Math 208, Section 3

Practice problems for Exam 2

1.
Find the integral of the function f(x,y) = x over the region R lying between the graphs of the curves
y = x-x2 and y = x-1 (see figure).

2.
Find the integral of the function f(x,y,z) = z over the region S bounded by the planes
x = 0, x = y, y = 1, z = 0 and

the surface z = x2+1 (see figure).

4.
Find the critical points of the function
f(x,y) = x2-y3+6xy

Describe what the Second Derivative Test says about each critical point.

5. (15 pts.) Calculate the first and second partial derivatives of the function

[(sin(x+y))/y]

1. (25 pts.) Find the critical points of the function

f(x,y) = x2-xy2-4x

and determine which of rel max, rel min, or saddle point, each is.

2. (20 pts.) Find the maximum value of the function

f(x,y) = x+2y

subject to the constraint g(x,y) = x2+2y2 = 5 .

5. Evaluate the following double integrals (10 pts. each):

(a): ò01 ò12 x2y-y2x dx dy

(b): ò01 òÖx1 xÖy dy dx

5. (20 pts.) Evaluate the integral

ò01ò0(1-y3)1/3 y(1-x3)1/3 dxdy

by changing the order of integration. (Trust me, you can't evaluate it in the order in which it is given!)

2. (20 pts.) Find the local extrema of the function

f(x,y) = x4-4xy+y2 ,

and determine, for each, if it is a local max. local min, or saddle point.

3. (20 pts.) Find the maximum and minimum values of the function

f(x,y) = 2x2-y+y2

subject to the constraint

g(x,y) = 4x2+y2 £ 4

4. (20 pts.) Evaluate the interated integral

ò02òx2 x2(y4+1)1/3 dy dx

by rewriting the integral to reverse the order of integration.

(Note: the integral cannot be evaluated in the order given....)

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