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台灣大學 106 年度 微積分(B)

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Integrals 3 題Infinite Series 3 題Derivatives 2 題Parametric equations and polar coordinates 2 題

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PART I: Multiple Choice (A), (B), (C), or (D)

You do not need to justify your answer. (40%; 4% each.)
第 1 題4 分Derivatives
(A) 232=82=642^{3^2} = 8^2 = 64.
(B) tan⁡−1(tan⁡π)=π\tan^{-1}(\tan \pi) = \pi.
(C) If f(x)=f−1(x)f(x) = f^{-1}(x) for all x∈Rx \in \mathbb{R}, then f(x)=xf(x) = x.
(D) (1+1n)n<3(1 + \frac{1}{n})^n < 3 for all n∈Nn \in \mathbb{N}.
第 2 題4 分Continuous functions
(A) If f(x)f(x) is continuous on (a,b)(a,b), then there is c∈(a,b)c \in (a,b) such that f(c)=max⁡(a,b)f(x)f(c) = \max_{(a,b)} f(x).
(B) Suppose that f(x)f(x) is continuous but not differentiable at x=0x = 0. The function g(x)=xf(x)g(x) = xf(x) must be differentiable at x=0x = 0.
(C) If f(x)f(x) is a differentiable function, then lim⁡x→af′(x)=f′(a)\lim\limits_{x \to a} f'(x) = f'(a).
(D) If f(x)f(x) is continuous on [a,b][a,b] and differentiable on (a,b)(a,b), then there is a unique c∈(a,b)c \in (a,b) such that f(b)−f(a)=f′(c)(b−a)f(b) - f(a) = f'(c)(b - a).
第 3 題4 分Applications of differentiation
Consider the function f(x)=1+e−x21−e−x2f(x) = \frac{1 + e^{-x^{2}}}{1 - e^{-x^{2}}}.
(A) f(x)f(x) does not have any horizontal asymptotes and any vertical asymptotes.
(B) f(x)f(x) has a horizontal asymptote, but f(x)f(x) does not have any vertical asymptotes.
(C) f(x)f(x) has a vertical asymptote, but f(x)f(x) does not have any horizontal asymptotes.
(D) f(x)f(x) has a horizontal asymptote and a vertical asymptote.
第 4 題4 分Integrals
(A) ∫−ππsin⁡mxsin⁡nx dx=0\int_{-\pi}^{\pi} \sin mx \sin nx \, dx = 0 for all m,n∈Nm, n \in \mathbb{N}.
(B) ∫−ππsin⁡mxcos⁡nx dx=0\int_{-\pi}^{\pi} \sin mx \cos nx \, dx = 0 for all m,n∈Nm, n \in \mathbb{N}.
(C) ∫−ππcos⁡mxcos⁡nx dx=0\int_{-\pi}^{\pi} \cos mx \cos nx \, dx = 0 for all m,n∈Nm, n \in \mathbb{N}.
(D) ∫−ππsin⁡mxcos⁡nx dx=0\int_{-\pi}^{\pi} \sin^m x \cos^n x \, dx = 0 for all m,n∈Nm, n \in \mathbb{N}.
第 5 題4 分Integrals
Consider the improper integral Ip=∫0∞1xpdxI_p = \int_0^\infty \frac{1}{x^p} dx, where pp is a positive number.
(A) IpI_p is convergent if p>1p > 1 and divergent if 0<p≤10 < p \leq 1.
(B) IpI_p is convergent if 0<p<10 < p < 1 and divergent if p≥1p \geq 1.
(C) IpI_p is convergent for all p>0p > 0.
(D) IpI_p is divergent for all p>0p > 0.
第 6 題4 分Parametric equations and polar coordinates
The enclosed area of the curve (x2+y2)2=x2−y2(x^2 + y^2)^2 = x^2 - y^2 can be written in the integral form as
(A) 2∫0π4cos⁡2θ dθ2 \int_0^{\frac{\pi}{4}} \cos 2\theta \, d\theta.
(B) 4∫0π4cos⁡2θ dθ4 \int_0^{\frac{\pi}{4}} \cos 2\theta \, d\theta.
(C) 2∫0π4cos⁡2θ dθ2 \int_0^{\frac{\pi}{4}} \sqrt{\cos 2\theta} \, d\theta.
(D) 12∫0π4cos⁡2θ dθ\frac{1}{2} \int_0^{\frac{\pi}{4}} \cos 2\theta \, d\theta.
第 7 題4 分Infinite Series
(A) If both ∑n=1∞an\sum_{n=1}^\infty a_n and ∑n=1∞bn\sum_{n=1}^\infty b_n are absolutely convergent, then ∑n=1∞anbn=(∑n=1∞an)⋅(∑n=1∞bn)\sum_{n=1}^\infty a_n b_n = \left(\sum_{n=1}^\infty a_n\right) \cdot \left(\sum_{n=1}^\infty b_n\right).
(B) If ∑n=1∞an\sum_{n=1}^\infty a_n is convergent and ∑n=1∞bn\sum_{n=1}^\infty b_n is divergent, then ∑n=1∞(an+bn)\sum_{n=1}^\infty (a_n + b_n) must be divergent.
(C) Suppose that f(x)f(x) is a positive and continuous function on [1,∞)[1, \infty), and the improper integral ∫1∞f(x)dx\int_1^\infty f(x) dx is convergent. Let an=f(n)a_n = f(n), then ∑n=1∞an\sum_{n=1}^\infty a_n is convergent.
(D) If an≤bna_n \leq b_n for all n∈Nn \in \mathbb{N} and ∑n=1∞bn\sum_{n=1}^\infty b_n is convergent, then ∑n=1∞an\sum_{n=1}^\infty a_n must be convergent.
第 8 題4 分Infinite Series
(A) ∑n=1∞(−1)n(1−cos⁡1n)\sum_{n=1}^\infty (-1)^n \left(1 - \cos \frac{1}{n}\right) is absolutely convergent.
(B) ∑n=1∞1n−ln⁡n\sum_{n=1}^\infty \frac{1}{n-\ln n} is convergent.
(C) ∑n=1∞1n1+1n\sum_{n=1}^\infty \frac{1}{n^{1+\frac{1}{n}}} is convergent.
(D) ∑n=2∞1nln⁡(n+1n−1)\sum_{n=2}^\infty \frac{1}{n} \ln \left(\frac{n+1}{n-1}\right) is divergent.
第 9 題4 分Derivatives of multi-variable functions
Consider the function f(x,y)=∣xy∣f(x,y) = \sqrt{|xy|}.
(A) f(x,y)f(x,y) is not continuous at (0,0)(0,0).
(B) f(x,y)f(x,y) is continuous at (0,0)(0,0), but partial derivatives fx(0,0)f_x(0,0) and fy(0,0)f_y(0,0) do not exist.
(C) Both partial derivatives fx(0,0)f_x(0,0) and fy(0,0)f_y(0,0) exist, but f(x,y)f(x,y) is not differentiable at (0,0)(0,0).
(D) f(x,y)f(x,y) is differentiable at (0,0)(0,0).
第 10 題4 分Multiple integrals
Consider the spherical coordinates system x=ρsin⁡ϕcos⁡θx = \rho \sin \phi \cos \theta, y=ρsin⁡ϕsin⁡θy = \rho \sin \phi \sin \theta, and z=ρcos⁡ϕz = \rho \cos \phi. The volume element dV=dx dy dzdV = dx \, dy \, dz can be changed as
(A) ρsin⁡ϕ dρ dθ dϕ\rho \sin \phi \, d\rho \, d\theta \, d\phi.
(B) ρcos⁡ϕ dρ dθ dϕ\rho \cos \phi \, d\rho \, d\theta \, d\phi.
(C) ρ2sin⁡ϕ dρ dθ dϕ\rho^2 \sin \phi \, d\rho \, d\theta \, d\phi.
(D) ρ2cos⁡ϕ dρ dθ dϕ\rho^2 \cos \phi \, d\rho \, d\theta \, d\phi.

PART II: Answer the following questions. (30%; 5% each.)

第 11 題15 分The Limit of a Function
Suppose that aa, bb and cc are constants and lim⁡x→−∞(ax2+bx+c+3x)=2\lim\limits_{x \to -\infty} \left(\sqrt{ax^2 + bx + c} + 3x\right) = 2, then (a,b)=(a,b) = ____.
第 12 題15 分Derivatives
Let f(x)=ex(ex−1)(ex−2)⋯(ex−106)f(x) = e^x(e^x - 1)(e^x - 2) \cdots (e^x - 106). Find f′(0)=f'(0) = ____.
第 13 題15 分Integrals
Suppose that f(t)f(t) is a continuous function on (1,∞)(1, \infty) satisfying ∫11+x2f(t)dt=ln⁡x\int_1^{1+x^2} f(t) dt = \ln x, then f(10)=f(10) = ____.
第 14 題15 分Techniques of integration
Evaluate the definite integral ∫121sin⁡−1xx2dx=\displaystyle\int_{\frac{1}{2}}^1 \frac{\sin^{-1} x}{x^2} dx = ____.
第 15 題15 分Parametric equations and polar coordinates
The length of the curve C:(x(t),y(t))=(t3+1,32t2−1)C : (x(t), y(t)) = \left(t^3 + 1, \frac{3}{2}t^2 - 1\right), 0≤t≤10 \leq t \leq 1 is ____.
第 16 題15 分Multiple integrals
Reverse the order of the iterated integral ∫0π∫sin⁡x1f(x,y)dydx\int_0^\pi \int_{\sin x}^1 f(x,y) dy dx : ____.

PART III: Solve the following problems. You need to write down complete arguments. (30%; 10% each.)

第 17 題10 分Applications of integration
Given a family of parabolas Pn:y=nx2+1nP_n : y = nx^2 + \frac{1}{n}, where n∈Nn \in \mathbb{N}. Let AnA_n be the area between PnP_n and Pn+1P_{n+1}. Find the limit lim⁡n→∞Ann3\lim\limits_{n \to \infty} \frac{A_n}{n^3}.
第 18 題10 分Infinite Series
Use the Taylor series method to find the limit lim⁡x→01−x4−e2x4(1−cos⁡x)sin⁡2x\lim\limits_{x \to 0} \frac{\sqrt{1 - x^4} - e^{2x^{4}}}{(1 - \cos x) \sin^2 x}.
第 19 題10 分Differential equations
Suppose that f(t)f(t) is a continuous function on [0,∞)[0, \infty) satisfying
f(t)=eπt2+∬x2+y2≤t2f(x2+y2)dA.f(t) = e^{\pi t^{2}} + \iint_{x^2+y^2 \leq t^2} f\left(\sqrt{x^2 + y^2}\right) dA.

Find f(t)f(t).