homework homework assignment #8 read section 5.8 page 355, exercises: 1 – 69(eoo) quiz next time...

Homework

Homework Assignment #8 Read Section 5.8 Page 355, Exercises: 1 – 69(EOO) Quiz next time

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Example, Page 355Evaluate the definite integral.


2

1

11. dx

x

22

1 1

2

1

1ln ln 2 ln1 ln 2

1ln 2

dx xx

dxx

Example, 55Evaluate the definite integral.


2

15. e

e dtt

2 2

2

2

1ln

ln ln 1 2 1

11

ee

e e

e

e

dt tt

e e

dtt

x

y



23

2 2

19.

1dx

x x

2 213 3

2 22

23

2 2

1sec

6 3 61

1

61

dx xx x

dxx x



2

0 2

113.

4dx

x

2

2 2 2 10 0 0 22

2 0

1 1

2

0 2

111 1 1 14 tan

14 4 4 214 441 1

tan 1 tan 0 04 4 4 16

1

4 16

xdx dx dx

xx x

dxx

Example, Page 355Evaluate the indefinite integral.


217.

25 4

dt

t

1

2 22

1

2

11 1 25 sin

1 5 5 5425 4 125 4 255

1 2sin

5 525 4

dtdt dt tC

t tt

dt tC

t



2

121.

1

x dx

x

2

2 2 2

21 1

2 2

1 2

2

11 , 2 ,

21 1 1

1 1 1sin sin

12 21 1 2

1sin 1

1

x dx xdx dx du duu x x xdx

dxx x x

xdx dx du xx C x C

ux x

x dxx x C

x



1

2

tan25.

1

xdx

x

12 2

1 221

2

121

2

1tan , ,

1 1

tantan

1 2

tantan

1

du dxu x du

dx x x

xdx uudu C x C

x

xdxx C

x



4log 3

029. 4x dx

4 4

4

4

log 3 log 3 0log 3

0

0

log 3

0

4 4 4 3 1 24

ln 4 ln 4 ln 4 ln 4 ln 4 ln 4

24

ln 4

xx

x

dx

dx

Example, Page 355Evaluate the integral using methods covered so far.


33. 2xe dx

2 2 2

2 2

x x x

x x

e dx e dx dx e x C

e dx e x C



437. 1xe dx

4 4

4 4

4 4

1 4 , 4,4

1 1 1

4 4 4

11

4

x x

x u u x

x x

du due dx e dx dx u x dx

dx

e dx dx e du x C e x C e x C

e dx e x C



241.

1 16

dx

x

1 1

2 2

1

2

4 , 4,4

1 1 1sin sin 4

4 4 41 16 1

1sin 4

41 16

du duu x dx

dxdx du

u C x Cx u

dxx C

x



1045. 7 xe dx

10

10

10 , 1010

17 7 7

10 10

7 10 710

ux u

xx

du duu x dx

dx

ee dx dx e du x C

ee dx x C



2449. xxe dx

2 2

2 2

2

4 4

4 4

4 , 8 ,8

1 1 1

8 8 8

1

8

x u u x

x x

du duu x x xdx

dx

xe dx e du e C e C

xe dx e C



53. 2 4

dx

x

2 4, 2,2

1 1 1ln ln 2 4

2 4 2 2 2

1ln 2 4

2 4 2

du duu x dx

dxdx du

u C x Cx u

dxx C

x



57. tan 4 1x dx

4 1, 4,4

cos1 1 sin

tan 4 1 tan4 4 cos sin

1 sin 1 1 1ln ln cos

4 cos 4 4 41

ln cos 4 14

1tan 4 1 ln cos 4 1

4

du duu x dx

dxv u

ux dx udu du dv

u udu

u dvdu v C u C

u v

x C

x dx x C



4ln 561.

xdx

x

22

2

4ln 5 4ln 5 1ln , ,

4 ln 54 5 4 ln 2 ln ln

2

4ln 52 ln ln

x x du dxdx dx dx u x du

x x x dx x x

x dx udx udu x C x x C

x x

xdx x x C

x



ln ln65.

ln

xdx

x x

22

2

ln ln 1ln ln , ,

ln ln lnln ln 1

ln lnln 2 2

ln ln 1ln ln

ln 2

x du dxdx u x du

x x dx x x x xx u

dx udu C x Cx x

xdx x C

x x



sin69. cos 3 xx dx

sin

sinsin

sinsin

cos 3 sin , cos , cos

3 3cos 3 3

ln 3 ln 3

3cos 3

ln 3

x

u xx u

xx

dux dx u x x du xdx

dx

x dx du C C

x dx C


Chapter 5: The IntegralSection 5.8: Exponential Growth and Decay

Jon Rogawski

Calculus, ETFirst Edition


There are many examples in biology, physics, and economics where“populations “ grow or decay at exponential rates. The general formula describing this growth or decay is:

If k > 0, then the population is growing and if k < 0, the population is decaying.


The e. coli bacteria shown in Figure 1 multiply exponentially. That is why, within hours of ingesting contaminated food or drink, a person becomes severely ill from this bacteria.

As can be extrapolated from the graph in Figure 2, in less than 24hours, an initial population of 1,000 bacteria would have grown to over 1,600,000, more than enough to affect the healthiest person.


If a quantity is changing in proportion to the amount currently present, then it is increasing/decreasing exponentially


The rate a which penicillin leaves the bloodstream is proportional to the amount present. If an initial amount of 450 mg is administered and 50 mg remains after seven hours, what is the decay constant? At what time after the injection, did 200 mg remain?


A frequently asked question is: “When will this process produce twice the original amount?” This question can be answered bysolving 2 = 1 ekt for t.

Example, Page 3662. A quantity P obeys exponential growth law P = e5t (t in years).

(a) At what time t is P = 10?

(b) At what time t is P = 20?

(c) What is the doubling time for P?



Figure 4 shows a graph of the number of bachelor’s degrees awardedin physics each year from 1955 to about 1978. The figure is markedto emphasize the doubling that took place about every seven years.


If the number of microscopic plants shown in Figure 5 doubles every30 hours, what is the k value we should use?


In the case of exponential decay, the half-life of the substance is calculated by using:

Archeologists use the half-life of carbon-14 to date organic materials.The process was developed following World War II.

homework homework assignment #8 read section 5.8 page 355, exercises: 1 – 69(eoo) quiz next time...

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