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台灣聯合大學系統 105 年度 微積分 A3/A4/A6

原始 PDF
壓縮檔內：微積分(A3 A4 A6).pdf
中央大學圖書館考古題頁
第 1 題10 分
Let
I=∫Cyx2+y2dx−xx2+y2dyI = \int_C \frac{y}{x^2+y^2} dx - \frac{x}{x^2+y^2} dy
where CC is a circle oriented counterclockwise.
(a)
Evaluate II if CC is given by (x−2016)2+(y−2016)2=1(x-2016)^2 + (y-2016)^2 = 1.
(b)
Evaluate II if CC is given by x2+y2=1x^2 + y^2 = 1.
第 2 題10 分
Find the maximum and minimum values of the function f(x,y,z)=x2−y2f(x,y,z) = x^2 - y^2 on the surface x2+2y2+3z2=1x^2 + 2y^2 + 3z^2 = 1.
第 3 題10 分
Compute
lim⁡x→∞(x+x+x−x)\lim\limits_{x \to \infty} \left( \sqrt{x + \sqrt{x + \sqrt{x - \sqrt{x}}}} \right)
第 4 題10 分
For what positive xx does the following series converge?
∑n=1∞(xn−1)\sum\limits_{n=1}^{\infty} (\sqrt[n]{x} - 1)
第 5 題10 分
Let B={(x,y,z)∈R3:x2+y2+z2≤1}B = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \leq 1\}. Evaluate the integral
∭Bx4+2y4x4+4y4+z4dV.\iiint_B \frac{x^4 + 2y^4}{x^4 + 4y^4 + z^4} dV.
第 6 題10 分
The nn-th derivative of 1x2+6x−1\frac{1}{x^2+6x-1} has the form Pn(x)(x2+6x−1)n+1\frac{P_n(x)}{(x^2+6x-1)^{n+1}} where Pn(x)P_n(x) is a polynomial. Find Pn(1)P_n(1) for all n≥0n \geq 0.
第 7 題20 分
(a)
Prove that
∫0∞(sin⁡xx)2dx=∫0∞sin⁡xxdx.\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx = \int_0^\infty \frac{\sin x}{x} dx.
(b)
Evaluate the improper integral
∫0∞sin⁡xxdx.\int_0^\infty \frac{\sin x}{x} dx.
第 8 題10 分
For each continuous function f:[0,1]→Rf : [0,1] \to \mathbb{R}, let I(f)=∫01xf(x)(x−f(x))dxI(f) = \int_0^1 xf(x)(x - f(x)) dx. Find the maximum value of I(f)I(f) over all such functions ff.
第 9 題10 分
Evaluate
∫0∞(x−x32+x52⋅4−x72⋅4⋅6+⋯ )(1+x222+x422⋅42+x622⋅42⋅62+⋯ )dx.\int_0^\infty \left( x - \frac{x^3}{2} + \frac{x^5}{2 \cdot 4} - \frac{x^7}{2 \cdot 4 \cdot 6} + \cdots \right) \left( 1 + \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 4^2} + \frac{x^6}{2^2 \cdot 4^2 \cdot 6^2} + \cdots \right) dx.