Math 212, Fall 2011
Exam 1
1. [24 points (6 pts each)]
(a) Find a b
if a b cos B .
Name: __________________________________
`D
(b) Find if
`B
D œ B ln
ˆ
B  C
#
‰

Math 212, Fall 2011 Exam 1
(c) Find the equation of the tangent plane to the surface D œ B C at the point a
$ß # b
.
2
`D
(d) Find if
`C
BD  D/ œ " .

Math 212, Fall 2011 Exam 1
2. [8 points] Find the absolute maximum value of the function a b # #
.
3
3. [6 points] Let 1 Bß C b
be a function, and suppose that linear approximation to estimate a b
.
a b
B a b
, and
C a b
. Use a

Math 212, Fall 2011 Exam 1
4. [20 points (5 pts each)] The following figure shows a contour plot for a differentiable function 0 Bß C b
:
5
0 1 2 3 4 5 6 7
5
4
3
20
18
16
14
8
10
12
14
4
3
16
2 2
4
6
8
10 12
14
18
1
2
1
0
0 1 2 3 4 5 6 7
0
(a) At which of the points a
#ß " , %ß " b
, and a
&ß $ b
`0
is the value of the greatest? Explain.
`B
4
(b) Estimate the value of a b
. Your answer must be correct to within 15% to receive full credit.

Math 212, Fall 2011 Exam 1 5
5
0
4
3
1 2
20
18
16
14
3 4
8
10
12
14
16
5
18
6 7
5
4
3
2
1
2
4
6
8
10 12
14
2
1
0
0 1 2 3 4 5 6 7
0
(c) The function 0 Bß C b
has three critical points on the rectangle c
!ß ( ‚ !ß & d
. Estimate the positions of these points, and identify each critical point as a local min, local max, or saddle point.
(d) Estimate
# #
( ( a b
" !
0 Bß C .C .B
. Your answer must be correct to within 10% to receive full credit.

Math 212, Fall 2011 Exam 1
5. [16 points] The figure to the right shows a polyhedron with six vertices, nine edges, and five faces. Use a double integral to find the volume of this polyhedron. You must show your work to receive full credit.
z
H 0,0,3 L
H 0,2,3 L
6
H 2,0,1 L
H 2,0,0 L x
H 0,0,0 L
H 0,2,0 L y

Math 212, Fall 2011 Exam 1
6. [12 points] Let be the region inside the circle B  C œ # and above the parabola C œ B #
.
Evaluate ((
V
C .E
.
7

Math 212, Fall 2011 Exam 1
7. [14 points] Let be the region defined by the following inequalities:
B  C Ÿ % , C € B , B € !
Use polar coordinates to evaluate (( È
V
"  B  C .E
.
8