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台灣聯合大學系統 104 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、填充題：共 8 題，每題 8 分，共 64 分。請將答案依題號順序寫在答案卷上。

第 1 題8 分
Find the interval of convergence of the power series ∑n=1∞2n(x−3)n(n+1)!\sum\limits_{n=1}^{\infty} \frac{2n(x-3)^n}{(n+1)!}. Answer: _______
第 2 題8 分
Evaluate ∫01xe2x(2x+1)2dx\int_0^1 \frac{xe^{2x}}{(2x+1)^2} dx. Answer: _______
第 3 題8 分
If lim⁡x→2f(x)x2=2\lim\limits_{x \to 2} \frac{f(x)}{x^2} = 2, find the limit: lim⁡x→2(f(x)+f(x)x)\lim\limits_{x \to 2} \left(f(x) + \frac{f(x)}{x}\right). Answer: _______
第 4 題8 分
Find d20dx20(sin⁡x2)(cos⁡x2)\frac{d^{20}}{dx^{20}}\left(\sin \frac{x}{2}\right)\left(\cos \frac{x}{2}\right). Answer: _______
第 5 題8 分
Find the limit: lim⁡x→∞1xln⁡x∫1xln⁡tdt\lim\limits_{x \to \infty} \frac{1}{x \ln x} \int_1^x \ln t dt. Answer: _______
第 6 題8 分
Find an equation of the tangent line to the graph of the function x2y3−y2+xy−1=0x^2y^3 - y^2 + xy - 1 = 0 at (1,1)(1,1). Answer: _______
第 7 題8 分
What value of aa makes f(x)=x2+(a/x)f(x) = x^2 + (a/x) have a point of inflection at x=1x = 1? Answer: _______
第 8 題8 分
We say that the two commodities are substitute commodities if a decrease in the demand for one results in an increase in the demand for the other. Conversely, two commodities are referred to as complementary commodities if a decrease in the demand for one results in a decrease in the demand for the other as well. Suppose that the demand functions of two commodities are x=f(p,q)=3p1+px = f(p,q) = \frac{3p}{1+\sqrt{p}}, and y=g(p,q)=11+q4y = g(p,q) = \frac{1}{1+q^4}. Determine whether these two commodities are substitute, complementary, or neither. 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
Test the series ∑n=0∞e2/nn2\sum\limits_{n=0}^{\infty} \frac{e^{2/n}}{n^2} for convergence or divergence.
第 2 題12 分
Evaluate ∬Rxex2dA\iint_R xe^{x^2} dA where RR is the plane region bounded by the yy-axis, x=0x = 0, the horizontal line y=4y = 4, and the graph of y=x2y = x^2.
第 3 題12 分
A manufacturer finds that it takes xx units of labor and yy units of capital to produce f(x,y)=100x3/4y1/4f(x,y) = 100x^{3/4}y^{1/4} units of product. If a unit of labor costs 100,aunitofcapitalcosts100, a unit of capital costs 200, and $200,000 is budgeted for production, determine how many units should be expended on labor and how many units should be expended on capital in order to maximize production.