Slide 1

Download Report

Transcript Slide 1

Homework

 Homework Assignment #5   Read Section 5.6

Page 341, Exercises: 1 – 19(Odd)

Homework, Page 341

Water flows into an empty reservoir at the rate of 3000 + 5

gal/hr. What is the quantity of water in the reservoir after 5 hrs?



  0 5  

tdt

   3000

 5

2 2   5 0 15000  125 2

gal

0.25

Homework, Page 341

A population of insects increases at a rate of 200 + 10

+ insects/day. Find the insect population after 3 days, assuming that there are 35 insects at

= 0.

I I

   0 5  0 5   



 0.25

2  0.25

2  

dt dt

  35 35  



  200

 10

2 2  0.25



3 3   3 0  647 insects

Homework, Page 341

A factory produces bicycles at a rate of 95 + 0.1

produced from day 8 to day 21?

2 bicycles per week (

in weeks). How many bicycles were –

t B



  1 3  

2     95

 0.1

3 3 

2 2   1 3   9 2      1 2 

Homework, Page 341

A cat falls from a tree (with zero initial velocity) at time

How far does the cat fall between

= 0.5 and

= 0. = 1 s? Use Galileo’s formula

(

) = –32

ft/s.



  1 0.5



 32



 16

16 2  1 0.5

  

12 ft

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

 4

2 2   5 0

 

2              t = 5, displacement = 10 t = 3, displacement = 18              0 5 

t dt

 26 ft        x

Homework, Page 341

11.

 2 



 y  2 0.5



 2  1 

  

 1  1 

  2 0.5

distance = 1, t = 2      2    2    0 m 1 0.5

 1 2  distance = 0.5, t = 1    2 0.5

 2  1

 1 m  distance = 0, t = 0.5

   x

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

1.0

1.5

2.0

2.5

3.0

L N



R N

      135

 129

15. Let

 

Explain why 

Homework, Page 341

is the net change in velocity over 

t t

1 , 2  .

 

 

 24



2 2 3 ft/s .

The integral 

2 is the net change in velocity over 

t t

1 , 2  since, by the FTC, 

2  1 6  24

 3

2   12

2   



 

   

v t

1 , the net change in velocity.

 6 1



 216



 12 6     1 2 1 6  3   

   

3 205 ft/s 

Homework, Page 341

17.

The traffic flow past a certain point on a highway is

(

) = 3,000 + 2,000

+300

2 , where

is in hours and

= 0 is 8 AM. How many cars pass by during the time interval from 8 to 10 AM?



  0 2  

 300

2 

       3000



 1000

2   100

3  10000 2 2   2 0 3  

 

3 



9, 200 cars

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

(

)

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

Jon Rogawski

Calculus, ET

First Edition Chapter 5:

The Integral

Section 5.6:

Substitution Method

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 sec 2  

The Substitution Method is formally stated in Theorem 1.

Breaking down the integral as follows: We see that the antiderivative of

(

)

(

) +

Example, Page 349

Calculate

 2

4  for the given function.

Example, Page 349

Write the integral in terms of

 1

dx u



2  1 and

du.

Then evaluate.

Example, Page 349

Write the integral in terms of

12.  

3  1  4



4  1 and

du.

Then evaluate.

Example, Page 349

Write the integral in terms of

16. 



 4

 1 and

du.

Then evaluate.

Example, Page 349

Show that the integral is equal to a multiple of sin(

(x)) +

for an appropriate choice of

(

). 30. 

2 cos 

3  1 

Example, Page 349

Evaluate the indefinite integral.

34. 

2 

3  1  3

Example, Page 349

Evaluate the indefinite integral.

48. 



 1



Example, Page 349

Evaluate the indefinite integral.

54. 

 

x dx

Homework

 Homework Assignment #6   Review Section 5.6

Page 349, Exercises: 1 – 57(EOO)  Quiz next time

Slide 1

Transcript Slide 1

Homework

Homework, Page 341

Homework, Page 341





Homework, Page 341

Homework, Page 341





  

Homework, Page 341





 

Homework, Page 341



Homework, Page 341

 

Homework, Page 341

 



   





   

Homework, Page 341



 



Homework, Page 341

 

Calculus, ET

Example, Page 349

Example, Page 349

Example, Page 349

Example, Page 349

Example, Page 349

Example, Page 349

Example, Page 349





Example, Page 349

 

Homework

Directory