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成功大學 107 年度 微積分C

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第 1 題10 分
Differentiate the function. f(x)=x5+5x+log⁡2x+log⁡x7f(x) = x^5 + 5^x + \log_2 x + \log_x 7
第 2 題10 分
Find the maximum value of the function. f(x)=ln⁡xx3f(x) = \frac{\ln x}{\sqrt[3]{x}}
第 3 題10 分
Evaluate the limit. lim⁡x→0sin⁡(2tan⁡−1x)−tan⁡(sin⁡−12x)x3\lim\limits_{x \to 0} \frac{\sin(2\tan^{-1} x) - \tan(\sin^{-1} 2x)}{x^3}
第 4 題10 分
Evaluate the limit. lim⁡n→∞1n(n+1)(n+3)(n+5)⋯(3n−1)n\lim\limits_{n \to \infty} \frac{1}{n} \sqrt[n]{(n+1)(n+3)(n+5)\cdots(3n-1)}
第 5 題10 分
Evaluate the integral. ∫dx(x2+2)32dx\int \frac{dx}{(x^2 + 2)^{\frac{3}{2}}} dx Hint: sin⁡3θ=3sin⁡θ−4sin⁡3θ\sin 3\theta = 3\sin \theta - 4\sin^3 \theta, cos⁡3θ=4cos⁡3θ−3cos⁡θ\cos 3\theta = 4\cos^3 \theta - 3\cos \theta.
第 6 題10 分
Use ∑n=1∞1n2=π26\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} to evaluate the integral ∫01ln⁡(1+x)xdx\int_0^1 \frac{\ln(1+x)}{x} dx.
第 7 題10 分
Find the area of the region bounded by the polar curve r=32+sin⁡θr = \frac{3}{2 + \sin \theta}. Hint: sketch the curve first.
第 8 題10 分
Let F(x,y,z)=xy+z3+xyz−5F(x, y, z) = xy + z^3 + xyz - 5. Suppose a function f(x,y)f(x, y) is defined in a neighborhood of (2,1)(2, 1) such that F(x,y,f(x,y))=0F(x, y, f(x, y)) = 0. Estimate f(1.97,1.04)f(1.97, 1.04) by linear approximation.
第 9 題10 分
Evaluate the double integral ∬Rxyy4−x4dA\iint_R \frac{xy}{y^4 - x^4} dA, where RR is the region in the first quadrant bounded by x2+y2=4x^2 + y^2 = 4, x2+y2=9x^2 + y^2 = 9, y2−x2=1y^2 - x^2 = 1, y2−x2=4y^2 - x^2 = 4.
第 10 題10 分
Evaluate the line integral ∫C(x−y)dx+(x+y)dyx2+y2\int_C \frac{(x-y)dx + (x+y)dy}{x^2 + y^2}, where CC is the curve r(t)=(2−t,1+2t)\mathbf{r}(t) = (2-t, 1+2\sqrt{t}), 0≤t≤10 \leq t \leq 1.
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