73)
Find the indefinite integral –7
(3x5)2dx.
73)
74)
Determine: (x3–x+ 1)(x4– 2x2+ 4x)–5dx
74)
75)
Suppose f(x) =
x
5
et2–et
et+tdt, then find f’(x)
75)
76)
The supply equation for a company is q= 4p2+ 3. Find dp
dq from dq
dp .
76)
77)
Find the exact area of the region bounded by the graphs of y= – x, y= 2x, y= 1, and y= 2.
Also sketch the region.
77)
78)
If the marginal cost (in dollars) for a company is C’(x) = 200, and the cost for producing 8
units is $2250, that is C(8) = 2250, then find C(x).
78)
79)
Scientists use the formula S=kW2/3 to determine an animal’s surface area S (in square
meters) from its weight W (in kilograms), where k is a constant that varies from animal to
animal. Suppose the scientist studies a certain animal with k= 0.3. Use differentials to
approximate the surface area if the animal weighs 65 kilograms. (Hint: Use W= 64.)
79)
80)
A company has determined that its marginal revenue function (in dollars) is given by
C’(x) = 1000 + 0.4x2, where x is the number of units made. Find the total revenue for
making 6 units by evaluating
6
0
(1000 + 0.4x2) dx .
80)
81)
Find the indefinite integral (2x+ 1)(3x– 2)
6dx.
81)
82)
Determine 8x7/3 – 11x5/3
x1/3 dx
82)
83)
Determine: 5x3dx
83)
84)
If the rate of change of a company’s revenue can be modeled by dr
dt = 0.5t, then find r(t) the
revenue function.
84)
85)
Determine: x23x3+ 8 dx
85)
86)
Determine: x2(2x3–5)4dx
86)
87)
Determine (x2+ 4x)ln(x3+ 6x2+ 1)
x3+ 6x2+ 1 dx
87)
88)
Find the exact area of the region bounded by the graphs of y= 9 –x2 and y= 5 – 3x. Also
sketch the region.
88)
89)
Determine: (x2+1)2dx
89)
90)
Use Simpson’s rule with n= 4 to find an approximate value of
2
0
1
4 +x2dx.
90)
91)
An oil company estimates that for the next few months, oil will be pumped (in thousands
of gallons per month) from a producing field at the rate of R(t) =2t
(t2+1)3+ 5, where t is in
months. Find 2t
(t2+1)3+ 5 dt.
91)
92)
A company has determined that its marginal revenue function (in dollars) is given by
C’(x) = 400 + 2x, where x is the number of units made. Find the total revenue for making 12
units by evaluating
12
0
(400 + 2x) dx .
92)
93)
The marginal price for a weekly demand of x bottles of shampoo is given by p’(x) =
–8x
(4x2+70)2. Find p(x).
93)
94)
The marginal cost function for a manufacturer’s product is given by dc
dq = 2q+q3+eq,
where c is in dollars. Find the cost function if fixed costs are $100.
94)
95)
Determine: (3x2+ 4x)(2x3+ 4x2+5)6dx
95)
96)
Determine: ex/2 dx
96)
97)
Suppose the surface area of a 2 meter–tall person changes at the rate of ds
dx = 0.145x–0.625
square meters per kilogram, where x is the weight in kilograms. Find the decrease in
surface area for a person whose weight decreases from 70 kilograms to 75 kilograms by
evaluating
70
75
0.145x–0.625 dx.
97)
98)
Use differentials to approximate ln(0.99).
98)
99)
Determine 4x2 + 4x + 2
2x – 1 dx
99)
100)
Suppose the surface area of a 1.7 meter–tall person changes at the rate of ds
dx = 0.13775
x–0.58 square meters per kilogram, where x is the weight in kilograms. Find the increase in
surface area for a person whose weight increases from 70 kilograms to 75 kilograms by
evaluating
75
70
0.13775x–0.58 dx.
100)
101)
Determine: 43/2 dx
101)
102)
Find dy if y=xex.
102)
103)
Evaluate:
1
0
x2+ 5x+ 4
x2+ 3x+ 1 dx
103)
104)
Determine: 1
4x
dx
104)
105)
Find the differential in terms of x and dx.
y=48 –x2
105)
106)
The marginal revenue for a product is given by dr
dx =
–x
(x2+12)2+ 998. Find r.
106)
107)
The length of a side of a square is increasing at a rate of ds
dt =4
3t1/3(t2/3 + 9) . Find s as a
function of t.
107)
108)
Find the differential of the function in terms of x and dx.
y= ln(x3– 3x+ 1)
108)
109)
Determine: p
5–1
2dp
109)
110)
For the region in the first quadrant bounded by f(x) = 2x+ 1, x= 0, y= 0, and x= 1,
approximate the area by evaluating S4. (Use the right–hand endpoint of each
sub–interval.)
110)
26
111)
Determine: (0.3)2dy
111)
112)
Find the exact area of the region bounded by y=x2– 4x and the x–axis. Also sketch the
region.
112)
113)
A bacteria population is increasing at a rate of dp
dt = 8t7t2. Find p as a function of t.
113)
114)
Determine: 65y+ 1 dy
114)
115)
Determine:
1/4
0
(1 – 4x)4dx
115)
116)
If the marginal cost function is given by C‘(q) =3
4–1
2 3q, find the cost function if C(0) =
1000.
116)
117)
The demand equation for a certain product is p= 200 –q2, where p is the price per unit (in
dollars) for q units. If its supply equation is p= 6q+ 160, find the consumers’ surplus and
the producers‘ surplus when market equilibrium has been established.
117)
118)
Determine: (2x)–1dx
118)
119)
Find dy if y= (5x2+4)3.
119)
120)
Find the differential in terms of x and dx.
y=(4x5– 7x3 + 4)4
120)
121)
Determine: e7–2x dx
121)
122)
After weeks of television promotion, an electronics store finds that its sales change at a rate
given by S’(t) = – t8t2+ 9 + 20t+ 20, where t is the number of days after the advertising
ends and S is in thousands of dollars. Find the total sales for the fourth week after the
advertising ends. Find
4
3
(–t8t2+ 9 + 20t + 20) dt.
122)
123)
Use differentials to approximate the change in the volume V of a sphere if the radius r is
increased from 5 cm to 5.3 cm. (Hint: Use V=4
3r3.)
123)
124)
Determine: e2dx
124)
125)
Use your graphing calculator to find the area between the x–axis and f(x) = – x2+ 7x– 10
and the x–axis on the interval 2 x 5. Verify your result by finding
5
(–x2+ 7x– 10) dx.
125)
126)
A managerial service determines that the rate of increase in maintenance costs (in dollars
per year) for a particular house is given by M’(x) = 2x(3x2+ 70), where x is the age of the
house in years and M(x) is the total (accumulated) cost of maintenance for x years. Find the
total cost for the first 3 years.
126)
127)
Use the trapezoidal rule with n= 3 to find an approximate value of
3
0
1
4 +x2dx.
127)
128)
Determine: 4
x–x
4dx
128)
129)
Determine: 4s+1ds
129)
130)
Determine: –1
2 – 3xdx
130)
131)
Determine: (5x4+ 4x2– 6x+ 5) dx
131)
29
132)
Determine:
4
0
e4x
4dx
132)
133)
The marginal revenue function for a manufacturer’s product is given by dr
dq =450
(q+5)2.
Determine the demand equation for the product.
133)
134)
If the marginal revenue function for a manufacturer’s product is dr
dq = 1000 – 10q– 6q2, find
the demand function.
134)
135)
Determine: x2– 2x+ 1
x3– 3x2+ 3x– 4 dx
135)
136)
Express the area of the shaded region in terms of an integral (or integrals). Do not evaluate
the expression:
136)
30
137)
Use differentials only to find an approximate value of 327.1.
137)
138)
If y” =x2+ex+1 and y’(–1) = 0 and y(–1) = – 1
4, find y.
138)
139)
Use differentials only to find an approximate value of 3.9.
139)
140)
A marketing group is studying population trends. It predicts that the population P of a
certain city will increase at the rate of dP
dt people per year, where dP
dt =15,000
t+ 4 and t is the
number of years past 1980. What is the expected increase in population from 1980 to 2001?
140)
141)
For the region bounded by f(x) = 3x– 1, y= 0, x= 1, and x= 3, approximate the area by
evaluating S4. (Use the right–hand endpoint of each subinterval.)
141)
142)
Find the area of the region enclosed by the graphs of y=x3–x2 and y= 2x.
142)
143)
If y” =ex– 2, and y'(0) =3 and y(0) = 4, find y.
143)
144)
Determine (5x2+ 4x)e5x3+6x2+7dx
144)
31
145)
Determine:
1
0
x(x2–1)5dx
145)
146)
Determine: e2x+e–2x+e2dx
146)
147)
A manufacturer’s marginal cost function is dc
dq = .1q+ 7. Determine the cost involved to
increase production from 40 to 60 units.
147)
148)
Determine: x2+ 6x– 3
xdx
148)
149)
The rate of change in cost of producing a product is given by C’(x) =2x
x2+ 2
+ 1. Find the
increase in cost from producing 20 units to producing 80 units.
149)
150)
The base of a triangle is decreasing at a rate of db
dt = – 3e–t
t. Find b as a function of t.
150)
151)
Determine y2(y2/3 + 3y) dy
151)
32
152)
The producers of a long running television show expect the number of viewers (in
thousands) to decrease at a rate of V‘(x) = – 20 – 3x, where x is the number of weeks after
the television season begins. Find the total decrease in viewers in the first three weeks by
evaluating
3
0
(–20 – 3x) dx .
152)
153)
Find the area of the region bounded by the given equations:
y=x2–x – 9 and y= – x2 + 3x + 7
153)
154)
Determine: 3t3– 4t2–t+ 1
t2dt
154)
155)
The income (in dollars) from a clothing chain is increasing at a rate of f(t) = 500e0.08t where
t is in weeks. Find
6
0
500e0.08tdt, the total income for the chain over the first six weeks.
155)
156)
When an object is moved from one environment to another, its temperature T changes at a
rate given by dT
dt =kCekt, where t is the time (in hours) after changing environments, C is
the temperature difference (original – new) between the environments, and k is a constant.
If the original environment is 40°, the new environment is 100°, and k= – 2, then find T(t).
156)
157)
If y=x, use differentials to approximate the change in y if x changes from 25 to 25.3.
157)
158)
The function f(x) =k(2x–x2) where k is a constant and 0 x 2, is a density function if
2
0
f(x) dx = 1. Find k so that this condition is met.
158)
33
159)
Find the area of the region bounded by the curve y=f(x) =x3–x and the x–axis on the
interval –1,1 . Use your graphing calculator to see a graph of this area.
159)
160)
For the region in the first quadrant bounded by f(x) =x2+ 2, y= 0, x= 0, and x= 1,
approximate the area by evaluating S2. (Use the right–hand endpoint of each
sub–interval.)
160)
161)
Determine: (x+ 1)(x2+2x –5)10 dx
161)
162)
Find the exact area of the region bounded by y=f(x) = 8 – 2x–x2 and the x–axis.
162)
163)
Determine: 1
3x+ 5 –(x7– 5x3)11(7x6– 15x2) dx
163)
164)
The supply equation for a company is q= 4p3– 2p. Find dp
dq from dq
dp .
164)
165)
Suppose that in the United States the rate of change of the percentage of workers who
belong to unions can be modeled by dy
dx = 0.24(0.1x+8)2– 0.02(0.1x+ 8) where x is the
number of years after 1940. Find y.
165)
166)
Scientists use the formula S=kW2/3 to determine an animal’s surface area S (in square
meters) from its weight W (in kilograms), where k is a constant that varies from animal to
animal. Suppose the scientist studies a certain animal with k= 0.1. Use differentials to
approximate the surface area if the animal weighs 124 kilograms. (Hint: Use W= 125.)
166)
167)
Find the area of the region bounded by the curves y=x2– 4 and y= 2x+x2 from x= 1 to x
= 2.
167)
168)
Find the area of the region bounded by the curve y=f(x) =x2– 4x+ 3 and the x–axis on the
interval 0,3 . Use your graphing calculator to see a graph of this area.
168)
169)
Use differentials to approximate 38.1.
169)
170)
Determine: e4x+3dx
170)
171)
A company has determined that its marginal cost function (in dollars) is given by
C’(x) = 1000 + 0.4x2, where x is the number of units made. Find the total cost for making 6
units by finding the area in the first quadrant bounded by y=C’(x) = 1000 + 0.4x2 and the
lines y= 0, x= 0, and x= 6.
171)
172)
Find the indefinite integral 3 3x2+ 5 5x3dx.
172)
173)
If y” = 6x+ 2 and y’(1) = 2 and y(1) = 2, find y.
173)
174)
Determine:
1
–1
(x2– 2x+ 2) dx
174)
35
175)
A factory is dumping garbage into a river at a rate given by dx
dt =et
200 tons per month,
where t is the time in months since the dumping began and x is the number of tons of
garbage. Find the function x(t) which gives the total number of tons of garbage dumped.
175)
176)
Determine:
2
2
3x
x2– 1 dx
176)
177)
A company has determined that its marginal revenue function (in dollars) is given by
R’(x) = 10,000 –x2, where x is the number of units sold. Find the total revenue for selling 4
units by finding the area in the first quadrant bounded by y=R’(x) = 10,000 –x2 and the
lines y= 0, x= 0, and x= 4.
177)
178)
Use differentials and R= 300x+ 50x2–x3 to approximate the change in revenue R (in
dollars) from selling x pounds if the number of pounds increases from 10 to 10.2.
178)
179)
The supply equation for a certain radio is given by p= 0.8 x+ 17 where p is the price in
dollars and x is the number of radios supplied. Use differentials to approximate the price
when 620 radios are supplied. (Hint: Use x = 625.)
179)
180)
The income (in dollars) from a clothing store is increasing at a rate of f(t) = 500e0.08t where
t is in weeks. Find
12
6
500e0.08tdt, the total income for the store between the sixth and
twelfth weeks.
180)
181)
For the region bounded by f(x) = 4 –x, y= 0, x= 0, and x= 3, approximate the area by
evaluating S6. (Use the left–hand endpoint of each subinterval.)
181)
182)
For the region bounded by f(x) = 3x– 1, y= 0, x= 1, and x= 3, approximate the area by
evaluating S4. (Use the left–hand endpoint of each subinterval.)
182)
183)
Determine 7x6– 8x4 + 9x3
2x2dx
183)
184)
Determine: x6x7– 8 dx
184)
185)
Determine: x
x2– 5 dx
185)
186)
Determine:
4
1
x x dx
186)
187)
Determine: (4x3– 6x2+ 7x– 2) dx
187)
188)
Suppose the surface area of a 1.8 meter–tall person changes at the rate of ds
dx = 0.138x–0.6
square meters per kilogram, where x is the weight in kilograms. Find the increase in
surface area for a person whose weight increases from 70 kilograms to 75 kilograms by
evaluating
75
70
0.138x–0.6 dx.
188)
189)
Suppose the rate of savings in a country is given by dS
dt = 3.3t2– 900 t + 32,891.60, where t
is the time in years, and S is the amount of money saved in billions of dollars. Find the
function S(t) which gives the total amount of money saved. You may assume that $0 is
saved at t= 0.
189)
190)
The supply equation for a company is q=300 +p. Find dp
dq from dq
dp .
190)
191)
Evaluate
1
–1
x dx
191)
192)
Determine 37x–4dx
192)
193)
A company has determined that its marginal revenue function (in dollars) is given by
R’(x) = 600 – 0.5x, where x is the number of units sold. Find the total revenue for selling 10
units by evaluating
10
0
(600 – 0.5x) dx .
193)
194)
Determine:
1
x
2–2
xdx
194)
195)
The rate of production of a new line of products is given by dx
dt = 300 +300
5t+ 4 . Find x(t).
195)
196)
A factory is dumping garbage into a river at a rate given by dx
dt =t3/4
400 tons per month,
where t is the time in months since the dumping began and x is the number of tons of
garbage. Find the function x(t) which gives the total number of tons of garbage dumped.
196)
197)
It is estimated that service industries will grow at the rate of R’(t) = 5e1/(t+1) where t is the
time in decades and R(t) is the percent of (non–farming) workers who are employed in the
service industry. If 30 percent of the (non–farming) workers are now in the service
industry, then use Simpson’s rule and n= 4 to approximate
30 +
2
0
5e1/(t+1) dt, the percentage of workers who are working in the service industry in
2 decades.
197)
198)
Find the area between the graph of y=f(x) =x3, and the x–axis on the interval 0,2 . Use
your graphing calculator to see a graph of this area.
198)
199)
Determine: 8(1 – 4x) dx
199)
200)
Determine: x+ 2
x2+ 4x– 2 dx
200)
201)
The supply equation for a company is q= 0.5ep– 3p. Find dp
dq from dq
dp .
201)
202)
Find the differential of the function in terms of x and dx.
y=ex3–x+1
202)
39
203)
Find the exact area of the region bounded by the graphs of x=y2 and x–y– 2 = 0. Also
sketch the region.
203)
204)
Determine: ln(x2+ 3x + 11) 2x+ 3
x2+ 3x+ 11 dx
204)
205)
Determine: 1
43x+ 2
dx
205)
206)
Determine: 23q+4dq
206)
207)
Determine 58x– 7
8x– 7 dx
207)
208)
If the marginal cost for a company is f(x) = 6, find 6 dx, the cost function.
208)
40