PastExamLabPastExamLab

成功大學 86 年度 微積分

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第 1 題15 分
Evaluating limits.
(a)8 分
Evaluate the limits. (i) lim⁡x→01[x]−x\lim\limits_{x \to 0} \frac{1}{[x]} - x. (ii) lim⁡Q→∞1n∑i=1n(2in−1)\lim\limits_{Q \to \infty} \frac{1}{n} \sum\limits_{i=1}^{n} (\frac{2i}{n} - 1).
(b)7 分
Show that there exists a number between π2\frac{\pi}{2} and π\pi such that tan⁡x=−x\tan x = -x.
第 2 題10 分
Let f(x)=xne−xf(x) = x^n e^{-x} for x≥0x \geq 0 (nn is a fixed but arbitrary positive integer).
(a)5 分
Show that the maximum value of f(x)f(x) is f(n)f(n).
(b)5 分
Show that (1+1n)n<e<(1−1n)−n(1 + \frac{1}{n})^n < e < (1 - \frac{1}{n})^{-n}.
第 3 題10 分
Sketch the graph of f(x)=2ex+e−xf(x) = 2e^x + e^{-x}. Indicate local extrema, inflection points, and concave structure.
第 4 題15 分
Let Γ(t)=∫0∞xt−1e−xdx\Gamma(t) = \int_0^\infty x^{t-1} e^{-x} dx, t>0t > 0.
(a)5 分
Show that ∫0∞xn−1exp⁡(−x2)dx=12Γ(n2)\int_0^\infty x^{n-1} \exp(-x^2) dx = \frac{1}{2} \Gamma(\frac{n}{2}), n∈Nn \in N.
(b)8 分
Evaluate ∫0∞e−x2dx\int_0^\infty e^{-x^2} dx.
(c)2 分
What is the value of Γ(12)\Gamma(\frac{1}{2})?
第 5 題30 分
Integration problems involving trigonometric functions.
(a)2 分
Show that ∫01sin⁡−1x1−x2dx=π28\int_0^1 \frac{\sin^{-1} x}{\sqrt{1-x^2}} dx = \frac{\pi^2}{8}.
(b)4 分
Show that ∫0π2sin⁡2n+1xdx=2⋅4⋅6⋯2n3⋅5⋯(2n+1)\int_0^{\frac{\pi}{2}} \sin^{2n+1} x dx = \frac{2 \cdot 4 \cdot 6 \cdots 2n}{3 \cdot 5 \cdots (2n+1)}, n=1,2,3,…n = 1,2,3,\ldots
(c)2 分
What is the value of ∫01x2n+11−x2dx\int_0^1 \frac{x^{2n+1}}{\sqrt{1-x^2}} dx, n=1,2,3,…n = 1,2,3,\ldots
(d)6 分
Use 11−x=1+∑n=1∞1⋅3⋅5⋯(2n−1)2⋅4⋅6⋯(2n)xn\frac{1}{\sqrt{1-x}} = 1 + \sum\limits_{n=1}^\infty \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} x^n, ∣x∣<1|x| < 1, to find a power series representation for sin⁡−1x\sin^{-1} x.
(e)4 分
Show that ∫01sin⁡−1x1−x2=∑n=1∞1(2n−1)2\int_0^1 \frac{\sin^{-1} x}{\sqrt{1-x^2}} = \sum\limits_{n=1}^\infty \frac{1}{(2n-1)^2}.
(f)2 分
Use (a) and (e) to show that ∑n=1∞1n2=π26\sum\limits_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}.
第 6 題15 分
Polar coordinate problems.
(a)7 分
Find the perimeter ss of the cardioid with polar equation r=1+cos⁡θr = 1 + \cos \theta.
(b)8 分
Find the surface area AA generated by revolving the cardioid in (a) around the xx-axis.
第 7 題15 分
Multiple integration problems.
(a)5 分
Evaluate ∫02∫y2yex2dxdy\int_0^2 \int_y^2 y e^{x^2} dx dy.
(b)10 分
Find the volume of the solid region that is interior to both the sphere x2+y2+z2=4x^2 + y^2 + z^2 = 4 of radius 2 and the cylinder (x−1)2+y2=1(x-1)^2 + y^2 = 1.
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