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台灣聯合大學系統 111 年度 微積分 A2

原始 PDF
壓縮檔內：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做序作答。請注意：本（甲）部分，共 8 題，含題型態為簡答題，不必詳列計算過程，僅答答案被包含在演算過程，將被視為試算流程，不另行批出計分。
第 1 題8 分
Find the slope of the tangent line to the graph of f(x)=xln⁡xf(x) = x^{\ln x} at the point (e,e)(e, e).
第 2 題8 分
Find the limit lim⁡x→0sin⁡x−xcos⁡xtan⁡3x\lim\limits_{x \to 0} \frac{\sin x - x \cos x}{\tan^3 x}.
第 3 題8 分
Find the volume of the solid bounded above by the surface z=f(x,y)=ex+2yz = f(x,y) = e^{x+2y} and below by the plane region RR, where RR is the triangle with vertices (0,0)(0,0), (1,0)(1,0), and (0,1)(0,1).
第 4 題8 分
Find the maximum value of f(x)=x12(1−x)3f(x) = x^{\frac{1}{2}}(1-x)^3 on the closed interval [0,1][0,1].
第 5 題8 分
Find the limit lim⁡n→∞∑k=1n1nln⁡(1+kn)\lim\limits_{n \to \infty} \sum\limits_{k=1}^{n} \frac{1}{n} \ln\left(1 + \frac{k}{n}\right).
第 6 題8 分
Evaluate the integral ∫0ln⁡2x−2e−1/xdx\int_{0}^{\ln 2} x^{-2} e^{-1/x} dx.
第 7 題8 分
The production for a certain country in the early years following World War II is described by the function f(x,y)=30x2/3y1/3f(x,y) = 30x^{2/3}y^{1/3} units, when xx units of labor and yy units of capital were utilized. Find the approximate change in output if the amount expended on labor had been decreased from 125 units to 123 units and the amount expended on capital had been increased from 27 to 29 units.
第 8 題8 分
Find the radius of convergence of the power series ∑n=1∞(−1)n+1(x+2)nn2n\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}(x+2)^n}{n2^n}.

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
Determine if the series converges or diverges.
(a)6 分
∑n=0∞(ln⁡(4en−1)−ln⁡(2en+1))\sum\limits_{n=0}^{\infty} (\ln(4e^n - 1) - \ln(2e^n + 1)).
(b)6 分
∑n=2∞1n(ln⁡n)3/2\sum\limits_{n=2}^{\infty} \frac{1}{n(\ln n)^{3/2}}.
第 2 題12 分
Find the critical point(s) of the function f(x,y)=ex2−y2f(x,y) = e^{x^2-y^2}. Then use the second derivative test to classify the nature of the point.
第 3 題12 分
Sketch the region of integration and evaluate the integral ∫01∫y12πsin⁡(πx2)x2dxdy\int_{0}^{1} \int_{\sqrt{y}}^{1} \frac{2\pi \sin(\pi x^2)}{x^2} dx dy.