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台灣聯合大學系統 109 年度 微積分 A2

原始 PDF
壓縮檔內：轉學考/微積分A2.pdf
中央大學圖書館考古題頁

甲、簡答題

共 8 題，每題 8 分，共 64 分。請在答案卷上列出題號依序作答。請注意：本（甲、）部分，共 8 題，命題型態為簡答題，不必詳列計算過程，做答時請直接算流程，將最後結果寫在答案流程，不予另行扣出計分。
第 1 題8 分
Find the value of lim⁡x→∞x+cos⁡xx−cos⁡x\lim\limits_{x \to \infty} \frac{x + \cos x}{x - \cos x}. Answer: ______
第 2 題8 分
Find all horizontal asymptotes of graph of the function f(x)=∣x∣xx2+1f(x) = \frac{|x|x}{x^2 + 1}. Answer: ______
第 3 題8 分
Find the smallest positive (x>0)(x > 0) inflection point of F(x)=∫0xcos⁡(t3/2)dtF(x) = \int_0^x \cos(t^{3/2}) dt. Answer: ______
第 4 題8 分
A building in the shape of a rectangular box is to have a volume of 12,000 ft³. It is estimated that the annual heating and cooling costs will be 2/squarefootforthetop,2/square foot for the top, 4/square foot for the front and back, and $3/square foot for the sides. What is the minimal annual heating and cooling cost? Answer: ______
第 5 題8 分
Find the values of pp for which the function f(x,y)={(xy)px4+y4,if (x,y)≠(0,0)0,if (x,y)=(0,0)f(x,y) = \begin{cases} \frac{(xy)^p}{x^4+y^4}, & \text{if } (x,y) \neq (0,0) \\ 0, & \text{if } (x,y) = (0,0) \end{cases} is discontinuous at (0,0)(0,0). Answer: ______
第 6 題8 分
Evaluate ∫0ln⁡5∫ex51ln⁡ydydx\int_0^{\ln 5} \int_{e^x}^5 \frac{1}{\ln y} dy dx. Answer: ______
第 7 題8 分
How many local extreme values does the function f(x,y)=10xye−(x2+y2)f(x,y) = 10xye^{-(x^2+y^2)} have? Answer: ______
第 8 題8 分
Let f(x,y)=kxye−(x2+y2)f(x,y) = kxye^{-(x^2+y^2)} be a joint probability density function on D={0<x<∞,0<y<∞}D = \{0 < x < \infty, 0 < y < \infty\}, then k=?k = ? Answer: ______

乙、計算、證明題

共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。
第 1 題12 分
Goods 1 and 2 are available at prices (in dollars) of p1p_1 per unit of good 1 and p2p_2 per unit of good 2. A utility function U(x1,x2)U(x_1, x_2) is a function representing the utility or benefit fo consuming xjx_j units of good jj. The marginal utility of the jjth good is ∂U/∂xj\partial U/\partial x_j, the rate of increase in utility per unit increase in the jjth good. Prove the following law of economics: Given a budget of LL dollars, utility is maximized at the consumption level (a,b)(a,b) where the ratio of marginal utility is equal to the ratio of prices:
Marginal utility of good 1Marginal utility of good 2=∂U/∂x1∂U/∂x2=p1p2\frac{\text{Marginal utility of good 1}}{\text{Marginal utility of good 2}} = \frac{\partial U/\partial x_1}{\partial U/\partial x_2} = \frac{p_1}{p_2}
第 2 題12 分
(a.)6 分
Determine whether the series ∑n=1∞(−1)nln⁡(1+1n)\sum\limits_{n=1}^{\infty} (-1)^n \ln\left(1 + \frac{1}{n}\right) diverges or converges conditionally or converges absolutely and give reasons for your answer. (6 points)
(b.)6 分
Show that if ∑n=1∞an\sum\limits_{n=1}^{\infty} a_n converges, then ∑n=1∞(3+sin⁡(an)5)n\sum\limits_{n=1}^{\infty} \left(\frac{3 + \sin(a_n)}{5}\right)^n converges. (6 points)
第 3 題12 分
A trough with a trapezoidal cross section is to be constructed with a 1-foot base and sides that are 20 feet long and 1 foot wide, as shown in the figure. Only the angle θ\theta can be varied. What value of θ\theta will maximize the trough's volume?
第 3 題圖表