[試題] 108-1 張樹城 微積分乙 期末考
課程名稱︰微積分乙
課程性質︰必修
課程教師︰張樹城
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰109/01/07
考試時限（分鐘）：110
試題 :
1.(a)(5 points) Find lim g(x,y) is defined by
(x,y)→(0,0)
2(x^2)y
------------ (x,y)≠(0,0)
x^4+y^2 ,
g(x,y)=
0 , (x,y)=(0,0)
(b)(5 points) Find lim h(x,y) is defined by
(x,y)→(0,0)
2(x^2)y
---------
x^2+y^2 , (x,y)≠(0,0)
h(x,y)=
0 , (x,y)=(0,0)
2.(5+5 points) Compute fxy(0,0) and fyx(0,0). Here f(x,y) is defined by
xy(x^2-y^2)
---------------
x^2+y^2 , (x,y)≠(0,0)
f(x,y)=
0 , (x,y)=(0,0)
(You should compute both of them separately in whole detail. Do not skip any steps.)
3.(a)(5 points) Find the local and absolute extreme values with their locations (if there is any) of the function f defined on R^2 by
f(x,y)= e^3x+y^3-3ye^x
(b)(5 points) Find the abosulte extreme values with their locations (if there is any) of the function
f(x,y)=x^3+y-y^2
over R ={(x,y)| x^2+y^2≦1 and y≧0}
(Remark. In these two questions, don't forget to explain why such values exist(or not), and why they are the answer.)
4.(a)(5 points) Find the tangent plane to the surface y+z^2=-x^2+9 at (1,4,2).
(b)(10 points) Find (z對x偏微分) and (z對y偏微分) at (0,0,0) for x^3+z^2+y(e^xz)+zcosy=0.
5.(a)(5 points) Evaluate
∫∫sinx
Ω ---- dxdy
x
Here Ω ={(x,y)|y≦x≦1 and 0≦y≦1 }.
(b)(5 points) Evaluate
∫∫ln(x^2+y^2)dxdy
D
Here D = {(x,y)|1/2≦ √(x^2+y^2)≦1}.
6.(a)(5 points) Find the area of the region enclosed by the ellipse x^2/8 +y^2/2 = 1.
(b)(10 points) Find the volume of the region in the first octant bounded by the coordinate planes and the plane x + y/2 + z/3 = 1.
(c)(10 points) Find the volume of the solid ellipsoid x^2/16 +y^2/25 +z^2/9 ≦ 1.
(d)(10 points) Find the volume of the solid region enclosed by the surface z =x^2 +3y^2 and z = 8 -x^2 -y^2.
7.(a)(5 points) Evaluate
∫∫e^-(x^2+y^2) dxdy
R^2
(b)(5 points) Evaluate
∞
∫ e^-(x^2) dx.
0
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