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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
1
142 Student Review Problems for Final Exam
NOTE: The following problems are to serve merely as practice for your final exam. The problems cover the
chapters discussed this semester except for the material of 7.1-7.3. The final exam for MATH 142 is NOT a
common exam; each instructor makes up his/her own final exam as they did for the other tests of the semester.
It is advised that you work your old exams, quizzes, and assignments as well as any old week-in-review problems
which you feel might be beneficial to you.
CHAPTER 1
√
1. If f (x) = x3 + ( x)2 − 47, find f (4).
f (x+h)−f (x)
.
h
√
√ x+5
.
(x−3)(x−1)
2. If f (x) = 4x2 + 2, find
3. Find the domain of
4. Graph the following piecewise functions:

2

 x
(a) f (x) =
,x≤0
x ,0<x≤1


3x , x > 1


 x+5
, x < −2
(b) f (x) =
+ 3 , −2 < x ≤ 1

 5
,x>1
x2


x+1



(c) f (x) =
(d) f (x) =
,x<1
0
,x=1
3x − 1 , 1 < x ≤ 2
6 − x2 , x > 2






 3
, x < −2
x2 − 4 , −2 ≤ x ≤ 2


2x + 1 , x > 2


 x
(e) f (x) =
,x≤0
2x2
,0<x<2

 x+4 , x >2
5. A furniture manufacturer makes bookcases and small desks, each requiring the use of a cutting machine.
Each bookcase requires 3 hours on the machine, and each desk requires 4 hours on the machine. The machine
is available 150 hours each week.
(a) If x is the number of bookcases and y is the number of desks that are manufactured each week, find
the equation that x and y must satisfy if all available time is used on the machine.
(b) The graph of the equation in the previous part is a straight line. Find the slope of this line and interpret
its meaning.
6. You buy a new house that costs $250,000. It depreciates linearly over 30 years with a scrap value of $50,000.
What is the house worth after 19 years?
7. A certain type of sports utility vehicle is valued at $30,000 brand new. It is expected to depreciate linearly
over 25 years. If the s.u.v. has a scrap value of $3,500, how many years will it take before it is valued at
$20,000?
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
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8. Suppose 2000 items are sold per week at $75/unit, but 6000 of the same item can be sold at $35/unit.
(a) Find the demand equation.
(b) What would the price of each item be if 4500 units were sold?
9. Suppose that 500 bottles of propane are sold per day at a price of $20 per bottle, and that 1500 bottles can
be sold per day at a price of $15 per bottle. Find the demand equation for the number of bottles of propane
sold per day.
10. Suppose 20,000 units of an item are sold per day at $200/unit and 25,000 units are sold per day at $150/unit.
(a) Find the demand equation.
(b) Find the revenue equation.
11. Suppose a retail store can sell 12,000 units of an item per day at $60/unit and 10,000 units can be sold per
day at $75/unit.
(a) Find the demand equation.
(b) Find the revenue equation.
12. Suppose 5000 units of an item are sold per day at $75/unit, and 2000 units can be sold per day at $125/unit.
(a) Find the demand equation.
(b) Find the revenue equation.
13. A manufacturer found that it costs $48,000 to produce 900 items in one month. The next month it costs
$42000 to produce 700 items. What are the fixed costs of the manufacturer?
14. Coke bottles cost $0.50 per bottle to produce. These bottles also have a fixed cost of $250 no matter how
many bottles are produced. Cokes sell for $0.75 per bottle.
(a) Find the cost equation.
(b) Find the revenue equation.
(c) Find the profit equation.
15. Joe Bob’s saddle shop makes saddles that cost $164 each to make. Fixed costs are $6500. The shop profits
$123,000 when 125 saddles are made and sold. Find the cost, revenue, and profit equations.
16. The demand for TVs is given by p = 100 − 0.5x, where p is the price in dollars and x is quantity. It costs
$50 to produce each TV it makes with $500 start-up costs. What are the cost and revenue functions for the
company?
17. A company makes candles. The fixed costs are $300 and the candles cost $0.75 each to produce. The candles
sell for $5 each.
(a) Find the cost equation.
(b) Find the revenue equation.
(c) Find the break-even quantity.
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
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18. It costs Pizza Palace $0.37 to produce each cheese pizza they sell (and they only sell cheese pizzas). Their
fixed costs are $450. The demand for the pizzas is given by p = 40 − 0.031x, where p is the price in dollars
and x is the quantity produced and sold.
(a) Find the cost and revenue functions.
(b) Find how many pizzas need to be produced in order to maximize profit.
(c) What price should be charged for each pizza in order to maximize profit?
19. A company manufactures gadgets. Their total cost to produce 200 gadgets is $1200 and their fixed costs
are $800. When the gadgets are sold at $5 each, the company can sell 300 gadgets. When the selling price
is decreased by a dollar, the company sells 50 more gadgets.
(a) What are the cost, revenue, and profit equations?
(b) How many gadgets should they sell in order to maximize profit?
(c) What is the maximum profit?
(d) What price should they sell the gadgets at in order to achieve this maximum?
20. If the demand equation for a firm is given by p = −0.8x + 39, find the values at which the revenue is a
maximum.
21. Given the cost function C(x) = 6x + 15 and the demand equation p = −5x + 46, find where the profit is
maximized and the maximum profit.
22. It costs a sports manufacturing company $20 to produce a solid oak baseball bat. There are no fixed costs.
The demand for the bat is given by p = 36 − 0.02x, where p is the price and x is the quantity. How many
bats need to be sold to maximize profit?
23. Find the amount you would need to invest at the beginning of 10 years to accumulate $80,000, if your money
was invested at 8% compounded
(a) annually.
(b) monthly.
(c) weekly.
(d) continuously.
24. How much money would accumulate after 10 years, if you invested $10,000 at 3.9% compounded
(a) continuously?
(b) quarterly?
(c) weekly?
25. How much money would accumulate after 35 years, if you invested $10,000 at 10.0% compounded
(a) annually?
(b) quarterly?
(c) monthly?
(d) weekly?
(e) daily?
(f) continuously?
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
26. You put $10,000 in an account paying 8% compounded quarterly.
(a) How long will it take to have $25,000 in the account?
(b) What is the effective yield of the account?
27. How many years will it take your money to become 8 times the amount you started with if you put your
money into an account that pays interest at a rate of 6.5% compounded continuously?
28. Suppose $50,000 is invested at an annual rate of 6% compounded monthly. Determine the time it takes for
this account to double.
29. Paul invested $1200 into a bank account paying 10% compounded monthly when he graduated college. How
many years would it take to double his investment?
30. A man has $5000 that he wants to put into a high interest CD account. It is compounded continuously at
a rate of 12%.
(a) How much is in the account after 20 years?
(b) What is the effective yield of this account?
31. Solve the following for x EXACTLY:
53+3x
25x
= 625
32. Under certain conditions, soda from an unopened can will begin to evaporate exponentially. After 8 days,
7/8 of the can will remain. How many days will it take 7/8 of the can to evaporate?
33. A beverage begins evaporating exponentially. After 5 days, 3/4 of the beverage will remain. How many
days will it take the beverage to totally evaporate at this rate?
√
34. If f (x) = x + 1 and g(x) = x − 2, find the following functions and the domains of each:
(a) (f + g)(x)
(b) (f − g)(x)
(c) (f ∗ g)(x)
(d) (f /g)(x)
35. Find (f ◦ g)(x), (g ◦ f )(x), (f ◦ f )(x), and (g ◦ g)(x) if
(a) f (x) = 2x2 + x − 1 and g(x) = x2 + 1
√
(b) f (x) = 3x3 + 2x + 1 and g(x) = x2
(c) f (x) = x + 4 and g(x) = 4x3
36. If f (x) = 6x + 9 and g(x) = x2 , find
(a) (f ◦ g)(x) and its domain.
(b) (g ◦ f )(x) and its domain.
37. If f (x) = 2x2 + 3x + 6 and g(x) =
x
3
+ 7, find (f ◦ g)(8).
38. Write equivalent statements for
(a) log(xy)
(b) log(y/x)
39. Write the given quantity as one simplified logarithm:
1
4
log y − 2 log x +
√
3 log z
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
40. Write the following in terms of log x, log y, and log z:
(a) log
(b) log
(c) log
x5
y3 z
2 y3
x√
z
x4
y5 z 2
41. Solve the following for x:
(a) (2)103x = 5
(b) (3)104x = 2
(c) 12 = (9)2x
(d) 56x+9 =
(e)
log25 5
40x
12
5x
= 65
42. Solve the following for x:
(a) log5 (x − 27) = 12
(b) log2 3x + log2 x = 5
43. Simplify the following:
(a) log(log b (xlogx b ))
(b) e5 ln 2
44. Find the best-fitting LINE and correlation coefficient for the following sets of data:
(a) Number of Cattle in Texas (in millions) from 1995 to 2000.
Year (x)
# of cattle(y)
1995
12
1996
17
1997
19
1998
23
1999
28
2000
34
(Let x = 0 correspond to the year 1995.)
(b) Number of Yard Signs vs Number of Votes Received
# of Signs (x)
# of Votes(y)
1
5
2
8
3
17
4
19
5
28
45. Find the best fitting model for the following. Justify your answer.
(a)
(x)
(y)
12
0.22
15
4.1
17
5.93
19
12.69
21
9.52
23
8.55
25
13.44
27
9.58
29
9.01
31
0.64
(b) Number of Bacteria (in thousands) Growing on Raw Meat Within an Hour vs Temperature
Temp. in ◦ C (x)
Bacteria (y)
14
2.14
17
4.37
19
6.91
21
8.43
23
11.25
25
13.07
27
15.71
46. Find the best fitting model for the following data. Justify your answer.
Day (x)
Population(y)
1
4
2
8
4
20
Estimate the day in which the population will reach 4000.
6
45
9
57
11
114
29
14.32
31
12.11
33
10.53
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
CHAPTER 2
47. Compute the following limits:
5x12 + 2x4 − 4
x→∞
7x16 + 53
5x5 − 2x3 + 9x2 − 6
(b) lim
x→∞
2x4 − 6x5 + 24
x3 + 5x + 54
(c) lim
x→−∞
x−9
2
3x − 4x
(d) lim
x→∞ 2x2 + 1
6e4x − 2e−x
(e) lim
x→∞ 2e−x + e4x
6e4x − 2e−x
(f) lim
x→−∞ 2e−x + e4x
x5 + x 3 + 6
(g) lim
x→−2 5x5 − 2x + 12
2ex − 1
(h) lim
x→∞ 1 + ex
ex − e−x + 200
(i) lim
x→−∞ e−x + ex − 3
ex − e−x
(j) lim x
x→∞ e + e−x
5x2 + 6
(k) lim
x→−∞ 2x + 4
2
(l) lim
x→∞ 1 + ex
(a) lim
(m) lim
x→∞
p
x2 + 1
48.
f(x)
6
4
2
−4
−2
2
4
Find:
(a) lim+ f (x)
x→2
(b) lim f (x)
x→2−
(c) lim f (x)
x→2
(d) Is f (x) continuous at x = 2?
49. Find the vertical and horizontal asymptotes of f (x) =
5x2 +4(x−7)
(x−2)(x−3) .
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
50. Find the value(s) of a that make f (x) continuous everywhere.
(
(a) f (x) =
(
(b) f (x) =
(
(c) f (x) =
x2 + x + a , x ≥ 2
ex
,x<2
4x2 − ax + 6 , x < 4
6x − a
,x≥4
4x2 − ax + 6 , x ≤ 0
6x − a
,x>0

2

 0.125x + 2 , x ≥ 3
(d) f (x) =
a

 3(x + 4)
(
(e) f (x) =
(
(f) f (x) =
(
(g) f (x) =
(
(h) f (x) =
(
(i) f (x) =
, −3 < x < 3
, x ≤ −3
x2 − 4ax − 6 , x ≤ 6
x − 8a
,x>6
x3 − 3x2 + 5 , x ≤ 1
ax + 4
,x>1
−ax + 5
,x≤3
2
ax + 2x + 4 , x > 3
ax2 + 5x − 7 , x ≤ 2
2x + 3a
,x>2
x2 + 6ax − 14 , x < 0
x − 10a
,x≥0
51. Find where the following functions are discontinuous.

2

 x
(a) f (x) =
2

 x
( √
(b) f (x) =
,x<1
,x=1
,x>1
x+8 , x ≤1
,x>1
x3 −x+6
x+1
52. Find the IROC of
√
(a) f (x) = 4x2 + 10 at x = 6.
(b) f (x) = ln(3x3 − 6x + 7) at x = 2.
(c) f (x) = ln(3x3 + 2x2 + 5) at x = 5.
53. Find the equation of the tangent line to
(a) f (x) = x2 + 12x − 4 at x = 2.
(b) f (x) = 2x2 − 3x + 2 at x = 2.
(c) f (x) = 8x3 + 2x2 + 3 at x = 4.
54. Use the limit definition of derivative to find the derivative of the following functions.
(b) f (x) =
3
x3 −4
−3x2
(c) f (x) =
3x2
(d) f (x) =
x
(a) f (x) =
√
+ 2x − 4
− 4x + 2
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
55. What three things on a graph indicate that a derivative will fail to exist?
56.
f(x)
6
2
−6
−4
−2
2
4
6
−3
−5
For what value(s) of x does f 0 (x) not exist?
CHAPTER 3
57. Find the derivatives of the following:
√
(a) f (x) = (2x)3 + e
(b) f (x) =
(c) f (x) =
x+3
4−2ex
1−3x
8x3 −3x2 +7
(7e3x+1 −7x3 +1)2
(d) f (x) = ln(2x2 + 3x − x−1 )3
(e) f (x) = (7x)13(5x
(f) f (x) =
4 −3x3 +x−23)
e2x+1 +12x2 +3x−3
(x3 +x)
x
(g) f (x) = ln x+1
(h) f (x) = (e(−3x
2 −2)
)(ln |800x−4 |)
(i) f (x) = (4x4 + 56x2 + 5)4 (x3 + 2x)
q
(j) f (x) = ln x+1
x+2
√
(k) f (x) = 36x +
√2
6x
+
√
3
8x2
p
(l) f (x) = (6x3 + 3x2 − x)4 ( e(2x+7) − 2ex + ex )
(m) f (x) = ex
2 +4x2 +3
(2x + 8x)2
(n) f (x) = x2 + 6x + (x + 1)4 + log 3x2
2
(p) f (x) =
6xex +3x2
x4
−3x2 + x
(q) f (x) =
2
e(4x +2x+7)
(r) f (x) =
ln(x2 )
ln(x4 +x2 +3)
(o) f (x) =
−4
√
x−3 + 2x2 − x
(s) f (x) = (ln |x2 + x + 1|)1/2
(t) f (x) =
(u) f (x) =
5x4 +4x2
ex
2 +2
3
10x
(e
)(e3x −2 )
(v) f (x) = 2(x
3 +2x+x+5)
58. Section 3.2 - #40 (pg.191)
59. Find the marginal cost when C(x) =
√
x − 4 + 5x2 + 10x − 100. Predict the cost of the 10th item.
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
60. The price of tomatoes is given by p =
x3 −2x2
x2 +4x+3 .
Find the marginal revenue function.
61. If the revenue in dollars for a product is given by R(x) = −4x2 + 60x, where x is the number of items sold,
find the marginal revenue for any x. Find R0 (2) and R0 (15) and interpret the meaning of these numbers.
62. A company figured their demand to be represented by the equation p = −0.5x + 3. They figured their profit
using the equation P (x) = −9.5x2 + 23.
(a) Find the marginal profit equation.
(b) Estimate the cost of the 10th item using the marginal cost equation.
CHAPTER 4
63. Given f (x) = 3x4 + 2x3 − 6x + 2, f 0 (x) = 12x3 + 6x2 − 6, and f 00 (x) = 36x2 + 12x, find
(a) Intervals where f (x) is increasing/decreasing
(b) All relative extrema
(c) Intervals where f (x) is concave up/down
(d) All inflection points
64. Find where f (x) = 2x3 − 3x2 − 6x + 10 is increasing/decreasing and concave up/down.
65. For the following functions, find the domain, VAs, HAs, intervals of increasing/decreasing, relative extrema,
intervals of concave up/down, and inflection points:
(a) f (x) = 1 − 8x + 24x2 + 36x3
(b) f (x) = x3 + 5x2 + 4x + 10
(c) f (x) = (2/3)x3 − 6x2 + 16x + 3
(d) f (x) =
x2 +6x+14
x+5
66. #84 on page 248
67. Find the points of absolute extremum (if any exist) of the following functions on the given intervals:
(a) f (x) = 3x3 + 2x2 + 3x + 9; [−16, 16]
(c) f (x) =
x−1
(x−1)2 ; [−10, 10]
4x3 − 25x2 + 10x
(d) f (x) =
x3
(e) f (x) =
3x2
(f) f (x) =
x
; (−∞, ∞)
3+x4
(b) f (x) =
+
3x2
− 5; [−13, 13]
+ 2x + 5; [−5, 2]
+ 6x + 7; [−3, 2]
68. Using calculus, find two non-negative numbers, x and y, such that 2x + y = 40 and the sum of their squares
is maximized.
69. Using calculus, find two non-negative numbers, x and y, such that x + y = 420 and xy is minimized.
70. A man earns $40 for every car he washes when he has 10 cars to wash. Because of a lack of time, the amount
of revenue he receives per car is reduced by $5 when he washes additional cars. The cost of washing a car
is $10. How many cars should be washed to maximize total profit?
71. During Spring Break, a resort in Jamaica has 25 cabanas that it can rent out to tourists. All 25 cabanas
will be full if the resort charges its special vacation rate of $250/day. However, for every $80 increase in the
rent/day, 2 cabanas will become vacant. What price should the resort charge per day in order to maximize
its revenue?
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
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72. A gardener sells 1 pound of squash for $0.75 and can sell 20 pounds at this price. The gardener thinks that
for every $0.05 increase in price per pound of squash, he will lose 2.3 pounds in sales. What price per pound
will maximize the gardener’s revenue?
73. A bus company charges $80 per person for sight-seeing and obtains 60 people for the trip. The company
shows that a $2 increase in the price above $80 results in the loss of one customer. The company has fixed
costs of $400 for each trip. Also, there are costs of $6 per customer. What should the company charge in
order to maximize its profits?
74. A poster is to have an area of 400 square inches with 2 inch margins at the bottom and sides and a 4 inch
margin at the top. What dimensions will give the largest printed area?
75. A box with no top is formed by cutting out 4 square corners from cardboard of the following dimensions
and folding up the sides. What should the width of the squares you are cutting out be in order to maximize
the volume of the box?
(a) 10-inch by 20-inch
(b) 12-inch by 7-inch
76. A 20-inch by 24-inch piece of cardboard is going to have squares cut out of the corners so that it can be
folded up and taped to be used as a litterbox for kittens. What should the dimensions of the box be if you
want the box to hold as much cat litter as possible?
77. #50 on page 261
78. #47 on page 261
79. A farmer is building a fence with 2 wooden sides and 2 stone sides. The farmer also wants a central divider,
but he wants to build it with the cheapest material and parallel to the sides built with the cheapest material.
He is buying the materials from his old calculus teacher whose prices are $2π per foot for stone and $6 per
foot for wooden material. If the farmer wants to enclose a pasture that is 500 square feet, what dimensions
will minimize his total costs?
80. A farmer wants to build a fence to enclose a rectangular area of 1600 square feet. The fence along 2 opposing
sides is to be made of chicken wire costing $2/foot. The third side is to be made with a rock wall costing
$14/foot, and the fourth side is to be made of wood costing $20/foot. Find the dimensions of the rectangle
that will allow the cheapest fence to be built.
81. JenJoy wants to construct a garden. She only has 500 feet of fencing to make this rectangular structure
with maximum area. What should the dimensions be?
82. A fence is to be built to enclose a rectangular area of 800 square feet. The fence along 3 sides is to be made
of material that costs $6/foot. The material for the fourth side costs $18/foot. Find the dimensions of the
rectangle that will allow the cheapest fence to be built.
83. A fence is to be built with an area of 500 square feet. The material for 3 sides costs $9/foot, and the material
for the fourth side costs $12/foot. Find the dimensions of the fence that have the least cost.
84. A fence is to be built around a 350 square foot rectangular courtyard. Three sides are to be made of stone
costing $25/foot, while the other side is to be made of steel costing $15/foot. Find the dimensions of the
enclosure that minimize total cost.
85. A wall is being built around a rectangular courtyard with 1000 square feet of area. One of the sides will not
have a wall. The cost of the wall is $10/foot. Find the dimensions of the courtyard that will cost the least
to enclose.
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Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
86. A freshman dorm is being built in the middle of nowhere on A&M’s campus during the summer of 2004. This
rectangular building is made to hold 200,400 cubic feet and has a square floor and ceiling. The material for
the floor and ceiling is $150 per square foot nd the sides cost $200 per square foot. Determine the dimensions
of the building that will minimize the cost of building the freshman dorm.
CHAPTER 5
87. Integrate the following:
Z
√
3
(a) (5ex + √ + x3 + 2x2 ) dx
x3
!
Z
2eln 2x
(b)
dx
x
Z
(c)
3x5/4 dx
Z 8t − 7
√
(d)
dt
t
Z
4x + 2
√
(e)
dx
x
Z
(f)
(x3 + 7x)(x2 + 5) dx
Z (g)
Z
(h)
Z Z
1
dx
dx
ex−4
2
5xe0.15x dx
Z
2
e −y
(k)
Z (l)
4(2x + 2)5 dx
(i)
(j)
1
x ln x2
−3/2
x
√
5
2x2 + 5
2
12y 2 − 12y + 10
+
− ye3y
3
2
3/4
(2y − 3y + 5y)
!
dy
dx
88. Given f 00 (x) = 4x3 + 5x2 − 0.5x + 2e2x − 12, find:
(a) f 0 (x), if f 0 (x) crosses the y-axis at -2.
(b) f (x) if f (x) crosses the y-axis at 3.
89. Given f 00 (x) = 20x3 − 10, f 0 (1) = 1, and f (1) = −5, find f (x).
90. Find the demand function for a shoe manufacturer if marginal demand , in dollars, is given by
p(1) = 8, where x is the number of thousands of shoes sold.
32x
(4x2 +1)2
and
91. Given the velocity of an object, v(t) = 3t2 + 4t − 3 ft/sec, use Riemann sums to find an upper and lower
estimate of the distance traveled during the first 4 seconds.
92. Calculate the following:
Z
5 x2
− 2x + 1
dx
ex + 5
10
Z 2
2
(b)
e3x +
dx
x
1
(a)
12
c
Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
Z
(c)
5
−3
Z 5
(d)
Z
(e)
Z
(f)
0
70
−x2
1
√ e( 2 ) dx
2π
0
93. T/F:
x2 + 3x + 1 dx
(x − 1)(x2 + (x + 1)3 )4 dx
50
1
Z
x2 − 3x + 5 dx
b1
a
x
Z
94. Evaluate
Z
95. If
97. If
4
a
(x2 + 50) dx
Z
f (x) dx = 5 and
Z
0
g(x) dx = 7, then evaluate
4
Z
7
f (x) dx = 23 and
5
Z
b
4
0
Z
96. If
dx = ln(b/a) where a < 0 and a < b
f (x) dx = 8 and
g(x) dx = 21, then evaluate
Z
4
8
[2f (x) + 4g(x)] dx.
Z
5
7
Z
8
0
4
g(x) dx = 3, then evaluate
4
5
7
[2f (x) − 3g(x)] dx.
8
[5f (x) − 10g(x)] dx.
98. Suppose the number of items produced on a certain piece of machinery by an average employee is increasing
1
at a rate given by 2t+1
, where t is in hours since placed on the machine. How many items are produced by
an average employee in the first 3 hours?
99. The rate of underage alcohol violations (in thousands of citations) in Texas is given to be 5e0.2t beginning
in 1996. How many tickets were given from the beginning of 1998 (Jan.1) to the end of 2000 (Dec.31)?
100. A certain radioactive substance initially weighing 250 grams decays at a rate given by −15e−0.5t , where t is
in years. How much of the substance is left after 10 years?
101. The bird population at A&M is increasing at a rate given by 1000et , where t is time in years measured from
the beginning of 1993 when the bird population was 15,000. Find the bird population at the end of 2001.
102. Aggie Joe decides to go out and have a couple of drinks with his buddies. After a couple of drinks and some
time groovin’ on the dance floor, his blood alcohol concentration after t hours is given by 0.16t
. What is the
t2 +1
average concentration of alcohol in Joe’s bloodstream during the 3 hours he spends at the club? According
to Texas’ legal statutes on drinking, will Aggie Joe spend the night in jail if he attempts to drive home and
is stopped by the police?
2
103. The value of a car purchased for $20,000 is decreasing at the rate of −1000te−0.25t , where t is the number
of years since the purchase of the car. How much will the car be worth in 5 years?
0.04
104. Annual sales growth (in millions of dollars) of a particular drug is given by 0.0025+160e
−2.7t , where t represents
the number of years since the drug’s approval by the FDA in 1987. Estimate (to the nearest 10 million dollars)
the value of total sales of the drug from Jan. 1987 through Dec. 1997.
105. Find the area of the region enclosed by the following functions.
(a) y = −x2 + 6 and y = x2 − 4
(b) y = −x2 + 5 and y = 2x2 − 7
(c) y = x2 + 3x − 2 and y = −2x2 − 4x + 3
106. Find the area of the region enclosed by y = x4 + 3 and y = x3 on [0,4].
c
Kathryn
Bollinger and her Spring 2001 Math 142 Students, May 2, 2001
107. Find the area of the region bounded by y = 2x2 + 5x + 3 and y = 5x + 3 on [-5,5].
Z
108. Find the exact value of
2
−2
p
4 − x2 dx
109. Find the equilibrium point, consumers’ surplus, and producers’ surplus of the following markets.
(a) S(x) = 5x + 10, D(x) = −5x + 50
(b) S(x) = 0.12x2 + 1.5x, D(x) = 45 − 0.18x2
(c) S(x) = 6x, D(x) = 20 − 4x
13