70)
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.
70)
Explanation:
71)
Determine: (7x+2)4dx
71)
Explanation:
72)
The demand equation for a product is p=49 – 6q and the supply equation is p= 1 +q.
Determine the consumers’ surplus and the producers’ surplus.
72)
Explanation:
73)
The velocity of an object is given by V(t) =12
t t2– 1
, where t is the time in seconds and V(t)
is the speed in feet/sec. Use n= 6 and Simpson’s rule to approximate
5
2
12
t t2– 1
dt, the
distance moved over the interval 2, 5 .
73)
Explanation:
74)
Determine x– 8
x + 4 dx
74)
Explanation:
75)
If q=p3– 6p2+ 3p– 8, find dp
dq when p= 1.
75)
Explanation:
21
76)
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.
76)
77)
Find the differential of the function in terms of x and dx.
y=ex3–x+1
77)
78)
Find the area between the graph of y=f(x) =e0.3x, and the x–axis on the interval 0,5 . Use
your graphing calculator to see a graph of this area.
78)
79)
If the rate of change of a company’s revenue can be modeled by dr
dt = 3t2, then find r(t) the
revenue function.
79)
80)
Evaluate:
11
5
e3dx
80)
81)
Determine: ln(xe2x)
xdx
81)
82)
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.
82)
83)
Determine: (2x+ 1)e(2x+1)2dx
83)
22
84)
Determine: (0.3)2dy
84)
85)
Determine 4t5(t2– 6t4) dt
85)
86)
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.)
86)
87)
The velocity of an object is given by V(t) =12
t t2– 1
, where t is the time in seconds and V(t)
is the speed in feet/sec. Use n= 6 and the trapezoidal rule to approximate
5
2
12
t t2– 1
dt,
the distance moved over the interval 2, 5 .
87)
88)
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.
88)
89)
The rate of production of a new line of products is given by dx
dt = 300 +300
5t+ 4 . Find x(t).
89)
90)
Find dy if y= (5x2+4)3.
90)
91)
Determine: z2– 3z+ 5
z+ 2 dz
91)
23
92)
The radius of a circle is growing at a rate of dr
dt =5
(t+ 1) ln(t+ 1) . Find r as a function of t.
92)
93)
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.)
93)
94)
Use differentials to approximate ln(0.99).
94)
95)
Determine:
1
–1
(x2– 2x+ 2) dx
95)
96)
If the marginal cost for a company is f(x) = 6, find 6 dx, the cost function.
96)
97)
Determine: 2(x2+ 5)
3dx
97)
98)
Determine:
4
–1
(2x+ 3) dx
98)
99)
A bacteria population is increasing at a rate of dp
dt = 4t39t4. Find p as a function of t.
99)
24
100)
Find dy if y=xex.
100)
101)
The marginal price for a weekly demand of x bottles of shampoo is given by p’(x) =
–8x
(4x2+70)2. Find p(x).
101)
102)
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.)
102)
103)
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.
103)
104)
Find the area of the region bounded by the curves y=x2– 4 and y= 2x+x2 from x= 1 to x
= 2.
104)
105)
Determine x2– 8x + 6
x– 4 dx
105)
106)
An oil tanker is losing oil at a rate of R’(t) =60
t2+ 9
where t is the time in minutes and R(t)
is the radius of the oil slick in feet. Use n= 6 and Simpson’s rule to approximate
5
0
60
t2+ 9
dt,the size of the radius after 5 seconds.
106)
107)
If y= ( x2+ 3x)4, find dx
dy .
107)
108)
Determine: x23x3+ 8 dx
108)
109)
Use differentials to approximate: 25.6.
109)
110)
Find the exact area of the region bounded by the graphs of y= 8 – 2x–x2 and y= 3x+ 2.
Also sketch the region.
110)
111)
For the region bounded by f(x) = 4 –x, y= 0, x= 0, and x= 3, approximate the area by
evaluating S6. (Use the right–hand endpoint of each subinterval.)
111)
112)
If the marginal cost function is given by C‘(q) =3
4–1
2 3q, find the cost function if C(0) =
1000.
112)
26
113)
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.
113)
114)
Determine 2x2 + 13x + 13
x + 5 dx
114)
115)
Determine: e2dx
115)
116)
Find the indefinite integral –7
(3x5)2dx.
116)
117)
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.
117)
118)
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 a rate of R(t) =3t2
(t3+1)4+ 9, where t is in
months. Find 3t2
(t3+1)4+ 9 dt.
118)
27
119)
The demand equation for a certain product is p= 30 – 0.1q, where p is the price per unit (in
dollars) for q units. If its supply equation is p= 0.15q+ 5, find the consumers’ surplus and
the producers‘ surplus when market equilibrium has been established.
119)
120)
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 evaluating
4
0
(10,000 –x2) dx .
120)
Explanation:
121)
If y’ =x+ 2 and y(2) = 5, find y.
121)
Explanation:
122)
The supply equation for a company is q= 0.5ep– 3p. Find dp
dq from dq
dp .
122)
Explanation:
123)
Determine:
8
4
8 –xdx
123)
Explanation:
124)
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.
124)
Explanation:
125)
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.
125)
Explanation:
Explanation:
126)
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.
126)
127)
Find the area of the region enclosed by the graphs of y=x3–x2 and y= 2x.
127)
Explanation:
128)
Determine 5xdx
128)
Explanation:
129)
Find the exact area of the region bounded by y=f(x) =2x + 5, x= – 2, x= 2, and the
x–axis.
129)
Explanation:
130)
Determine: x2(x3+ 2x2– 5) dx
130)
Explanation:
131)
Suppose the surface area of a 1.9 meter–tall person changes at the rate of ds
dx = 0.14x–0.62
square meters per kilogram, where x is the weight in kilograms. Find the decrease in
surface area for a person whose weight decreases from 75 kilograms to 70 kilograms by
evaluating
70
75
0.14x–0.62 dx.
131)
Explanation:
132)
A company has determined that its marginal cost function (in dollars) is given by
C’(x) = 400 + 2x, where x is the number of units made. Find the total cost for making 12
units by finding the area in the first quadrant bounded by y=C’(x) = 400 + 2xand the lines
y= 0, x= 0, and x= 12.
132)
Explanation:
Explanation:
133)
The acceleration of an object after t seconds is given by y” = 32, the velocity at 0 seconds is
given by y’(0) = 0 feet/sec, and the position at 2 seconds is given by y(2) = 64 feet. Find y(t).
133)
134)
Determine: (2x+ 3)ex2+3x+5dx
134)
135)
Use the trapezoidal rule with n= 3 to find an approximate value of
3
0
1
4 +x2dx.
135)
136)
Find the exact area of the region bounded by y=f(x) =ex + x, x= 0, x= 2, and the x–axis.
136)
137)
Determine 8x7/3 – 11x5/3
x1/3 dx
137)
138)
Determine: (x+ 1)(x2+2x –5)10 dx
138)
139)
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.
139)
140)
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).
140)
141)
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 .
141)
142)
Suppose that the points (3, 2), (3.25, 2.5), (3.5, 2.25), (3.75, 2)and (4, 1.5) lie on the graph of
the continuous function f, where f(x) 0. Use Simpson‘s rule and all of the given points to
approximate the area between the graph of f and the x–axis on the interval 3, 4 .
142)
143)
Determine: 4
x–x
4dx
143)
144)
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.
144)
145)
If the marginal revenue function for a manufacturer’s product is dr
dq = 1000 – 10q– 6q2, find
the demand function.
145)
146)
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.
146)
31
147)
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.
147)
148)
Determine: x2– 4x+ 7
x+ 1 dx
148)
149)
The rate of change of the value of a house that cost $250,000 to build can be modeled by dV
dt
= 7.5e0.075t, where t is the time in years since the house was built, and V is the value (in
hundreds of thousands of dollars) of the house. Find V(t).
149)
150)
If the rate of change of a company’s revenue can be modeled by dr
dt = 712t, then find r(t) the
revenue function.
150)
151)
After weeks of radio promotion, a shoe store finds that its sales change at a rate given by
S’(t) = – 12t2+ 300t, where t is the number of days after the advertising ends and S is in
dollars. Find the total sales for the first week after the promotion ends.
Find
7
0
(–12t2+ 300t) dt.
151)
152)
Evaluate:
1
0
x2+ 5x+ 4
x2+ 3x+ 1 dx
152)
153)
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.)
153)
32
154)
Evaluate:
2
1
2xdx
2
–
2
1
(2x)2dx
154)
155)
Use the trapezoidal rule with n= 3 to estimate the value of
3
0
1
x2+ 1 dx
155)
156)
Use differentials only to find an approximate value of 327.1.
156)
157)
If y” =x2+ex+1 and y’(–1) = 0 and y(–1) = – 1
4, find y.
157)
158)
Evaluate:
1
0
x2+ 3x+ 2
x2+ 2x+ 1 dx
158)
159)
Determine: 1
43x+ 2
dx
159)
160)
After weeks of radio promotion, a shoe store finds that for the next 30 days its sales change
at a rate given by S‘(t) = – 12t2+ 300t, where t is the number of days after the advertising
ends and S is in dollars. Find the total sales for the second week after the promotion ends.
Find
14
7
(–12t2+ 300t) dt.
160)
161)
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.
161)
162)
Determine:
4
3zez2+ 3 dz
162)
163)
Evaluate the definite integral
2
0
2xdx by first finding out Sn and then taking limit as n
. (Use the right–hand endpoint of each sub–interval.)
163)
164)
Use differentials to approximate e0.02.
164)
165)
Determine –2x·79–x2dx
165)
166)
Determine: (x3+3x) dx
166)
167)
Determine: x+ 2
x2+ 4x– 2 dx
167)
34
168)
Determine: ln(x2+ 3x + 11) 2x+ 3
x2+ 3x+ 11 dx
168)
169)
Determine:
4
0
e4x
4dx
169)
170)
Find the indefinite integral (2x+ 1)(3x– 2)
6dx.
170)
171)
Determine: (y2+ 1)(y2– 1) dy
171)
172)
Determine: x
4x2+ 1
dx
172)
173)
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?
173)
174)
Find the exact area of the region bounded by y=x2– 4x and the x–axis. Also sketch the
region.
174)
175)
Use differentials and C= 250 + 0.30x to approximate the change in the cost C (in dollars) to
produce x pounds of candy if the number of pounds of candy increases from 10 pounds to
10.6 pounds.
175)
176)
Determine: (y– 1)(y2+y+ 1) dy
176)
177)
The supply equation for a company is q= 4p2+ 3. Find dp
dq from dq
dp .
177)
178)
Use differentials to approximate: ln(0.95).
178)
179)
Use differentials only to find an approximate value of 4.1.
179)
180)
Determine: 1
3x+ 5 –(x7– 5x3)11(7x6– 15x2) dx
180)
181)
Determine: 4s+1ds
181)
182)
Determine: 2x+ 3
4dx
182)
183)
Find y subject to the given conditions: y” = 6x– 2; y’(1) = 2; y(1) = 2.
183)
184)
Determine: 65y+ 1 dy
184)
185)
Determine: 43/2 dx
185)
186)
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.
186)
187)
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.)
187)
188)
The producers of a long running radio show expect the number of listeners (in hundreds)
to decrease at a rate of L’(x) = – 5 –x2, where x is the number of weeks after the first day of
the year. Find the total decrease in listeners in the first 2 weeks by evaluating
2
0
(–5 –x2) dx .
188)
189)
Determine: (3x2+ 4x)(2x3+ 4x2+5)6dx
189)
190)
Suppose the rate of savings in a country is given by dS
dt = 5.1et + 400t2 + 6000, 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.
190)
191)
Find the exact area of the region bounded by the graphs of x=y2 and x–y– 2 = 0. Also
sketch the region.
191)
192)
Determine: 5x3dx
192)
193)
If y” = 6x+ 2 and y’(1) = 2 and y(1) = 2, find y.
193)
194)
The height of a cylinder is decreasing at a rate of dh
dt = – 12t2et3. Find h as a function of t.
194)
38
195)
Find y subject to y’ = (1 + 7x)2; y(0) = 1.
195)
196)
Find the indefinite integral 5x–3
5x
dx.
196)
Explanation:
197)
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).
197)
Explanation:
198)
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.)
198)
Explanation:
199)
Determine: ex/2 dx
199)
Explanation:
200)
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.
200)
Explanation:
201)
Determine: 4x9 – 3x2dx
201)
Explanation:
39
Explanation:
202)
Determine: (2x)–1dx
202)
203)
Determine e
–2
x2
x3dx
203)
204)
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 .
204)
205)
Find the differential in terms of x and dx.
y=48 –x2
205)
206)
Determine 5x4– 7x2 + 9
x2dx
206)
207)
Use Simpson’s rule with n= 4 to find an approximate value of
2
0
1
4 +x2dx.
207)
208)
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
2
(–x2+ 7x– 10) dx.
208)
40