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台灣大學 · 生命科學系 · 轉學考考古題 · 民國111年（2022年）

111 年度 微積分(C)

台灣大學 · 生命科學系 · 轉學考

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第 1 題10 分
Evaluate the limits.
(•)
lim⁡x→0ln⁡(1−3x2)e−x+x−cos⁡x=\lim\limits_{x \to 0} \frac{\ln(1-3x^2)}{e^{-x} + x - \cos x} = __(1)__.
(•)
lim⁡x→∞ln⁡(1−9x2)e−x+x−cos⁡x=\lim\limits_{x \to \infty} \frac{\ln(1-9x^2)}{e^{-x} + x - \cos x} = __(2)__.
第 2 題10 分
Consider the graph of the function f(x)=x2+2x3−x4−x3x2−5xf(x) = \frac{\sqrt{x^2 + 2x^3} - \sqrt{x^4 - x^3}}{\sqrt{x^2 - 5x}}. Find all vertical asymptotes. __(3)__. (Hint: find the domain) Find all horizontal asymptotes. __(4)__.
第 3 題10 分
Consider the curve given by the equation x3+y=9xyx^3 + y = 9x\sqrt{y}. Find an equation of the tangent line at the point (4,8)(4,8). __(5)__. Find d2ydx2\frac{d^2y}{dx^2} at the point (4,8)(4,8). __(6)__.
第 4 題10 分
Let ff be a smooth function and F(x)=∫xxtf(t)etdtF(x) = \int_{\sqrt{x}}^{\sqrt{x}} \frac{tf(t)}{e^t} dt. Find F′(x)F'(x). __(7)__. (Your answer would contain ff) Suppose that F(x)=f(2x)F(x) = f(\sqrt{2x}). Solve the integral equation for ff. __(8)__.
第 5 題10 分
Let RR be the region under y=xy = \sqrt{x}, above y=ln⁡xy = \ln x, and between x=1x = 1 and x=2x = 2. Find the volume of the solid obtained by rotating RR about the xx-axis. __(9)__. Find the volume of the solid obtained by rotating RR about the line x=4x = 4. __(10)__.
第 6 題10 分
Evaluate ∫(1x+tan⁡−1x−x2)dx=\int \left(\frac{1}{x} + \tan^{-1} x - \frac{x}{2}\right) dx = __(11)__. Determine if the improper integral ∫1∞(1x+tan⁡−1x−x2)dx\int_1^{\infty} \left(\frac{1}{x} + \tan^{-1} x - \frac{x}{2}\right) dx is convergent or divergent. Evaluate the improper integral if it is convergent. __(12)__.
第 7 題10 分
Evaluate the given double integrals.
(•)
∫01∫03x−2y2ycos⁡(x2−9y2)dxdy=\int_0^1 \int_0^{\sqrt{3x-2y^2}} y\cos(x^2 - 9y^2) dx dy = __(13)__.
(•)
∫01∫03x−2y2sin⁡(x2+2y2)dxdy=\int_0^1 \int_0^{\sqrt{3x-2y^2}} \sin(x^2 + 2y^2) dx dy = __(14)__.
第 8 題10 分
Let f(x)=ex+e−x2f(x) = e^x + e^{-x^2}. Find the Taylor Series of f(x)f(x) at x=0x = 0. __(15)__. Use the Taylor Series to find the value of f(2022)(0)f^{(2022)}(0). __(16)__.
第 9 題10 分
Sketch the curve y=(x−4)x4y = (x - 4)\sqrt[4]{x}. Label the following information: (a) Intervals of Increase/Decrease (b) Concavity (c) Local Extrema.
第 10 題10 分
Use the method of Lagrange Multipliers to find the point(s) on the surface y2=16+xzy^2 = 16 + xz that are closest to the origin.
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