PastExamLabPastExamLab

成功大學 107 年度 微積分A

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第 1 題10 分
Evaluate:
(a)5 分
lim⁡n→∞nn+1\lim\limits_{n \to \infty} \frac{n}{n + 1}
(b)5 分
lim⁡x→0e−1x(1x6−3x3)\lim\limits_{x \to 0} e^{-\frac{1}{x}} \left(\frac{1}{x^6} - \frac{3}{x^3}\right)
第 2 題10 分
Find the point on the curve y=x3/2y = x^{3/2} that is closest to the point (52,0)(\frac{5}{2}, 0).
第 3 題10 分
Find the length of the parametric curve
x=3t2,y=2t3,0≤t≤1.x = 3t^2, \quad y = 2t^3, \quad 0 \leq t \leq 1.
第 4 題10 分
Determine whether the improper integral
∫1∞1−cos⁡xx2dx\int_1^{\infty} \frac{1 - \cos x}{x^2} dx
is convergent or divergent.
第 5 題10 分
Find the radius of convergence of the power series ∑n=1∞n2xn\sum\limits_{n=1}^{\infty} n^2 x^n and evaluate ∑n=1∞n22n\sum\limits_{n=1}^{\infty} \frac{n^2}{2^n}.
第 6 題10 分
Find the equation of the tangent plane and the normal line to the surface
S:x2y+exyz−2cos⁡(xz)=0S : x^2 y + e^{xyz} - 2\cos(xz) = 0
at (1,1,0)(1, 1, 0).
第 7 題10 分
Let f:R2→Rf : \mathbb{R}^2 \to \mathbb{R} be a function. Assume that all second partial derivatives of ff exist and continuous and
fxx+fyy=0,for all (x,y)∈R2.f_{xx} + f_{yy} = 0, \quad \text{for all } (x, y) \in \mathbb{R}^2.
Define g:R2→Rg : \mathbb{R}^2 \to \mathbb{R} by g(u,v)=f(u2−v2,2uv)g(u, v) = f(u^2 - v^2, 2uv). Find guu+gvvg_{uu} + g_{vv}.
第 8 題10 分
Evaluate the double integral ∬Dsin⁡(x+y)dA\iint_D \sin(x + y) dA where DD is the region bounded by x+y=πx + y = \pi, x+y=0x + y = 0 and x−y=πx - y = \pi and x−y=0x - y = 0.
第 9 題10 分
Evaluate the triple integral ∭Ex2+y2dV\iiint_E \sqrt{x^2 + y^2} dV, where EE is the solid region in R3\mathbb{R}^3 bounded by the surface z=1z = 1 and z=x2+y2z = x^2 + y^2.
第 10 題10 分
Evaluate the line integral ∫C(yex+sin⁡y)dx+(ex+xcos⁡y)dy\int_C (ye^x + \sin y)dx + (e^x + x \cos y)dy along the curve C:r(t)=(t2+1)i+(t2−1)jC : r(t) = (t^2 + 1)\mathbf{i} + (t^2 - 1)\mathbf{j}, 0≤t≤20 \leq t \leq 2. Hint: you may find a potential function ff of the vector field F(x,y)=(yex+sin⁡y)i+(ex+xcos⁡y)j\mathbf{F}(x, y) = (ye^x + \sin y)\mathbf{i} + (e^x + x \cos y)\mathbf{j}, i.e. find ff so that F=∇f\mathbf{F} = \nabla f.
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