PastExamLabPastExamLab

成功大學 113 年度 微積分B

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There are 10 questions worth 10 points each. Show all your works. Simplify and highlight your final answers. Answers without work shown will NOT receive credits.
第 1 題10 分
Given function f(x)=(x2−x+1)100f(x) = (x^2 - x + 1)^{100}.
(a)5 分
Find the first derivative f′(0)f'(0).
(b)5 分
Find the second derivative f′′(0)f''(0).
第 2 題10 分
Evaluate the limit. lim⁡x→03+cos⁡3x−2x2\lim\limits_{x \to 0} \frac{\sqrt{3 + \cos 3x} - 2}{x^2}
第 3 題10 分
(a)5 分
Given function f(x)=2x2f(x) = 2^{x^2}. Find the derivative f′(2)f'(2).
(b)5 分
Given function g(x)=x2xg(x) = x^{2x}. Find the derivative g′(2)g'(2).
第 4 題10 分
Evaluate the definite integral. ∫23xx2−5x+4dx\int_2^3 \frac{x}{x^2 - 5x + 4} dx
第 5 題10 分
Evaluate the improper integral. ∫3∞e−3xdx\int_3^{\infty} e^{-\sqrt{3x}} dx
第 6 題10 分
Given the Taylor series of the function as below. Find the values of c1,c2,c3c_1, c_2, c_3.
32+x=2+c1x+c2x2+c3x3+⋯\sqrt{32 + x} = 2 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots
第 7 題10 分
(a)5 分
Given function f(x)=sin⁡(3x)f(x) = \sin(3x). Find the higher derivative f(111)(0)f^{(111)}(0).
(b)5 分
Given function g(x)=sin⁡(x3)g(x) = \sin(x^3). Find the higher derivative g(111)(0)g^{(111)}(0).
第 8 題10 分
Given function f(x,y)=(4x2+y2)e−2xf(x,y) = (4x^2 + y^2)e^{-2x}. Find all critical points and determine their types (local maximum/local minimum/saddle point).
第 9 題10 分
The Cobb-Douglas production function is P(x,y)=x1/3y2/3P(x,y) = x^{1/3}y^{2/3} (xx: capital, yy: labour) subject to budget constraint 3x1/2+5y1/2=453x^{1/2} + 5y^{1/2} = 45. Use the method of Lagrange multiplier to find the values of x,yx, y such that PP is maximized.
第 10 題10 分
Evaluate the double integral. ∫04∫x2sin⁡(y4)y2dydx\int_0^4 \int_{\sqrt{x}}^2 \frac{\sin(y^4)}{y^2} dy dx
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