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台灣大學 · 數學系 · 轉學考考古題 · 民國109年（2020年）

109 年度 微積分(A)

台灣大學 · 數學系 · 轉學考

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第 1 題15 分
Suppose that ff is a continuous function on a closed interval [a,b][a, b]. Show that ff is uniformly continuous on [a,b][a, b].
第 2 題20 分
Let N\mathbb{N} be the set of all natural numbers and n∈Nn \in \mathbb{N}.
(a)
Consider a sequence {an}n=1∞\{a_n\}_{n=1}^\infty defined by {a1=2an+1=2+an\begin{cases} a_1 = \sqrt{2} \\ a_{n+1} = \sqrt{2 + a_n} \end{cases} for n≥1n \geq 1. Show that lim⁡n→∞an\lim\limits_{n \to \infty} a_n exists and evaluate the limit.
(b)
Prove the relation lim⁡n→∞1nk+1∑i=1nik=1k+1\lim\limits_{n \to \infty} \frac{1}{n^{k+1}} \sum\limits_{i=1}^{n} i^k = \frac{1}{k+1} for any nonnegative integer kk.
第 3 題15 分
(a)
If y′=ayy' = ay, show that y=ceaxy = ce^{ax} for some cc.
(b)
If f(x+y)=f(x)f(y)f(x + y) = f(x)f(y) and ff is differentiable, show that either f(x)=0f(x) = 0 or f(x)=eaxf(x) = e^{ax} for some aa.
第 4 題15 分
Show that ∫−∞∞e−x2dx=π\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}.
第 5 題15 分
Calculate ∬Szdx∧dy−xdy∧dz\iint_S z dx \wedge dy - x dy \wedge dz, where SS is the spherical cap x2+y2+z2=1x^2 + y^2 + z^2 = 1, x>1/2x > 1/2, oriented positively with respect to the normal pointing to infinity.
第 6 題20 分
Prove that the function f(x,y)=e−y2+2xyf(x, y) = e^{-y^2 + 2xy} can be expanded in a series of the form
∑n=0∞Hn(x)n!yn,\sum_{n=0}^\infty \frac{H_n(x)}{n!} y^n,
that converges for all values of xx and yy and that the polynomials Hn(x)H_n(x) satisfy
(a)
Hn(x)H_n(x) is a polynomial of degree nn
(b)
Hn′(x)=2nHn−1(x)H_n'(x) = 2nH_{n-1}(x)
(c)
Hn+1(x)−2xHn(x)+2nHn−1(x)=0H_{n+1}(x) - 2xH_n(x) + 2nH_{n-1}(x) = 0
(d)
Hn′′(x)−2xHn′(x)+2nHn(x)=0H_n''(x) - 2xH_n'(x) + 2nH_n(x) = 0
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