Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
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.
1)
A)
4
B)
6
C)
8
D)
10
E)
12
2)
If g(x) =x4(2x–1)10, then g’(1) =
2)
A)
80.
B)
0.
C)
24.
D)
1.
E)
14.
3)
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)?
3)
A)
14
B)
16
C)
4
D)
12
E)
9
4)
If f(x) =x2+ 1, then the percentage rate of change of f(x) when x= 3 is
4)
A)
10%.
B)
20%.
C)
30%.
D)
40%.
E)
50%.
1
5)
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.
5)
A)
200
B)
–100
C)
100
D)
–200
E)
none of the above
6)
An equation of the tangent line to the curve y= 4x2– 6x– 5 at the point (–1, 5) is
6)
A)
y= 8x– 6.
B)
y= – 14x+ 19.
C)
y= – 14x– 9.
D)
y= 14x+ 71.
E)
y= (8x– 6)(x– 1) + 5.
7)
If f(x) =x2– 5x+ 2
3x+ 2 , then f’(x) =
7)
A)
3x2+ 4x– 16
(3x+2)2.
B)
16 – 4x– 3x
(3x+2)2.
C)
4 + 26x– 9x2
(3x+2)2.
D)
2x– 5
3.
E)
9x2– 26x– 4
(3x+2)2.
2
8)
A value of x for which the slope of the curve y=x3
3–3x2
2+ 2x+ 1 is zero is
8)
A)
3.
B)
–1.
C)
2.
D)
–2.
E)
0.
9)
If f(x) = (x+1)2(x+2)3, then f’(x) =
9)
A)
(x+ 1)(x+2)2(3x+ 2).
B)
6(x+ 1)(x+2)2.
C)
(x+ 1)(x+2)2(9x+ 5).
D)
(x+ 1)(x+ 2)(4x+ 3).
E)
(x+ 1)(x+2)2(5x+ 7).
E
10)
If y=1
35x2– 3
, then dy
dx =
10)
A)
–10x2
33(5x2–3)4
.
B)
1
10x– 3 .
C)
–10x
33(5x2–3)4
.
D)
10x
33(5x2–3)2
.
E)
–1
3(5x2– 3) 35x2– 3
.
C
C
11)
If y=u5– 8u2+ 2u– 1 and u=x+ 10, find dy
dx when x= – 9.
11)
A)
1
B)
–9
C)
–9
2
D)
0
E)
–1
12)
If y=8(9 –3x)5
5, then y‘ =
12)
A)
–24(9 – 3x)4.
B)
8(9 –3x)4.
C)
8(9 –3x)4
25 .
D)
24(9 –3x)4.
E)
–8(9 –3x)4
25 .
13)
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
13)
A)
0.68 and 0.32.
B)
0.8 and 0.2.
C)
0.6 and 0.4.
D)
0.72 and 0.28.
E)
0.2 and 0.8.
4
14)
An equation of the tangent line to the curve y=x2– 9 at the point where x= 5 is
14)
A)
y= 2x+ 6.
B)
y= 2x– 6.
C)
y=1
4x+11
4.
D)
y=5
4x–9
4.
E)
y=5
4x+9
4.
15)
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.
15)
A)
4
B)
8
C)
10
D)
16
E)
20
16)
If f(x) =x2– 3x–2/3
x, then f(x) =
16)
A)
2x+ 2x–1
x2
B)
1 +5x–8/3
C)
3x2+2x–5/3 –3x–3/2
x2
D)
2x+2x–5/3
E)
none of the above
17)
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
17)
A)
8.
B)
9.
C)
6.
D)
5.
E)
7.
18)
If y=x3– 2x2, the relative range of change y with respect to x when x= – 1 is
18)
A)
–7.
B)
–3.
C)
0.
D)
–7
3.
E)
–1
3.
D
19)
If f(x) =x2+ 4
x2– 2 , then f(x) =
19)
A)
12x
(x2+2)2
B)
1
C)
4x+ 26x– 9x2
(3x+2)2
D)
6
(x2–2)2
E)
–12x
(x2–2)2
E
6
E
20)
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?
20)
A)
1
B)
2
C)
3
D)
4
E)
5
21)
By direct use of the definition of a derivative, the derivative of f(x) =1
x is
21)
A)
lim
h
0
1
x+h.
B)
lim
h
0
1
h.
C)
lim
h
0
1
x+h+1
x.
D)
1
x2.
E)
lim
h
0
1
x +h–1
x
h.
22)
If y=x4x+ 3, then dy
dx =
22)
A)
2
2 4x+ 3 .
B)
(4x+ 1) 4x+ 3.
C)
2x
4x+ 3
+4x+ 3.
D)
x
2 4x+ 3
–4x+ 3.
E)
x
2 4x+ 3
+4x+ 3.
23)
By the definition of a derivative, the derivative of f(x) =x is
23)
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–x
h.
E)
lim
h
0
x+h
h.
24)
If p=8m2– 9m+ 3, then the rate of change of p with respect to m when m= 1 is
24)
A)
1.
B)
7.
C)
–9.
D)
2.
E)
16.
8
25)
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.
25)
A)
1120
B)
112
C)
82.4
D)
12
E)
14
26)
If y=22+x, then dy
dx =
26)
A)
22.
B)
5.
C)
9.7.
D)
1.
E)
0.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
27)
The revenue R from the sale of x units of a product is R=9
x+ 18x. The number of units
sold after t weeks of advertising is x= 20 +6
t. Find dR
dt when t= 6.
27)
28)
Find all values of x for which the curve y= 2x4–x2 has a horizontal tangent line.
28)
29)
True or False: If a function is continuous at x =a, then it is differentiable at x =a.
29)
30)
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.
30)
9
31)
If the output function is given by Q(x) = 150 +3x2, find the slope of the curve at the point
where x= 25.
31)
32)
The demand for a product is given by D(x) =850
x
– 1. Rewrite this function so that all
terms have the form xn.
32)
33)
Find the slope of the curve y=x+ 4 at the point where x= 5.
33)
34)
Find y’ if y=x2+ 1.
2x3– 1 .
34)
35)
If the output function is given by Q(x) =x2+ 1320, find the slope of the curve at the point
where x= 1.
35)
36)
The demand for a product is given by the equation p=300
q3/2 . Find the marginal demand
without using the quotient rule.
36)
37)
Find y if y=e.
37)
38)
Find the rate of change of y=x(x2+ 9x+ 3) with respect to x.
38)
39)
Find y if y=1
4–x
2.
39)
10
40)
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.
40)
41)
Find the derivative of y=x5+7x+ 3
x3+5x+ 1 .
41)
42)
Find all values of x for which the curve y=x2+ 6x– 4 has a horizontal tangent line.
42)
43)
Find v if v=t5/7(t2/7+ 3t).
43)
44)
The revenue R from the sale of x units of a product is R=2
x+ 10x. The number of units
sold after t weeks of advertising is x= 7 +12
t. Find dR
dt when t= 3.
44)
45)
Suppose a person learns y items in x hours, as given by y= 50 x. Find dy
dx .
45)
46)
Find y if y= (0.003)4.
46)
47)
If y=(6u2–7)3 and u=(9 – 2x)5, then by direct use of the chain rule find dy
dx and evaluate
when x= 5.
47)
48)
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.
48)
49)
The demand D for a product at price x is given by D=35
x– 1 +40. Find d
dx
35
x– 1 + 40 and
discuss what happens at x= 1.
49)
50)
Suppose the position function of an object moving along a number line is given by s=f(t) =
2t2+ 11, where t is in seconds and s is in meters.
(a) Find the average velocity over the interval 3, 3.1 .
(b) Find the velocity when t= 3.
50)
51)
The total cost (in dollars) of producing x portable radios per day is
C(x) = 1000 + 100x–0.5x2. The marginal cost of producing x radios is found by taking the
derivative of C(x). What is the marginal cost of producing x radios?
51)
52)
The weekly output of a certain product is Q(x) = 200x+6x2 where x is the number of
workers. Find the equation of the tangent line to this equation when there are 60 workers.
52)
53)
Let p= 225 –3q2 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= 10? Assume that p is in dollars.
53)
54)
Suppose that the cost C of processing the contaminated water at an industrial site so that
only p percent of the contaminants escape is given by C(p) =750,000 – 7500p
p. Find dC
dp
without using the quotient rule.
54)
12
55)
If the profit for a cat food company (in hundred of dollars) is given by the equation P(x) =
178x–3x2, use the derivative function on your graphing calculator to find the slope of the
tangent to this curve at x= 243.
55)
56)
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.
56)
57)
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
57)
58)
Suppose the number of calories of heat required to raise 1 gram of water (or ice) from
–40°C to x°C is given by f(x) =
1
2x+ 20, if –40 x< 0
x+ 100, if 0 x
. Is this function differentiable for
all values of x?
58)
59)
If an object moves horizontally according to x = 5t– 2 and vertically according to y= 2x2+
x, find its vertical speed, dy
dt .
59)
60)
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’.
60)
13
61)
If a manufacturer‘s average cost equation is c=350
q– 422q+17q2, find the total cost
function c, and the marginal cost function dc
dq . What is the marginal cost when 20 units are
produced?
61)
62)
When a ball is thrown downward at a speed of 35 feet/s from a height of 1000 feet, its
height H in feet after t seconds is given by H= 1000 – 35t –16t2. Find dH
62)
63)
Differentiate: g(x) =3x
x + 4
63)
64)
Find the slope of the curve y=4
2 –x at the point (3, –4).
64)
65)
Find the equation of the tangent line to the graph of the curve y=
7(x2–8)3 at the point
(3, 1).
65)
66)
Differentiate: f(x) = (3x2 + x)(8x– 7)
66)
14
67)
The reaction R to the injection of a drug is a function of the dosage x according to the
equation R(x) =x2450 –x
4. Find the value of the derivative of this function when x= 10.
67)
68)
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
8.
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 3
milligrams.
68)
69)
The demand for a product is given by the equation p=220
q. Find the marginal demand
without using the quotient rule.
69)
70)
If the revenue function is given by R(x) = 11x+ 9, find the derivative of the revenue
function (this is called the marginal revenue).
70)
71)
By direct use of the definition of a derivative, find d
dx f(x) if f(x) =2x2– 3x.
71)
72)
By direct use of the definition of derivative, find d
dx (f(x)) if f(x) =1
x + 6 .
72)
73)
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?
73)
15
74)
Find y if y=x2+ 2x + 3
x.
74)
Explanation:
75)
If the profit for a car company (in thousands of dollars) is given by the equation 400 –6x2,
find the slope of the tangent to this curve at (1, 394).
75)
Explanation:
76)
Differentiate: f(x) =2x– 9
x + 4
4
76)
Explanation:
77)
A stereo company sells 300 car stereos per month at a price of $280 per car stereo. Market
research indicates that they can sell one additional stereo for each $1 they reduce the price.
In this case the total revenue is given by: R(x) = (300 +x)(280 –x) where x is the number of
additional stereos above 300. Find the marginal revenue function using the product rule.
77)
Explanation:
78)
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.
78)
Explanation:
79)
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.
79)
Explanation:
80)
If the output function is given by Q(x) = 5x+ 122, find the slope of the curve at the point
where x= 7.
80)
Explanation:
81)
By direct use of the definition of a derivative, find d
dx f(x) if f(x) =x2+ 4.
81)
82)
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.
82)
83)
The typing speed of a certain computer student varies with the number of hours of practice
x according to S= 12 0.6x+ 5. Find S’.
83)
84)
Find y if y=x–2–x+x–4/7.
84)
85)
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.
85)
86)
The average cost c of producing q units of a product is given by
c= 0.001q2– 0.2q+ 11 +15,000
q. Find the marginal cost function.
86)
87)
Differentiate: g(x) =(3x– 9)3
(5x– 7)7
87)
88)
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.
88)
89)
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 .
89)
90)
Find the slope of the curve y=2x+ 5
x– 3 at the point (4, 13).
90)
91)
If f(x) is defined by f(x) =x2,if x< 1
x, if 1 x, then use your graphing calculator to graph the
functions g(x) =x2 and h(x) =x to determine:
(a) if f(x) is continuous for all values of x.
(b) if f(x) is differentiable for all values of x.
91)
92)
Find y’ if y= (2x+ 1) x– 2.
92)
93)
If the revenue function is given by R(x) = 10x, find the derivative of the revenue function
(this is called the marginal revenue).
93)
94)
A soda stand usually sells 400 sodas per day at $0.70 each. A business student’s research
tells her that for every $0.10 decrease in price, the stand will sell 55 more sodas per day.
The revenue function for the soda stand is given by: R(x) = (0.70 – 0.1x)(400 + 55x) where x
is the number of $0.10 reductions in price. Find dR
dx , the marginal revenue.
94)
95)
A manufacturer will sell N(x) cars after spending $x thousand on advertising, where N(x) =
1000 –3780
x. Find the equation of the tangent line to this curve when x= 20.
95)
18
96)
The average cost c of producing q units of a product is given by c=4q
q+ 2 +10,000
q. Find the
marginal cost function.
96)
Explanation:
97)
Explanation:
By direct use of the definition of derivative, find d
97)
98)
Explanation:
If the cost C of removing p percent of the particulate pollution from the exhaust gases at an
industrial site is given by C(p) =6200p
98)
99)
Find y if y=1
x3.
99)
Explanation:
100)
By direct use of the definition of a derivative, find d
dx f(x) if f(x) = 4x– 3.
100)
Explanation:
101)
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.
101)
Explanation:
102)
Find y if y=3
4x4/3 –1
5x1/4 +x–5/6.
102)
Explanation:
103)
If the revenue function is given by R(x) = 6x+ 4, find the derivative of the revenue function
(this is called the marginal revenue).
103)
104)
Find dy
dx where y=
79x3– 8x+ 5.
104)
105)
If the revenue for a certain company is given by R(x) = 500, find dR
dx .
105)
106)
If the demand equation is D(x) = 2 +4
x, then find an equation of the tangent to this curve
when x= 3. Use your graphing calculator to verify that this line is tangent to D(x) when x=
3.
106)
107)
Find y’ if y=
4(2x+5)3.
107)
108)
If the output function is given by Q(x) =3x2+ 1, find the slope of the curve at the point
where x= 4.
108)
109)
The sales s of a product vary with the number of weeks of an advertising campaign
according to s= 1 +5
t+ 2 . Use the power rule to find ds
dt .
109)
20