PastExamLabPastExamLab

台灣大學 · 物理學系 · 轉學考考古題 · 民國112年（2023年）

112 年度 微積分(B)

台灣大學 · 物理學系 · 轉學考

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PART I. Fill in the blanks.

Each blank is worth 7 points. Only the final clearly labeled answer will be graded.
第 1 題7 分
Evaluate lim⁡x→0(x+4−2tan⁡−1(πx))=\lim\limits_{x \to 0} \left(\frac{\sqrt{x + 4} - 2}{\tan^{-1}(\pi x)}\right) = _____(1)_____.
第 2 題7 分
The curve described by x−xy+y=3x - \sqrt{xy} + y = 3 passes through the point (4,1)(4, 1). Find an equation for the tangent line to the curve at (4,1)(4, 1). _____(2)_____.
第 3 題7 分
The absolute maximum value of the function f(x)=x6e−2x2f(x) = x^6 e^{-2x^2} is _____(3)_____.
第 4 題7 分
The graph of the function g(x)=ln⁡(x3)xg(x) = \frac{\ln(x^3)}{\sqrt{x}} has an inflection point at x=x = _____(4)_____.
第 5 題7 分
Solve for the function ff that satisfies \int_\sqrt{x} ^4 \frac{f(t^2)}{\ln t} dt = e^{(x-16)^2} - \frac{x}{16}, x>1x > 1. f(x)=f(x) = _____(5)_____.
第 6 題7 分
The volume generated by rotating the region under the curve y=1x(x+1)y = \frac{1}{\sqrt{x(\sqrt{x} + 1)}} from x=1x = 1 to x=4x = 4 about the xx-axis is _____(6)_____.
第 7 題7 分
Determine if the improper integral ∫2∞dxx(x2−1)3/2\int_2^\infty \frac{dx}{x(x^2 - 1)^{3/2}} is convergent or divergent. Evaluate the improper integral if it is convergent. _____(7)_____.
第 8 題7 分
Find the point on the surface given by x+8y+27z=14\sqrt{x} + \sqrt{8y} + \sqrt{27z} = 14 that is closest to the origin. _____(8)_____.
第 9 題7 分
Evaluate ∫02∫04−x2∫2−4−x2−y22+4−x2−y2x2+y2+z2 dz dy dx=\int_0^2 \int_0^{\sqrt{4-x^2}} \int_{2-\sqrt{4-x^2-y^2}}^{2+\sqrt{4-x^2-y^2}} \sqrt{x^2 + y^2 + z^2} \, dz \, dy \, dx = _____(9)_____.
第 10 題7 分
Solve the initial value problem. xdydx=3x2−2yx \frac{dy}{dx} = 3x^2 - 2y, y(1)=2y(1) = 2. y=y = _____(10)_____.

PART II. Show ALL your work and justify your answer.

Each problem is worth 15 points. Only the clearly laid out steps of solving the problem will be graded.
第 11 題15 分
Evaluate the integral by making an appropriate change of variables. ∬Rcos⁡[(y−xy+x)2]dA\iint_R \cos\left[\left(\frac{y - x}{y + x}\right)^2\right] dA where RR is the trapezoidal region with vertices (2,0)(2, 0), (3,0)(3, 0), (0,3)(0, 3), and (0,2)(0, 2).
第 12 題15 分
Sketch the graph of the function f(x)=x2/3(12−x)1/3f(x) = x^{2/3}(12 - x)^{1/3}. Label the following objectives on your graph: (a) Asymptotes (b) Local extrema (points and values) (c) Intervals of increase/decrease (d) Concave up/down intervals.
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