※ [本文轉錄自 NTU-Exam 看板]
作者: taipower (快點達到16吋吧) 站內: NTU-Exam
標題: [試題] 92下 李聰成 微積分乙下 期末考
時間: Sat Jun 25 16:56:12 2005
1.By considering different paths of approach,show that the function
f(x,y)=xy/(x^2+y^2) has no limit as (x,y)→(0,0)
2.
(a)If f(x,y)=xe^y,find the rate of change of f at the point p(2,0) in the
direction from p to q(0.5,2)
(b)In what direction does f have the maximum rate of change? What is this
maximum rate of change?
3.Find the tangent plane and normal line of the surface
sin(xyz)=x+2y+3z at the point (2,-1,0)
4.Use a double integral to find the area of the region within both of the
circles r=cosθ and r=sinθ
5.Find the absolute maximum and minimum values of f(x,y)=e^(-xy) on the
region x^2+4y^2≦1
6.The plane x+y+z=1 cut the cylinder x^2+y^2=1 in an ellipse. Find the points
on the ellipse that lie closest to and farthest from the origin.
7.Find the volume of the solid in the first octant bounded by the cylinder
z=9-y^2 and the plane x=2
8.Evaluate the integral
1 1
∫∫e^(x^2)dxdy
0 y
9.Find the local maximum and minimum values and saddle points of the fuction.
f(x,y)=sinx+siny+sin(x+y), 0＜x＜2π, 0＜y＜2π
10.Evaluate the integral by making an appropriate change of variables
∫∫cos((y-x)/(y+x))dA
R
where R is the trapezoidal region with vertices (1,0)(2,0)(0,2)(0,1)
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