1. Let . Find each of the following.
a. b.
2. Find the domain of each function.
a. b.
3. The graph of is the same as the graph of except that it is moved how?
4. For the functions and find and the domains of each.
a. and b. and
5. Find the intercepts of the following functions. Also, determine whether the graphs of the functions are symmetric with respect to the -axis or the origin.
a. b. c.
6. For each of the following, find
a. b.
7. Calculate the following limits.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o.
p.
q.
r.
8. For each of the following, define such that the given function is continuous at 3.
a. b.
9. Determine the intervals on which the functions defined below are continuous.
a. b.
10. Identify all asymptotes of the following.
a. b. c. d.
11. Give a specific example to show that it is possible for to exist if is undefined.
12. Determine when if
13. Determine if
14. Determine when if
15. Determine when if
16. Find the equation of the tangent line to the curve defined by when
17. At what point is the tangent line to the curve parallel to the line
18. Find the equation of the line tangent to the graph of at the point -
19.
20. If find
21. Determine if
22.
23. Determine at if
24.
25. Determine if
26. If find
27.
28. Let and . Determine the intervals on which is increasing.
29. Determine the intervals on which is decreasing if and
30. Determine the intervals on which is concave upward if
31. Determine the intervals on which is concave downward if and
32. Determine all points of inflection for
33. Determine all points of inflection for
34. Let . Find all local extrema for
35. Let . Find all local extrema for
36. For each of the following, find the maximum and minimum values of the given function on the indicated interval.
a. b. -
37. A rock thrown from the top of a cliff is feet above the ground seconds after being thrown.
a. Determine the height of the cliff.
b. Determine the time it takes the rock to reach the ground.
c. Find the velocity of the rock when it strikes the ground.
38. A rock is thrown vertically upward from the roof of a house 32 feet high with an initial velocity of 128 ft/sec.
a. What is the speed of the rock at the end of 2 seconds?
b. What is the maximum height the rock will reach?
39. What is the maximum area which can be enclosed by 200 ft of fencing if the enclosure is in the shape of a rectangle and one side of the rectangle requires no fencing?
40. A woman throws a ball vertically upward from the ground. The equation of its motion is given by - where is the initial velocity of the ball. If she wants the ball to reach a maximum height of 100 ft, find
41. A rectangular open tank is to have a square base, and its volume is to be 125 yd. The cost per square yard for the base is $8 and for the sides is $4. Find the dimensions of the tank in order to minimize the cost of the material.
42. A power station is on one side of a river which is mile wide, and a factory is 1 mile downstream on the other side of the river. It costs $300 per foot to run power lines overland and $500 per foot to run them under water. Find the most economical way to run the power lines from the power station to the factory.
43. A cardboard box manufacturer wishes to make open boxes from pieces of cardboard 12 in square by cutting equal squares from the four corners and turning up the sides. Find the length of the side of the square to be cut out in order to obtain a box of the largest possible volume. What is the largest possible volume?
44. A train leaves a station traveling north at the rate of 60 mph. One hour later, a second train leaves the same station traveling east at the rate of 45 mph. Find the rate at which the trains are separating 2 hours after the second train leaves the station.
45. A street light hangs 24 ft above the sidewalk. A man 6 ft tall walks away from the light at the rate of 3 ft/sec. At what rate is the length of his shadow increasing?
46. A barge is pulled toward a dock by means of a taut cable. If the barge is 20 ft below the level of the dock, and the cable is pulled in at the rate of 36 ft/min, find the speed of the barge when the cable is 52 ft long.
47. Find the values of and if and
48. Use differentials to approximate the maximum possible error that can be produced when calculating the volume of a cube if the length of an edge is known to be ft.
49. Approximate using each of the following.
a. Differentials b. A linearization
50. The moment of inertia of an annular cylinder is where is the mass of the cylinder, is its outer radius, and is its inner radius. If and changes from to use differentials to estimate the resulting change in the moment of inertia.
51. The range of a shell shot from a certain ship is meters, where is the angle above horizontal of the gun when it is shot. If the gun is intended to be fired at an angle of radians to hit its target, but due to waves it actually shot radians too low, use differentials to estimate how far short of its target the shell will fall.
52. For each of the following, determine whether the Intermediate Value Theorem guarantees the equation has a solution in the specified interval.
a.
b.
c.
d.
53. Find if and -
54. Find if and - -
For problems 55 and 56, let be the function defined by the graph shown.
55. Estimate each of the following. Round numbers to the nearest integer.
a. The instantaneous rate of change of at
b. The average rate of change of over the interval
c. The intervals where is increasing and where it is decreasing.
d. The inflection point or points of
56. Find the intervals where is increasing and where it is decreasing. Round numbers to the nearest integer.
57. Let as shown in the graph below, be the velocity of a car in meters per second at time in seconds, where positive velocity means the car is moving forward. Round your answers to the nearest integer.
a. When did the car stop?
b. Approximately how far did the car travel in the time interval 8 to 12 seconds?
c. Approximately how far did it travel in the time interval 12 to 14 seconds?
d. Approximately how far did it travel in the time interval 8 to 14 seconds?
e. At the time 2 seconds, is the car moving forward or backward? Is the driver's foot on the gas or the brake?
f. At the time 16 seconds, is the car moving forward or backward? Is the driver's foot on the gas or the brake?
58. Integrate.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m. n.
59.
Differentiate.
a.
b.
60. Find the area of the region bounded by and
61. Find the area of the region bounded by and
62. Find the area of the region bounded by and
63. Find the volume of the solid generated by revolving the region bounded by the -axis, and the line about the -axis.
64. Find the volume of the solid generated by revolving the region bounded by and about the -axis.
65. Find the volume of the solid generated by revolving the region in the first quadrant bounded by , and the -axis about each of the following.
a. the -axis b.
66. Find the volume of the solid generated by revolving the region bounded by the -axis, and the lines and about the -axis.
67. A solid has as its base the region in the first quadrant bounded by . Every plane section of the solid taken perpendicular to the -axis is a square. Find the volume of the solid.
68. A solid has as its base the region in the -plane bounded by the graphs of and . Find the volume of the solid if every cross section by a plane perpendicular to the -axis is a semicircle with diameter in the -plane.
69. Find the average value of the function on the interval -
70. Find the average value of the function on the interval
71. Assume that the density of water is 62.5 lb/ ft. A cylindrical water tank with a circular base has radius 3 feet and height 10 feet. Find the work required to empty the tank by pumping the water out of the top for each of the following situations.
a. The tank is full. b. The tank is half full.
72. A bucket with 24 lb of water is raised 30 feet from the bottom of a well. Find the work done in each of the following cases.
a. The weight of the empty bucket is 4 lb and the weight of the rope is negligible.
b. The bucket weighs 4 lb and the rope weighs 4 oz/ft.
c. The bucket weighs 4 lb, the rope weighs 4 oz/ft, and water is leaking out of the bucket at a constant rate so that only 18 lb of water remain in the bucket when it reaches the top.
73. Match each numbered item with a lettered item. (There are more lettered items than numbered items. Some lettered items do not match any numbered item.)
1. Definition of
2. Definition of
3. Definition of
4. Definition of `` is continuous at ''.
5. The Intermediate Value Theorem.
6. Definition of the derivative of at
7. Definition of a function being differentiable at
8. Theorem relating differentiability and continuity.
9. The power rule for differentiation.
10. Definition of the differential.
11. Definition of a function having an absolute maximum at
12. Definition of a function having a local maximum at
13. The Extreme Value Theorem.
14. Definition of a function that is increasing on an interval
15. The Mean Value Theorem.
16. Definition of an antiderivative of on an interval
a.
b. For every there is a corresponding number such that whenever
c. If is continuous on the closed interval and is a number strictly between and then there exists a number in such that
d. The limit exists.
e. If is differentiable at then is continuous at
f. If is continuous at then is differentiable at
g.
h. The function has the property for all in
i.
j. for all in the domain of
k. For every there is a corresponding number such that whenever
l. For every there is a corresponding number such that whenever
m. There is an open interval containing such that for all in
n. for all in
o. If is continuous on then there are numbers and in such that is an absolute maximum for in and is an absolute minimum for in
p. If is differentiable,
q. whenever and and are in
r. If is continuous on and differentiable on then there is a number in such that
1. a.
b.
2. a. Dom: --
b. Dom: ---
3. Left 3 units and down 4 units.
4. a.
Dom:
Dom: --
b.
Dom: -
Dom: -
5. a. -intercept:
-intercept:
Symmetry: none
b. -intercept: none
-intercepts: none
Symmetry: origin
c. -intercept:
-intercepts: -
Symmetry: -axis
6. a. DNE
b.
7. a. -
b. -
c.
d. -
e.
f. DNE
g.
h. -
i. DNE
j. 0
k.
l. DNE
m. 0
n.
o.
p.
q.
r. -
8. a.
b.
9. a. -
b. -- -
10. a. HA:
VA:
OA: none
b. HA:
VA: -
OA: none
c. HA: -
VA:
OA: none
d. HA:
VA:
OA: none
11.
There are infinitely many correct answers.
12. -
13.
14.
15. -
16.
17.
18.
19.
20.
21. -
22.
23. -
24.
25.
26. -
27. -
28. Inc:
29. Dec:
30. CU: -
31. CD: --
32. IP:
33. IP: none
34. Local max: at -
Local min: at -
35. Local max: at 0
Local min: - at - - at
36. a. Max: 0
Min: -
b. Max: 16
Min: 0
37. a. 192 ft
b. 6 sec
c. - ft/sec
38. a. 64 ft/sec
b. 288 ft
39. 5000 ft
40.
41. Dimensions:
42. Overland mile
Underwater mile
43. 2 2 square
Volume of 128 in
44. mph
45. 1 ft/sec
46. 39 ft/min
47.
48. ft
49. a.
b.
50.
51. meters
52. a. Yes
b. Yes
c. Yes
d. No
53.
54.
55. a. 4
b.
c. Dec:
Inc:
d. IP:
56. Dec:
Inc:
57. a. seconds
b. meters
c. meters
d. meters
e. Forward
Accelerator
f. Backward
Brake
58. a.
b.
c.
d. -
e.
f.
g.
h.
i.
j.
k. -
l.
m. -
n.
59. a.
b.
60.
61.
62.
63.
64.
65. a.
b.
66.
67.
68.
69.
70.
71. a. foot-pounds
b. foot-pounds
72. a. foot-pounds
b. foot-pounds
c. foot-pounds
73.
1. l 2. k 3. b 4. a 5. c 6. i 7. d 8. e
9. g 10. p 11. j 12. m 13. o 14. q 15. r 16. h
1. Let . Find each of the following.
a.
b.
2. Find the domain of each function.
a.
Dom: --
b.
Dom: ---
3. The graph of is the same as the graph of except that it is moved how?
Since the graph of is the graph of shifted left 3 units and down 4 units.
4. For the functions and find and the domains of each.
a. and
Dom:
Dom: --
b. and
Dom: -
Dom: -
5. Find the intercepts of the following functions. Also, determine whether the graphs of the functions are symmetric with respect to the -axis or the origin.
a.
-intercept:
-intercept:
-
Neither even nor odd.
Symmetry: none
b.
undefined
-intercept: none
No solution.
-intercepts: none
-
-
So is odd.
Symmetry: origin
c.
-intercept:
-intercepts:
-
-
-
So is even.
Symmetry: -axis
6. For each of the following, find
a.
So does not exist.
b.
7. Calculate the following limits.
a.
-
b.
-
c.
Divide the leading coefficients.
d.
If then
-
e.
If then
f.
-
DNE
g.
If then
h.
If then
-
i.
-
DNE
j.
0
k.
k.
If - then
By the Squeeze Theorem,
l.
If - - then
DNE
m.
If then -
- 0
0
By the Squeeze Theorem,
n.
o.
o.
Let
p.
Let
q.
Let
r.
Let
-
8. For each of the following, define such that the given function is continuous at 3.
a.
b.
9. Determine the intervals on which the functions defined below are continuous.
a.
-
-
- -
-
So is continuous at 4.
Hence, is continuous on -
b.
-
DNE
So is not continuous at -
Note that is left continuous at -
Hence, is continuous on
--
and
-
10. Identify all asymptotes of the following.
a.
HA:
-
VA:
OA: none
b.
HA:
-
VA: -
OA: none
c.
-
-
HA: -
-
VA:
OA: none
d.
HA:
0
-
VA:
OA: none
11. Give a specific example to show that it is possible for to exist if is undefined.
DNE
There are infinitely many correct answers.
12. Determine when if
-
-
-
13. Determine if
14. Determine when if
15. Determine when if
-
-
-
-
16. Find the equation of the tangent line to the curve defined by when
17. At what point is the tangent line to the curve parallel to the line
18. Find the equation of the line tangent to the graph of at the point -
-
-
-
19. Let . Find when
20. If find
21. Determine if
-
22. Determine if
-
23. Determine at if
-
24. Let . Find at
.
-
25. Determine if
26. If find
-
-
27. If . Find at
-
-
-
28. Let and . Determine the intervals on which is increasing.
Inc:
29. Determine the intervals on which is decreasing if and
Dec:
30. Determine the intervals on which is concave upward if
CU: -
31. Determine the intervals on which is concave downward if and
CD: --
32. Determine all points of inflection for
CD: -
CU: --
IP:
33. Determine all points of inflection for
CD: never
CU: -
IP: none
34. Let . Find all local extrema for
-
Dec: --
Inc: -- -
Local max: at -
Local min: at -
35. Let . Find all local extrema for
Dec: --
Inc: -
Local max: at 0
Local min: - at - - at
36. For each of the following, find the maximum and minimum values of the given function on the indicated interval.
a.
(max)
-
- (min)
Note that -
b. -
-
(min)
- (max)
Note that -
37. A rock thrown from the top of a cliff is feet above the ground seconds after being thrown.
a. Determine the height of the cliff.
192 ft
b. Determine the time it takes the rock to reach the ground.
-
6 sec
c. Find the velocity of the rock when it strikes the ground.
-
-
-
- ft/sec
38. A rock is thrown vertically upward from the roof of a house 32 feet high with an initial velocity of 128 ft/sec.
a. What is the speed of the rock at the end of 2 seconds?
-
-
64 ft/sec
b. What is the maximum height the rock will reach?
-
-
288 ft
39. What is the maximum area which can be enclosed by 200 ft of fencing if the enclosure is in the shape of a rectangle and one side of the rectangle requires no fencing?
-
-
-
(max)
5000 ft
40. A woman throws a ball vertically upward from the ground. The equation of its motion is given by - where is the initial velocity of the ball. If she wants the ball to reach a maximum height of 100 ft, find
-
-
-
-
-
41. A rectangular open tank is to have a square base, and its volume is to be 125 yd. The cost per square yard for the base is $8 and for the sides is $4. Find the dimensions of the tank in order to minimize the cost of the material.
Dec:
Inc:
Absolute min: at
Length of base side: 5
Height: 5
Dimensions:
42. A power station is on one side of a river which is mile wide, and a factory is 1 mile downstream on the other side of the river. It costs $300 per foot to run power lines overland and $500 per foot to run them underwater. Find the most economical way to run the power lines from the power station to the factory.
Recall: 1 mile = 5280 feet
Solve
(min)
Overland mile
Underwater mile
43. A cardboard box manufacturer wishes to make open boxes from pieces of cardboard 12 in square by cutting equal squares from the four corners and turning up the sides. Find the length of the side of the square to be cut out in order to obtain a box of the largest possible volume. What is the largest possible volume?
(max)
2 2 square
Volume of 128 in
44.
A train leaves a station traveling north at the
rate of 60 mph. One hour later, a second train
leaves the same station traveling east at the rate of
45 mph. Find the rate at which the trains are separating
2 hours after the second train leaves the station.
Let be the amount of time the second train has traveled.
mph
Alternatively,
mph
45. A street light hangs 24 ft above the sidewalk. A man 6 ft tall walks away from the light at the rate of 3 ft/sec. At what rate is the length of his shadow increasing?
1 ft/sec
46. A barge is pulled toward a dock by means of a taut cable. If the barge is 20 ft below the level of the dock, and the cable is pulled in at the rate of 36 ft/min, find the speed of the barge when the cable is 52 ft long.
39 ft/min
47. Find the values of and if and
48. Use differentials to approximate the maximum possible error that can be produced when calculating the volume of a cube if the length of an edge is known to be ft.
ft
49. Approximate using each of the following.
a. Differentials
b. A linearization
50. The moment of inertia of an annular cylinder is where is the mass of the cylinder, is its outer radius, and is its inner radius. If and changes from to use differentials to estimate the resulting change in the moment of inertia.
51. The range of a shell shot from a certain ship is meters, where is the angle above horizontal of the gun when it is shot. If the gun is intended to be fired at an angle of radians to hit its target, but due to waves it actually shot radians too low, use differentials to estimate how far short of its target the shell will fall.
-
-
-
-
meters
52. For each of the following, determine whether the Intermediate Value Theorem guarantees the equation has a solution in the specified interval.
a.
-
Yes.
b.
Yes.
c.
-
Yes.
d.
No.
Note that is not continuous on
53. Find if and -
-
-
-
54. Find if and - -
- -
- -
For problems 55 and 56, let be the function defined by the graph shown.
55. Estimate each of the following. Round numbers to the nearest integer.
a. The instantaneous rate of change of at
Note that the slope of the tangent line at the point is approximately 4. So
b. The average rate of change of over the interval
-
c. The intervals where is increasing and where it is decreasing.
Dec:
Inc:
d. The inflection point or points of
IP:
56. Find the intervals where is increasing and where it is decreasing. Round numbers to the nearest integer.
Recall that is increasing where is concave upward and that is decreasing where is concave downward.
The function
CD:
CU:
The function
Dec:
Inc:
57. Let as shown in the graph below, be the velocity of a car in meters per second at time in seconds, where positive velocity means the car is moving forward. Round your answers to the nearest integer.
a. When did the car stop?
seconds
b. Approximately how far did the car travel in the time interval 8 to 12 seconds?
Let be the position at time
(area under the curve)
meters
c. Approximately how far did it travel in the time interval 12 to 14 seconds?
Let be the position at time
(area above the curve)
meters
d. Approximately how far did it travel in the time interval 8 to 14 seconds?
From time to time the car travels forward 16 meters.
From time to time the car travels backward 4 meters.
The total distance that the car travels is 20 meters.
The displacement of the car is 12 meters.
e. At the time 2 seconds, is the car moving forward or backward? Is the driver's foot on the gas or the brake?
Forward
Accelerator
f. At the time 16 seconds, is the car moving forward or backward? Is the driver's foot on the gas or the brake?
Backward
Accelerating
Brake
The above answer may be counterintuitive. When you are driving a car in reverse, the velocity of the car is negative. When you apply the brake, the speed decreases but the velocity increases. This is the reason that the car is accelerating when your foot is on the brake.
58. Integrate.
a.
Let and
b.
Let and
c.
-
-
-
d.
-
-
e.
-
-
-
f.
Let and
g.
-
h.
Let
and
-
-
i.
Let and
j.
Let and -
Note that
-
k.
Let and -
-
-
-
-
-
l.
Let and
m.
Let and -
-
-
-
n.
Let and
59.
Differentiate.
Use the Fundamental Theorem of Calculus Part I.
a.
b.
60. Find the area of the region bounded by and
-
-
-
61. Find the area of the region bounded by and
-
-
-
62. Find the area of the region bounded by and
63. Find the volume of the solid generated by revolving the region bounded by the -axis, and the line about the -axis.
Disk Method
Shell Method
64. Find the volume of the solid generated by revolving the region bounded by and about the -axis.
First, find the points of intersection.
Intersection Points:
Find the volume using either the washer method or the shell method.
Washer Method
Shell Method
65. Find the volume of the solid generated by revolving the region in the first quadrant bounded by , and the -axis about each of the following.
a. the -axis
Disk Method
Shell Method
-
-
-
b.
Disk Method
Shell Method
66. Find the volume of the solid generated by revolving the region bounded by the -axis, and the lines and about the -axis.
Shell Method
67. A solid has as its base the region in the first quadrant bounded by . Every plane section of the solid taken perpendicular to the -axis is a square. Find the volume of the solid.
-
-
-
68. A solid has as its base the region in the -plane bounded by the graphs of and . Find the volume of the solid if every cross section by a plane perpendicular to the -axis is a semicircle with diameter in the -plane.
(semicircle)
69. Find the average value of the function on the interval
- -
70. Find the average value of the function on the interval
- -
71. Assume that the density of water is 62.5 lb/ ft. A cylindrical water tank with a circular base has radius 3 feet and height 10 feet. Find the work required to empty the tank by pumping the water out of the top for each of the following situations.
a. The tank is full.
Partition the tank into equal slices.
The height of each slice is ft.
The volume of each slice is ft
The weight of each slice is lb.
The work required to ``lift'' out the slice is foot-pounds.
foot-pounds
Alternatively,
foot-pounds
b. The tank is half full.
Partition half of the tank into equal slices.
The height of each slice is ft.
The volume of each slice is ft
The weight of each slice is lb.
The work required to ``lift'' out the slice is foot-pounds.
foot-pounds
Alternatively,
foot-pounds
72. A bucket with 24 lb of water is raised 30 feet from the bottom of a well. Find the work done in each of the following cases.
a. The weight of the empty bucket is 4 lb and the weight of the rope is negligible.
foot-pounds
b. The bucket weighs 4 lb and the rope weighs 4 oz/ft.
-
-
-
foot-pounds
c. The bucket weighs 4 lb, the rope weighs 4 oz/ft, and water is leaking out of the bucket at a constant rate so that only 18 lb of water remain in the bucket when it reaches the top.
-
-
-
foot-pounds
73. Match each numbered item with a lettered item. (There are more lettered items than numbered items. Some lettered items do not match any numbered item.)
1. Definition of
2. Definition of
3. Definition of
4. Definition of `` is continuous at ''.
5. The Intermediate Value Theorem.
6. Definition of the derivative of at
7. Definition of a function being differentiable at
8. Theorem relating differentiability and continuity.
9. The power rule for differentiation.
10. Definition of the differential.
11. Definition of a function having an absolute maximum at
12. Definition of a function having a local maximum at
13. The Extreme Value Theorem.
14. Definition of a function that is increasing on an interval
15. The Mean Value Theorem.
16. Definition of an antiderivative of on an interval
a.
b. For every there is a corresponding number such that whenever
c. If is continuous on the closed interval and is a number strictly between and then there exists a number in such that
d. The limit exists.
e. If is differentiable at then is continuous at
f. If is continuous at then is differentiable at
g.
h. The function has the property for all in
i.
j. for all in the domain of
k. For every there is a corresponding number such that whenever
l. For every there is a corresponding number such that whenever
m. There is an open interval containing such that for all in
n. for all in
o. If is continuous on then there are numbers and in such that is an absolute maximum for in and is an absolute minimum for in
p. If is differentiable,
q. whenever and and are in
r. If is continuous on and differentiable on then there is a number in such that
1. l 2. k 3. b 4. a 5. c 6. i 7. d 8. e
9. g 10. p 11. j 12. m 13. o 14. q 15. r 16. h