PastExamLabPastExamLab

臺灣綜合大學系統 108 年度 微積分B

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第 1 題10 分極限
Evaluate the followings.
(a)5 分
lim⁡x→01cos⁡x\lim\limits_{x \to 0} \frac{1}{\cos x}.
(b)5 分
lim⁡x→01−cos⁡xex−1−x\lim\limits_{x \to 0} \frac{1 - \cos x}{e^x - 1 - x}.
第 2 題10 分微分
Water is pumped into a spherical balloon so that its volume increases at a rate 5cm3/s5cm^3/s. How fast is the radius of the balloon increasing when the diameter is 4cm4cm? Here the volume of a ball of radius rr is 43πr3\frac{4}{3}\pi r^3.
第 3 題10 分微分
Let f:(−δ,δ)→Rf : (-\delta, \delta) \to \mathbb{R} be a function so that f(0)=12f(0) = \frac{1}{2} for δ>0\delta > 0. Assume that f(x)f(x) satisfies the equation
x2+(f(x))2=(2x2+2(f(x))2−x)2x^2 + (f(x))^2 = (2x^2 + 2(f(x))^2 - x)^2
for all x∈(−δ,δ)x \in (-\delta, \delta). Suppose we know that ff is differentiable at 0. Compute the tangent line to the curve y=f(x)y = f(x) at the point (0,12)\left(0, \frac{1}{2}\right).
第 4 題10 分積分
Find the minimum of the function
f(x)=∫1xettdtf(x) = \int_1^{\sqrt{x}} \frac{e^t}{t} dt
on [1,∞)[1, \infty). Explain how you obtain the minimum and find the point where the minimum of ff occurs.
第 5 題10 分積分
Find the volume of the solid SS where the base of SS is a circular disk of radius 11 and parallel cross sections perpendicular to its base are squares.
第 5 題圖表
第 6 題10 分積分
Evaluate the improper integral
∫0∞xtan⁡−1x(1+x2)2dx\int_0^{\infty} \frac{x \tan^{-1} x}{(1 + x^2)^2} dx
if it exists. Here tan⁡−1x=arctan⁡x\tan^{-1} x = \arctan x.
第 7 題10 分級數
Find the radius of convergence and the interval of convergence of the power series
∑n=1∞(−2)nn2xn.\sum\limits_{n=1}^{\infty} \frac{(-2)^n}{n^2} x^n.
第 8 題10 分偏微分
Let ff be the function
f(x,y)={xy2+12y3x2+y2if (x,y)≠(0,0)0if (x,y)=(0,0).f(x, y) = \begin{cases} \frac{xy^2 + \frac{1}{2}y^3}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0). \end{cases}
Evaluate fx(0,0)f_x(0, 0) and fy(0,0)f_y(0, 0) if they exist.
第 9 題10 分偏微分
Use Lagrange multipliers to find the extremum of the function
f(x,y,z)=xyzf(x, y, z) = xyz
subject to the constraint xy+2xz+2yz=12xy + 2xz + 2yz = 12.
第 10 題10 分多重積分
Let a>0a > 0. Evaluate the double integral
∫−aa∫−a2−x2a2−x2cos⁡(x2+y2)dydx.\int_{-a}^{a} \int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}} \cos(x^2 + y^2) dy dx.