PastExamLabPastExamLab

成功大學 88 年度 微積分

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第 1 題10 分
Please state the fundamental theorem of calculus.
第 2 題10 分
Find the Mclaurin series for the function f(x)=arctan⁡xf(x) = \arctan x.
第 3 題10 分
Determine whether the improper integral ∫0∞sin⁡xxdx\int_0^\infty \frac{\sin x}{x} dx converges or diverges.
第 4 題10 分
Evaluate lim⁡x→1(3x1x+5sin⁡(x−1)x−1)\lim\limits_{x \to 1} \left(3x^{\frac{1}{x}} + \frac{5\sin(x-1)}{x-1}\right).
第 5 題10 分
Assume that f:[0,2]→Rf : [0,2] \to R is a continuous function and ∫02f(x)dx=2\int_0^2 f(x) dx = 2. Prove that there is a c∈[0,2]c \in [0,2] such that f(c)=cf(c) = c.
第 6 題10 分
Let an>0a_n > 0 for all integer nn. Assume that the series ∑n=1∞an\sum\limits_{n=1}^\infty a_n converges. Do the following series ∑n=1∞an\sum\limits_{n=1}^\infty \sqrt{a_n} and ∑n=1∞(an)2\sum\limits_{n=1}^\infty (a_n)^2 converge? diverge? or neither? If they are in the first two cases, please prove it. Otherwise please give examples.
第 7 題10 分
Evaluate ∫01∫x21yeydydx\int_0^1 \int_{x^2}^1 ye^y dydx.
第 8 題10 分
Find the equation of the tangent plane for the surface z=xyz = xy at (1,1,1)(1,1,1).
第 9 題10 分
Evaluate ∬Rex+ydxdy\iint_R e^{x+y} dxdy, where RR is the region bounded by the xx-axis, yy-axis, and x+y=4x + y = 4.
第 10 題10 分
Find the extreme value of the function f(x,y)=2x2+y2−4x−2y+1f(x,y) = 2x^2 + y^2 - 4x - 2y + 1 in the region S={(x,y)∣x29+y216≤1}S = \{(x,y) | \frac{x^2}{9} + \frac{y^2}{16} \leq 1\}.
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