PastExamLabPastExamLab

成功大學 103 年度 微積分B

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※ 考生請注意：本試題不可使用計算機。請於答案卷(卡)作答，於本試題紙上作答者，不予計分。
第 1 題18 分
Find the following limits:
(a)6 分
lim⁡x→∞x+3⋅(x+2−x+1)\lim\limits_{x \to \infty} \sqrt{x + 3} \cdot (\sqrt{x + 2} - \sqrt{x + 1})
(b)6 分
lim⁡x→0x−sin⁡x2tan⁡(x2)−x\lim\limits_{x \to 0} \frac{x - \sin x}{2 \tan(\frac{x}{2}) - x}
(c)6 分
lim⁡n→∞(nn2+1+nn2+4+⋯+n2n2)\lim\limits_{n \to \infty} (\frac{n}{n^2 + 1} + \frac{n}{n^2 + 4} + \cdots + \frac{n}{2n^2})
第 2 題10 分
Let f(x)=sin⁡−1x2+4−x2f(x) = \sin^{-1} \frac{x}{2} + \sqrt{4 - x^2}, for each x∈[−2,2]x \in [-2, 2]
(a)5 分
Find f′(x)f'(x)
(b)5 分
Find extreme values of f(x)f(x)
第 3 題12 分
Evaluate the following integrals:
(a)6 分
∫0π2sin⁡2x⋅cos⁡2x dx\int_0^{\frac{\pi}{2}} \sin 2x \cdot \cos^2 x \, dx
(b)6 分
∫01dxex+1\int_0^1 \frac{dx}{e^x + 1}
第 4 題10 分
Find the convergence set of ∑n=1∞(−1)n(x−2)nn⋅4n\sum\limits_{n=1}^{\infty} \frac{(-1)^n(x-2)^n}{n \cdot 4^n}.
第 5 題10 分
Gauss' probability integral is I0(α)=∫0∞e−αx2dx=12παI_0(\alpha) = \int_0^{\infty} e^{-\alpha x^2} dx = \frac{1}{2}\sqrt{\frac{\pi}{\alpha}} Evaluate I1(α)=∫0∞xe−αx2dxI_1(\alpha) = \int_0^{\infty} xe^{-\alpha x^2} dx and I2(α)=∫0∞x2e−αx2dxI_2(\alpha) = \int_0^{\infty} x^2 e^{-\alpha x^2} dx.
第 6 題10 分
Determine and classify the stationary points of the function f(x,y)=xy+(x−y)3f(x,y) = xy + (x-y)^3.
第 7 題10 分
Evaluate ∫01∫y1sin⁡(x3)dxdy\int_0^1 \int_{\sqrt{y}}^1 \sin(x^3) dxdy.
第 8 題10 分
Evaluate ∬Rxy dA\iint_R xy \, dA, where R is the region bounded by y=2x−x2y = \sqrt{2x - x^2} and y=0y = 0, by converting to Polar Coordinates.
第 9 題10 分
A certain gas satisfies the law pV=T−4pTpV = T - \frac{4p}{T} where p=pressure, V=volume, and T=temperature. Calculate ∂T∂p\frac{\partial T}{\partial p} and ∂T∂V\frac{\partial T}{\partial V} at the point where p=V=1p = V = 1 and T=2T = 2.
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