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台灣大學 · 地理環境資源學系 · 轉學考考古題 · 民國109年（2020年）

109 年度 微積分(C)

台灣大學 · 地理環境資源學系 · 轉學考

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第 1 題10 分
lim⁡x→∞x−1x=?\lim\limits_{x \to \infty} \frac{\sqrt{x} - 1}{x} = ?
第 2 題10 分
Find f′(2)f'(2) if f(x)=eg(x)f(x) = e^{g(x)} and g(x)=∫1x11+tdtg(x) = \int_1^x \frac{1}{1+t}dt.
第 3 題10 分
Trochoid x=2θ−sin⁡θx = 2\theta - \sin \theta, y=2−cos⁡θy = 2 - \cos \theta. Find the tangent line of the curve at θ=π2\theta = \frac{\pi}{2}.
第 4 題10 分
Trochoid x=2θ−sin⁡θx = 2\theta - \sin \theta, y=2−cos⁡θy = 2 - \cos \theta. Find the area under the curve and above the xx-axis for 0≤θ≤2π0 \leq \theta \leq 2\pi.
第 5 題10 分
Let the region RR be enclosed by the curves y=x2y = x^2 and y=2−x2y = 2 - x^2. Find the volume of the solid obtained by rotating the region RR about x=1x = 1.
第 6 題10 分
Let the region RR be enclosed by the curves y=x2y = x^2 and y=2−x2y = 2 - x^2. Find the arc length of the region RR.
第 7 題10 分
Evaluate ∬Rcos⁡(y−xy+x)dA\iint_R \cos\left(\frac{y-x}{y+x}\right)dA where RR is the trapezoidal region with vertices (1,0)(1,0), (2,0)(2,0), (0,2)(0,2) and (0,1)(0,1).
第 8 題10 分
Let f(x,y)=x4+y4−4xy+1f(x,y) = x^4 + y^4 - 4xy + 1. Find local maxima, local minima, and saddle points of f(x,y)f(x,y).
第 9 題10 分
Find the absolute maximum value and absolute minimum value of f(x,y)=x4+y4−4xy+1f(x,y) = x^4 + y^4 - 4xy + 1 on the disk x2+y2≤1x^2 + y^2 \leq 1.
第 10 題10 分
Solve the differential equation xy′=y+x2sin⁡xxy' = y + x^2 \sin x with y(π)=2πy(\pi) = 2\pi.
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