Practice Problems: Final Exam
Solutions
1. (a) Using the Product Rule and the Chain Rule,
`D
sinaBCb BC cosaBCb .
`B
(b) We must take the antiderivative once with respect to B and twice with respect to C, resulting in a
"
function like 0 aB Cb B$ C# . It
Math 212, Fall 2011
Name: _
Exam 2
1. [18 points (9 pts each)]
(a) Find the equation of the tangent plane to the surface B# CD " at the point a$ # '.
(b) Find parametric equations for the line tangent to the curve ra>b sin > $/> /#> at the point a! $ "b.
Math 212
Name: _
Homework 6
1. The figure to the right shows a tiling of the
plane by congruent regular hexagons. Find
the coordinates of the point T .
P
0, 0
1, 0
2. In the figure to the right, a rectangle with a
length of "! units and a width of & units
Math 212
Name: _
Homework 5
y
0, 2
1. Let W be the shaded region shown in the figure to the right.
(a) Find the equation for the horizontal line C " in polar
coordinates. Express your answer as an equation of the
form < 0 a)b.
1, 1
1, 1
x
(b) Find a simpl
Practice Problems: Exam 2
#
1. Evaluate ( B# C# .E, where V is the shaded region shown in the figure below.
V
r
1
y
x
1
, 0
1, 0
2. Evaluate ( B .Z , where T is the polyhedron shown in the figure below.
T
z
0,0,3
0,2,3
2,0,1
0,0,0
0,2,0
2,0,0
y
x
3. Let I
Math 212
Name: _
Homework 2
1. A copper rod with a length of )!.! cm is placed along the B-axis, with the left end at B !. The
middle portion of the rod is heated, and then the rod is thermally insulated from its surroundings.
(a) After being heated, the
Math 212
Name: _
Homework 7
1. The figure to the right shows a regular hexagon in
three-dimensional space. Find the coordinates of the
point T .
7, 1, 2
P
2, 5, 1
3, 7, 4
0, 0, 1
2. The figure to the right shows a a pyramid in
three-dimensional space.
(a)
Math 212
Name: _
Homework 1
1. Though the resistance of an ideal resistor is constant, the resistance of a store-bought resistor
typically depends on both the temperature of the resistor and the amount of current flowing through it.
(a) A certain resistor
Math 212, Fall 2011
Exam 1
1. [24 points (6 pts each)]
(a) Find 0C a$ !b if 0 aB Cb B# C B/C B cos B.
(b) Find
`D
if D B lnB# C#
`B
Name: _
Math 212, Fall 2011
Exam 1
(c) Find the equation of the tangent plane to the surface D B# C at the point a$ #b.
(d
Math 212
Name: _
Homework 8
1. The figure below shows a unit circle rolling along the B-axis. While rolling, the circle moves to the
right with a speed of " unitsec, and simultaneously rotates clockwise at a rate of " radiansec.
y
t
2
t
t
3 2
t
2
P
x
A po
Math 212
Name: _
Homework 6
1. The figure to the right shows a tiling of the
plane by congruent regular hexagons. Find
the coordinates of the point T .
P
0, 0
1, 0
2. In the figure to the right, a rectangle with a
length of "! units and a width of & units
Math 212
Name: _
Homework 5
y
0, 2
1. Let W be the shaded region shown in the figure to the right.
(a) Find the equation for the horizontal line C " in polar
coordinates. Express your answer as an equation of the
form < 0 a)b.
1, 1
1, 1
x
(b) Find a simpl
Math 212
Name: _
Homework 7
1. The figure to the right shows a regular hexagon in
three-dimensional space. Find the coordinates of the
point T .
7, 1, 2
P
2, 5, 1
3, 7, 4
0, 0, 1
2. The figure to the right shows a a pyramid in
three-dimensional space.
(a)
Practice Problems: Final Exam
1. (a) Find
`D
if D B sinaBCb.
`B
` $0
(b) Find a function 0 aB Cb such that
B# .
#
`B `C
%
(c) Evaluate ( (
"
"B
B/BC .C .B.
!
(d) Compute Hu 0 aB Cb, where 0 aB Cb /&B sina"!Cb and u $& %&.
2. The following figure shows a