Survey of Calculus—Departmental Review Sheet
Fall 2005
(Brief Applied Calculus, 3rd Edition. Berresford, Rockett. Houghton Mifflin, 2004.)
All work should be shown and all answers should be exact unless stated otherwise. It is fine to leave
radicals in the denominator of a fraction, but they should be reduced. The departmental formula
sheet and an acceptable graphing calculator are the only resources allowed for use during the final
exam. Each problem on this review sheet is matched with its corresponding objective in the course
outline and the section of the textbook in which it appears.
[Objective #1, Section 2.1]
1. Evaluate the following limits without using a graphing calculator or making tables.
1 1
1 

3
2

3
x

3
x

6
x
x 2
lim
a. lim t  5 t 2 
b. lim
c.
x  1
x2 x  2
t  25

x2  x

[Objective #1, Sections 2.1, 2.7]
2. Use the given graph to answer the following:
a. lim f ( x )
b. lim f ( x )
x0
d.
e.
f.
g.
c. lim f ( x)
x0
x0
x-values where the function is discontinuous
x-values where the function is nondifferentiable
x-values where the slope of the tangent line is zero.
x-values where the slope of the tangent line is undefined
[Objective #2, Section 2.2]
3. Use the definition of derivative to find f ' ( x) .
a. f ( x)  2x 2  3x  2
1
x
b. f ( x ) 
[Objective #2, Sections 2.3, 2.4, 2.6]
4. Find the equation for the tangent line to the curve of f ( x)  3x 2  6x  2 at the point (0,2). Write your
answer in slope-intercept form.
5.
dy
for each of the following:
dx
2
1
a. y  3x 5  4 x  2 
4
x
x
Find



c. y  3x 2  5 x  1 2 x  5
1
x
e. y  x2
x 1
h. y  1  x
b. y  6 x
d. y=
4
3
3x
2
3

1
3x 2  2 x  1
x

g. y  e 2 x 1  ln x 2  3x
f. 2xy  3x 2  4x  5 y 3


3
i. y   x 2  1  x 


4

[Objective #4, Sections 3.1, 3.2]
6. For each of the following find:
a. critical points
b. intervals where the function is increasing and decreasing
c. relative extrema
d. possible points of inflection
e. intervals where the function is concave up and concave down
f. graph by hand
i) f ( x)  x 3  3x 2  9x  7
ii) f ( x)  5x 4  x 5
iii) f ( x)  5 x 2  1
[Objective #5, Sections 5.1, 5.2, 5.6]
7. Integrate the following:
a.
d.
g.

(x 2 
 (21

3
3
 5x 2  4)dx
b.
x2
t 
5
6
t
)dt
3xe x dx


2x
(6e 3
f.
x
x 2  2dx

e
x
i.

5x 2  6 x  2

h.
 1 x
1
j.
c.
e.
5
2

x
1
(3 x 2  2)dx
k.
0
x2  4
dx
x2
x 2 3x  4dx
x
0
dx
2
x
2
5
 )dx
x
1
dx
l.
dx
x
dx
x
[Objective #9, Section 7.2]
8. Find f x , f y , f xx , f yy , and f xy for each of the following:
a. f ( x, y)  x 4 y 3  3x 3  4 y 2  2xy
b. f ( x, y)  e xy
c. f ( x, y)  y ln x  xe y
d. f ( x, y )  93 x y  4 xy 2
[Objective #3, Section 2.5]
9.
After t hours a car is a distance s (t )  60 t 
100
miles from its starting point. Find the velocity after 2 hours.
t 3
[Objective #3, Section 2.3]
10. A company’s cost function is C ( x)  20  3x 
54
dollars (for 5  x  20 ) . Find the marginal cost function.
x
Find the marginal cost at x=9 and interpret your answer.
[Objective #3, Section 2.6]
11. Blood flowing through an artery encounters a resistance of R( x)  0.25(0.01x  1) 4 , where x is the distance
(in centimeters from the heart. Find the instantaneous rate of change of the resistance 100 centimeters from the
heart.
[Objective #3, Sections 3.3, 3.4]
12. A homeowner wants to enclose three adjacent rectangular pens of equal size, as in the diagram below. What is the
largest total area that can be enclosed using only 240 feet of fence?
13. A open-top box with a square base is to have a volume of exactly 500 cubic inches. Find the dimensions of the
box that can be made with the smallest amount of materials.
14. A computer dealer can sell 12 personal computers per week at a price of $2000 each. He estimates that each $400
price decrease will result in three more sales per week. If the computers cost him $1200 each, what price should
he charge to maximize his profit? How many will he sell at that price?
15. A store can sell 12 telephone answering machines per day at a price of $200 each. The manger estimates that for
each $10 price reduction she can sell 2 more per day. The answering machines cost the store $80 each. Find the
price and quantity sold per day to maximize the company’s profit.
[Objective #3, Section 3.6]
16. A cube of ice is melting so that each edge is decreasing at the rate of 2 inches per hour. Find how fast the volume
of the ice is decreasing at the moment when each edge is 10 inches long.
[Objective #3, Section 4.3]
17. A patient receives an injection of 1.2 cubic centimeters of a drug, and the amount remaining in the bloodstream t
hours later is A(t )  1.2e 0.05t . Find the rate of change of this amount just after the injection (at t=0) and after 2
hours.
[Objective #6, Section 4.4]
18. A videotape store works out a demand function for its videotape rentals and finds it to be D( p)  120  20 p
where p is the price per rental. Find the elasticity of demand at the price of $2. Should the store raise or lower the
price to increase revenue?
19. The demand function for a newspaper is
D( p)  80 ,000 75  p , where p is the price in cents. The publisher
currently charges 50 cents, and it plans to raise the price to increase revenues. Will this strategy succeed? Why?
[Objective #7, Section 5.3]
20. Approximate the area below f ( x)  x 2  1 from a=1 to b=3 by calculating a Riemann sum with the 4 rectangles.
Then use integration to find the exact area below the curve from a to b.
21. Find the area under each curve between the given x-values.
a. f ( x)  4  x from x=0 to x=4
b. f ( x)  e x from x=0 to x=1
[Objective #5, Section 5.4]
t
22. A student can memorize words at the rate of 6e 5 words per minute after t minutes. Find the total number of
words that the student can memorize in the first 15 minutes.
23. Find the average value of f ( x)  36  x 2 on the interval 2,2.
24. A company’s marginal cost function is MC ( x)  x
cost of the first 400 units.
1
2
 4 dollars, where x is the number of units. Find the total
[Objective #8, Section 5.5]
25. For the demand function d ( x)  120  0.16 x and the supply function s( x)  0.08 x Find the market demand. Find
the consumers’ surplus at the market demand. Find the producers’ surplus at the market demand.
[Objective #7, Section 5.4]
26. Find the area bounded by each pair of curves
a. y  x 2  3x and y  3x  1
b.
y  x 3 and y  x 2
c. y  x 3 and y  8x
[Objective #11, Section 7.3]
27. A firm that produces two kinds of baseball glove, one that sells for $18 each and the other for $25 each. The total
revenue from the sale of x thousand gloves at $18 each and y thousand at $25 each is given by
R( x, y)  18 x  25 y . The company determines that the total cost, in thousands of dollars to produce these gloves
is C( x, y)  4x 2  6xy  3 y 2  20x  19 y 12 . Find the amount of each type of glove that must be produced and
sold in order to maximize profit.
28. The number of office workers near a beach resort who call in “sick” on a warm summer day is
f ( x, y)  xy  x 2  y 2  110x  50 y  5200 where x is the air temperature (70  x  100 ) and y is the water
temperature 60  y  80  . Find the air and water temperatures that maximize the number of absentees.
[Objective #10, Section 7.5]
29. Find the maximum value of f ( x, y)  3xy subject to the constraint 2 x  y  8
30. Find the minimum value of f ( x, y)  2 y 2  6x 2 subject to the constraint 2 x  y  4
31. A metal box with a square base is to have a volume of 45 cubic inches. If the top and bottom cost 50 cents per
square inch and the sides cost 30 cents per square inch, find the dimensions that minimize cost.