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台灣聯合大學系統 114 年度 微積分 A3/A4/A6

原始 PDF
壓縮檔內：轉學考/微積分A3A4A6.pdf
中央大學圖書館考古題頁

甲、填充題

共 10 題，每題 8 分，共 80 分。請在答案卷上列出題號依序作答。請注意：本（甲）部分，共 10 題，命題型態為填充題，請依題號順序獨立列出，勿同時陳列出計算過程。倘若答案被包含在演算過程，將被視為演算流程，不另行扣出計分。
第 1 題8 分
Evaluate the limit lim⁡x→01+tan⁡3x−1+sin⁡3xx3\displaystyle\lim\limits_{x \to 0} \frac{\sqrt{1 + \tan 3x} - \sqrt{1 + \sin 3x}}{x^3}.
第 2 題8 分
If f(x)=∫0g(x)11+t3dtf(x) = \int_0^{g(x)} \frac{1}{\sqrt{1 + t^3}} dt, where g(x)=∫0cos⁡2x[1+sin⁡(t2)]dtg(x) = \int_0^{\cos 2x} [1 + \sin(t^2)] dt, find f′(π/4)f'(\pi/4).
第 3 題8 分
Find the volume of the solid obtained by rotating about the xx-axis the region enclosed by the curves y=4x2+4y = \frac{4}{x^2 + 4}, y=0y = 0, x=0x = 0, and x=2x = 2.
第 4 題8 分
Find the absolute maximum value of f(x,y)=e−x2−y2(x2+2y2)f(x,y) = e^{-x^2-y^2}(x^2 + 2y^2) on the set DD, where DD is the disk x2+y2≤4x^2 + y^2 \leq 4.
第 5 題8 分
Suppose z=f(x,y)z = f(x,y), where x=g(s,t)x = g(s,t), y=h(s,t)y = h(s,t), g(1,2)=3g(1,2) = 3, gs(1,2)=−1g_s(1,2) = -1, gt(1,2)=4g_t(1,2) = 4, h(1,2)=6h(1,2) = 6, hs(1,2)=−5h_s(1,2) = -5, ht(1,2)=10h_t(1,2) = 10, fx(3,6)=7f_x(3,6) = 7, and fy(3,6)=8f_y(3,6) = 8. Find ∂z∂s\frac{\partial z}{\partial s} when s=1s = 1 and t=2t = 2.
第 6 題8 分
Find the directional derivative of the function f(x,y)=xx2+y2f(x,y) = \frac{x}{x^2 + y^2} at the point P(1,2)P(1,2) in the direction of v=(3,4)\mathbf{v} = (3,4).
第 7 題8 分
Evaluate the integral ∬R(1+yx)dA\displaystyle\iint_R \left(1 + \frac{y}{x}\right) dA, where RR is the region enclosed by the lines x+y=1x + y = 1, x+y=3x + y = 3, y=2xy = 2x, and y=x/2y = x/2.
第 8 題8 分
Find the average value of the function f(x)=∫x2cos⁡(t2)dtf(x) = \int_x^2 \cos(t^2) dt on the interval [0,2][0,2].
第 9 題8 分
Evaluate the integral ∫−22∫04−y2∫−4−x2−y24−x2−y2y2x2+y2+z2 dzdxdy\displaystyle\int_{-2}^2 \int_0^{\sqrt{4-y^2}} \int_{-\sqrt{4-x^2-y^2}}^{\sqrt{4-x^2-y^2}} y^2\sqrt{x^2 + y^2 + z^2} \, dz dx dy.
第 10 題8 分
Evaluate ∫CF⋅dr\int_C \mathbf{F} \cdot d\mathbf{r}, where F(x,y)=(x2+1,tan⁡−1x)\mathbf{F}(x,y) = (\sqrt{x^2 + 1}, \tan^{-1} x) and CC is the triangle from (0,0)(0,0) to (1,1)(1,1) to (0,1)(0,1) to (0,0)(0,0).

乙、計算、證明題

共 2 題，每題 10 分，共 20 分。須詳細寫出計算及證明過程，否則不予計分。
第 1 題10 分
(a)5 分
Use the integral test to determine whether the series ∑n=2∞1nln⁡n\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{n\sqrt{\ln n}} is convergent or divergent.
(b)5 分
Find the interval of convergence of the series ∑n=1∞(x+2)nn4n\displaystyle\sum\limits_{n=1}^{\infty} \frac{(x+2)^n}{n4^n}.
第 2 題10 分
(a)5 分
Evaluate the integral ∫1∞1x+x3dx\int_1^{\infty} \frac{1}{x + x^3} dx.
(b)5 分
Evaluate the integral ∫1∞tan⁡−1xx2dx\int_1^{\infty} \frac{\tan^{-1} x}{x^2} dx.