PastExamLabPastExamLab

臺灣綜合大學系統 113 年度 微積分B

原始 PDF
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 分鏈鎖律・高階導數
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 分極限・Taylor 展開
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 分指數函數微分
(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 分定積分・部分分式
Evaluate the definite integral. ∫23xx2−5x+4dx\displaystyle\int_2^3 \frac{x}{x^2 - 5x + 4} dx
第 5 題10 分瑕積分・換元法
Evaluate the improper integral. ∫3∞e−3xdx\displaystyle\int_3^{\infty} e^{-\sqrt{3x}} dx
第 6 題10 分Taylor 級數係數・二項式展開
Given the Taylor series of the function as below. Find the values of c1,c2,c3c_1, c_2, c_3.
32+x5=2+c1x+c2x2+c3x3+⋯\sqrt[5]{32 + x} = 2 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots
第 7 題10 分高階導數・Taylor 級數對應
(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 分多變數函數・臨界點分類
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Lagrange 乘子法・最佳化
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 分二重積分・交換積分順序
Evaluate the double integral. ∫04∫x2x⋅sin⁡(y2)y2dydx\displaystyle\int_0^4 \int_{\sqrt{x}}^2 \sqrt{x}\cdot\frac{\sin(y^2)}{y^2} dy dx