PastExamLabPastExamLab

成功大學 114 年度 微積分B

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There are 10 questions, each worth 10 credit points. • Show all your work clearly to receive full credit. • Simplify and highlight your final answers for easy review. • Answers submitted without supporting work will NOT earn any credit.
第 1 題10 分
Let g(x)g(x) be the function
g(x)=e−x2.g(x) = e^{-x^2}.
Compute g′(1)g'(1) and g′′(1)g''(1).
第 2 題10 分
Find the extreme values of the function
f(x)=x2−3x3f(x) = \frac{x^2 - 3}{x^3}
on the interval [1,5][1, 5].
第 3 題10 分
Evaluate the following definite integral:
∫01xe2xdx.\int_0^1 xe^{2x} dx.
第 4 題10 分
Evaluate the following definite integral:
∫011x2+3dx.\int_0^1 \frac{1}{\sqrt{x^2 + 3}} dx.
第 5 題10 分
Let f(x)=2x−1f(x) = 2x^{-1}, and T=∑n=0∞an(x−2)nT = \sum\limits_{n=0}^{\infty} a_n(x - 2)^n the Taylor series of f(x)f(x). What is a100a_{100}?
第 6 題10 分
Suppose that xx and yy satisfy the equation
y3−x2=4.y^3 - x^2 = 4.
Find dy/dxdy/dx and d2y/dx2d^2y/dx^2 when (x,y)=(2,2)(x, y) = (2, 2).
第 7 題10 分
Find the length LL of the space curve given by
x(t)=3cos⁡t+1,y(t)=3sin⁡t+1andz(t)=4t+1x(t) = 3\cos\sqrt{t + 1}, \quad y(t) = 3\sin\sqrt{t + 1} \quad \text{and} \quad z(t) = 4\sqrt{t + 1}
for t=0t = 0 to 11.
第 8 題10 分
Let
f(x,y)=x+sin⁡(x+2y).f(x, y) = x + \sin(x + 2y).
Find the unit vector in the direction in which ff increases most rapidly at the point (0,0)(0, 0) and give the rate of change of ff in that direction.
第 9 題10 分
Solve the following optimization problem using Lagrange multipliers:
f(x,y)=x2subject tox+y=1.f(x, y) = x^2 \quad \text{subject to} \quad x + y = 1.
第 10 題10 分
Evaluate the double integral
∬R(x+y)dA,\iint_R (x + y) dA,
where RR is the region bounded by y=xy = x and y=x2y = x^2.
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