PastExamLabPastExamLab

成功大學 102 年度 微積分A

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第 1 題10 分
Find the limit: lim⁡x→0(1x⋅arcsin⁡x−1x3)\lim\limits_{x \to 0} \left( \frac{1}{x \cdot \arcsin x} - \frac{1}{x^3} \right).
第 2 題10 分
Let f(x)=∫1xt⋅arctan⁡(t2)dtf(x) = \int_1^x t \cdot \arctan(t^2) dt. Find f′(1)f'(1) and f′′(1)f''(1).
第 3 題10 分
Evaluate ∫0∞dx(x+4)x\int_0^{\infty} \frac{dx}{(x+4)\sqrt{x}}.
第 4 題10 分
Solve the differential equation: y′=(1+x)−1+sec⁡2xy' = (1+x)^{-1} + \sec^2 x, y(0)=1,y′(0)=0y(0) = 1, y'(0) = 0.
第 5 題12 分
For ∑n=1∞3nn!xn−1\sum\limits_{n=1}^{\infty} \frac{3n}{n!} x^{n-1}, determine the interval of convergence and also find its sum.
第 6 題12 分
Let f(x)=x4+ax3+bf(x) = x^4 + ax^3 + b, and have a local extreme value of −17-17 at x=3x = 3.
(a)
Fine the value of aa and bb.
(b)
Is that local extreme value maximal or minimal?
(c)
Is there any inflection point of the graph of ff?
第 7 題12 分
Two particles are free to move on the curves y=x2y = x^2 and x−y=1x - y = 1, respectively. What are their positions when they are closest together?
第 8 題12 分
Determine the values of aa and bb such that the ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 contains the circle (x−1)2+y2=1(x-1)^2 + y^2 = 1 and has least area.
第 9 題12 分
Evaluate ∬R(y−xy+x)1/2dA\iint_R \left( \frac{y-x}{y+x} \right)^{1/2} dA, where RR is the trapezoid in the xyxy plane bounded by the lines x=0x = 0, y=xy = x, x+y=1x + y = 1 and x+y=2x + y = 2.
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