PastExamLabPastExamLab

成功大學 98 年度 微積分

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第 1 題10 分
Show that
lim⁡n→∞∑k=0n−1n2−k2n2=π4\lim\limits_{n \to \infty} \sum\limits_{k=0}^{n-1} \frac{\sqrt{n^2 - k^2}}{n^2} = \frac{\pi}{4}
(Hint: Riemann sum)
第 2 題10 分
Let f(x)=xxf(x) = x^x, x>0x > 0
(a)5 分
Calculate f′(x)f'(x)
(b)5 分
Find lim⁡x→0+f(x)\lim\limits_{x \to 0^+} f(x)
第 3 題10 分
Find ∫xx2−1dx\int \frac{\sqrt{x}}{x^2 - 1} dx (Hint: x=u2x = u^2)
第 4 題10 分
Find ∫cos⁡(ln⁡x)dx\int \cos(\ln x) dx (Hint: integration by parts twice)
第 5 題10 分
Find lim⁡n→∞(en∫n2ne−x2dx)\lim\limits_{n \to \infty} \left( e^n \int_n^{2n} e^{-x^2} dx \right) (Hint: pinching or sandwich theorem)
第 6 題10 分
Find the radius of convergence of the power series
∑3k(k!)3(3k)!xk\sum \frac{3^k(k!)^3}{(3k)!} x^k
第 7 題10 分
Use the method of Lagrange multiplier to find the extreme values of f(x,y)=xy3f(x,y) = xy^3 subject to the side condition x2+3y2=16x^2 + 3y^2 = 16.
第 8 題10 分
Find the area of the surface by revolving the cycloid x(t)=t−sin⁡tx(t) = t - \sin t, y(t)=1−cos⁡ty(t) = 1 - \cos t, 0≤t≤2π0 \leq t \leq 2\pi about the xx-axis.
第 9 題10 分
Let
g(x,y)={x2y2x4+y4,(x,y)≠(0,0)0,(x,y)=(0,0)g(x,y) = \begin{cases} \frac{x^2y^2}{x^4 + y^4}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}
Show that lim⁡(x,y)→(0,0)f(x,y)\lim\limits_{(x,y) \to (0,0)} f(x,y) does not exist.
第 10 題10 分
Use polar coordinates to compute
∫121∫−x2+y2x2+y21x2+y2dydx.\int_\frac{1}{2}^1 \int_{-\sqrt{x^2 + y^2}}^{\sqrt{x^2 + y^2}} \frac{1}{\sqrt{x^2 + y^2}} dy dx.
(Hint: cos⁡θ+sin⁡θ=2sin⁡(θ+π4)\cos \theta + \sin \theta = \sqrt{2}\sin(\theta + \frac{\pi}{4}))
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