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Hprec5.2

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• (a) P(35)=314.3 ; (b) approx 2065
• Hprec5.2

1. 1. 5-2: Exponential Functions © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Graph and identify transformations of exponential functions. •Use exponential functions to solve application problems.
2. 2. Definition For all real numbers a, ,1a ≠ there is an exponential function with base a whose domain is all real numbers and whose rule is: ( ) x f x a=
3. 3. Try This How does the exponential function differ from other functions such as ( ) x f x a= 2 ( )f x x= and ?3 ( )f x x=
4. 4. Important Idea The graph of looks like: ( ) x f x a= 1a > 0 1a< < Exponential Grow Exponential Decay (0,1)
5. 5. Important Idea A good investment will grow exponentially due to compounding. years \$ Exponential Grow
6. 6. Example Graph y=2x using a table of values... •What happens to y as x → ∞ •What happens to y as x → −∞ •What is the value of y when x=0.
7. 7. Try This Graph using your calculator What happens to y as x → ∞ What happens to y as x → −∞ What is the value of y when x=0. 1 2 x y   =    
8. 8. Solution Exponential Decay 1 2 x y   =    
9. 9. 1 2 x y   =     (0,1) Exponential Grow Compare: Exponential Decay Important Idea 2xy =
10. 10. Important Idea For 0<a<1: y=ax models exponential decay For a>1: y=ax models exponential growth
11. 11. Example Graph on the same axes and describe behavior: y=2x y=4x y=8x
12. 12. Try This 2 1 4 x y   =  ÷   Graph on the same axes and describe behavior: 1 1 2 x y   =  ÷   3 1 8 x y   =  ÷  
13. 13. Solution 1 1 2 x y   =  ÷   2 1 4 x y   =  ÷   3 1 8 x y   =  ÷  
14. 14. Try This Graph on the same axes and describe behavior: 1 2x y = 3 2 2x y + = 3 3 2 4x y − = −
15. 15. Solutions 1 2x y = 3 2 2x y + = 3 3 2 4x y − = −
16. 16. Example If you invest \$5000 in a stock that increases at an average rate of 8% per year, then the value of your stock is given by the function: ( ) 5000(1.08)x f x = where x is measured in years. What is your investment worth in 10 years?
17. 17. Example If you invest \$5000 in a stock that increases at an average rate of 8% per year, then the value of your stock is given by the function: ( ) 5000(1.08)x f x = where x is measured in years. When will your investment be worth \$15000?
18. 18. Try This ( ) 5000(1.08)x f x = How would you change the equation from the last problem, , if your investment was \$6000 and your investment increases at 10% per year? ( ) 6000(1.10)x f x =
19. 19. Definition The natural exponential function is a variation of ( ) x f x a= and is written ( ) x f x e= . e ≈ 2.718.
20. 20. Important Idea The number e is located on your calculator in 2 places
21. 21. Example If the population of the U.S. continues to grow as it has since 1980, then the pop. (in millions) in year t where t=0 corresponds to 1980 is given by: .0093 ( ) 227 t p t e= a. Estimate the population in 2015.
22. 22. Example If the population of the U.S. continues to grow as it has since 1980, then the pop. (in millions) in year t where t=0 corresponds to 1980 is given by: .0093 ( ) 227 t p t e= b. when will the population reach 500 million?
23. 23. Try This The amount of 1 kg. of plutonium that remains after t years is ( ) .99997t m t = How much of the original 1 kg. of plutonium remains after 10,000 years? .74 kg
24. 24. Important Idea In real world applications, most things cannot grow forever as suggested by the exponential growth model. The Logistic Model is designed to model situations that have limited future growth.
25. 25. Example The population of certain bacteria in a beaker at time t hours is given by 2 100,000 ( ) 1 50 t p t e − = + Graph and find the upper limit on the bacteria population.
26. 26. Lesson Close We will examine other applications of exponential functions in future lessons.
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