Transcript Slide 1
Homework
Homework Assignment #5 Read Section 5.6
Page 341, Exercises: 1 – 19(Odd)
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
1.
Homework, Page 341
Water flows into an empty reservoir at the rate of 3000 + 5
t
gal/hr. What is the quantity of water in the reservoir after 5 hrs?
F
F
0 5
tdt
3000
t
5
t
2 2 5 0 15000 125 2
gal
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
3.
0.25
t
2
Homework, Page 341
A population of insects increases at a rate of 200 + 10
t
+ insects/day. Find the insect population after 3 days, assuming that there are 35 insects at
t
= 0.
I I
0 5 0 5
t
t
0.25
t
2 0.25
t
2
dt dt
35 35
200
t
10
t
2 2 0.25
t
3 3 3 0 647 insects
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
5.
Homework, Page 341
A factory produces bicycles at a rate of 95 + 0.1
produced from day 8 to day 21?
t
2 bicycles per week (
t
in weeks). How many bicycles were –
t B
B
1 3
t
2 95
t
0.1
t
3 3
t
2 2 1 3 9 2 1 2
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Homework, Page 341
7.
A cat falls from a tree (with zero initial velocity) at time
t
How far does the cat fall between
t
= 0.5 and
t
= 0. = 1 s? Use Galileo’s formula
v
(
t
) = –32
t
ft/s.
D
D
1 0.5
32
16
t
16 2 1 0.5
12 ft
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Homework, Page 341
Assume that a particle moves in a straight line with given velocity. Find the total displacement and total distance traveled over the time interval, and draw a motion diagram, with distance y and time labels.
0 5
12
t
4
t
2 2 5 0
2 t = 5, displacement = 10 t = 3, displacement = 18 0 5
t dt
26 ft x
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Homework, Page 341
Assume that a particle moves in a straight line with given velocity. Find the total displacement and total distance traveled over the time interval, and draw a motion diagram, with distance and time labels.
11.
t
2
y 2 0.5
t
2 1
dt
t
1 1
t
2 0.5
distance = 1, t = 2 2 2 0 m 1 0.5
1 2 distance = 0.5, t = 1 2 0.5
t
2 1
dt
1 m distance = 0, t = 0.5
x
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Homework, Page 341
13.
The rate (in liters per minute) at which water drains from a tank is recorded at half-minute intervals. Use the average of the left- and right endpoint approximation to estimate the amount of water drained in the first 3 min.
t 0
l/min 50
0.5
48
1.0
46
1.5
44
2.0
42
2.5
40
3.0
38
L N
R N
135
l
129
l
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
15. Let
Explain why
t
1
t
2
Homework, Page 341
is the net change in velocity over
t t
1 , 2 .
24
t
t
2 2 3 ft/s .
The integral
t
1
t
2 is the net change in velocity over
t t
1 , 2 since, by the FTC,
t
1
t
2 1 6 24
t
3
t
2 12
t
2
t
3
2
v t
1 , the net change in velocity.
6 1
216
12 6 1 2 1 6 3
3 205 ft/s
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Homework, Page 341
17.
The traffic flow past a certain point on a highway is
q
(
t
) = 3,000 + 2,000
t
+300
t
2 , where
t
is in hours and
t
= 0 is 8 AM. How many cars pass by during the time interval from 8 to 10 AM?
Q
Q
0 2
t
300
t
2
dt
3000
t
1000
t
2 100
t
3 10000 2 2 2 0 3
3
9, 200 cars
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Homework, Page 341
19.
To encourage manufacturers to reduce pollution, a carbon tax on each ton of CO 2 released into the atmosphere has been proposed. To model the effects of such a tax, policymakers study the
marginal cost of abatement B
(
x
)
,
defined as the cost of increasing CO 2 reduction from x to x + 1 tons (in units of 10,000 tons – Figure 4). Which quantity is represented by 0 3
?
0 3 represents the tens of thousands of dollars spent to reduce CO emissions by 2 30,000 tons
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Jon Rogawski
Calculus, ET
First Edition Chapter 5:
The Integral
Section 5.6:
Substitution Method
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
When differentiating functions, we sometimes need to use the Chain Rule. We will now cover the Substitution Method of integration which is the Chain Rule “in reverse.”
x
2 sec 2
dx
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
The Substitution Method is formally stated in Theorem 1.
Breaking down the integral as follows: We see that the antiderivative of
f
(
u
)
du
is
F
(
u
) +
C
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Example, Page 349
Calculate
du
4.
u
2
x
4 for the given function.
8
x
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Example, Page 349
Write the integral in terms of
u
1
dx u
x
2 1 and
du.
Then evaluate.
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Example, Page 349
Write the integral in terms of
u
12.
x
4
x
3 1 4
dx
,
u
x
4 1 and
du.
Then evaluate.
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Example, Page 349
Write the integral in terms of
u
16.
x
4
x
u
4
x
1 and
du.
Then evaluate.
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Example, Page 349
Show that the integral is equal to a multiple of sin(
u
(x)) +
C
for an appropriate choice of
u
(
x
). 30.
x
2 cos
x
3 1
dx
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Example, Page 349
Evaluate the indefinite integral.
34.
x
2
x
3 1 3
dx
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Example, Page 349
Evaluate the indefinite integral.
48.
x
2
x
1
7
dx
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Example, Page 349
Evaluate the indefinite integral.
54.
x dx
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Homework
Homework Assignment #6 Review Section 5.6
Page 349, Exercises: 1 – 57(EOO) Quiz next time
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company