PastExamLabPastExamLab

成功大學 82 年度 微積分

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第 1 題8 分
Let f:[0,+∞)→Rf : [0, +\infty) \to \mathbb{R} such that f(x)=cos⁡xf(x) = \cos\sqrt{x}.
(1)4 分
Find the derivative of ff;
(2)4 分
Evaluate ∫cos⁡xdx\int \cos\sqrt{x} dx.
第 2 題8 分
Find the limit lim⁡x→0+xln⁡x\lim\limits_{x \to 0^+} x^{\ln x}.
第 3 題16 分
(1)8 分
Find the Maclaurin series of f(x)=(1+x)αf(x) = (1 + x)^\alpha, where α∈R\alpha \in \mathbb{R}. Show that the function f(x)f(x) is analytic at x=0x = 0.
(2)8 分
Find the 4-th order Taylor's expansion of sin⁡(x+2y)\sin(x + 2y) at the point (0,0)(0, 0).
第 4 題16 分
(1)8 分
Find lim⁡(x,y)→(0,0)sin⁡(x2+y2)x2+y2\lim\limits_{(x,y) \to (0,0)} \frac{\sin(x^2 + y^2)}{x^2 + y^2}.
(2)8 分
Is the function f(x,y)={(x2+y2)sin⁡1x2+y2if (x,y)≠(0,0),0,if (x,y)=(0,0),f(x,y) = \begin{cases} (x^2 + y^2)\sin \frac{1}{\sqrt{x^2 + y^2}} & \text{if } (x,y) \neq (0,0), \\ 0, & \text{if } (x,y) = (0,0), \end{cases} continuously partially differentiable at (0,0)(0,0). Is ff differentiable at (0,0)(0,0)?
第 5 題16 分
(1)8 分
Show that ∫0+∞e−x2dx\int_0^{+\infty} e^{-x^2} dx is convergent. Also find the value which the improper integral converges to.
(2)8 分
Let RR be the region between the graph of the curve y=exp⁡(−x2)y = \exp(-x^2) and its asymptote. Find the volume of the solid generated by revolving the region RR about the YY-axis.
第 6 題18 分
(1)8 分
Let a1>0a_1 > 0, an+1=6(1+an)7+ana_{n+1} = \frac{6(1 + a_n)}{7 + a_n}. Show that the sequence {an}\{a_n\} is convergent and find its limit.
(2)10 分
Is the series ∑n=1+∞ann!nn\sum\limits_{n=1}^{+\infty} \frac{a^n n!}{n^n} convergent? If it is convergent, find its sum; otherwise, show the reason why it is divergent.
第 7 題18 分
(1)8 分
Let RR be a connected compact region in R2\mathbb{R}^2 and let γ\gamma be the boundary of RR such that it is a smooth closed oriented simple curve. Show that the area of RR is 12∫γ−ydx+xdy\frac{1}{2} \int_{\gamma} -y dx + x dy.
(2)10 分
Consider the extrema of the function f(x,y)=(y−x2)(y−x3)f(x, y) = (y - x^2)(y - x^3).
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