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台灣聯合大學系統 111 年度 微積分 A3/A4/A6

原始 PDF
壓縮檔內：微積分A3A4A6.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第一頁上。

請注意：本（甲）部分，共8題，命題型態為填充題，請依題號順序獨立出，勿同時併列出計算過程。倘若答案被包含在演算過程，將被視為試算流程，不予另行批出分。
第 1 題8 分
Evaluate lim⁡n→∞(1nn+1+1nn+2+⋯+1nn+n)\lim\limits_{n\to\infty} \left(\frac{1}{\sqrt{n\sqrt{n+1}}} + \frac{1}{\sqrt{n\sqrt{n+2}}} + \cdots + \frac{1}{\sqrt{n\sqrt{n+n}}}\right). Answer: _______
第 2 題8 分
Find the volume of the smaller region cut from the solid sphere ρ≤2\rho \leq 2 by the plane z=1z = 1. Answer: _______
第 3 題8 分
Evaluate ∫4x2−3x+24x2−4x+3dx\int \frac{4x^2 - 3x + 2}{4x^2 - 4x + 3} dx. Answer: _______
第 4 題8 分
Evaluate the iterated integral ∫01∫x1sin⁡(y2)dydx\int_0^1 \int_x^1 \sin(y^2) dy dx. Answer: _______
第 5 題8 分
Find the area of the portion of the plane y+2z=2y + 2z = 2 inside the cylinder x2+y2=4x^2 + y^2 = 4. Answer: _______
第 6 題8 分
Suppose that F(x)F(x) is an antiderivative of f(x)=cos⁡xxf(x) = \frac{\cos x}{x}, x>0x > 0. Express ∫13cos⁡3xxdx\int_1^3 \frac{\cos 3x}{x} dx in terms of FF. Answer: _______
第 7 題8 分
Along all triangles in the first quadrant formed by the xx-axis, the yy-axis, and tangent lines to the graph of y=3x−x2y = 3x - x^2, what is the smallest possible area? Answer: _______
第 8 題8 分
A space probe in the shape of the ellipsoid 4x2+y2+4z2=164x^2 + y^2 + 4z^2 = 16 enters Earth's atmosphere and its surface begins to heat. After 1 hour, the temperature at the point (x,y,z)(x, y, z) on the probe's surface is T(x,y,z)=8x2+4yz−16z+600T(x, y, z) = 8x^2 + 4yz - 16z + 600. Find the hottest point on the probe's surface. 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
Find the Taylor polynomials of orders 2 generated by f(x)={e−1/x2,if x≠00,if x=0f(x) = \begin{cases} e^{-1/x^2}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} at a=0a = 0.
第 2 題12 分
Let aa and bb be constants with 0<a<b0 < a < b. Does the sequence {(an+bn)1/n}\{(a^n + b^n)^{1/n}\} converge? If it does converge, what is the limit?
第 3 題12 分
Find the limit of ff or show that the limit does not exist.
(a.)6 分
lim⁡(x,y)→(1,0)xey−1xey−1+y\lim\limits_{(x,y)\to(1,0)} \frac{xe^y - 1}{xe^y - 1 + y}
(b.)6 分
lim⁡(x,y)→(0,0)sin⁡(x−y)∣x∣+∣y∣\lim\limits_{(x,y)\to(0,0)} \frac{\sin(x - y)}{|x| + |y|}