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成功大學 94 年度 微積分

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Show all your work. Explanation is required for each problem. No calculator is allowed.
第 1 題10 分
The line y=mx+by = mx + b (m≠0m \neq 0) is called a slant asymptote of the graph y=f(x)y = f(x) if
lim⁡x→∞[f(x)−(mx+b)]=0orlim⁡x→−∞[f(x)−(mx+b)]=0.\lim\limits_{x \to \infty} [f(x) - (mx + b)] = 0 \quad \text{or} \quad \lim\limits_{x \to -\infty} [f(x) - (mx + b)] = 0.
Find all (horizontal, vertical, slant) asymptotes of the graph of y=ln⁡(1+ex)y = \ln(1 + e^x).
第 2 題10 分
Let f(x)f(x) be defined by
f(x):={sin⁡x,if x<0;ln⁡(1+2x),if x≥0.f(x) := \begin{cases} \sin x, & \text{if } x < 0; \\ \ln(1 + 2x), & \text{if } x \geq 0. \end{cases}
(Note: ln⁡x=log⁡ex\ln x = \log_e x.) Find f′(x)f'(x).
第 3 題10 分
Find the length of the curve y=x3/2y = x^{3/2} for 0≤x≤40 \leq x \leq 4.
第 4 題10 分
Evaluate the improper integral
∫01x−1/2(1−x)−1/2 dx.\int_0^1 x^{-1/2}(1-x)^{-1/2} \, dx.
第 5 題10 分
Determine whether the series
∑n=2∞1n(ln⁡n)2\sum\limits_{n=2}^{\infty} \frac{1}{n(\ln n)^2}
is convergent or divergent.
第 6 題10 分
Find the third Taylor polynomial of f(x)=sin⁡(sin⁡x)f(x) = \sin(\sin x) at 00 (i.e., the Taylor polynomial of f(x)f(x) up to the term x3x^3).
第 7 題10 分
Let F(x1,x2)F(x_1, x_2) be a real-valued function defined on R2\mathbb{R}^2 with partial derivatives Fi:=∂F∂xiF_i := \frac{\partial F}{\partial x_i} for i=1,2i = 1, 2. Suppose that the variable vv is implicitly defined as a function of uu by the equation F(u−v,u+v)=0F(u - v, u + v) = 0. Find dvdu\frac{dv}{du} in terms of F1F_1 and F2F_2.
第 8 題10 分
The plane x+y+2z=2x + y + 2z = 2 intersects the paraboloid z=x2+y2z = x^2 + y^2 in an ellipse. Find the point on the ellipse that is farthest from the origin.
第 9 題10 分
Find the volume of the solid bounded by the paraboloid z=1−x2−y2z = 1 - x^2 - y^2 and the plane z=0z = 0.
第 10 題10 分
Evaluate the line integral ∫CF⋅dr\int_C \mathbf{F} \cdot d\mathbf{r}, where F(x,y,z)=−y2i+xj+z2k\mathbf{F}(x, y, z) = -y^2\mathbf{i} + x\mathbf{j} + z^2\mathbf{k} and CC is the curve of intersection of the plane z=2z = 2 and the cylinder x2+y2=1x^2 + y^2 = 1. (Orient CC to be counterclockwise when viewed from above.)
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