课程名称︰微积分乙下
课程性质︰必修
课程教师︰李聪成
开课学院：
开课系所︰
考试日期（年月日）︰2016年6月23日
考试时限（分钟）：160分钟
试题 :
请将每一步骤表达清楚，不可以只写答案。
1. Find the local maximum and minimum values and saddle point(s) (鞍点)
of the function
f(x,y)=2x^4-xy^2+2y^2
2. Evaluate the limit or show that it does not exist.
x^2-y^2
lim ───────
(x,y)→(0,0) x^2+y^2
3. The derivative (导数) of f(x,y) at (1,2) in the direction (方向) of i+j is
2√2 and in the direction of -2j is -3. What is the derivative of f in the
direction -i-2j. Give reasons for your answer.
4. Find the volume (体积) of the solid that lies under the paraboloid (抛物面)
z=x^2+y^2, above the xy-plane, and inside the cylinder x^2+y^2=2x.
5. Evaluate the double integral.
1 1
∫ ∫ cos(x^3) dxdy
0 √y
6. The plane x+y+z=12 intersects the paraboloid (抛物面) z=x^2+y^2 in an
ellipse. Find the points on the ellipse that are
(a) closest to (最接近) and
(b) farthest from (离最远) the origin.
7. Find an equation (方程式) for the tangent plane (切平面) and parametric (参
数) equations for the normal line (法线) to the surface with the equation
x^2+y^2+z^2=6
at the point (-1,2,1).
8. Let R be the region (区域) bounded by the graphs of y=√x, x=0, and y=3.
Evaluate
∫ ∫ (2xy^2+2ycosx) dA.
R
9. Use polar coordinates to evaluate the integral.
2 √(8-x^2) x 2
∫ ∫ (──) dydx
0 x y
10. Evaluate the integral
e^(y-4x)
∫ ∫ ──── dA,
R y+2x
where R is the region (区域) bounded by
y=4x+2, y=4x+5, y=3-2x, and y=1-2x.