PastExamLabPastExamLab

成功大學 99 年度 微積分

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第 1 題14 分
Suppose f:(0,∞)→Rf: (0, \infty) \to \mathbb{R} such that f(x)=ln⁡x1+ln⁡2xf(x) = \frac{\ln x}{1 + \ln^2 x}.
(a)6 分
Find the asymptotic lines of the graph y=f(x)y = f(x), if exist.
(b)8 分
Find the absolute extreme values of ff, if exist.
第 2 題12 分
Define the function E(x)=2π∫0xe−t2dtE(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt.
(a)8 分
Show that E(x)=2π∑n=0∞(−1)nx2n+1n!(2n+1)E(x) = \frac{2}{\sqrt{\pi}} \sum\limits_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{n!(2n+1)}.
(b)4 分
What is the domain of E(x)E(x)? Explain your answer.
第 3 題10 分
Let f(x)=∫0xtcos⁡(t2)dtf(x) = \int_0^x t \cos(t^2) dt. Find f′(0)f'(0) and f′′(0)f''(0).
第 4 題14 分
For the double integral ∫01∫sin⁡−1yπcos⁡x1+cos⁡2x dxdy\int_0^1 \int_{\sin^{-1} y}^{\pi} \cos x \sqrt{1 + \cos^2 x} \, dx dy,
(a)6 分
change the order of integration to be dydxdy dx;
(b)8 分
and then evaluate the integral.
第 5 題14 分
Let f(x,y)=x3−6xy+y3f(x, y) = x^3 - 6xy + y^3.
(a)6 分
Find the critical points of f(x,y)f(x, y).
(b)8 分
Determine whether the critical points are points of maximum, minimum values or saddle points.
第 6 題12 分
For a differential equation x2y′′−3xy′+4y=0x^2 y'' - 3xy' + 4y = 0,
(a)6 分
use z=ln⁡xz = \ln x to transform such an equation into an equation with constant coefficients;
(b)6 分
find the general solution of (a) in terms of xx.
第 7 題10 分
Let the function z=f(x−y,y−x)z = f(x - y, y - x). Prove that ∂z∂x+∂z∂y=0\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0.
第 8 題14 分
Let Q be the solid region cut from the sphere x2+y2+z2=4x^2 + y^2 + z^2 = 4 by the cylinder r=2sin⁡θr = 2\sin\theta.
(a)6 分
List the double integral to find the volume of Q using polar coordinate system.
(b)8 分
Evaluate the double integral at (a).
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