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台灣聯合大學系統 104 年度 微積分 A3/A4/A5/A7

原始 PDF
壓縮檔內：微積分A3A4A5A7&951.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 d20dx20[sin⁡x2cos⁡x2]\frac{d^{20}}{dx^{20}}\left[\sin\frac{x}{2}\cos\frac{x}{2}\right]. Answer: _______
第 2 題8 分
If f(x)=ln⁡x+tan⁡−1xf(x) = \ln x + \tan^{-1} x, find (f−1)′(π4)(f^{-1})'(\frac{\pi}{4}). Answer: _______
第 3 題8 分
Evaluate ∫012e2x(2x+1)2dx\int_0^1 \frac{2e^{2x}}{(2x+1)^2} dx. Answer: _______
第 4 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 5 題8 分
Find the limit: lim⁡x→∞1xln⁡x∫1xln⁡t dt\lim\limits_{x \to \infty} \frac{1}{x \ln x} \int_1^x \ln t \, dt. Answer: _______
第 6 題8 分
Find the maximum rate of change of f(x,y)=x2y+exysin⁡yf(x,y) = x^2 y + e^{xy} \sin y at (1,0)(1,0). Answer: _______
第 7 題8 分
Find lim⁡(x,y)→(0,0)cos⁡(x2+y2)−1x2+y2\lim\limits_{(x,y) \to (0,0)} \frac{\cos(x^2 + y^2) - 1}{x^2 + y^2}. Answer: _______
第 8 題8 分
Evaluate ∫01∫01−xx+y(y−2x)2 dy dx\int_0^1 \int_0^{1-x} \sqrt{x + y(y - 2x)^2} \, dy \, dx by applying the transformation u=x+yu = x + y, v=y−2xv = y - 2x and integrating over a appropriate region in the uvuv-plane. 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
Sketch the region of integration, reverse the order of integration, and evaluate the integral ∫03∫x/31ey3 dy dx\int_0^3 \int_{\sqrt{x/3}}^1 e^{y^3} \, dy \, dx.
第 2 題12 分
Find the counterclockwise circulation for the field F=(x+exsin⁡y)i+(x+excos⁡y)j\mathbf{F} = (x + e^x \sin y)\mathbf{i} + (x + e^x \cos y)\mathbf{j} along the curve CC: The right-hand loop of the lemniscate r2=cos⁡2θr^2 = \cos 2\theta.
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.