Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If y= ln 2x+ 7, then y’ =
1)
A)
(2x+7)3.
B)
1
2x+ 7 .
C)
2
2x+ 7 .
D)
1
2x+ 7 .
E)
1
(2x+7)3.
2)
If y=(3x2+ 5)(8x–9)2
(1 +x)4(4 +x2), logarithmic differentiation gives y’ =
2)
A)
6x
3x2+ 5
+16
8x– 9 +4
1 +x+2x
4 +x2.
B)
(3x2+ 5)(8x– 9)2
(1 +x)4(4 +x2)
6x
3x2+ 5
+16
8x– 9 –4
1 +x–2x
4 +x2.
C)
(3x2+ 5)(8x–9)2
1 +x4(4 +x2)
6x
3x2+ 5
+16
8x– 9 +4
1 +x+2x
4 +x2.
D)
6x
3x2+ 5
+16
8x– 9 –4
1 +x–2x
4 +x2.
E)
8x3– 3x2+ 4x– 1
(1 +x)8(4 +x2) 2.
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)
–1.5.
B)
–4.
C)
–0.5.
D)
–6.
E)
–1.
4)
If 2x2–xy +y2= 4, then dy
dx =
4)
A)
2(2x+y)
x.
B)
x2=y2.
C)
2y+ 4x
x.
D)
4x– 1 + 2y.
E)
y –4x
2y – x .
5)
If y= ln x2– 4x– 5
x+ 2 , then dy
dx =
5)
A)
x2– 4x– 5
x+ 2
x+ 2
x2– 4x– 5 .
B)
x+ 2
x2– 4x– 5 .
C)
2(x– 2)
x2– 4x– 5
–1
x+ 2 .
D)
x2– 4x– 5
x+ 2
2(x – 2)
x2– 4x– 5
–1
x+ 2 .
E)
eln(x2–4x–5) – ln(x+2) .
2
6)
If f(x) =e(4x+1)2, then f’(x) =
6)
A)
2(4x+ 1)e(4x+1)2.
B)
e2(4x+1) .
C)
e8(4x+1) .
D)
8(4x+ 1)e(4x+1)2.
E)
(4x+ 1)e(4x+1)2– 1.
7)
If f(x) =x3x + 1, then f’(x) =
7)
A)
3x+ 1
x+ 3 ln x.
B)
x3x + 1 3x+ 1
x+ 3 ln x.
C)
(3x+ 1)x3x.
D)
(2x+ ln x)x3x + 1.
E)
(ln x)x3x.
8)
If 7x2+ 4y2= 1, then dy
dx =
8)
A)
1 – 14x
8y.
B)
–7x
4y.
C)
7x+ 4y.
D)
14x+ 8y.
E)
4y
7x.
3
9)
If f(x) =x2+ 1
e3x, then f’(x) =
9)
A)
2x– 3x2– 3
e6x.
B)
2x–x2– 1
e6x.
C)
2x–x2– 1
e3x.
D)
2x– 3x2– 3
e3x.
E)
2x
e3x+2.
10)
An approximation of one root of x2– 3x+ 1 = 0 is x1= 2. Apply Newton’s method twice to find the
approximation x3.
10)
A)
2.66667
B)
2.61820
C)
3.00000
D)
2.23814
E)
2.49125
11)
If y= (x2+ 5x– 1)e2x, then y’ =
11)
A)
(3x2– 10x+ 7)e2x.
B)
(x2+ 7x+ 4)e2x.
C)
(2x+ 5)e2x–1
2x.
D)
(2x+ 5)32x.
E)
(2x2+ 12x+ 3)e2x.
4
12)
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
12)
A)
100
9.
B)
283
4.
C)
0.87.
D)
0.87e.
E)
100e
9.
13)
If x2+y2= 5, find y” when x= 1 and y= 2.
13)
A)
3
4
B)
–7
4
C)
0
D)
–5
8
E)
1
14)
An approximation of one root of x4– 2x– 3 = 0 is x1= 1. Apply Newton’s method twice to find the
approximation x3.
14)
A)
2.58138
B)
2.32075
C)
2.46182
D)
1.46328
E)
1.51926
15)
If y= (ln 2)2, then dy
dx =
15)
A)
0.
B)
2eln 2.
C)
2 ln 2.
D)
1
(ln 2)2.
E)
eln 2.
5
16)
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
16)
A)
–13
2.
B)
–9
2500 .
C)
–9.
D)
–2500
9.
E)
–2
13 .
17)
If y= (x2+ 3x+1)2, then y” =
17)
A)
8(2x+3)2.
B)
2(6x2+ 18x+ 11).
C)
2(3x2+ 9x+ 5).
D)
(2x+3)2.
E)
4.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
18)
Find dy
dx if 4x2y– xey=x–y + 2
18)
19)
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.
19)
20)
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.
20)
6
21)
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 – 30x+ 14y= 0. Find dy
dx .
21)
22)
Find dy
dx if y2x + 4 + ln(x + 2) =xy3
22)
23)
Find an equation of the tangent line to the curve y= ln(x+ 3) when x= – 2.
23)
24)
Determine the point elasticity of the demand equation p=100
q+ 2 , where p> 0 and q> 0.
24)
25)
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.
25)
26)
If f(x) = 8x3– 6x2+ 7x– 2, find f”’(x).
26)
27)
Find y’ if y= ln(2x2– 3).
27)
7
28)
If y=x+ 1
(x+3)33x+ 4
, use logarithmic differentiation to find y’.
28)
29)
Differentiate: g(x) = 4xe5x– 7
29)
30)
If y=(x+ 2)3(x2– 7)
(3x+1)2(x+ 8) , use logarithmic differentiation to find y’.
30)
31)
If C(x), the cost to produce x units of a product, is C(x) = 4x3+ 13x+ 8 and the marginal
cost function is C’(x), then find the rate of change of the marginal function with respect to x
when x= 9.
31)
32)
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.
32)
33)
Find y’ if y=e2x+1
(1 – 2x)2.
33)
34)
Determine the point elasticity of the demand equation pq +p+ 50q = 10,000 when p= 200.
34)
8
35)
Find y’ if y= ln(x2) +ln3x.
35)
36)
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).
36)
37)
The location of an Earth–orbiting satellite in a grid centered at the center of the Earth is
given by 8x2+ 11y2= 4995. Find d2y
dx2.
37)
38)
If y=x3+3x, then find y’.
38)
39)
If y=x ln x, then find y(4).
39)
40)
Find y’ if y=ln x
x.
40)
41)
If f(x) = 4x3+ 7x2– 5x– 3, find f”’(x).
41)
42)
Find y’ if y=x3x.
42)
9
43)
An approximation of one root of x4– 8x+ 4 = 0 is x1= 1. Apply Newton’s method twice to
find the approximation x3.
43)
44)
If f(x) =ex2+1, find f”(x).
44)
45)
For an isosceles right triangle, the length of the hypotenuse h is related to the length of a
side x by
2x2=h2. Find dh
dx , the rate of change of the hypotenuse with respect to to a side.
45)
46)
If y=f(x) =ex2+ 3, find the (a) relative and (b) percentage rate of change of y when x= 0.5.
46)
47)
At a volleyball game concession stand, the profit P from selling x number of T–shirts (in
hundreds) is given by P= 8x– 4x 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.
47)
48)
If f(x) = ln x2, find d4
dx4f(x) .
48)
49)
Suppose a population is growing according to the equation P=100et. Find the rate of
growth of the population, dP
dt .
49)
50)
Find an equation of the tangent line to the curve x2+y2+xy = 16 at the point (0, 2).
50)
51)
Find y’ if y=10x2+ 1.
51)
52)
Find y’ if y=e2x+1.
52)
53)
The total revenue from the sales of a certain product are given by R(x) =2000x
ln(3x+ 10) . Find
the marginal revenue.
53)
54)
The altitude (in feet) of a rocket t seconds into flight is given by h= – t3+ 80t2+ 750t+ ln t+
200, t
0. Use Newton’s method to approximate when the rocket will hit the ground.
54)
55)
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?
55)
56)
Suppose that the supply of q units of a product at price $p is given by q= 10 + 25 ln p. Find
dq
dp .
56)
11
57)
If x2+y2= 4, use implicit differentiation to find d2y
dx2 and simplify your answer. Note: dy
dx
should not appear in your final answer.
57)
58)
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.
58)
59)
Find y’ if y=x2 ln(2x– 3)
2x– 3 .
59)
60)
Determine the point elasticity of the demand equation p= – q+ 30, where p> 0 and q>
0.
60)
61)
Find y’ if xy =y2+ 1.
61)
62)
If y=e2x2–3, then find y” at x= 0.
62)
63)
The altitude (in feet) of a rocket t seconds into flight is given by h= – t3+ 20t2+ 900t+ 10, t
0. Use Newton’s method to approximate when the rocket will hit the ground.
63)
12
64)
Find the second derivative y” from y= (11 – 3x– 5x3)7/5 .
64)
65)
Determine the point elasticity of the demand equation p2+ 3p + q = 72, when p= 5.
65)
66)
Find y’ if y=(2x+ 3)3x– 4
66)
67)
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.
67)
68)
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.
68)
69)
Find y’ if y=3(4x– 5)7(9 –x)6
(8x– 3)5x6e–2x
69)
70)
In psychology, the Weber–Fechner law for stimulus response is R=k ln S
S0 where R is the
response, S is the stimulus, and S0 is the lowest level of stimulus that can be detected. Find
dR
ds .
70)
13
71)
Use Newton’s method to approximate the root of x3+ 5x– 1 = 0 that lies between 0 and 1.
Continue the approximation procedure until the difference of two successive
approximations is less than 0.0001.
71)
72)
If y= ln x3+ 3x– 1
x2+ 2x– 1
, then find dy
dx .
72)
73)
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).
73)
74)
Suppose the demand equation for the manufacturer’s product is p= 100e–0.04q, where p is
the price per unit for q units. Find the marginal revenue function.
74)
75)
An approximation of one root of x4– 6x+ 3 = 0 is x1= 1. Apply Newton’s method twice to
find the approximation x3.
75)
76)
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.
76)
77)
If p=qe2q – 4, find the rate of change of p with respect to q when q= 2.
77)
78)
Find y’ if y=(3x– 7)(9x– 4)(2x +7)
(3x + 4)(5x + 4)(2x + 5)
78)
79)
If f(x) = (2x–5)4, find f”’(x).
79)
80)
The total revenue from the sales of a certain product are given by R(x) =3000x
ln(5x+ 20) . Find
the marginal revenue.
80)
81)
Find y’ if ln(xy)+ y = 2.
81)
82)
Differentiate: f(x) =610x3
82)
83)
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.
83)
84)
If y=x2(x+1)4
x2+ 4 , use logarithmic differentiation to find y’.
84)
15
85)
Suppose a population is growing according to the equation P=50et. Find the rate of
growth of the population, dP
dt .
85)
86)
If the total profit (in dollars) from the sale of x lawn mowers is P(x) = 30x– 0.03x2– 750 +
ln(x), use Newton’s method to approximate the break–even points for profit. (Note: There
are 2 break–even points; one is between 0 and 40, and the other is between 900 and 1000.)
86)
87)
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 .
87)
88)
If y= (4x–3)432x+ 1, use logarithmic differentiation to find y’.
88)
89)
Find y’ if y=x2e3x.
89)
90)
The total cost for a product is given by C(x) = 36 + 5 ln(3x+ 9). Find the rate of change of
C”(x).
90)
91)
Find y’ if y=x3 ln(4x+ 5).
91)
92)
Determine the point elasticity of the demand equation 3p2q= 5000 + 2000p2, when p=
50.
92)
16
93)
The location of an Earth–orbiting satellite in a grid centered at the center of the Earth is
given by 5x2+ 7y2= 3000. Find d2y
dx2.
93)
94)
If f(x) =x2+ 25, find f”(x).
94)
95)
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.
95)
96)
If y=eln(x3+ 2x + 1), then find y’.
96)
97)
The total cost for a product is given by C(x) = 1750 + 300 ln(x+ 2). Find the rate of change
of C”(x).
97)
98)
The intensity of an earthquake is measured on the Richter scale. The reading R is given by
R= log I
I0, where I is the intensity and I0 is a standard minimum intensity. If I0= 5, find
dR
dl , the rate of change of the Richter scale reading with respect to the intensity.
98)
99)
Find y’ if y= – 3e4x2– 5x + 3.
99)
17
100)
If the position of a particle moving along the x–axis at a given time is given by x(t) =t4+ 5
t3– 7t+ 19, find all the higher–order derivatives of this function.
100)
101)
Find y’ if 3x2– 7y2= 8.
101)
102)
Find y’ if y= ln ln(2x+ 3) .
102)
103)
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.
103)
104)
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?
104)
105)
Find y’ if y=xex
x+ 1 .
105)
106)
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.
106)
18
107)
Determine the point elasticity of the demand equation: q=p2– 10p+ 200.
107)
108)
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 .
108)
109)
Suppose that a company can produce 15,000 units when the number of hours of skilled
labor y and unskilled labor x satisfy 500 = (x+1)1/4(y+9)1/5. Find dy
dx , the rate of change of
skilled labor hours with respect to unskilled labor hours.
109)
110)
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 .
110)
111)
Use logarithmic differentiation to find dy
dx from y=xx3.
111)
112)
Find dy
dx if 3x2+ 7xy +y2= 19
112)
113)
If y=x2+ 1
x+ ln x, then find y’.
113)
19