Practice Problems: Exam 1
1. (a) Find 0B a"ß %b if 0 aBß Cb œ B$ C#  ÈBC.
(b) Find
` #D
if D œ B# /BC .
`B#
ÈC
$
(c) Evaluate ( (
!
!
BÈB#  " .B .C.
2. Let X be the triangular region with vertices a!ß !b, a1ß !b, and a!ß 1b. Find the average value of
sinaB  Cb on X .
3. Let D œ 0 aBß Cb, and suppose that 0 a"ß "b œ %, 0B a"ß "b œ #, and 0C a"ß "b œ $.
(a) Use a linear approximation to estimate the value of D when aBß Cb œ a"Þ"ß "Þ#b.
(b) Assuming B œ < cos ) and C œ < sin ), use the Chain Rule to evaluate
`D
at the point a"ß "b.
`<
4. The following picture shows a polyhedron with eight vertices, twelve edges, and six faces:
z
H0,0,3L
H0,2,2L
H2,0,2L
H0,0,0L
H0,2,0L
H2,0,0L
H2,2,1L
x
H2,2,0L
y
(a) Find a formula for the plane that contains the points a#ß #ß "b, a#ß !ß #b, a!ß #ß #b, and a!ß !ß $b.
(b) Use a double integral to find the volume of this polyhedron. You must show your work to
receive full credit.
5. Find all critical points of the function 0 aBß Cb œ #B$  $C#  'BC, and classify each critical point as
a local maximum, a local minimum, or a saddle point.
6. Let V be the region defined by B € ! and B#  C# Ÿ %. Use polar coordinates to evaluate the
integral (( aB#  C# b
$Î#
.E.
V
7. (a) Find the equation of the tangent plane to the surface BD $  CD œ % at the point a"ß #ß #b.
(b) Use your answer to part (a) to estimate the value of D for which "Þ!'D $  #Þ!%D œ %.
8. Write each of the following double integrals as an iterated integral with appropriate bounds for B
and C.
(a) (( 0 aBß Cb .E, where V is the region bounded by the parabola C œ B# and the line C œ B  #.
V
(b) (( 0 aBß Cb .E, where W is the region defined by aB  "b#  C# Ÿ " and B Ÿ ".
W
(c) (( 0 aBß Cb .E, where X is the triangular region with vertices at a!ß !b, a"ß "b, and a!ß $b.
X
9. The following figure shows several cross-sections for the graph of a function 0 aBß Cb:
30
30
30
25
25
25
20
20
20
15
15
15
10
10
10
5
5
5
0
0
0.25
0.5
0.75
1
0
0
C œ !Þ%
(a) Estimate
`0
at the point a!Þ&ß !Þ&b.
`B
(b) Estimate
`0
at the point a!Þ&ß !Þ&b.
`C
0.25
0.5
C œ !Þ&
0.75
1
0
0
0.25
0.5
C œ !Þ'
0.75
1
10. Let V be the region shown in the following figure:
y
3
2
1
x
0
0
Evaluate (( B# .E.
V
1
2
3
4