PastExamLabPastExamLab

成功大學 114 年度 微積分A

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There are 10 questions worth 10 points each. Show all your works. Simplify and highlight your final answers. Answers without work shown will NOT receive credits.
第 1 題10 分
Given functions f(x)=log⁡a(x2)f(x) = \log_a(x^2) and g(x)=log⁡x(2x)g(x) = \log_x(2^x). Find f′(2)f'(2) and g′(2)g'(2).
第 2 題10 分
Evaluate the limit. lim⁡x→0sin⁡(2arcsin⁡(3x))−6xx3\lim\limits_{x \to 0} \frac{\sin(2 \arcsin(3x)) - 6x}{x^3}
第 3 題10 分
Let function g(x)g(x) be the inverse of f(x)=x1+2x2f(x) = x\sqrt{1 + 2x^2}. Find g′(6)g'(6).
第 4 題10 分
Evaluate the limit. lim⁡n→∞[2arctan⁡(2n)π]n\lim\limits_{n \to \infty} \left[\frac{2\arctan(2n)}{\pi}\right]^n
第 5 題10 分
Evaluate the definite integral. ∫0π/3(3+tan⁡θsec⁡θ)2dθ\int_0^{\pi/3} (3 + \tan \theta \sec \theta)^2 d\theta
第 6 題10 分
Given function f(x)=(1−x)3e−x4f(x) = (1 - x)^3 e^{-x^4}. Find the higher derivative f(2025)(0)f^{(2025)}(0).
第 7 題10 分
Given polar curve r=e2θr = e^{2\theta}. Find the slope of tangent line of curve at θ=π4\theta = \frac{\pi}{4}. Also find the arc length of curve for 0≤θ≤π20 \leq \theta \leq \frac{\pi}{2}.
第 8 題10 分
Let CC be the curve of intersection of surfaces xy+yz+zx=−14xy + yz + zx = -14 and x2+y2+z2=29x^2 + y^2 + z^2 = 29. The tangent line of curve CC at point (2,3,−4)(2, 3, -4) is given by x−2a=y−3b=z+4\frac{x - 2}{a} = \frac{y - 3}{b} = z + 4. Find the values of a,ba, b.
第 9 題10 分
Use the method of Lagrange multiplier to find the shortest and longest distance from the origin to curve 9x2+16xy+21y2=1259x^2 + 16xy + 21y^2 = 125.
第 10 題10 分
Let DD be the region in xyxy-plane bounded by x2=yx^2 = y, x2=3yx^2 = 3y, y2=xy^2 = x, y2=3xy^2 = 3x. Use the transformation u=x2yu = \frac{x^2}{y}, v=y2xv = \frac{y^2}{x} to evaluate the double integral ∬Dy2x4+3y2dA\iint_D \frac{y^2}{x^4 + 3y^2} dA
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