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成功大學 113 年度 微積分C

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Show all your work and justify all your answers.
第 1 題10 分
(a)5 分
Evaluate lim⁡x→2∣6x−17∣−∣6x−7∣3x−6\lim\limits_{x \to 2} \frac{|6x - 17| - |6x - 7|}{3x - 6}.
(b)5 分
Evaluate lim⁡x→0[cos⁡(2x)]−x2\lim\limits_{x \to 0} [\cos(2x)]^{-x^2}.
第 2 題10 分
Let CC be the curve defined by the parametric equations x=t3+1x = t^3+1, y=t4+ty = t^4+t. Find the slope of the tangent line to CC at the point (0,0)(0, 0).
第 3 題10 分
If y5+5xy+1=0y^5 + 5xy + 1 = 0, find the value of d2ydx2\frac{d^2y}{dx^2} at the point (0,−1)(0, -1).
第 4 題10 分
Find the absolute maximum value of the function f(x)=x8e1−x2f(x) = x^8 e^{1-x^2}.
第 5 題10 分
Evaluate ∫036x−x2dx\int_0^3 \sqrt{6x - x^2} dx.
第 6 題10 分
Let RR be the region bounded by the curve y=x2−x7y = x^2 - x^7 and the xx-axis. Find the volume of the solid obtained by rotating RR about the line x=3x = 3.
第 7 題10 分
Find the interval of convergence of the series ∑n=1∞n(x−3)n2n(n2+1)\sum\limits_{n=1}^{\infty} \frac{n(x-3)^n}{2^n(n^2+1)}.
第 8 題10 分
Let f(x,y,z)=x2+y2−4xy+z+1f(x, y, z) = x^2 + y^2 - 4xy + z + 1. Find all possible numbers a,b,c∈Ra, b, c \in \mathbb{R} such that the direction in which ff increases most rapidly at the point (a,b,c)(a, b, c) is in the direction of i+j+2k\mathbf{i} + \mathbf{j} + 2\mathbf{k}.
第 9 題10 分
Evaluate ∬Dcos⁡(x+y)dA\iint_D \cos(x + y) dA, where D={(x,y)∈R2∣∣x∣+∣y∣≤π6}D = \left\{(x, y) \in \mathbb{R}^2 \mid |x| + |y| \leq \frac{\pi}{6}\right\}.
第 10 題10 分
Let F\mathbf{F} be the vector field defined by F(x,y,z)=(9x2z)i+(8sin⁡(x3)+e2z)j+(x3yln⁡(x2+1))k\mathbf{F}(x, y, z) = (9x^2 z)\mathbf{i} + (8\sin(x^3) + e^{2z})\mathbf{j} + (x^3 y \ln(x^2 + 1))\mathbf{k}, and let SS be the surface of the solid bounded by the planes x+3z=6x + 3z = 6, y=3y = 3, x=0x = 0, y=0y = 0, and z=0z = 0. Suppose SS is given with positive (outward) orientation. Evaluate the flux of F\mathbf{F} across SS.
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