1. Quiz 3-1a 1.Write the equation that models the data under the column labeled g(x). 2. Write the equation that models the data under the column labeled f(x) above. labeled f(x) above. 3.Without using your calculator, determine if the following function growth or decay? function growth or decay? 4. Without using your calculator, determine if the following function growth or decay? function growth or decay?

2. 3.1B Applications of Exponential Functions

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

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

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

6. Population Growth Quadratic equation!

7. Population Growth Quadratic equation!

8. Population Growth Special cubic!

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

10. Word problems There are 4 quantities in the equation: 2. Initial population 3. Growth rate 1. Population “t” years/min/sec in the future 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 quantities. You just have to “plug them in” to the equation and solve for the unknown quantity. and solve for the unknown quantity.

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

12. Simple Interest (savings account) Amount (as a function of time) function of time) Initial amount (“principle”) (“principle”)Interest rate rate time 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

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

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

15. 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. unknown exponent algebraically.

16. 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.

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

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

19. Finding an Exponential Function $500 was deposited into an account that pays “simple interest” (interest paid at the end of the year). $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

20. 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?

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

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

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

24. 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. P(0) = 100 P(t) = 350000 What is the growth factor?

25. 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. Where do the two graphs cross? t = 11 hours + 0.77hrs t = 11 hours + 46 min

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

27. 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. “t” years after 1900. Enter the data into your calculator and use calculator and use exponential regression exponential regression to determine the model (equation). to determine the model (equation).

28. Exponential Regression Stat p/b  gives lists Enter the data: Let L1 be years since initial value Let L2 be population 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’. values for ‘a’ and ‘b’.

29. 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? actual value of 290.8 million?

30. Find the amout of material after ‘20’ days if the initial mass is 5 grams and it doubles every 4 days: The issue is units !!! Initial value ‘a’  units of grams Can the exponent have any units? This doubles every 4 days. How many times does it double in 20 days? double in 20 days? The mass (# of grams) at some time “t” in the future is the initial mass (# of grams) times some number. the initial mass (# of grams) times some number. NO !!!

31. Units of the exponent The input value is time (with units of seconds, minutes, hrs, etc.). How can the input value have units and the exponent not have any units (since that is where the input value is inserted into the any units (since that is where the input value is inserted into the equation)? equation)? IF the input value has the units of time in seconds, then the exponent really has the units of “# of times the base is used as a factor / day” really has the units of “# of times the base is used as a factor / day” to make the units work out. to make the units work out. Since the base is a 3, then this could be shortened to “# of triples/ day” “# of triples/ day” This could be shortened to “per day” which in math is “1/day”

32. Find the amout of material after 20 days if the initial mass is 5 grams and it doubles every 4 days: Initial mass = 5 grams mass doubles every 4 days (amount (grams) as a function of time) No units remain in the exponent. exponent.

33. Find the amout of material after 20 days if the initial mass is 5 grams and it doubles every 4 days: Initial mass = 5 grams mass doubles every 4 days (amount (grams) as a function of time) So you could just write it as:

34. Your turn: 17. The crowd in front of the Tunisian parlament building increased by a factor of 4 every 3 hours. If the initial crowd had 500 people in it, how many If the initial crowd had 500 people in it, how many people would there be after 12 hours? people would there be after 12 hours? 18. The amount of radioactive Rubidium 88 decreases by a factor of 2 every 8 minutes. If there was 5 grams of the material at the start, how much would there be after 30 minutes?

35. HOMEWORK