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台灣大學 107 年度 微積分(B)

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Applications of differentiation 2 題Multiple integrals 2 題Infinite Series 2 題Continuous functions 1 題

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Part I Multiple Choice

Choose the most suitable answer among the choices (A), (B), (C) and (D) and put it into the "Multiple Choice Answer" section of your ANSWER SHEET.
第 1 題5 分Applications of differentiation
Let ff be a smooth function on R\mathbb{R}.
(a) If f′′>0f'' > 0 on R\mathbb{R}, then f′f' must be increasing on R\mathbb{R}.
(b) If f′′>0f'' > 0 on R\mathbb{R}, then ff must be concave upward on R\mathbb{R}.
(c) If f′′(a)=0f''(a) = 0, then the point (a,f(a))(a, f(a)) must be an inflection point.

Among the above three statements, how many of them are true?
A) Only one. B) Only two. C) All of them. D) None of them.
第 2 題5 分Continuous functions
A function ff on R\mathbb{R} is odd if f(−x)=−f(x)f(-x) = -f(x) for all x∈Rx \in \mathbb{R} while it is even if f(−x)=f(x)f(-x) = f(x) for all x∈Rx \in \mathbb{R}. Suppose that gg and hh are two functions on R\mathbb{R}.
(a) If gg is an even function, then the composite function h∘gh \circ g must be even.
(b) If gg is an odd function and lim⁡x→0+g(x)=3\lim\limits_{x \to 0^+} g(x) = 3, then the limit lim⁡x→0−g(x)\lim\limits_{x \to 0^-} g(x) must be −3-3.
(c) If gg is an even function and lim⁡x→0+g(x)=3\lim\limits_{x \to 0^+} g(x) = 3, then the limit lim⁡x→0g(x)\lim\limits_{x \to 0} g(x) must be 33.

Among the above three statements, how many of them are true?
A) Only one. B) Only two. C) All of them. D) None of them.
第 3 題5 分Multiple integrals
Consider the integral Ip,r:=∬D1(x2+y2)p/2dAI_{p,r} := \iint_D \frac{1}{(x^2+y^2)^{p/2}} dA, where DD is the region bounded by two concentric circles centred at the origin with radii rr and 11 respectively, 0<r<10 < r < 1. Let Jp:=lim⁡r→0+Ip,rJ_p := \lim\limits_{r \to 0^+} I_{p,r}.
(a) JpJ_p is convergent if p<1p < 1.
(b) JpJ_p is convergent if 1<p<21 < p < 2.
(c) JpJ_p is convergent if 2<p2 < p.

Among the above three statements, how many of them are true?
A) Only one. B) Only two. C) All of them. D) None of them.
第 4 題5 分Derivatives of multi-variable functions
Let f(x,y)={xsin⁡y21−ex2+y2if (x,y)≠(0,0)0if (x,y)=(0,0)f(x,y) = \begin{cases} \frac{x \sin y^2}{1 - e^{x^2+y^2}} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{cases}
(a) lim⁡x→0lim⁡y→0f(x,y)=0\lim\limits_{x \to 0}\lim\limits_{y \to 0} f(x,y) = 0.
(b) lim⁡x→0lim⁡y→0f(x,y)=lim⁡y→0lim⁡x→0f(x,y)\lim\limits_{x \to 0}\lim\limits_{ y \to 0} f(x,y) = \lim\limits_{y \to 0}\lim\limits_{ x \to 0} f(x,y).
(c) lim⁡(x,y)→(0,0)f(x,y)=0\lim\limits_{(x,y) \to (0,0)} f(x,y) = 0.

Among the above three statements, how many of them are true?
A) Only one. B) Only two. C) All of them. D) None of them.
第 5 題5 分Infinite Series
(a) The series ∑n=1∞ncos⁡(nπ)sin⁡1n\sum_{n=1}^{\infty} n \cos(n\pi) \sin \frac{1}{n} is absolutely convergent.
(b) The series ∑n=1∞sin⁡(n)n2+15n\sum_{n=1}^{\infty} \sin(n) \frac{n^2 + 1}{5^n} is absolutely convergent.
(c) The series ∑n=2∞(−1)nn(ln⁡n)2−1\sum_{n=2}^{\infty} \frac{(-1)^n}{n(\ln n)^{\sqrt{2}-1}} is absolutely convergent.

Among the above three statements, how many of them are true?
A) Only one. B) Only two. C) All of them. D) None of them.

Part II Fill in the blanks

Find a suitable answer to fill in each of the blanks below. Write the LABEL ON THE BLANK as well as YOUR ANSWER clearly in your ANSWER SHEET. Please write your answers in the order of the numbers of the labels. Explanation to your answer is NOT needed.
第 6 題5 分The Limit of a Function
lim⁡n→∞12+22+⋯+n2(1+2+⋯+n)2=\lim\limits_{n \to \infty} \frac{1^2 + 2^2 + \cdots + n^2}{(\sqrt{1} + \sqrt{2} + \cdots + \sqrt{n})^2} = _____(6)_____.
第 7 題5 分Derivatives
Let f(x)=∫0π2+2sin⁡xsin⁡(y+xcos⁡y) dyf(x) = \int_0^{\frac{\pi}{2}+2\sin x} \sin (y + x \cos y) \, dy. It follows that f′(0)=f'(0) = _____(7)_____.
第 8 題5 分Applications of differentiation
Let f(x)=arctan⁡(x3)+x2−1x2+1f(x) = \arctan (x^3) + \frac{x^2 - 1}{x^2 + 1}. The graph of ff has a horizontal asymptote represented by the equation _____(8)_____ and the global minimum value of ff is _____(9)_____.
第 9 題5 分Techniques of integration
∫011(x2+1)2dx=\int_0^1 \frac{1}{(x^2 + 1)^2} dx = _____(10)_____.
第 10 題5 分Infinite Series
The third term of the Maclaurin series (i.e. the Taylor series centred at x=0x = 0) of arcsin⁡(3x)\arcsin(3x) is _____(11)_____ (note that the answer should be a monomial in xx, the term of x0x^0 is counted as the 0-th term). The radius of convergence of the series is _____(12)_____.
第 11 題5 分Differential equations
If y=y(x)y = y(x) satisfies the differential equation x2y′+3xy=2ln⁡xx^2 y' + 3xy = 2 \ln x for x>0x > 0 with y(1)=2y(1) = 2, then y(x)=y(x) = _____(13)_____.
第 12 題5 分Applications of integration
Let CC be a variable path in the xyxy-plane of arc-length 11 starting at the point (3,1)(\sqrt{3}, 1) and ending at the point (a,b)(a, b). Suppose that G(a,b):=∫Cf drG(a, b) := \int_C \sqrt{f} \, dr, where f(x,y):=arctan⁡yxf(x, y) := \arctan \frac{y}{x}, is a function in aa and bb. Then, GG attains its maximum at a=a = _____(14)_____ and b=b = _____(15)_____, and the maximum value of GG is _____(16)_____.
第 13 題5 分Multiple integrals
If aa and bb are positive constants and if max⁡{p,q}\max \{p, q\} denotes the maximum between the numbers pp and qq, the iterated integral ∫0a∫0bemax⁡{b2x2,a2y2}dy dx=\int_0^a \int_0^b e^{\max\{b^2 x^2, a^2 y^2\}} dy \, dx = _____(17)_____.
第 14 題5 分Vector calculus
Let EE be a tetrahedron (四面體) in R3\mathbb{R}^3 bounded by the planes x+y+z=3x + y + z = 3, x=2zx = 2z, y=0y = 0 and z=0z = 0. Let also F:=(x−y)i+(y2+z2)j+ex3k\mathbf{F} := (x - y) \mathbf{i} + (y^2 + z^2) \mathbf{j} + e^{x^{3}} \mathbf{k}.
(a) curl F=\text{curl } \mathbf{F} = _____(18)_____.
(b) If S1S_1 is the boundary surface of EE (including all faces) endowed with the outward orientation, one has ∬S1curl F⋅dS=\iint_{S_1} \text{curl } \mathbf{F} \cdot d\mathbf{S} = _____(19)_____.
(c) If S2S_2 is the surface obtained from S1S_1 by removing the face in the xyxy-plane while keeping the orientation from S1S_1 on all other faces, one then has ∬S2curl F⋅dS=\iint_{S_2} \text{curl } \mathbf{F} \cdot d\mathbf{S} = _____(20)_____.