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台灣大學 · 數學系 · 轉學考考古題 · 民國108年（2019年）

108 年度 微積分(A)

台灣大學 · 數學系 · 轉學考

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第 1 題20 分
Prove L'Hospital's rule in the following case: Suppose f(x),g(x)f(x), g(x) are differentiable with continuous derivatives f′(x),g′(x)f'(x), g'(x) on the interval (−1,1)(-1,1) such that f(0)=g(0)=0f(0) = g(0) = 0 and g′(0)≠0g'(0) \neq 0. Then
lim⁡x→0f(x)g(x)=f′(0)g′(0).\lim\limits_{x \to 0} \frac{f(x)}{g(x)} = \frac{f'(0)}{g'(0)}.
第 2 題30 分
(a)15 分
Give an example of differentiable functions fn(x),n≥0f_n(x), n \geq 0, on (−1,1)(-1,1) such that ∑n=0∞fn(x)\sum_{n=0}^{\infty} f_n(x) converges to a function on (−1,1)(-1,1) which is not differentiable. You need to justify your answer.
(b)15 分
Suppose the power series ∑n=0∞anxn\sum_{n=0}^{\infty} a_n x^n converges to a function f(x)f(x) on the interval (−1,1)(-1,1). Show that f(x)f(x) is differentiable with derivative f′(x)=∑n=1∞nanxn−1f'(x) = \sum_{n=1}^{\infty} na_n x^{n-1} on (−1,1)(-1,1).
第 3 題20 分
Let f(x,y)f(x,y) be a function on R2\mathbb{R}^2 whose partial derivatives of any order are continuous. Suppose fx(0,0)=fy(0,0)=0f_x(0,0) = f_y(0,0) = 0 and fxx(0,0)=fyy(0,0)=2,fxy(0,0)=1f_{xx}(0,0) = f_{yy}(0,0) = 2, f_{xy}(0,0) = 1. Prove that there exists ϵ>0\epsilon > 0 such that f(x,y)>f(0,0)f(x,y) > f(0,0) for all 0<x2+y2<ϵ20 < x^2 + y^2 < \epsilon^2 (so f(x,y)f(x,y) has a local minimum at (0,0)(0,0)).
第 4 題20 分
Let f(x)f(x) be a continuous function on the closed interval [−1,1][-1,1]. Show that
lim⁡n→∞∫−11f(x)sin⁡nx dx=0.\lim\limits_{n \to \infty} \int_{-1}^{1} f(x) \sin nx \, dx = 0.
第 5 題20 分
Let TT be the surface in R3\mathbb{R}^3 given by the equation
(x2+y2−2)2+z2=1(\sqrt{x^2 + y^2} - 2)^2 + z^2 = 1
and n⃗\vec{n} the outward unit normal vector. Evaluate the flux integral ∬TF⃗⋅n⃗ dS\iint_T \vec{F} \cdot \vec{n} \, dS of the vector field
F⃗(x,y,z)=(0,0,∣z∣).\vec{F}(x, y, z) = (0, 0, |z|).
(Here dSdS denotes the element of surface.)
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