1. Quiz 7-1: 1. Where does the graph cross the y-axis? 2. f(1) = ? 3. Horizontal asymptote = ? 4. How was the function transformed to get f(x) above? to get f(x) above? 5. Domain = ? 6. range = ?

2. 7-2, and 7-3 Exponential Decay And The growth/decay factor (base) “e”

3. Putting it all together: If negative: Reflect across x-axis Initial value: Crosses y-axis here Growth factor: If negative: Reflect across y-axis Horizontal shift vertical shift

4. Population Growth We can rewrite the change in the population as some percentage of the original population. as some percentage of the original population. The population at the end of one period of time equals the initial population plus the growth/decay of the population. initial population plus the growth/decay of the population. If “r” is the % change in population (decimal equivalent), then we can rewrite that as: then we can rewrite that as: Factoring out the common factor results in:

5. Population Growth over “t” time periods: Population (as a function of time) function of time) Initial population population Growth rate rate time It’s just a formula!!! The initial population of a colony of bacteria is 1000. The population increases by 50% is 1000. The population increases by 50% every hour. What is the population after 5 hours? every hour. What is the population after 5 hours? Percent rate of change (in decimal form) (in decimal form)

6. Interest (savings account) Amount (as a function of time) function of time) Initial amount (“principle”) (“principle”) Interest rate time In problems like this, the words in the problem will give you all but one of the quantities in the equation. The quantities are: A(t) P r k t Number of times the interest is paid per year interest is paid per year(compounded) It will be up to you to solve for the missing quantity.

7. A bank account pays 3.5% interest per year compounded monthly. If you initially invest $200, how much money will you have after 5 years? will you have after 5 years? What is the unknown quantity? A(t) P r k t A(5) = ?

8. A bank account pays quarterly compounded interest. If you initially invested $1000 and after 10 years had $2200 in your account, what was the annual interest rate? What is the unknown quantity? Divide left/right by 1000 40 th root left/right Subtract 1, multiply by 4

9. Your turn: A bank account pays 14% interest per year compounded quarterly. If you initially invest $2500, compounded quarterly. If you initially invest $2500, how much money will you have after 7 years? how much money will you have after 7 years? 1. 2. Five years after you made a single deposit in an earning 3% compounded monthly, it contains $580.81. earning 3% compounded monthly, it contains $580.81. What was your initial deposit? What was your initial deposit?

10. Exponential Decay ‘a’ is the initial value  f(0) = ‘a’ ‘b’ is called the decay factor 0 < ‘b’ < 1 Table of values x f(x) 0 4 1 2 2 1 3 0.5 4 0.25 ½ ½ ½ ½ 8 -216 ‘d’ shifts everything up or down

11. Exponential Growth and Decay exponential growth: growth factor > 1 exponential decay: growth factor 0 < b < 1

12. Your turn: For each of the following what is the: a. “initial value”? a. “initial value”? b. “decay factor”? b. “decay factor”? c. “horizontal asymptote” c. “horizontal asymptote” d. Any reflections (across x-axis or y-axis) d. Any reflections (across x-axis or y-axis) 3. 4. 5.

13. Identifying the Parts of the function: ‘a’ is the initial value  f(0) = ‘a’ ‘b’ is called the decay factor (if 0 < b < 1) Initial value: 10 y-intercept: f(0) = 10 + 2 = 12 decay factor: 0.3 ‘d’ shifts graph up/down and is the horizontal asymptote Horizontal asymptote: y = 2

14. Graphing Exponential Decay Use the “power of the calculator” or: f(1) = ? 3. Horizontal asymptote asymptote 1. f(0) = ? 2. Some other point f(0) = 6 f(1) = 3 y = 0 y = 0 Domain = ? Range = ? All real #’s y > 0

15. Where does the number ‘e’ come from? Named after the Swiss mathematician Leonard Euler (1707 – 1783). Leonard Euler (1707 – 1783). The number ‘e’ can be found on your calculator by “2nd” + “ln” + “1” = 2.718 281828459045 … by “2nd” + “ln” + “1” = 2.718 281828459045 … ‘e’ is an irrational number like pi or the square root of a prime number. The numbers after the root of a prime number. The numbers after the decimal point go on forever without any repetition decimal point go on forever without any repetition of number patterns. of number patterns.

16. The slope of the tangent line at any point on the line at any point on the curve is curve is The number ‘e’ (Named after Leonard Euler, a Swiss mathematician) ‘e’ is a very unique number The slope of the tangent line at x = 0 is line at x = 0 is The slope of the tangent line at x = 1 is line at x = 1 is

17. Exponential Functions and the number ‘e’ Any exponential function of the form: Can be written in the form: Exponential Growth: k > 0 Exponential Decay: k < 0

18. Exponential Functions and ‘e’

19. What processes does ‘e’ have to do with? Think of a bacteria cell. Over a certain period of time it splits a certain period of time it splits from one cell into two cells. The from one cell into two cells. The number of bacteria doubles. number of bacteria doubles. This type of growth occurs in “spurts”. At one instant of time it is a single bacteria. The next instant it is two bacteria. The number of bacteria doubles. A(t) = 2 A(t) = 2 In “t” time periods it doubles ‘t’ times. t = 1 t = 1

20. What processes does ‘e’ have to do with? The number e (2.718…) represents the maximum compound rate of growth from a process that grows at 100% for one time period. Sure, you start out expecting to grow from 1 to 2. But with each tiny step forward you create a little “dividend” that starts growing on its own. When all is said and done, you end up with e (2.718…) at the end of 1 time period, not 2. Instead of in “spurts,” this type of growth occurs continuously  growth of money in an account that pays interest continuously  growth of money in an account that pays interest continuously  the decay of radioactive material which occurs continuously  the decay of radioactive material which occurs continuously t = 1 t = 1 e= 2.718… e= 2.718…

21. The “natural” number ‘e’ works perfectly with natural processes Exponential growth of populations Exponential decay of radioactive material

22. Your turn: 7. simplify 6. simplify 6. simplify

23. Exponential Function (base ‘e’) What is the “initial value” ? Describe the transformation: What is the horizontal asymptote ? Is it growth or decay? Where does g(x) cross the y-axis ?

24. Putting it all together: If negative: Reflect across x-axis Initial value: Crosses y-axis here Growth factor: If negative: decay Horizontal asymptote vertical shift

25. Your turn: For each of the following what is the: a. “initial value”? a. “initial value”? b. Growth or decay? b. Growth or decay? c. “horizontal asymptote” c. “horizontal asymptote” d. Any reflections (across x-axis) d. Any reflections (across x-axis) 8. 9. 10.

26. Your turn: 11. The population of Detroit decreases by by 3% every 11. The population of Detroit decreases by by 3% every year. In 2000 the population was 1.5 million. year. In 2000 the population was 1.5 million. What was the population in 2009? 12. The population of a small town can be modeled by: What is the % change in population for every time period ‘t’ ? period ‘t’ ?

27. Value of a depreciating asset. According to tax law, the value of a piece of equipment can be “depreciated” and the depreciation can be used as a “business “depreciated” and the depreciation can be used as a “business expense” to reduce the amount of taxes that you pay. expense” to reduce the amount of taxes that you pay. A company buys a car for $20,000. It can depreciate the value of the car by 20% per year. What is the value of the car after 2 years?

28. Your turn: A company is in debt for $300,000. It is reducing its debt at the annual rate of 15%. What is the company’s debt at the annual rate of 15%. What is the company’s debt after 10 years? debt after 10 years? 13. $59,062.32

29. continuous compounding Remember that base “e” is used for things that grow (or decrease) continuously instead of in “spurts.” The more frequently the interest is paid in a savings account the faster the money will grow for a given interest rate. the faster the money will grow for a given interest rate. The fastest possible rate of growth for a given interest rate occurs with continuous compounding. occurs with continuous compounding. Periodic compounding continuous compounding PERT

30. Your turn: You put $500 into an account earning 5% interest which is compounded continuously. How much which is compounded continuously. How much money will be in the account after 6 years? money will be in the account after 6 years? 14. $675.93 15. Radioactive material “decays” continuously (to some other element). The “decay constant” of (to some other element). The “decay constant” of Carbon-14 is k = -0.0001258. If 50 grams of C-14 Carbon-14 is k = -0.0001258. If 50 grams of C-14 decays for 2000 years, how much carbon-14 will decays for 2000 years, how much carbon-14 will remain? remain? 38.9 gms

31. HOMEWORK  Section 7-1 (page 482) 2, 4, 16, 18, 20, 24, 28 2, 4, 16, 18, 20, 24, 28 Section 7-2 (page 489) 4, 6, 8, 16, 18, 30a 4, 6, 8, 16, 18, 30a Section 7-3 (page 495) 6, 8, 20, 22, 32, 38, 40, 56 6, 8, 20, 22, 32, 38, 40, 56 (21 total problems)