PastExamLabPastExamLab

成功大學 81 年度 微積分

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第 1 題20 分
Question 1
(i)10 分
Suppose u=f(x,y,z)u = f(x,y,z), x=g(x,y,t)x = g(x,y,t) and y=h(x,t)y = h(x,t) are differentiable real-valued functions in suitable domains. Find ∂u∂x(x,t)=?\frac{\partial u}{\partial x}(x,t) =? ∂u∂t(x,t)=?\frac{\partial u}{\partial t}(x,t) =?
(ii)10 分
Suppose f:R2→Rf : \mathbb{R}^2 \to \mathbb{R} is continuous and g:R→Rg : \mathbb{R} \to \mathbb{R} is defined by g(t)=∫0t∫0xf(x,y)dydxg(t) = \int_0^t \int_0^x f(x,y) dy dx. Determine g′(t)=?g'(t) = ?
第 2 題20 分
Question 2
(i)10 分
Is the integral ∫1∞sin⁡1xdx\int_1^{\infty} \sin \frac{1}{x} dx convergent? Justify your answer.
(ii)10 分
Let an=1⋅3⋅5⋯(2n−1)2⋅4⋅6⋯2na_n = \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots 2n}. Is {an}\{a_n\} convergent? Justify your answer.
第 3 題20 分
Question 3
(i)10 分
Find the work done by the force F(x,y,z)=(yz,xz,xy)\mathbf{F}(x,y,z) = (yz, xz, xy) in moving an object from (0,0,0)(0,0,0) to (1,2,3)(1,2,3) along the curve r(t)=(t,2t,3t)\mathbf{r}(t) = (t, 2t, 3t).
(ii)10 分
Use Green's Theorem to evaluate ∫Cx2ydx+3xydy\int_C x^2 y dx + 3xy dy, where CC is the positively oriented simple closed curve determined by the graphs of y=x2y = x^2 and y=xy = \sqrt{x}.
第 4 題20 分
Let ff be differentiable for x>0x > 0. Prove or disprove
(i)10 分
If lim⁡x→0+f(x)=0\lim\limits_{x \to 0^+} f(x) = 0, then lim⁡x→∞f′(x)=0\lim\limits_{x \to \infty} f'(x) = 0.
(ii)10 分
If lim⁡x→0+f(x)=∞\lim\limits_{x \to 0^+} f(x) = \infty, then lim⁡x→∞f′(x)=−∞\lim\limits_{x \to \infty} f'(x) = -\infty.
第 5 題20 分
Question 5
(i)10 分
Evaluate ∫01∫y1sin⁡y dy dx\int_0^1 \int_y^1 \sin y \, dy \, dx.
(ii)10 分
Evaluate ∫012dx1+(tan⁡x)3\int_0^{\frac{1}{2}} \frac{dx}{1 + (\tan x)^{\sqrt{3}}}.
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