PastExamLabPastExamLab

成功大學 100 年度 微積分

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第 1 題14 分
Compute the following limits, if exist:
(a)7 分
lim⁡x→0sin⁡xx−[x]\lim\limits_{x \to 0} \frac{\sin x}{x - [x]}, where [x][x] is the Gauss function,
(b)7 分
lim⁡x→1∫1xe−t2dtln⁡x\lim\limits_{x \to 1} \frac{\int_1^{\sqrt{x}} e^{-t^2} dt}{\ln x}.
第 2 題12 分
Find the local extrema of f(x)=x2ln⁡xf(x) = x^2 \ln x for x>0x > 0, discuss concavity and find the point of inflection.
第 3 題14 分
Calculate the following integrals:
(a))7 分
∫24dxxln⁡x\int_2^4 \frac{dx}{x \ln \sqrt{x}}
(b))7 分
∫01x⋅arctan⁡x2dx\int_0^1 x \cdot \arctan x^2 dx
第 4 題12 分
Find the area of the surface generated by revolving the curve 6xy=x4+36xy = x^4 + 3 from x=1x = 1 to x=3x = 3 about the xx-axis.
第 5 題12 分
Use ∑n=0∞xnn!=ex\sum\limits_{n=0}^{\infty} \frac{x^n}{n!} = e^x, prove ∑n=1∞n2xnn!=(x2+x)ex\sum\limits_{n=1}^{\infty} \frac{n^2 x^n}{n!} = (x^2 + x)e^x, and find the sum ∑n=1∞n2n!\sum\limits_{n=1}^{\infty} \frac{n^2}{n!}.
第 6 題12 分
Find the average value of f(x,y)=xyf(x,y) = xy over the quarter circle x2+y2≤1x^2 + y^2 \leq 1 in the first quadrant.
第 7 題12 分
Evaluate F(x,y)=∫0∞e−xt−e−yttdtF(x, y) = \int_0^{\infty} \frac{e^{-xt} - e^{-yt}}{t} dt for x>0x > 0, y>0y > 0.
第 8 題12 分
Find the points on the curve 17x2+12xy+8y2=10017x^2 + 12xy + 8y^2 = 100 that are closest to and farthest away from the origin.
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