Quiz 3-1a This data can be modeled using an exponential equation  Find ‘a’ and ‘b’ Where does cross the y-axis ? 3. Is r(x) an exponential growth or decay function? 4. Convert r(x) to exponential base ‘e’ notation:

Applications of Exponential Functions and the Logistic Function 3.1B Applications of Exponential Functions and the Logistic Function

Exponential Function Initial value Input variable Growth factor: What does ‘b’ equal In order for it to be “growth”? What is the value of ‘y’ where the graph crosses the y-axis? What does ‘b’ equal In order for it to be “decay”?

Your turn: 1. Where does it cross the y-axis? Graph the functions: 1. Where does it cross the y-axis? 2. What is the “intial value of f(t) ?

Population Growth If population grows at a constant percentage rate over a year time frame, (the final population is the initial population plus a percentage of the orginial population) then the population at the end of the first year would be: Percent rate of change (in decimal form) At the end of the second year the population would be:

Population Growth Quadratic equation!

Population Growth Quadratic equation!

Population Growth Special cubic!

Population Growth Percent rate of change (in decimal form) time Population (as a function of time) Initial population Growth rate Growth factor: Initial value

Word problems There are 4 quantities in the equation: 1. Population “t” years/min/sec in the future 2. Initial population 3. Growth rate 4. time The words in the problem will give you three of the four quantities. You just have to “plug them in” to the equation and solve for the unknown quantity.

Population Growth Percent rate of change (in decimal form) time Population (at time “t”) in the future Initial population Growth rate The initial population of a colony of bacteria is 1000. The population increases by 50% every hour. What is the population after 5 hours? Unknown value

Simple Interest (savings account) time Amount (as a function of time) Initial amount (“principle”) Interest rate A bank account pays 3.5% interest per year. If you initially invest $200, how much money will you have after 5 years? Unknown value

Your turn: 3. A bank account pays 14% interest per year. If you initially invest $2500, how much money will you have after 7 years? The population of a small town was 1500 in 1990. The population increases by 3% every year. What is the population in 2009? 4.

Solve by graphing San Jose, CA Year Population 1990 782,248 1990 782,248 2000 895,193 Assuming exponential growth, when will the population equal 1 million? Let ‘t’ = years since 1990 We must find the growth factor ‘b’ ‘b’ = 1.0136 Unknown value

Example 1,000,000 ‘t’ = approximately 18 18 years AFTER 1990  2008 Later in the chapter we will learn how to solve for the unknown exponent algebraically.

Your Turn: 5. When did the population reach 50,000 ? The population of “Smallville” in the year 1890 was 6250. Assume the population increased at a rate of 2.75% per year. 5. When did the population reach 50,000 ?

Your turn: 6. Assuming exponential growth, when will Year Population 1990 248,709,873 2009 307,006,550 USA Your turn: 6. Assuming exponential growth, when will the population exceed 400 million? We must find the growth factor ‘b’ ‘b’ = 1.0111 43 yrs after t = 0 (1990) 2033

Your turn: 7. Assuming exponential growth, when will Year Population 1900 76.21 million 2000 248.71 million USA Your turn: 7. Assuming exponential growth, when will the population exceed 400 million? We must find the growth factor ‘b’ ‘b’ = 1.0119 140.2 yrs after t = 0 (1900) 2040.2

Finding an Exponential Function $500 was deposited into an account that pays “simple interest” (interest paid at the end of the year). 25 years later, the account contained $1250. What was the percentage rate of change? Unknown value

Your Turn: 8. The population of “Smallville” in the year 1890 was 6250. Assume the population increased at a rate of 2.75% per year. What is the population in 1915 ? 9. The population of “Bigville” in the year 1900 was 25,200. In 1955 the population was 37,200. What was the percentage rate of change? 10. The population of “Ghost-town” in the year 1900 was 3500. In 1935 the population was 200. What was the percentage rate of change?

Finding Growth and Decay Rates Is the following population model an exponential growth or decay function? Find the constant percentage growth (decay) rate. ‘r’ > 0, therefore this is exponential growth. ‘r’ = 0.0136 or 1.36%

Your turn: 11. Is it growth or decay? b = 1.5 b > 0 Growth! 12. Find the constant percentage growth (decay) rate. ‘r’ = 0.5 or 50%  % rate of growth is 50% ‘r’ > 0, therefore this is exponential growth.

Finding an Exponential Function Determine the exponential function with initial value = 10, increasing at a rate of 5% per year. ‘r’ = 0.05 or

Modeling Bacteria Growth Suppose a culture of 100 bacteria cells are put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350,000. What is the growth factor? P(0) = 100 P(t) = 350000

Modeling Bacteria Growth Suppose a culture of 100 bacteria cells are put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350,000. t = 11 hours + 0.77hrs Where do the two graphs cross? t = 11 hours + 0.77hrs t = 11 hours + 46 min

Your turn: 13. A family of 10 rabbits doubles every 2 years. When will the family have 225 members? b = 2 t = 7.8 years t = 7 years 6 months

Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003. (Don’t enter the 2003 value). Let P(t) = population, “t” years after 1900. Enter the data into your calculator and use exponential regression to determine the model (equation).

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: 14. What is your equation? 15. What is your predicted population in 2003 ? 16. Why isn’t your predicted value the same as the actual value of 290.8 million?

HOMEWORK