PastExamLabPastExamLab

成功大學 106 年度 微積分C

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第 1 題10 分
Find the limit
lim⁡x→0+[1ln⁡(1+x)−1x].\lim\limits_{x \to 0^+} \left[ \frac{1}{\ln(1 + x)} - \frac{1}{x} \right].
第 2 題10 分
Find all possible relative extrema and saddle points of the function
f(x,y)=2xy−12(x4+y4)+1.f(x, y) = 2xy - \frac{1}{2}(x^4 + y^4) + 1.
第 3 題10 分
Find the arc length of the polar curve:
r=7θ,0≤θ≤2π.r = 7^\theta, \quad 0 \leq \theta \leq 2\pi.
第 4 題10 分
Compute the integral ∫01sin⁡−1(x)dx\int_0^1 \sin^{-1}(x)dx.
第 5 題10 分
Let J(x)J(x) be a function satisfying the differential equation xJ′′(x)+J′(x)+xJ(x)=0xJ''(x) + J'(x) + xJ(x) = 0 for all values of xx and J(0)=1J(0) = 1. Find J′′(0)J''(0).
第 6 題10 分
Determine whether the following series is convergent or divergent.
lim⁡n→∞∑k=1n∣16k2n3−16kn2+3n∣\lim\limits_{n \to \infty} \sum\limits_{k=1}^n \left| \frac{16k^2}{n^3} - \frac{16k}{n^2} + \frac{3}{n} \right|
If it is convergent, find its sum. Otherwise, give a reason for your answer.
第 7 題10 分
Let u\mathbf{u} and v\mathbf{v} be the unit normal vectors of the tangent planes of the surfaces S1:x2+y2+z2+2x−4y−4z=12S_1 : x^2 + y^2 + z^2 + 2x - 4y - 4z = 12 and S2:4x2+y2+16z2=24S_2 : 4x^2 + y^2 + 16z^2 = 24 at the common point (1,−2,1)(1, -2, 1). Compute the inner product u⋅v\mathbf{u} \cdot \mathbf{v}.
第 8 題10 分
Evaluate the integral ∬xydS\iint xydS, where the surface S={(x,y,z)∣x2+y2=1,x≥0,y≥0,0≤z≤1}S = \{(x, y, z) | x^2 + y^2 = 1, x \geq 0, y \geq 0, 0 \leq z \leq 1\}.
第 9 題10 分
Compute the integral
∫01∫−x−x2x−x2x2+y2dydx.\int_0^1 \int_{-\sqrt{x-x^2}}^{\sqrt{x-x^2}} \sqrt{x^2 + y^2} dydx.
第 10 題10 分
Let RR be the region bounded by the graph of xy=1xy = 1, xy=4xy = 4, x=2x = 2, and x=3x = 3. Compute the integral
∬Re−xy1+x2dA.\iint_R \frac{e^{-xy}}{1 + x^2} dA.
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