Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If f(x) =x2+ 1
e3x, then f’(x) =
1)
A)
2x– 3x2– 3
e6x.
B)
2x– 3x2– 3
e3x.
C)
2x
e3x+2.
D)
2x–x2– 1
e6x.
E)
2x–x2– 1
e3x.
2)
If 2x2–xy +y2= 4, then dy
dx =
2)
A)
2(2x+y)
x.
B)
x2=y2.
C)
4x– 1 + 2y.
D)
y –4x
2y – x .
E)
2y+ 4x
x.
1
3)
The demand function for a manufacturer’s product is given by p= 500 – 5q–q2 where p is the price
per unit when q units are demanded. The point elasticity of demand when q = 5 is
3)
A)
–4.
B)
–6.
C)
–0.5.
D)
–1.
E)
–1.5.
4)
If y= ln 2x+ 7, then y’ =
4)
A)
1
2x+ 7 .
B)
(2x+7)3.
C)
1
(2x+7)3.
D)
2
2x+ 7 .
E)
1
2x+ 7 .
5)
If y= (ln 2)2, then dy
dx =
5)
A)
2 ln 2.
B)
0.
C)
2eln 2.
D)
1
(ln 2)2.
E)
eln 2.
6)
An approximation of one root of x4– 2x– 3 = 0 is x1= 1. Apply Newton’s method twice to find the
approximation x3.
6)
A)
2.32075
B)
1.46328
C)
2.58138
D)
1.51926
E)
2.46182
7)
If f(x) =e(4x+1)2, then f’(x) =
7)
A)
(4x+ 1)e(4x+1)2– 1.
B)
e2(4x+1) .
C)
2(4x+ 1)e(4x+1)2.
D)
8(4x+ 1)e(4x+1)2.
E)
e8(4x+1) .
8)
If f(x) =x3x + 1, then f’(x) =
8)
A)
x3x + 1 3x+ 1
x+ 3 ln x.
B)
(3x+ 1)x3x.
C)
(2x+ ln x)x3x + 1.
D)
(ln x)x3x.
E)
3x+ 1
x+ 3 ln x.
9)
If y= (x2+ 3x+1)2, then y” =
9)
A)
4.
B)
2(3x2+ 9x+ 5).
C)
2(6x2+ 18x+ 11).
D)
(2x+3)2.
E)
8(2x+3)2.
3
10)
If x2+y2= 5, find y” when x= 1 and y= 2.
10)
A)
1
B)
–7
4
C)
0
D)
–5
8
E)
3
4
11)
If y= (x2+ 5x– 1)e2x, then y’ =
11)
A)
(x2+ 7x+ 4)e2x.
B)
(2x2+ 12x+ 3)e2x.
C)
(2x+ 5)32x.
D)
(2x+ 5)e2x–1
2x.
E)
(3x2– 10x+ 7)e2x.
12)
If y= ln x2– 4x– 5
x+ 2 , then dy
dx =
12)
A)
2(x– 2)
x2– 4x– 5
–1
x+ 2 .
B)
x2– 4x– 5
x+ 2
2(x – 2)
x2– 4x– 5
–1
x+ 2 .
C)
eln(x2–4x–5) – ln(x+2) .
D)
x2– 4x– 5
x+ 2
x+ 2
x2– 4x– 5 .
E)
x+ 2
x2– 4x– 5 .
13)
If y=(3x2+ 5)(8x–9)2
(1 +x)4(4 +x2), logarithmic differentiation gives y’ =
13)
A)
(3x2+ 5)(8x–9)2
1 +x4(4 +x2)
6x
3x2+ 5
+16
8x– 9 +4
1 +x+2x
4 +x2.
B)
6x
3x2+ 5
+16
8x– 9 –4
1 +x–2x
4 +x2.
C)
6x
3x2+ 5
+16
8x– 9 +4
1 +x+2x
4 +x2.
D)
8x3– 3x2+ 4x– 1
(1 +x)8(4 +x2) 2.
E)
(3x2+ 5)(8x– 9)2
(1 +x)4(4 +x2)
6x
3x2+ 5
+16
8x– 9 –4
1 +x–2x
4 +x2.
14)
An approximation of one root of x2– 3x+ 1 = 0 is x1= 2. Apply Newton’s method twice to find the
approximation x3.
14)
A)
2.66667
B)
3.00000
C)
2.49125
D)
2.23814
E)
2.61820
15)
The demand function for a manufacturer’s product is given by p= 20 – 0.02q, where p is the price
per unit when q units are demanded. The point elasticity of demand when q = 100 is
15)
A)
–9
2500 .
B)
–2500
9.
C)
–2
13 .
D)
–13
2.
E)
–9.
16)
The average cost c for producing q units of product is given by c=10,000eq/900
q. At a production
level of 900 units, the marginal cost is
16)
A)
100
9.
B)
283
4.
C)
0.87e.
D)
0.87.
E)
100e
9.
17)
If 7x2+ 4y2= 1, then dy
dx =
17)
A)
–7x
4y.
B)
4y
7x.
C)
14x+ 8y.
D)
7x+ 4y.
E)
1 – 14x
8y.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
18)
Find the point elasticity of the demand equation p= 15 – 0.3q for the value q= 300, and
determine whether demand is elastic, inelastic or has unit elasticity.
18)
19)
If f(x) =2
x2, find f”(x).
19)
20)
Suppose that P, the proportion of people affected by a certain disease, is described by ln
P
1 –P= 0.9t, where t is the time in months. Find dP
dt , at the rate at which P grows with
respect to time.
20)
6
21)
The height h(t) of a rock dropped off of a 200 foot building is given by h(t) = 200 – 16t2,
where t is the time measured in seconds. Find d2h
dt2, the acceleration of the rock, when t= 3.
21)
22)
Suppose a population is growing according to the equation P=50et. Find the rate of
growth of the population, dP
dt .
22)
23)
If f(x) = ln x2, find d4
dx4f(x) .
23)
24)
Determine the point elasticity of the demand equation 3p2q= 5000 + 2000p2, when p=
50.
24)
25)
If y=eln(x3+ 2x + 1), then find y’.
25)
26)
Find y’ if y= ln(2x2– 3).
26)
27)
Determine the point elasticity of the demand equation (p+ 1) q+ 3 = 1000, when p= 24.
27)
28)
Find y’ if y=log2(4x+ 5).
28)
7
29)
Suppose that the supply of q units of a product at price $p is given by q= 10 + 25 ln p. Find
dq
dp .
29)
30)
If C(x), the cost to produce x units of a product is C(x) = 6x3+ 11x+ 124 and the marginal
cost function is C’(x), then find the rate of change of the marginal function with respect to
x.
30)
31)
Find dy
dx if 3x2+ 7xy +y2= 19
31)
32)
Determine the point elasticity of the demand equation pq +p+ 50q = 10,000 when p= 200.
32)
33)
If y=ex×ex2, then find y‘.
33)
34)
The demand function for a manufacturer’s product is given by p= 300 –q2, where p is the
price per unit when q units are demanded.
(a) Determine the point elasticity of demand when q= 5.
(b) For q= 5, is demand elastic, inelastic, or does it have unit elasticity?
(c) For what value of q does demand have unit elasticity?
34)
35)
The total sales x ( in hundred thousands) for a music CD t months after the CD is released
can be approximated by x=120t2
t2+ 90 . Use logarithmic differentiation to find dx
dt .
35)
8
36)
Suppose the number of fleas N (in thousands) in a given area is related to the number of
pounds of insecticide x sprayed on the nesting areas according to Nx – 10x+N= 300. Find
dN
dx , the rate of change of the number of fleas with respect to the number of pounds of
insecticide.
36)
37)
Find y’ if y= ln (x2+5)5(3 –4x)4.
37)
38)
If the point elasticity of demand for a product is –2.3 and the price of the product increases
3%, what is the approximate percentage decrease in demand?
38)
39)
The position of an object moving on the x–axis is given by x(t) = 12t3+ 32t+3, where t is the
time in seconds. Find x”(t), the acceleration of the rock, when t= 6.
39)
40)
An object moving along the circle x2+y2= 36. What is dy
dx , the rate of change of the
y–coordinate of the particle with respect to the x–coordinate of the particle?
40)
41)
The demand function for a manufacturer’s product is given by p=400
q+ 2 , where p is the
price per unit when q units are demanded.
(a) Find the point elasticity of demand when q= 100.
(b) For q= 100, is demand elastic, inelastic, or does it have unit elasticity?
41)
42)
If y=x2+ 3
x+ 4
3
, use logarithmic differentiation to find y’.
42)
9
43)
The altitude (in feet) of a rocket t seconds into flight is given by h= – 2t3+ 114t2+ 480t+ 1, t
0. Use Newton’s method to approximate when the rocket will hit the ground.
43)
44)
The total revenue from the sales of a certain product are given by R(x) =2255x
ln(7x+ 50) . Find
the marginal revenue.
44)
45)
If y=x2+ 1
x+ ln x, then find y’.
45)
46)
Find y’ if y= (x+1)5x.
46)
47)
Use Newton’s method to approximate the root of x3– 6x+ 1 = 0 that lies between 0 and 1.
Continue the approximation procedure until the difference of two successive
approximations is less than 0.0001.
47)
48)
Find y’ if xy =y2+ 1.
48)
49)
Find y’ if y=x3x.
49)
50)
Find y’ if y=10x2+ 1.
50)
10
51)
Suppose you deposit $1000 in a savings account with 3.6% interest compounded monthly.
Then A, the amount of money in the account after t years, is given by A= 1000(1.003)12t.
Use logarithmic differentiation to find dA
dt .
51)
52)
The location of an Earth–orbiting satellite in a grid centered at the center of the Earth is
given by 2x2+ 6y2= 5000. Find d2y
dx2.
52)
53)
If y=x+ 1
(x+3)33x+ 4
, use logarithmic differentiation to find y’.
53)
54)
If the cost to produce x units of a product is C(x) = 0.5x2+ 9x+ 440 and the marginal cost
function is C’(x), then find C”(x), the rate of change of the marginal function with respect
to x.
54)
55)
The volume of a company’s sales y (in thousands of dollars) is related to its advertising
expenditures x (in thousands of dollars) by the equation xy – 11x+ 25y= 0. Find dy
dx .
55)
56)
Find y’ if ln(xy)+ y = 2.
56)
11
57)
If y=(x+ 2)3(x2– 7)
(3x+1)2(x+ 8) , use logarithmic differentiation to find y’.
57)
58)
Suppose that c= (0.2q–20)3+ 20,000 is a total cost function, where c is the total cost (in
dollars) of producing q units of a product. How fast is marginal cost changing when q=
150?
58)
59)
If 2x2+ 3y2= 8, use implicit differentiation to find d2y
dx2 and simplify your answer. Note:
dy
dx should not appear in your final answer.
59)
60)
Suppose a population is growing according to the equation P=100et. Find the rate of
growth of the population, dP
dt .
60)
61)
Differentiate: f(x) =610x3
61)
62)
If the demand equation for a product is q=100
p+ 2 , where q is the number of units
demanded at price p per unit, find the elasticity of demand when p= 48.
62)
63)
Find y’ if y=x2e3x.
63)
64)
Determine the point elasticity of the demand equation pq = 81, where p> 0 and q> 0.
64)
65)
Medical research has found that between heartbeats the pressure in the aorta of a normal
adult is a function of time and can be modeled by the equation P= 95e–0.491t. Find dP
dt , the
rate at which the pressure changes with respect to time.
65)
66)
The total cost for a product is given by C(x) = 212 + 3 ln(2x+ 1). Find the rate of change of
C”(x).
66)
67)
Suppose the number of fleas N (in thousands) in a given area is related to the number of
pounds of insecticide x sprayed on the nesting areas according to Nx – 9x+N= 250. Find
dN
dx , the rate of change of the number of fleas with respect to the number of pounds of
insecticide.
67)
68)
Find an equation of the tangent line to the curve y= ln(x+ 3) when x= – 2.
68)
69)
Find y’ if y=3x(4x– 6)7
(5x– 9)2(7x– 5)9
3
69)
70)
The total revenue from the sales of a certain product are given by R(x) =3000x
ln(5x+ 20) . Find
the marginal revenue.
70)
13
71)
At a football game concession stand, the profit P from selling x number of T–shirts (in
hundreds) is given by P= 11x– 5x ln x. Find dP
dx . Use your graphing calculator to graph
both P and dP
dx on the same set of axes. Verify that when the graph of P reaches its peak, the
graph of dP
dx crosses the x–axis.
71)
72)
Differentiate: g(x) =e2x– 5
72)
73)
If y=x3+3x, then find y’.
73)
74)
Suppose p= 500 – 0.01q– 0.02 q is the demand equation for a product. Find the rate of
change of quantity with respect to price.
74)
75)
If y=e2, then find y’.
75)
76)
The loudness of sound (L, measured in decibels) perceived by the human ear depends
upon intensity levels (I) according to L= 10 log I
I0, where I0 is the standard threshold of
audibility. If I0= 1, find dL
dl , the rate of change of the loudness with respect to the standard
threshold of audibility.
76)
14
77)
Find y’ if y=e1–x
x2– 1 .
77)
78)
The location of an Earth–orbiting satellite in a grid centered at the center of the Earth is
given by 3x2+ 4y2= 7000. Find d2y
dx2.
78)
79)
Find an equation of the tangent line to the curve x2+y+y2= 13 at the point (–1, 3).
79)
80)
Suppose that a company can produce 12,000 units when the number of hours of skilled
labor y and unskilled labor x satisfy 384 = (x+2)3/4(y+3)1/3. Find dy
dx , the rate of change of
skilled labor hours with respect to unskilled labor hours.
80)
81)
An approximation of one root of x4
2– 2x3+ 5 is x1= 1.5. Apply Newton’s method twice to
find an approximation to the root of the polynomial.
81)
82)
Find all of the higher–order derivatives of x(t) =1
3x3–1
2x2– 4x. Use your graphing
calculator to graph x(t) and x’(t) on the same axes. Verify that when x(t) is increasing
(sloping upward), the value of x’(t) is positive. Repeat this for x’(t) and x”(t) and for x”(t)
and x”’(t).
82)
83)
The demand equation for a product is p= 400 – 0.2q2. Find the rate of change of quantity
with respect to price.
83)
84)
If y= ln x3+ 3x– 1
x2+ 2x– 1
, then find dy
dx .
84)
85)
If the position of a particle moving along the x–axis at a given time is given by x(t) =t3,
find all the higher–order derivatives of this function.
85)
86)
If f(x) =x2+ 25, find f”(x).
86)
87)
The total revenue from the sales of a certain product are given by R(x) =400x
ln(2x+ 7) . Find
the marginal revenue.
87)
88)
The total cost for a product is given by C(x) = 36 + 5 ln(3x+ 9). Find the rate of change of
C”(x).
88)
89)
At a soccer game concession stand, the profit P from selling x number of T–shirts (in
hundreds) is given by P= 7x– 3x ln x. Find dP
dx .
89)
16
90)
Differentiate: f(x) = (10x– 4) ·59x– 4
90)
91)
Use your graphing calculator to graph the equation y=x3–ex+x. Zoom in on the graph
to approximate the roots of the equation. Use Newton‘s method to verify your results.
91)
92)
Suppose a population is growing according to the equation P=99et. Find the rate of
growth of the population, dP
dt .
92)
93)
If y= (7x+5)11, then find y”’.
93)
94)
Find y’ if y= ln x– 1
x+ 1 .
94)
95)
Find y’ if y=e2x+1
(1 – 2x)2.
95)
96)
If f(x) =7
x, find the rate of change of f’.
96)
97)
If the position of a particle moving along the x–axis at a given time is given by x(t) =t3+ 3
t2– 5t+ 12, find all the higher–order derivatives of this function.
97)
17
98)
If f(x) = 8x3– 6x2+ 7x– 2, find f”’(x).
98)
99)
Use your graphing calculator to graph the equation y= ln(x2) – 2e3x+x3+ 11. Zoom in on
the graph to approximate the roots of the equation. Use Newton’s method to verify your
results.
99)
100)
Determine the point elasticity of the demand equation: q=p2– 10p+ 200.
100)
101)
If the total profit (in dollars) from the sale of x dishwashers is P(x) = 40x– 0.02x2– 1000 + 2
ln(x), use Newton’s method to approximate the break–even points for profit. (Note: There
are 2 break–even points; one is between 10 and 50, and the other is between 1900 and
2000.)
101)
102)
Determine the point elasticity of the demand equation p2+ 3p + q = 72, when p= 5.
102)
103)
Find all of the higher–order derivatives of x(t) =x4– 3x2–x. Use your graphing calculator
to graph x(t) and x’(t) on the same axes. Verify that when x(t) is decreasing (sloping
downward), the value of x’(t) is negative. Repeat this for x’(t) and x”(t), for x”(t) and x”’(t),
and for x”’(t) and x(4)(t).
103)
104)
Differentiate: f(x) =e
–2
x2
104)
–2
105)
An approximation of one root of x4– 6x+ 3 = 0 is x1= 1. Apply Newton’s method twice to
find the approximation x3.
105)
18
106)
If ey+y+x= 2, use implicit differentiation to find d2y
dx2 and simplify your answer. Note:
dy
dx should not appear in your final answer.
106)
107)
Use logarithmic differentiation to find dy
dx from y=
3x2+ 2x– 7
5x2– 1 73x– 5
.
107)
108)
Differentiate: f(x) =e6x3– 5x2 + 3x– 4
108)
109)
The height h(t) of a rock thrown upward from the ground at a speed of 132 feet/sec is given
by h(t) = 132t– 16t2, where t is the time measured in seconds. Find d2h
dt2, the acceleration of
the rock at time t.
109)
110)
If y=x
2x– 1 , find d2y
dx2.
110)
111)
Differentiate: f(x) =49 –x2
(x– 7)2
111)
19
Explanation: