Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If y=22+x, then dy
dx =
1)
A)
5.
B)
9.7.
C)
22.
D)
1.
E)
0.
2)
If p=8m2– 9m+ 3, then the rate of change of p with respect to m when m= 1 is
2)
A)
–9.
B)
16.
C)
1.
D)
2.
E)
7.
3)
If f(x) =x2– 5x+ 2
3x+ 2 , then f’(x) =
3)
A)
16 – 4x– 3x
(3x+2)2.
B)
9x2– 26x– 4
(3x+2)2.
C)
3x2+ 4x– 16
(3x+2)2.
D)
2x– 5
3.
E)
4 + 26x– 9x2
(3x+2)2.
1
4)
An equation of the tangent line to the curve y=x2– 9 at the point where x= 5 is
4)
A)
y=5
4x–9
4.
B)
y= 2x– 6.
C)
y=1
4x+11
4.
D)
y=5
4x+9
4.
E)
y= 2x+ 6.
5)
If y=u5– 8u2+ 2u– 1 and u=x+ 10, find dy
dx when x= – 9.
5)
A)
–1
B)
1
C)
0
D)
–9
E)
–9
2
6)
If g(x) =x4(2x–1)10, then g’(1) =
6)
A)
1.
B)
80.
C)
0.
D)
24.
E)
14.
7)
For a particular host–parasite situation, the number y of hosts that are parasitized when the host
density is x is given by y=500x
5 + 10x. At what rate is the number of hosts parasitized changing with
respect to host density when x= 2?
7)
A)
1
B)
2
C)
3
D)
4
E)
5
8)
If f(x) =x2+ 1, then the percentage rate of change of f(x) when x= 3 is
8)
A)
10%.
B)
20%.
C)
30%.
D)
40%.
E)
50%.
9)
A particle travels along a straight line path according to the equation of motion s=t2+ 3t+ 4,
where t is in seconds and s is in meters. The velocity (in meters per second) of the particle at t= 2 is
9)
A)
6.
B)
7.
C)
9.
D)
5.
E)
8.
10)
By direct use of the definition of a derivative, the derivative of f(x) =1
x is
10)
A)
lim
h
0
1
h.
B)
lim
h
0
1
x +h–1
x
h.
C)
lim
h
0
1
x+h+1
x.
D)
lim
h
0
1
x+h.
E)
1
x2.
11)
Suppose the demand function for a manufacturer’s product is given by p=400
q+ 10 , where p
represents the price per unit for q units. Find the marginal revenue when q= 10.
11)
A)
4
B)
6
C)
8
D)
10
E)
12
12)
If y=8(9 –3x)5
5, then y‘ =
12)
A)
8(9 –3x)4.
B)
8(9 –3x)4
25 .
C)
24(9 –3x)4.
D)
–8(9 –3x)4
25 .
E)
–24(9 – 3x)4.
13)
The average cost c for producing q units of a product is given by c= 0.01q2+ 11 +1000
q. Find the
marginal cost when q= 10.
13)
A)
14
B)
12
C)
112
D)
82.4
E)
1120
4
14)
For a certain group of births, the number 1x of people that survive to age x years is given by
1x= 2400 196 – 2x, 0 x
98. Find the rate of change of 1x with respect to x when x= 26.
14)
A)
–200
B)
–100
C)
200
D)
100
E)
none of the above
15)
An equation of the tangent line to the curve y= 4x2– 6x– 5 at the point (–1, 5) is
15)
A)
y= (8x– 6)(x– 1) + 5.
B)
y= 8x– 6.
C)
y= – 14x– 9.
D)
y= – 14x+ 19.
E)
y= 14x+ 71.
16)
By the definition of a derivative, the derivative of f(x) =x is
16)
A)
lim
h
0
x+h–x
h.
B)
lim
h
0
x+h–x
h.
C)
lim
h
0
x+h
h.
D)
lim
h
0
x+h
h.
E)
lim
h
0
x+h–x
h.
17)
If f(x) =x2– 3x–2/3
x, then f(x) =
17)
A)
1 +5x–8/3
B)
2x+ 2x–1
x2
C)
3x2+2x–5/3 –3x–3/2
x2
D)
2x+2x–5/3
E)
none of the above
18)
If f(x) =x2+ 4
x2– 2 , then f(x) =
18)
A)
6
(x2–2)2
B)
12x
(x2+2)2
C)
1
D)
–12x
(x2–2)2
E)
4x+ 26x– 9x2
(3x+2)2
D
6
A
19)
If f(x) = (x+1)2(x+2)3, then f’(x) =
19)
A)
(x+ 1)(x+2)2(3x+ 2).
B)
(x+ 1)(x+ 2)(4x+ 3).
C)
6(x+ 1)(x+2)2.
D)
(x+ 1)(x+2)2(9x+ 5).
E)
(x+ 1)(x+2)2(5x+ 7).
20)
A value of x for which the slope of the curve y=x3
3–3x2
2+ 2x+ 1 is zero is
20)
A)
–2.
B)
2.
C)
–1.
D)
3.
E)
0.
21)
If y=x4x+ 3, then dy
dx =
21)
A)
2
2 4x+ 3 .
B)
x
2 4x+ 3
–4x+ 3.
C)
x
2 4x+ 3
+4x+ 3.
D)
(4x+ 1) 4x+ 3.
E)
2x
4x+ 3
+4x+ 3.
7
22)
Suppose a person with x years of eduction before seeking regular employment can expect to
receive an average yearly income of y dollars per year, where y=100x3/2 + 5200, 4 x 16. For
what value of x is y increasing at the rate of 600 (dollars/year of education)?
22)
A)
16
B)
4
C)
12
D)
14
E)
9
23)
If y=x3– 2x2, the relative range of change y with respect to x when x= – 1 is
23)
A)
–3.
B)
–7.
C)
0.
D)
–7
3.
E)
–1
3.
24)
If y=1
35x2– 3
, then dy
dx =
24)
A)
10x
33(5x2–3)2
.
B)
–1
3(5x2– 3) 35x2– 3
.
C)
–10x2
33(5x2–3)4
.
D)
–10x
33(5x2–3)4
.
E)
1
10x– 3 .
25)
If C= 10 + 0.7I– 0.2 I is a consumption function, then the marginal propensity to consume and the
marginal propensity to save when I= 25 are respectively given by
25)
A)
0.2 and 0.8.
B)
0.8 and 0.2.
C)
0.6 and 0.4.
D)
0.68 and 0.32.
E)
0.72 and 0.28.
26)
Suppose the demand function for a manufacturer’s product is given by p= 20 – 0.8q, where p
represents the price per unit for q units. Find the marginal revenue when q= 10.
26)
A)
4
B)
8
C)
10
D)
16
E)
20
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
27)
Find the equation of the tangent line to the graph of the curve y= 6x5/3 + 3x1/3 + 1 at the
point (1, 10).
27)
28)
The concentration of salt in a solution is given by C(t) =t1/3. Find the rate of change of the
concentration, d
dt t1/3 .
28)
29)
Differentiate: g(x) =
57x2 + 6x– 4
29)
30)
By using the definition of a derivative, find f‘ (x) where f(x) =2
3x+ 5 .
30)
9
31)
Differentiate: g(x) =3x
x + 4
31)
32)
A bus company will plan a group trip for groups of 10 or larger. If the group is exactly 10,
the cost is $50 per person. The company offers to discount each person’s ticket by $2 for
each addition to the group above 10. The revenue for the bus company is given by: R(x) =
(10 +x)(50 – 2x) where x is the number of additional people above 10. Find the marginal
revenue function using the product rule.
32)
33)
The demand for a product is given by D(x) =850
x
– 1. Rewrite this function so that all
terms have the form xn.
33)
34)
An object moves along the ground according to the equation f(x) =x23. Find f(x).
34)
35)
The concentration of acid in a buffer solution is given by A= 0.8. Graph this function on
your graphing calculator and describe its slope.
35)
36)
The proportion P of voters who recognize the name of a certain candidate after t months of
advertising is given by P(t) =18t
t2+ 80 . Find the equation of the tangent line to this curve
when x= 1. Use your graphing calculator to verify that this line is tangent to P(t) when x=
1.
36)
37)
Suppose the position function of an object moving along a number line is given by s=f(t)=
t3+ 2t2+ 5, where t is in seconds and s is in meters.
(a) Find the average velocity over the interval 2, 2.01 .
(b) Find the velocity when t= 2.
37)
38)
If the distance in feet of a ball thrown from a bridge is given by D(t) = 30t+16t2 where t is
the number of seconds after the ball is thrown, then the velocity of the ball is found by D(t).
What is the velocity of the ball when t= 3?
38)
39)
If a manufacturer‘s average cost equation is c= 500 + 15q+0.3q2, find the total cost function
c, and the marginal cost function dc
dq . What is the marginal cost when 10 units are
produced?
39)
40)
When a ball is thrown upward at a speed of 60 feet/s from a height of 500 feet, its height H
in feet after t seconds is given by H= 500 + 60t –16t2. Find dH
dt .
40)
41)
By direct use of the definition of derivative, find d
dx (f(x)) if f(x) = 4 – 5x2.
41)
42)
Find the equation of the tangent line to the graph of the curve y=
7(x2–8)3 at the point
(3, 1).
42)
43)
Find y if y= 0.39x–2–0.17x–1+ 3.927 + 1.9x.
43)
44)
During the morning commute, the average speed of a vehicle on the highway is given by
f(t) = 20t– 40 t+ 50, where t is the number of hours after 6:00 A.M. Rewrite this function so
that all terms are in the form tn.
44)
11
45)
Find y’ if y=1
(7x2– 3x+11)2
45)
46)
Find the slope of the curve y=1
x– 10 at the point (11, 1).
46)
47)
Find all values of x for which the curve y= 2x4–x2 has a horizontal tangent line.
47)
48)
If the cost C of removing p percent of the impurities from the waste water in a
manufacturing process is given by C(p) =9800p
100 – 2p, find C’(p).
48)
49)
Suppose that the equation r= 430q–2q2 gives the total revenue r (in dollars) that a
manufacturer receives when q units of a product are sold. Determine the marginal revenue
when q= 100 and interpret your result.
49)
50)
The demand for a product is given by the equation p=300
q3/2 . Find the marginal demand
without using the quotient rule.
50)
51)
Find y if y=x2+ ln 2.
51)
52)
A particle travels along a straight line path according to the equation of motion s=4t3+
2t2+ 1, where t is in seconds and s is in meters. Find the velocity of the particle at t= 2.
52)
12
53)
If f(x) =
3x, then find f’ (x) and determine whether or not f’(x) exists for all values of x.
Use your graphing calculator to graph f(x) . Use this graph to describe the behavior of the
tangent line to f(x) at
x= 0.
53)
54)
The demand D for a product at price x is given by D=17
x+ 20. Find d
dx
17
x+ 20 and
graph the result on your graphing calculator. What is the behavior of the graph when x=
0?
54)
55)
Find y’ if y= (x3– 10x+ 2)(x2+ 7x+ 1).
55)
56)
In a predator–prey experiment, it was statistically determined that the number y of prey
consumed by an individual predator was a function of prey density x (the number of prey
per unit of area) where y=0.8x
1 + 0.03x. Determine the rate of change of prey consumed with
respect to prey density.
56)
57)
Find the slope of the curve y= 4x2– 2x + 1 at the point (3, 31).
57)
58)
Suppose you sell cups of caffe latte for $3 each. If you sell q cups of latte, the total revenue r
is given by r= 3q. Find the marginal revenue.
58)
59)
Find the rate of change of y=7
x with respect to x when x= 2.
59)
13
60)
Find y if y=7
3(9x+ 3).
60)
61)
A smoothie stand usually sells 150 smoothies per day at $4 each. A business student’s
research tells her that for every $0.10 decrease in the price, the stand will sell 5 more
smoothies per day. The revenue function for the smoothie stand is given by: R(x) = (4 –
0.1x)(150 + 5x) where x is the number of $0.10 reductions in price. Find dR
dx , the marginal
revenue.
61)
62)
Suppose that the profit p made by selling a certain product at a price x is given by p=f(x)
and the rate of change of that profit with respect to change in price is dp
dx = 10 at x= 7.
Estimate the change in the profit p if the price changes from 7.5 to 8.5.
62)
63)
When a ball is dropped from 256 feet above the ground, its height H in feet above the
ground after t seconds is given by H= 256 – 16t2. Find dH
dt .
63)
64)
If the height of an eaglet sitting in a nest is given by y= 324, find dy
dt .
64)
65)
One hour after x milligrams of a particular drug are given to a person, the change in body
temperature T(x), in the degrees Fahrenheit, is given approximately by T(x) =x21 –x
9.
The rate at which T changes with respect to size of dosage x, T’(x), is called the sensitivity of
the body to the dosage. Use the product rule to find the slope of T(x) when the dosage is 4
milligrams.
65)
66)
Find y’ if y=2x+ 3
7 – 5x.
66)
14
67)
An object moves along the ground according to the equation x=t4. Find dx
dt .
67)
68)
A hot dog vendor sells chili dogs for $2.75 each. If she sells q chili dogs, the total revenue r
is given by r= 2.75q. Find the marginal revenue.
68)
69)
Find y if y= (0.003)4.
69)
70)
The position of an object thrown upward from a height of 100 feet/sec from a height of 40
feet is given by: y(t) = 40 + 100t–16t2. Find the rate of change of y with respect to t and
evaluate it when t= 1 and t= 4. Interpret your results.
70)
71)
Find the rate of change of the area of a square with respect to its one side. Evaluate it when
side = 4 units.
71)
72)
A catering company will plan a group dinner for groups of 20 or larger. If the group is
exactly 20, the cost is $15 per person. The company offers to discount each person’s ticket
by $0.50 for each addition to the group above 20. The revenue for the catering company is
given by: R(x) = (20 +x)(15 –0.5x) where x is the number of additional people above 20.
Find the marginal revenue function using the product rule.
72)
73)
If an object moves horizontally according to x = 3t– 5 and vertically according to y= 3x2,
find its vertical speed, dy
dt .
73)
74)
The revenue R from the sale of x units of a product is R=5
x+ 50x. The number of units
sold after t weeks of advertising is x= 48 +12
t. Find dR
dt when t= 4.
74)
75)
Find the equation of the tangent line to the graph of the curve y= 0.1x3+ 0.2x+ 0.0095 at
the point (0, 0.0095).
75)
76)
If the velocity function is given by V(x) = 3x–x2, find an equation of the tangent line to the
graph of V(x) at (1, 2).
76)
77)
If the velocity function is given by V(x) = 12x–4x2, find an equation of the tangent line to
the graph of V(x) at (1, 8).
77)
78)
The total revenue r for selling x number of bicycles is given by r= 380x. Determine the
relative and percentage rates of change of revenue with respect to x when 4 bicycles are
sold.
78)
79)
If the velocity function is given by V(x) = 6x2+ 13, find an equation of the tangent line to
the graph of V(x) at (3, 67).
79)
80)
Find the derivative of y=x5+7x+ 3
x3+5x+ 1 .
80)
81)
For the consumption function C= 10 +5
8I–I
2,
(a) find the marginal propensity to consume when I= 16;
(b) find the marginal propensity to save with I= 16.
81)
16
82)
If the output function is given by Q(x) =x2+ 1320, find the slope of the curve at the point
where x= 1.
82)
83)
The cost of producing x number of objects is given by C(x) = 5x+12x2. Determine the
relative and percentage rates of change of cost with respect to x when 10 objects are made.
83)
84)
Find y if y=7 – 2x
5
84)
85)
By using the definition of a derivative, find f(x) =1
2x+ 1 .
85)
86)
Find y’ if y= 5(2x2– 3x+4)8.
86)
87)
Use the Chain Rule to find dy
dx where y=7u11 – 8u+ 5; u=9x2– 11x+ 5.
87)
88)
Let p= 20 +1
q be the demand function for a manufacturer’s product. Find the rate of
change of price p per unit with respect to quantity q. How fast is the price changing with
respect to q when q= 2? Assume that p is in dollars.
88)
89)
Find the equation of the tangent line to the graph of the curve y= 2x2+ 3x+ 11 at the point
(0, 11).
89)
17
90)
A fruit stand usually sells 85 apples per day at $0.60 each. A business student’s research
tells her that for every $0.05 decrease in price, the stand will sell 8 more apples per day.
The revenue function for the fruit stand is given by: R(x) = (0.60 – 0.05x)(85 + 8x) where x is
the number of $0.05 reductions in price. Find dR
dx , the marginal revenue.
90)
91)
The position of an object dropped from a height of 400 feet is given by: y(t) = 400 –16t2.
Find the rate of change of y with respect to t and evaluate it when t= 1 and t= 3. Interpret
your results.
91)
92)
If y=
3x2+ 3x+ 7, then find the rate of change of y with respect to x.
92)
93)
If y=3u3–2u2– 5u– 6 and u=4x2– 2x–13, then by direct use of the chain rule find dy
dx
and evaluate when x= 2.
93)
94)
Find y if y=x4+ 4
x2.
94)
95)
Suppose that the demand for x number of hats (in thousands) is determined by the price p
of the hats as follows: x= 10 +200
2p. Find dx
dp without using the quotient rule.
95)
96)
If the revenue function is given by R(x) = 10x, find the derivative of the revenue function
(this is called the marginal revenue).
96)
97)
The typing speed of a certain computer student varies with the number of hours of practice
x according to S= 15 30.9x+ 2. Find S‘.
97)
98)
Differentiate: f(x) =3x2 + 6
5x2– 2x
98)
99)
The volume V of a spherical cell is given by V=4
3r3, where r is the radius. Find the rate of
change of volume with respect to the radius when r= 3 ×10–5 cm. Give your answer in
terms of .
99)
100)
Suppose the cost of a product can be approximated by c=13
x– 2x+ 22. Find the derivative
of this function to determine whether or not this function is continuous for all values of x.
100)
101)
By direct use of the definition of a derivative, find d
dx f(x) if f(x) = 4x– 3.
101)
102)
The concentration of salt in a solution is given by C(t) =t1/2. Find the rate of change of the
concentration dC
dt .
102)
19
103)
The demand D for a product at price x is given by D= – 3x+ 100. Find d
dx –3x+ 100
and discuss what happens at x= 0.
103)
104)
The position of an object thrown upward at a speed of 8 feet/sec from a height of 10 feet is
given by: y(t) = 10 + 8t–16t2. Find the rate of change of y with respect to t and evaluate it
when t= 0.25. Use your graphing calculator to graph y(t). Use the graph to interpret the
behavior of the object when t= 0.25.
104)
105)
Find the derivative of y=
1
x–7x
x2+1
2
x–3x
x2+ 1
.
105)
106)
If the output function is given by Q(x) = 5x+ 122, find the slope of the curve at the point
where x= 7.
106)
107)
By direct use of the definition of a derivative, find d
dx f(x) if f(x) =x2+ 4.
107)
108)
If an object moves horizontally according to x = 2t+ 1 and vertically according to y= 2x3,
find its vertical speed, dy
dt .
108)
109)
If the output function is given by Q(x) =3x2+ 1, find the slope of the curve at the point
where x= 4.
109)
20