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臺灣綜合大學系統 108 年度 微積分C

本頁整理這份微積分考卷的題目、解答與詳解步驟，可直接對照每題內容與答案重點。

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這份試題整理

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PastExamLab Summary
題數
10
題
總分
100
分
已整理詳解
0%
0 / 10 題

微積分C 其他年度考古題

本科目共 9 個年度已整理題目，可直接切換查閱。

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第 1 題10 分
Find the following limits:
(a)5 分
lim⁡x→02x−14x−1\lim\limits_{x \to 0} \frac{2^x - 1}{4^x - 1}
(b)5 分
lim⁡x→0sin⁡−1(1−x1−x2)\lim\limits_{x \to 0} \sin^{-1}\left(\frac{1-x}{1-x^2}\right), where sin⁡−1\sin^{-1} is the inverse function of sine.
第 2 題10 分
Define f(x)=tan⁡2(x)f(x) = \tan^2(x) for x∈(0,π2)x \in (0, \frac{\pi}{2}) and let f−1f^{-1} be its inverse function. Find (f−1)′(3)(f^{-1})'(3).
第 3 題10 分
Compute the following integrals
(a)5 分
∫0π/2sin⁡2x2+cos⁡xdx\int_0^{\pi/2} \frac{\sin 2x}{2 + \cos x} dx
(b)5 分
∫12(ln⁡x)2x3dx\int_1^2 \frac{(\ln x)^2}{x^3} dx
第 4 題10 分
Find the arc length of the curve with equation x2/3+y2/3=1x^{2/3} + y^{2/3} = 1 within the region {(x,y)∣x≥0 and y≥0}\{(x,y) | x \geq 0 \text{ and } y \geq 0\}.
第 5 題10 分
Find the slope of the tangent line to the polar curve r=1+sin⁡(2θ)r = 1 + \sin(2\theta) at the point specified by θ=π/3\theta = \pi/3.
第 6 題10 分
Find the radius of convergence of the series ∑n=1∞n(x+3)n4n+1\sum\limits_{n=1}^{\infty} \frac{n(x+3)^n}{4^{n+1}}.
第 7 題10 分
Find the Maclaurin series of order 4 for the function f(x)=e−x2cos⁡xf(x) = e^{-x^2} \cos x, i.e., approximate f(x)f(x) by a polynomial a0+a1x+a2x2+a3x3+a4x4a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4.
第 8 題10 分
Find the maximum value of the function f(x,y,z)=x+2y+3zf(x,y,z) = x + 2y + 3z on the curve of intersection of the plane x−y+z=1x - y + z = 1 and the cylinder x2+y2=1x^2 + y^2 = 1.
第 9 題10 分
Let F(x,y)=−2yx2+y2i+2xx2+y2j\mathbf{F}(x,y) = \frac{-2y}{x^2 + y^2}\mathbf{i} + \frac{2x}{x^2 + y^2}\mathbf{j} and let D={(x,y)∣x2+y2=9}D = \{(x,y) | x^2 + y^2 = 9\}. Find
∫∂DF⋅Tds\int_{\partial D} \mathbf{F} \cdot T ds
, where we traverse the boundary ∂D\partial D in the counterclockwise direction and T\mathbf{T} is the unit tangent vector.
第 10 題10 分
Let F(x,y,z)=xi+yj+zk(x2+y2+z2)3/2\mathbf{F}(x,y,z) = \frac{x\mathbf{i} + y\mathbf{j} + z\mathbf{k}}{(x^2 + y^2 + z^2)^{3/2}} and let D={(x,y,z)∣x24+y24+z29=1}D = \{(x,y,z) | \frac{x^2}{4} + \frac{y^2}{4} + \frac{z^2}{9} = 1\}. Find the flux of F\mathbf{F} outward across the boundary of DD.