课程名称︰微积分乙 课程性质︰必修 课程教师︰张树城 开课学院：医学院 开课系所︰医学系 考试日期（年月日）︰109/01/07 考试时限（分钟）：110 试题 : 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对x偏微分) and (z对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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