqrt{9+\frac1{x^2}}}\cdot\sqrt{9+\frac1{x^2}}=1\cdot3=3
容易看出當 n>1 時應有 \displaystyle\lim_{x\to\infty}\left(3x^{n-1}+nx^{n-2}\right)=\infty;而當 n<1 時應有 \displaystyle\lim_{x\to\infty}\left(3x^{n-1}+nx^{n-2}\right)=0,故皆不合。而取 n=1 時能檢驗符合題意。訣竅
通分整理後使用羅必達法則即可;亦可在此使用泰勒級數觀察之,隨後應用至第二小題中。解法
訣竅
使用反函數的定義與連鎖律求解。解法
由反函數的定義有 g\left(f\left(x\right)\right)=x。使用連鎖律可得
g'\left(f\left(x\right)\right)f'\left(x\right)=1
可以觀察到 f\left(1\right)=\frac\pi4,故取 x=1 可得\displaystyle g'\left(\frac\pi4\right)=\frac1{f'\left(1\right)}
再者,我們有 \displaystyle f'\left(x\right)=\frac1x+\frac1{1+x^2},因此 \displaystyle f'\left(1\right)=\frac32,從而所求為 \displaystyle g'\left(\frac\pi4\right)=\frac23。進一步的,我們對 g'\left(f\left(x\right)\right)f'\left(x\right)=1 再求導一次可得
g''\left(f\left(x\right)\right)\left[f'\left(x\right)\right]^2+g'\left(f\left(x\right)\right)f''\left(x\right)=0
此時也有 \displaystyle f''\left(x\right)=-\frac1{x^2}-\frac{2x}{\left(1+x^2\right)^2}。故取 x=1 可得\displaystyle g''\left(\frac\pi4\right)\cdot\left(\frac32\right)^2+\frac23\cdot\left(-\frac32\right)=0
因此所求為 \displaystyle g''\left(\frac\pi4\right)=\frac49。訣竅
使用泰勒展開式來求高階導數值。解法
使用正弦函數的泰勒展開式可知\displaystyle f\left(x\right)=\frac{x-\sin x}{x^3}=\frac{\displaystyle x-\left(x-\frac{x^3}6+\frac{x^5}{120}-\frac{x^7}{5040}+\frac{x^9}{362880}-\cdots\right)}{x^3}=\frac16-\frac{x^2}{120}+\frac{x^4}{5040}-\frac{x^6}{362880}+\cdots
因此所求為 \displaystyle f^{\left(6\right)}\left(0\right)=6!\cdot-\frac1{362880}=-\frac1{504}。訣竅
第一小題為經典問題,分兩段後運用變數變換對消;第二小題則需使用兩次分部積分法處理。解法
訣竅
運用多變函數的連鎖律計算求解即可。解法一
首先留意到 r=\sqrt{x^2+y^2}、\displaystyle\theta=\tan^{-1}\frac{y}x,如此可使用多變函數連鎖律可知\displaystyle\begin{aligned}&\frac{\partial f}{\partial x}=\frac{\partial f}{\partial r}\frac{\partial r}{\partial x}+\frac{\partial f}{\partial\theta}\frac{\partial\theta}{\partial x}=\frac{x}{\sqrt{x^2+y^2}}\frac{\partial f}{\partial r}-\frac{y}{x^2+y^2}\frac{\partial f}{\partial\theta}=\cos\theta\frac{\partial f}{\partial r}-\frac{\sin\theta}r\frac{\partial f}{\partial\theta}\\&\frac{\partial f}{\partial y}=\frac{\partial f}{\partial r}\frac{\partial r}{\partial y}+\frac{\partial f}{\partial\theta}\frac{\partial\theta}{\partial y}=\frac{y}{\sqrt{x^2+y^2}}\frac{\partial f}{\partial r}+\frac{x}{x^2+y^2}\frac{\partial f}{\partial\theta}=\sin\theta\frac{\partial f}{\partial r}+\frac{\cos\theta}r\frac{\partial f}{\partial\theta}\end{aligned}
進一步的計算二階偏導函數有\displaystyle\begin{aligned}&\frac{\partial^2f}{\partial x^2}=\frac{\partial}{\partial x}\left(\cos\theta\frac{\partial f}{\partial r}-\frac{\sin\theta}r\frac{\partial f}{\partial\theta}\right)=\frac{\sin^2\theta}r\frac{\partial f}{\partial r}+\cos^2\theta\frac{\partial^2f}{\partial r^2}-\frac{2\sin\theta\cos\theta}r\frac{\partial^2f}{\partial\theta\partial r}+\frac{\sin^2\theta}{r^2}\frac{\partial^2f}{\partial\theta^2}\\&\frac{\partial^2f}{\partial y^2}=\frac{\partial}{\partial y}\left(\sin\theta\frac{\partial f}{\partial r}+\frac{\cos\theta}r\frac{\partial f}{\partial\theta}\right)=\frac{\cos^2\theta}r\frac{\partial f}{\partial r}+\sin^2\theta\frac{\partial^2f}{\partial r^2}+\frac{2\sin\theta\cos\theta}r\frac{\partial^2\theta\partial r}+\frac{\cos^2\theta}{r^2}\frac{\partial^2f}{\partial\theta^2}\end{aligned}
將兩式相加並使用 \sin^2\theta+\cos^2\theta=1 便有所欲證的等式,證明完畢。解法二
使用多變函數的連鎖律可知\begin{aligned}&\displaystyle\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r}=\cos\theta\frac{\partial f}{\partial x}+\sin\theta\frac{\partial f}{\partial y}\\&\frac{\partial f}{\partial\theta}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial\theta}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial\theta}=-r\sin\theta\frac{\partial f}{\partial x}+r\cos\theta\frac{\partial f}{\partial y}\end{aligned}
進一步地計算二階偏導函數有\displaystyle\begin{aligned}&\frac{\partial^2f}{\partial r^2}=\cos^2\theta\frac{\partial^2f}{\partial x^2}+2\cos\theta\sin\theta\frac{\partial^2f}{\partial x\partial y}+\sin^2\theta\frac{\partial^2f}{\partial y^2}\\&\frac{\partial^2f}{\partial\theta^2}=-r\cos\theta\frac{\partial f}{\partial x}-r\sin\theta\frac{\partial f}{\partial y}+r^2\sin^2\theta\frac{\partial^2f}{\partial x^2}-2r^2\sin\theta\cos\theta\frac{\partial^2f}{\partial x\partial y}+r^2\cos^2\theta\frac{\partial^2f}{\partial y^2}\end{aligned}
據此計算可以注意到\displaystyle\frac{\partial^2f}{\partial r^2}+\frac1{r^2}\frac{\partial^2f}{\partial\theta^2}+\frac1r\frac{\partial f}{\partial r}=\left(\cos^2\theta+\sin^2\theta\right)\frac{\partial^2f}{\partial x^2}+\left(\sin^2\theta+\cos^2\theta\right)\frac{\partial^2f}{\partial y^2}=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}
訣竅
根據題意使用拉格朗日乘子法求解,但僅能獲得一座標,由考慮極值的函數的特點可以注意到距離原點很遠處其函數值將無限至的小。解法
設定拉格朗日乘子函數如下F\left(x,y,z,\lambda\right)=y+xz-2x^2-y^2-z^2+\lambda\left(x+y+z-35\right)
據此解下列的聯立方程組\left\{\begin{aligned}&F_x\left(x,y,z,\lambda\right)=z-4x+\lambda=0\\&F_y\left(x,y,z,\lambda\right)=1-2y+\lambda=0\\&F_z\left(x,y,z,\lambda\right)=x-2z+\lambda=0\\&F_{\lambda}\left(x,y,z,\lambda\right)=x+y+z-35=0\end{aligned}\right.
將第一式與第三式相減可得 3z=5x,即 \displaystyle z=\frac{5x}3,從而第一式與第二式可得 \displaystyle\left(x,y,z\right)=\left(\frac{3\lambda}7,\frac{\lambda+1}2,\frac{5\lambda}7\right)。將此代入第四式有\displaystyle\frac{3\lambda}7+\frac{\lambda+1}2+\frac{5\lambda}7=35
故得 \displaystyle\lambda=\frac{161}9,故得座標 \displaystyle\left(x,y,z\right)=\left(\frac{23}3,\frac{85}9,\frac{115}9\right)。另一方面取 \left(x,y,z\right)=\left(0,t,35-t\right),那麼可以注意到
f\left(0,t,35-t\right)=t-t^2-\left(35-t\right)^2\to-\infty~~\mbox{當}~~t\to\infty
故函數 f 在 x+y+z=35 上無極小值,而在 \displaystyle\left(\frac{23}3,\frac{85}9,\frac{115}9\right) 處達到最大值,其值為 \displaystyle-\frac{21272}{81}。訣竅
按提示使用變數代換法處理,這裡由邊界去考慮代換的處理。解法
令 u=4x^2-y^2、v=4x^2+y^2,那麼觀察四段邊界可改寫如下\displaystyle\frac{\partial\left(x,y\right)}{\partial\left(u,v\right)}=\left|\begin{matrix}\displaystyle\frac{\partial x}{\partial u}&\displaystyle\frac{\partial x}{\partial v}\\\displaystyle\frac{\partial y}{\partial u}&\displaystyle\frac{\partial y}{\partial v}\end{matrix}\right|=\left|\begin{matrix}\displaystyle\frac{\partial u}{\partial x}&\displaystyle\frac{\partial u}{\partial y}\\\displaystyle\frac{\partial v}{\partial x}&\displaystyle\frac{\partial v}{\partial y}\end{matrix}\right|^{-1}=\left|\begin{matrix}8x&-2y\\8x&2y\end{matrix}\right|^{-1}=\frac1{32xy}=\frac1{8\sqrt{v^2-u^2}}
如此所求的重積分可用變數變換改寫並計算如下\displaystyle\begin{aligned}\iint_RxdA&=\int_4^{16}\int_u^{\frac{5u}3}\sqrt{\frac{v+u}8}\cdot\frac1{8\sqrt{v^2-u^2}}dvdu\\&=\frac{\sqrt2}{32}\int_4^{16}\int_u^{\frac{5u}3}\frac1{\sqrt{v-u}}dvdu=\frac{\sqrt2}{16}\int_4^{16}\sqrt{v-u}\Big|_{v=u}^{v=\frac{5u}3}du\\&=\frac{\sqrt3}{24}\int_4^{16}\sqrt{u}du=\left.\frac{\sqrt3}{36}u^{\frac32}\right|_4^{16}=\frac{14\sqrt3}{9}\end{aligned}
訣竅
根據題意使用 Stokes 定理即可;亦可按曲面積分的意義計算求解。解法一
考慮曲面 S 的邊界\partial S=\left\{\left(x,y,z\right)\in\mathbb{R}^3:~x^2+y^2=4,~z=0.\right\}
應用 Stokes 定理可知\displaystyle\iint_S\left(\mbox{curl}~F\cdot n\right)dS=\oint_{\partial S}F\cdot dr
現在將此邊界參數化為 \left\{\begin{aligned}&x=2\cos\theta\\&y=2\sin\theta\\&z=0\end{aligned}\right.,其中參數 \theta\in\left[0,2\pi\right),如此所求的面積分可改寫並計算如下\displaystyle\begin{aligned}\iint_S\left(\mbox{curl}~F\cdot n\right)dS&=\int_0^{2\pi}\left(0,2\cos\theta,2\sin\theta\right)\cdot\left(-2\sin\theta,2\cos\theta,0\right)d\theta\\&=4\int_0^{2\pi}\cos^2\theta d\theta=2\int_0^{2\pi}\left(1+\cos2\theta\right)d\theta=\left(2\theta+\sin2\theta\right)\Big|_0^{2\pi}=4\pi\end{aligned}
解法二【本題不可採用此法】
首先計算旋度如下
\mbox{curl}~F=\left|\begin{matrix}\vec{i}&\vec{j}&\vec{k}\\\partial_x&\partial_y&\partial_z\\z&x&y\end{matrix}\right|=\left(1,1,1\right)
接著將該曲面參數化有\left\{\begin{aligned}&x=\sqrt5\cos\theta\sin\phi\\&y=\sqrt5\sin\theta\sin\phi\\&z=-1+\sqrt5\cos\phi\end{aligned}\right.,那麼參數範圍為 \left\{\begin{aligned}&0\leq\theta\leq2\pi\\&0\leq\phi\leq\cos^{-1}\frac1{\sqrt5}\end{aligned}\right.。據此所求的面積分可改寫並計算如下
\displaystyle\begin{aligned}\iint_S\left(\mbox{curl}~F\cdot n\right)dS&=\int_0^{2\pi}\int_0^{\cos^{-1}\frac1{\sqrt5}}\left(1,1,1\right)\cdot\left(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi\right)\cdot5\sin\phi d\phi d\theta\\&=5\int_0^{2\pi}\int_0^{\cos^{-1}\frac1{\sqrt5}}\cos\theta\sin^2\phi d\phi d\theta+5\int_0^{2\pi}\int_0^{\cos^{-1}\frac1{\sqrt5}}\sin\theta\sin^2\phi d\phi d\theta+5\int_0^{2\pi}\int_0^{\cos^{-1}\frac1{\sqrt5}}\sin\phi\cos\phi d\phi d\theta\\&=0+0+\left.5\cdot2\pi\cdot\frac{-\cos^2\phi}2\right|_0^{\cos^{-1}\frac1{\sqrt5}}=5\pi\cdot\left(1-\frac15\right)=4\pi\end{aligned}
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