homework homework assignment #5 read section 5.6 page 341, exercises: 1 – 19(odd) rogawski...

Homework

Homework Assignment #5 Read Section 5.6 Page 341, Exercises: 1 – 19(Odd)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 3411. Water flows into an empty reservoir at the rate of 3000 + 5t gal/hr. What is the quantity of water in the reservoir after 5 hrs?


5

25

0

0

5 0 3000 5 3000 52

12515000 0 15062.5

2

tF F tdt t

gal

Homework, 413. A population of insects increases at a rate of 200 + 10t + 0.25t2 insects/day. Find the insect population after 3 days, assuming that there are 35 insects at t = 0.


5 20

5 20

32 3

0

3 0 200 10 0.25

3 0 200 10 0.25

35 200 10 0.252 3

35 600 5 9 0.25 9 647 insects

I I t t dt

I I t t dt

t tt

Homework, Page 3415. A factory produces bicycles at a rate of 95 + 0.1t2 – t bicycles per week (t in weeks). How many bicycles were produced from day 8 to day 21?


3 21

33 2

1

3 1 95 0.1

95 0.13 2

9 195 3 0.1 9 95 0.1

2 2

285 0.9 4.5 95 0.1 0.5 186 bicycles

B B t t dt

t tt

Homework, Page 3417. A cat falls from a tree (with zero initial velocity) at time t = 0. How far does the cat fall between t = 0.5 and t = 1 s? Use Galileo’s formula v(t) = –32t ft/s.


1

0.5

12

0.5

1 0.5 32

16

16 4 12 ft

D D t dt

t

Homework, Page 341Assume 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.


9. 12 4 ft/s, 0,5t

52

5

0

0

2

5

0

12 4 12 42

12 5 2 5 0 10 ft

12 4 26 ft

tt dt t

t dt

x

y

t = 5, displacement = 10

t = 3, displacement = 18

Homework, Page 341Assume 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.


211. 1 m/s, 0.5,2t

2

12 20.5

0.5

2 20.5

11

1 1 12

2 0.5 2

2.5 2.5 0 m

1 1 m

tt dt t

t dt

x

y

distance = 0, t = 0.5

distance = 0.5, t = 1

distance = 1, t = 2

Homework, Page 34113. 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 0.5 1.0 1.5 2.0 2.5 3.0

l/min 50 48 46 44 42 40 38

0.5 50 48 46 44 42 40 135

0.5 48 46 44 42 40 38 129

N

N

L l

R l

Homework, Page 341


2

1 1 2

2 2

15. Let be the acceleration of an object in linear motion at time .

Explain why is the net change in velocity over , .

Find the net change in velocity over 1,6 if 24 3 ft/s .

t

t

a t t

a t dt t t

a t t t

2

1

2

1

1 2

2 1

6 2 3 2 36 2 2 31 1

The integral is the net change in velocity over , since,

by the FTC, , the net change in velocity.

24 3 12 12 6 6 12 1 1

12 36 216 1

tt

tt

a t dt t t

a t dt v t v t

t t dt t t

2 1 1

432 216 11 216 11 205 ft/s

Homework, Page 34117. The traffic flow past a certain point on a highway is q(t) = 3,000 + 2,000t +300t2, 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?


2 20

22 3

0

2 3

32

10 8 3000 2000 300

3000 1000 100

3000 2 1000 2 100 2

3000 0 10000 100 0

6,000 4,000 800 0 9,200 cars

Q Q t t dt

t t t

Homework, Page 34119. To encourage manufacturers to reduce pollution, a carbon tax on each ton of CO2 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 CO2 reduction from x to x + 1 tons (in units of 10,000 tons – Figure 4). Which quantity is represented by


3

0 ?B t dt

3

0

2

represents the tens of

thousands of dollars spent to

reduce CO emissions by

30,000 tons

B t dt


Chapter 5: The IntegralSection 5.6: Substitution Method

Jon Rogawski

Calculus, ETFirst Edition

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.”


2 2 3Consider the integral 3 secx x dx


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

Example, Page 349Calculate du for the given function.


44. 2 8u x x

Example, Page 349Write the integral in terms of u and du. Then evaluate.


28. 2 1x x dx 2 28. 2 1 , 1x x dx u x



3

444

12. , 11

xdx u x

x



16. 4 1 , 4 1x x dx u x

Example, Page 349Show that the integral is equal to a multiple of sin(u(x)) + C for an appropriate choice of u(x).


2 330. cos 1x x dx

Example, Page 349Evaluate the indefinite integral.


32 334. 1x x dx



7248. 1x x dx



tan ln54.

xdx

x

Homework

Homework Assignment #6 Review Section 5.6 Page 349, Exercises: 1 – 57(EOO) Quiz next time


homework homework assignment #5 read section 5.6 page 341, exercises: 1 – 19(odd) rogawski...

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