Quiz 3-1B 1. When did the population reach 50,000 ? 2. The population of “Smallville” in the year 1890 was 6250. Assume the population increased at a rate of 2.75% per year. 1. When did the population reach 50,000 ? A bank account pays 14% interest per year. If you initially invest $2500, how much money will you have after 7 years? 2.

3.1C TheLogistic Function

Your turn: Or 1. Which of the two models best represents population as a function of time (bacteria, zebras, monkeys, etc.) in the real world? Or Exponential Logistic 2. Why? (justify your answer)

Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population. We must use Logistic function if the growth is limited !!! What factors can limit the size of the population?

Logistic Function y = 1 y = 0 huge

What does vertically stretched Logistic Function y = 1 Parent Function: y = 0 Vertical stretch by a factor of ‘3’ What does vertically stretched by a factor of ‘3’ mean? What happens to the horizontal asymptotes?

What does vertically stretched Logistic Function y =3 Parent Function: y = 0 Vertical stretch by a factor of ‘3’ What does vertically stretched by a factor of ‘3’ mean? What happens to the horizontal asymptotes?

Logistic Function y = c y = 0 y = 12 y = 12 y = 0 y = 0 Parent Function: y = 0 General Form: ‘c’ is the “limit to growth” Logistic Growth: a, c, and k > 0 0 < b < 1 y = 12 y = 12 y = 0 y = 0

Logistic Function y = c y = 0 y = 12 y = 0 y = 12 y = 0 ‘c’ is the “limit to growth” Logistic Decay: k < 0 or b > 1 y = 12 y = 0 y = 12 y = 0

Your turn: 3. Is it logistic growth or decay?

Logistic Function x = 0 y = 12 y = 0 1. Find the y-intercept: (0, 6) 1. Find the y-intercept: x = 0 What is the x-value of the y-intercept? 2. Find the asymptotes: y = 12 ‘12’ is the “limit to growth” y = 0 huge huge

Your turn: 5. Find the y-intercept for the following equation. 6. Find the horizontal asymptotes for the above equation. 7. Find the y-intercept for the following equation. 8. Find the horizontal asymptotes for the above equation.

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

Rumors at RHS How many students have heard the rumor by the end of day ‘0’ ? How long does it take for 1000 students to have heard the rumor ?

Rumors at RHS How long does it take for 1000 students to have heard the rumor ? Solve graphically 4.5 days

Your turn: 9. How many days does it take until In countries without a free press, people usually believe rumors more than the news. The above equation models the number of people who have heard the rumor “t”-days after the rumor was started. 9. How many days does it take until half the population has heard the rumor? 10. How people have heard the rumor by the end of the first day (day “0”).

Deriving the Logistics function The word problem will give you values to plug into the equation. There are 4 unknown quantities in the formula. Growth factor ‘k’ Limit to growth Coefficient ‘a’ Function value corresponding to a specific input value of ‘x’ Knowing the initial value and the limit to growth will allow you to find ‘a’. Once you know ‘a’, if you are given the function value for some input value of ‘x’, you will be able to find ‘k’.

Deriving the Logistics function Initial value is 10. Limit to growth is 40. Passes thru (1, 20) f(0) = 10 c = 40 a = 3

Deriving the Logistics function Initial value is 10. Limit to growth is 40. Passes thru (1, 20) f(0) = 10 c = 40 a = 3 We still do not have the “tools” to be able to solve for an unknown exponent algebraically! How can we solve for ‘k’? Graphically.

Two equation method (intersection two curves method): k = 1.1 One equation method (x-intercept method): Final equation: k = 1.1

Modeling Using Logistic Regression The following data is the population of the ebola virus population (in billions) in a petri dish at one hour intervals. Enter the data into your calculator and use Logistic regression to determine the model (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 Regression Stat p/b  gives lists Enter the data: Let L1 be years since initial value Let L2 be population Stat p/b  calc p/b scroll down to exponential regression “ExpReg” displayed: enter the lists: “L1,L2” The calculator will display the values for ‘a’ and ‘b’.

Your turn: 11. What is your equation? 12. What is the maximum possible population? 13. What was the population 3 ½ hours after the start of the experiment? 14. What was the initial population (t = 0) ?

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