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台灣聯合大學系統 107 年度 微積分 A3/A4/A7

原始 PDF
壓縮檔內：轉學考/微積分A3A4A7.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，命題型態為填答題，必須以填充題形式將答案寫在答案卷第一頁，倘若答案被包含在演算過程中，將被視為試算草稿，無法採計計分。
第 1 題8 分
Find the critical number of y=1−4π2(tan⁡−1x)2y = 1 - \frac{4}{\pi^2}(\tan^{-1} x)^2. Answer :_______
第 2 題8 分
Determine the limits of integration where a≤ba \leq b such that ∫ab(x2−16)dx\int_a^b (x^2 - 16) dx has minimal value. Answer :_______
第 3 題8 分
Evaluate ∫−∞∞ex1+e2xdx\int_{-\infty}^{\infty} \frac{e^x}{1 + e^{2x}} dx. Answer :_______
第 4 題8 分
Find the slope of the surface f(x,y)=(x3+y3)1/3f(x, y) = (x^3 + y^3)^{1/3} at the point (0,0)(0, 0) in the yy-direction. Answer :_______
第 5 題8 分
Find the surface area of the portion of the plane z=4−2x−2yz = 4 - 2x - 2y that lies above the circle x2+y2≤1x^2 + y^2 \leq 1 in the first quadrant. Answer :_______
第 6 題8 分
Find an equation of the tangent plane to the paraboloid r(u,v)=ui+vj+(u2+v2)k\mathbf{r}(u, v) = u \mathbf{i} + v \mathbf{j} + (u^2 + v^2) \mathbf{k} at the point (1,2,5)(1, 2, 5). Answer :_______
第 7 題8 分
Evaluate the integral ∫0∞∫0∞1(1+x2+y2)2dxdy\int_0^{\infty} \int_0^{\infty} \frac{1}{(1 + x^2 + y^2)^2} dx dy. Answer :_______
第 8 題8 分
Use a change of variables to find the volume of the solid region lying below the surface z=(x+4y)(x−y)z = \sqrt{(x + 4y)(x - y)} and above the plane region RR : region bounded by the parallelogram with vertices (0,0)(0, 0), (1,1)(1, 1), (5,0)(5, 0) and (4,−1)(4, -1). Answer :_______

乙、計算、證明題：共 3 大題，每大題 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
A heat-seeking particle is located at the point (−1,2)(-1, 2) on a metal plate whose temperature at (x,y)(x, y) is T(x,y)=64−2x2−y2T(x, y) = 64 - 2x^2 - y^2.
(a)6 分
In what direction from (−1,2)(-1, 2) does the temperature increase most rapidly? What is this rate of increase?
(b)6 分
Find the path of the particle as it continuously moves in the direction of maximum temperature increase.
第 2 題12 分
Determine if the given series converges or diverges. Explain your reasoning.
(a.)6 分
∑n=1∞(3n+2n+3)n\sum\limits_{n=1}^{\infty} \left(\frac{3n + 2}{n + 3}\right)^n
(b.)6 分
∑n=1∞e2/nn2\sum\limits_{n=1}^{\infty} \frac{e^{2/n}}{n^2}
第 3 題12 分
Find the maximum value of ∫Cy3dx+(27x−x3)dy\int_C y^3 dx + (27x - x^3) dy, where CC is any circle in the xyxy-plane, oriented counterclockwise.