PastExamLabPastExamLab

成功大學 111 年度 微積分B

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第 1 題10 分
Evaluate the following limits.
(a)5 分
lim⁡x→04x−1x\lim\limits_{x \to 0} \frac{4^x - 1}{x}.
(b)5 分
lim⁡x→610−x−4x+6\lim\limits_{x \to 6} \frac{\sqrt{10 - x} - 4}{x + 6}.
第 2 題10 分
Compute the integral ∫03∣x2−3x+2∣dx\int_0^3 |x^2 - 3x + 2|dx.
第 3 題10 分
Compute the integral ∫0πe−xsin⁡(π−x)dx\int_0^{\pi} e^{-x} \sin(\pi - x)dx.
第 4 題10 分
Compute the integral ∫191x(1+x)2dx\int_1^9 \frac{1}{\sqrt{x}(1 + \sqrt{x})^2}dx.
第 5 題10 分
Compute the integral ∫0216x2+7x+2dx\int_0^2 \frac{1}{6x^2 + 7x + 2}dx.
第 6 題10 分
Find the slope of the tangent line to the graph of y2(x2+y2)=x2y^2(x^2 + y^2) = x^2 at the point (22,22)(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}).
第 7 題10 分
Define u(x,y)=y+xexyu(x, y) = y + xe^{xy}, where x=2s+tx = 2s + t and y=2t+1y = 2t + 1.
(a)5 分
Find ∂u∂s\frac{\partial u}{\partial s} when (x,y)=(0,1)(x, y) = (0, 1).
(b)5 分
Find ∂u∂t\frac{\partial u}{\partial t} when (s,t)=(0,0)(s, t) = (0, 0).
第 8 題10 分
(a)5 分
Let {an}n∈N\{a_n\}_{n \in \mathbb{N}} be a sequence defined by an=2nn!a_n = \frac{2^n}{n!}. Find ∑n=1∞nan\sum\limits_{n=1}^{\infty} na_n.
(b)5 分
Use the Integral Test to determine whether the series ∑n=1∞1n(ln⁡n)\sum\limits_{n=1}^{\infty} \frac{1}{n(\ln n)} is convergent or divergent?
第 9 題10 分
Find the absolute maxima and absolute minima of f(x,y)=3x2+2y2−4yf(x, y) = 3x^2 + 2y^2 - 4y on the region RR in the xyxy-plane bounded by the graphs of y=x2y = x^2 and y=4y = 4.
第 10 題10 分
Use Lagrange multipliers to find the extreme values of f(x,y)=54x2+74y2−32xyf(x, y) = \frac{5}{4}x^2 + \frac{7}{4}y^2 - \frac{\sqrt{3}}{2}xy subject to the constraint x2+y2=1x^2 + y^2 = 1.
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