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6.1 Exponential Growth and Decay

6.1 Exponential Growth and Decay. With Applications. Exponential Expression. An expression where the exponent is the variable and the base is a fixed number. Multiplier. The base of an exponential expression. Growth vs. Decay. When b>1, f(x) = b x represents GROWTH

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6.1 Exponential Growth and Decay

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  1. 6.1 Exponential Growth and Decay With Applications

  2. Exponential Expression • An expression where the exponent is the variable and the base is a fixed number

  3. Multiplier • The base of an exponential expression

  4. Growth vs. Decay • When b>1, f(x) = bx represents GROWTH • When 0<b<1, f(x) = bx represents DECAY

  5. Applications • Exponential Growth and Decay can be found in many applications • Ex: population growth, stocks, science studies, compound interest, and effective yield

  6. Basic Growth/Decay Applications: • When dealing with most growth and decay apps, you have an equation such as:

  7. Base is your multiplier • Growth: multiplier = 100% + rate • Decay: multiplier = 100% - rate

  8. Growth/Decay App. WS Problem 1: • The population of the United States was 248,718,301 in 1990 and was projected to grow at a rate of about 8% per decade. Predict the population, to the nearest hundred thousand, for the years 2010 and 2025.

  9. Growth/Decay App. WS Problem 1: • Growth Application • Initial Population = 248,718,301 • Multiplier = 100% + 8% = 108% = 1.08 • Expression to model the problem:

  10. Growth/Decay App. WS Problem 1: • 2010: 2 decades after 1990 • n = 2

  11. Growth/Decay App. WS Problem 1: • Round to the nearest hundred thousand: • 290,100,000 = Population in 2010

  12. Growth/Decay App. WS Problem 1: • 2025: 3.5 decades after 1990 • n = 3.5

  13. Growth/Decay App. WS Problem 1: • Round to the nearest hundred thousand: • 325,600,000 = Population in 2025

  14. Population Formula WS • This is for homework, but it is for you to practice writing population formulas. • YOU DO NOT HAVE TO SOLVE ANYTHING….JUST WRITE THE FORMULAS

  15. Compound Interest Formula • Another application of exponential growth • The total amount of an investment A, earning compound interest is: P = Principal, r = annual interest rate, n = # of times interest is compounded per year, t = time in years

  16. Example • Find the final amount of a $500 investment after 8 years, at 7% interest compounded annually, quarterly, monthly, daily.

  17. Example (cont) • P = $500 • r = 7% = .07 • t = 8 years • Annually, n = 1

  18. Example (cont):Annually

  19. Example (cont):Quarterly • n = 4

  20. Example (cont):Monthly • n = 12

  21. Example (cont):Daily • n = 365

  22. Effective Yield • The annually compounded interest rate that yields the final amount of an investment. • Determine the effective yield by fitting an exponential regression equation to 2 points. • Effective Yield = b - 1

  23. Example • A collector buys a painting for $100,000 at the beginning of 1995 and sells it for $150,000 at the beginning of 2000. Write an equation to model this situation and then find the effective yield.

  24. Example (Cont) • When modeling the situation, you use the compounded interest formula, and you let n = 1 for compounded annually. • A(t) = ending value = $150,000 • P = initial = $100,000 • n = 1 and t = 5 years

  25. Example (Cont)

  26. Example (Cont) • Now we need to find the effective yield • First we need 2 points that would model the data: (0, 100000) and (5, 150000)

  27. Example (Cont) • Plug these points into your calc • STAT, EDIT

  28. Example (Cont) • Then generate the Exponential Regression: STAT, CALC, 0:ExpReg • y = abx • a = 100000 • b = 1.084 • Effective yield = 1.084 – 1 = .084 = 8.4% annual interest rate

  29. Homework: • Finish BOTH WS • Pg 358 #15, 18, 21, 37, 42, 47, 48 • Pg. 367 #17-23odd, 29-33odd, 47, 49

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