PastExamLabPastExamLab

成功大學 79 年度 微積分

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第 1 題20 分
Find the following two limits and give the reasons for your each step:
lim⁡x→0+cos⁡(sin⁡x)+sin⁡(1−cos⁡x)−1x2;\lim\limits_{x \to 0^+} \frac{\cos(\sin x) + \sin(1 - \cos x) - 1}{x^2};
lim⁡x→+∞∫0xexp⁡(t2)dtx2.\lim\limits_{x \to +\infty} \frac{\int_0^x \exp(t^2) dt}{x^2}.
第 2 題10 分
Given that aa and bb are two real roots of the equation f(x)=0f(x) = 0, where f(x)f(x) is a polynomial, prove that there is at least one real root of the equation f′(x)+f(x)=0f'(x) + f(x) = 0 which lies between aa and bb. (HINT: Consider the function g(x)=f(x)exp⁡xg(x) = f(x)\exp x)
第 3 題20 分
(A)10 分
Prove that the improper integral ∫0+∞ex−1exp⁡(−t)dt\int_0^{+\infty} e^{x-1} \exp(-t) dt is convergent for x>0x > 0.
(B)10 分
For x>0x > 0, define f(x)=∫0+∞ex−1exp⁡(−t)dtf(x) = \int_0^{+\infty} e^{x-1} \exp(-t) dt. Prove that f(x+1)=xf(x)f(x + 1) = xf(x) and f(n)=(n−1)!f(n) = (n-1)!, where (n−1)!(n-1)! is the factorial of n−1n-1 for a positive integer nn.
第 4 題20 分
(A)10 分
Is the following series convergent? If it is, also find its sum.
11⋅3⋅5+23⋅5⋅7+35⋅7⋅9+47⋅9⋅11+⋯\frac{1}{1 \cdot 3 \cdot 5} + \frac{2}{3 \cdot 5 \cdot 7} + \frac{3}{5 \cdot 7 \cdot 9} + \frac{4}{7 \cdot 9 \cdot 11} + \cdots
(HINT: Consider the rule for partial fractions.)
(B)10 分
Find the interval of convergence of the power series
∑n=1+∞n+1(n+2)(n+3)xn.\sum\limits_{n=1}^{+\infty} \frac{n+1}{(n+2)(n+3)} x^n.
第 5 題10 分
Find the length of the arc of the parametric curve x=(cos⁡t)+(t⋅sin⁡t)x = (\cos t) + (t \cdot \sin t), y=(sin⁡t)−(t⋅cos⁡t)y = (\sin t) - (t \cdot \cos t), for t∈[0,2π]t \in [0, 2\pi].
第 6 題10 分
Determine the dimensions of a rectangular box, without a top, to be made from a given amount of material for the box to have the greatest possible volume.
第 7 題10 分
Find the total work done in moving an object in the counterclockwise direction once around the circle x2+y2=a2x^2 + y^2 = a^2, if the motion is caused by the force field F⃗(x,y)=(sin⁡x−y)i⃗+(exp⁡y−x2)j⃗\vec{F}(x, y) = (\sin x - y)\vec{i} + (\exp y - x^2)\vec{j}. Assume the arc is measured in meters and the force is measured in newtons.
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