Quiz 3-1BQuiz 3-1B

1. 1. When did the population reach 50,000 ?When did the population reach 50,000 ?

The population of “Smallville” in the year 1890 was 6250. The population of “Smallville” in the year 1890 was 6250. Assume the population increased at a rate of 2.75% per year.Assume the population increased at a rate of 2.75% per year.

trPtP )1()(

A bank account pays 14% interest per year. If you initially A bank account pays 14% interest per year. If you initially invest $2500, how much money will you have after 7 years? invest $2500, how much money will you have after 7 years?

2. 2.

3.1C3.1CTheLogistic FunctionTheLogistic Function

Your turn:Your turn:

OrOr

Exponential LogisticExponential Logistic

2. 2. Why? (justify your answer) Why? (justify your answer)

1.1. Which of the two models best represents population Which of the two models best represents population as a function of time (bacteria, zebras, monkeys, etc.) as a function of time (bacteria, zebras, monkeys, etc.) in the real world?in the real world?

Maximum Sustainable Maximum Sustainable PopulationPopulation

Exponential growth is unrestrictedExponential growth is unrestricted, but population growth , but population growth often is not. For many populations, the growth begins often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit exponentially, but eventually slows and approaches a limit to growth called the to growth called the maximum sustainable populationmaximum sustainable population..

We must use Logistic function if the growth is limited !!!We must use Logistic function if the growth is limited !!!

What factors can limit the size of the population?What factors can limit the size of the population?

Logistic FunctionLogistic Functionxe

xf

1

1)(

xexf

11

1)(

? yx? )(limit xfx

00

01

1)(

xf

y = 1y = 1

? yx

? )(limit xfx

xexf

1

1)( hugehuge

y = 0y = 0

xexf

1

1)(

Logistic FunctionLogistic Function

xexf

1

1)(

Parent Function:Parent Function:

Vertical stretch by a factor of ‘3’Vertical stretch by a factor of ‘3’

What happens to the horizontal asymptotes?What happens to the horizontal asymptotes?

xexf

1

3)(3

y = 1y = 1

y = 0y = 0

What does vertically stretched What does vertically stretched by a factor of ‘3’ mean?by a factor of ‘3’ mean?

Logistic FunctionLogistic Function

xexf

1

1)(

Parent Function:Parent Function:

Vertical stretch by a factor of ‘3’Vertical stretch by a factor of ‘3’

What happens to the horizontal asymptotes?What happens to the horizontal asymptotes?

xexf

1

3)(3

y = 0y = 0

What does vertically stretched What does vertically stretched by a factor of ‘3’ mean?by a factor of ‘3’ mean?

y =3y =3

Logistic FunctionLogistic Function

xexf

1

1)(

kxae

cxf

1

)(

Parent Function:Parent Function:

General Form:General Form:

xab

cxf

1)(

a, c, and k > 0a, c, and k > 0

0 < b < 10 < b < 1

Logistic GrowthLogistic Growth::

‘‘c’ is the “c’ is the “limitlimit to growth” to growth”

y = 0y = 0 y = cy = c

xxf

8.01

12)(

y = 12y = 12

y = 0y = 0

xexf

21

12)(

y = 0y = 0

y = 12y = 12

Logistic FunctionLogistic Function

kxae

cxf

1

)(

xab

cxf

1)(

k < 0 or b > 1k < 0 or b > 1

Logistic DecayLogistic Decay::

‘‘c’ is the “c’ is the “limitlimit to growth” to growth”

y = 0y = 0 y = cy = c

xxf

2.11

12)(

y = 12y = 12

y = 0y = 0

xexf

21

12)(

y = 12y = 12

y = 0y = 0

Your turn:Your turn:

4. 4. Is it logistic growth or decay?Is it logistic growth or decay?

3.3. Is it logistic growth or decay? Is it logistic growth or decay?

xexf

521

3)(

xxf

5)8.1(51

3)(

Logistic FunctionLogistic Function1. Find the y-intercept:1. Find the y-intercept:

xxf

2.11

12)(

x = 0x = 0

y = 12y = 12

y = 0y = 0

What is the x-value of the y-intercept?What is the x-value of the y-intercept?

02.11

12)0(

f

11

12

2

12 6

2. Find the asymptotes:2. Find the asymptotes:

‘‘12’ is the “12’ is the “limitlimit to growth” to growth”

? )(limit xfx

xxf

2.11

1)(

hugehuge

(0, 6)(0, 6)

xxf

2.11

1)(

hugehuge

Your turn:Your turn:

6. 6. Find the horizontal asymptotes for the above equation.Find the horizontal asymptotes for the above equation.

5.5. Find the y-intercept for the following equation. Find the y-intercept for the following equation.

xexf

521

3)(

7.7. Find the y-intercept for the following equation. Find the y-intercept for the following equation.

xxf

2)2.1(21

4)(

8. 8. Find the horizontal asymptotes for the above equation.Find the horizontal asymptotes for the above equation.

Modeling a RumorModeling a Rumor

Roy High School has about 1500 students. 5 students start a Roy High School has about 1500 students. 5 students start a rumor, which spreads logistically so thatrumor, which spreads logistically so that

models the number of students who have heard the rumor by models the number of students who have heard the rumor by the end of ‘t’ days, where ‘t’ = 0 is the day the rumor began to the end of ‘t’ days, where ‘t’ = 0 is the day the rumor began to spread.spread.

)291(

1500)(

9.0 tetS

Rumors at RHSRumors at RHS)291(

1500)(

9.0 tetS

How many students have heard the rumorHow many students have heard the rumor by the end of day ‘0’ ?by the end of day ‘0’ ?

)291(

1500)0(

)0(9.0

eS

How long does it take for 1000 students toHow long does it take for 1000 students to have heard the rumor ?have heard the rumor ?

)291(

15001000

)(9.0 te

)291(

1500

50

Rumors at RHSRumors at RHS)291(

1500)(

9.0 tetS

How long does it take for 1000 students toHow long does it take for 1000 students to have heard the rumor ?have heard the rumor ?

)291(

15001000

)(9.0 te

Solve graphicallySolve graphically

)291(

1500)(9.01 te

y

10002 y 4.5 days4.5 days

Your turn:Your turn:)51(

million 80)(

05.0 tetS

In countries without a free press, people usually believe In countries without a free press, people usually believe rumors more than the news. The above equation modelsrumors more than the news. The above equation models the number of people who have heard the rumor “t”-days the number of people who have heard the rumor “t”-days after the rumor was started. after the rumor was started.

)51(

8040

)(05.0 te

million 3.13

9. 9. How many days does it take until How many days does it take until half the population has heard the rumor?half the population has heard the rumor?

10. 10. How people have heard the rumor How people have heard the rumor by the end of the first day (day “0”).by the end of the first day (day “0”).

days 2.32)51(

80)0(

)0(05.0

eS

)51(

80

kxae

cxf

1

)(

Deriving the Logistics Deriving the Logistics functionfunction

The word problem will give you values to plug into the equation.The word problem will give you values to plug into the equation.

There are 4 unknown quantities in the formula. There are 4 unknown quantities in the formula.

Limit to growthLimit to growth

Coefficient ‘a’Coefficient ‘a’

Growth factor ‘k’Growth factor ‘k’

Function value corresponding toFunction value corresponding to a specific input value of ‘x’a specific input value of ‘x’

Knowing the Knowing the initial value initial value and the and the limit to growth limit to growth will allow youwill allow you to find to find ‘a’‘a’. .

Once you know ‘a’, if you are Once you know ‘a’, if you are given the function value for somegiven the function value for some input value of ‘x’, input value of ‘x’, you will be able to find you will be able to find ‘k’.‘k’.

Deriving the Logistics Deriving the Logistics functionfunction

kxae

cxf

1

)( Initial value is 10.Initial value is 10.Limit to growth is 40.Limit to growth is 40.Passes thru (1, 20)Passes thru (1, 20)

)0(1

4010)0(

kaef

c = 40c = 40

a

1

4010

10

401 a a = 3a = 3

)1(31

4020)1(

kef

20

4031 ke 13 ke

3

1 ke

f(0) = 10f(0) = 10

Deriving the Logistics Deriving the Logistics functionfunction

kxae

cxf

1

)( Initial value is 10.Initial value is 10.Limit to growth is 40.Limit to growth is 40.Passes thru (1, 20)Passes thru (1, 20)

)0(1

4010)0(

kaef

c = 40c = 40

a

1

4010

10

401 a a = 3a = 3

)1(31

4020)1(

kef

20

4031 ke 13 ke

3

1 ke

f(0) = 10f(0) = 10

We still do not have the “tools” to be able to solve for anWe still do not have the “tools” to be able to solve for an unknown exponent algebraically! How can we solve for ‘k’? unknown exponent algebraically! How can we solve for ‘k’?

Graphically.Graphically.3

1 21 yey k

3

1 ke

3

1 21 yey k

Two equation methodTwo equation method(intersection two curves method):(intersection two curves method):

k = 1.1k = 1.1

kxexf

31

40)(

One equation methodOne equation method (x-intercept method):(x-intercept method):

ke3

10

key 3

11

k = 1.1k = 1.1

3

1 ke

Final equation:Final equation:

)(1.131

40)(

xexf

Modeling Using Logistic Modeling Using Logistic RegressionRegressionThe following data is the population of the ebola virus The following data is the population of the ebola virus

population (in billions) in a petri dish at one hour intervals.population (in billions) in a petri dish at one hour intervals.

Enter the data into yourEnter the data into your calculator and use calculator and use Logistic regressionLogistic regression to determine the model to determine the model (equation).(equation).

Time (hrs) Population(billions)

1 1.2

2 1.7

3 2.5

4 3.3

5 3.8

6 3.9

7 3.97

Exponential RegressionExponential RegressionStat p/b Stat p/b gives lists gives lists

Enter the data: Enter the data: Let L1 be years since initial valueLet L1 be years since initial value

Let L2 be populationLet L2 be population

Stat p/b Stat p/b calc p/b calc p/bscroll down to exponential regressionscroll down to exponential regression

““ExpReg” displayed:ExpReg” displayed:enter the lists: “L1,L2”enter the lists: “L1,L2”

xabxf )( The calculator will display the The calculator will display the values for ‘a’ and ‘b’.values for ‘a’ and ‘b’.

Your turn:Your turn:11. 11. What is your equation?What is your equation?

12. 12. What is the maximum possible population? What is the maximum possible population?

13. 13. What was the population 3 ½ hours after the start What was the population 3 ½ hours after the start of the experiment?of the experiment?

14. 14. What was the initial population (t = 0) ? What was the initial population (t = 0) ?

HOMEWORKHOMEWORKPage 296:Page 296: Even Even Problems: Problems: 24-28, 46a, 46b, 46c, 48, 50, 56,24-28, 46a, 46b, 46c, 48, 50, 56, 58b, and 58c.58b, and 58c. 11 problems11 problems