2. MAT 150 Module 8 – Exponential Functions Lesson 1 – Exponential functions and their applications

3. Standard Form of an Exponential Function An exponential function is a function in the form. k is the initial value of the function the initial value is the value of the function when x = 0. a n is the base of the function. The base is the rate at which the function increases (grows) or decreases (decays).

4. Important thing to remember: Every time x increases by 1 y is multiplied by a n

5. as n approaches ∞ The number e e is a special number e is approximately 2.7181828

6. Calculator Note To raise to a power, use ^ or y x Put parentheses around the entire power!!! 2^(3*4)+5 = 4101 Example: Compute 2 3x + 5 for x = 4 2^3*4+5 = 37 Correct!Incorrect!

7. Exponential Growth vs. Decay Two Cases Exponential Growth Exponential Decay F(x) increases as x increases F(x) decreases as x increases

8. Exponential Growth vs. Decay Exponential Growth Exponential Decay X increases Y increases X increases Y decreases

9. Exponential Growth vs Decay Base: a n >1 Base: a n < 1 Growth Function Decay Function Compound interest Population Growth Depreciation of an asset Radioactive Decay

10. Exponential Growth Analyze the function y = 5(2.5) 3x

11. Exponential Growth Example: Analyze the function y = 5(2.5) 3x. The base of the function is 2.5 3 ≈ 15.625 Base is >1: Growth Function The Initial Value is 5Y = 5 when x = 0

12. Exponential Growth

13. Exponential Decay Analyze the function y = 2e -0.1x

14. Exponential Decay Analyze the function y = 2e -0.1x The base of the function is e -0.1 ≈ 0.905 Base is <1: Decay Function The Initial Value is 2Y = 2 when x = 0

15. Exponential Decay

16. Applications of exponential functions Exponential functions have lots of real-world uses in modeling: Growth of investments Depreciation of assets in accounting Radioactive decay Population growth Let’s explore some of these applications!

17. Example 1- Applications The function f(x) = 700(0.5) 0.014286x models the amount in pounds of a particular radioactive material stored in a concrete vault, where x is the number of years since the material was put into the vault. a.How many pounds of radioactive material were initially stored in the vault? b.To the nearest pound, how many pounds of the amount of radioactive material will be left in the vault after 50 years? c.When will there be only 10 grams remaining?

18. Example 1 - Solution a.How many pounds of radioactive material were initially stored in the vault? The initial value is the value of the function when x = 0. f(0) = 700 There were 700 pounds initially stored in the vault. X = 0 Year Zero Initial amount

19. Example 1 - Solution b.To the nearest pound, how many pounds of the amount of radioactive material will be left in the vault after 50 years? The amount remaining after 50 years = f(50) Calculate the equation for x = 50 f(50) = 700(0.5) (0.014286*50) ≈ 426.65 ≈ 427 Approx. 427 pounds remain after 50 years.

20. Example 1 - Solution c.When will there be only 10 grams remaining? In this case, we want to find an x value so that y = 10. We can solve this by graphing.

21. Example 1 - Solution

22. Example 2 - Applications Jim invests $2000 in a savings account that pays 0.3% compounded continuously. The function that gives us his account balance after t years is A(t) = 2000e 0.003t. a.How much money will he have after five years? b.How many years will it take to double his initial investment? (Use the graphing calculator to approximate to the nearest year.)

23. Example 2 - Solution a. How much money will he have after five years? The amount remaining after 5 years = f(5) Calculate the equation for x = 5 f(5) = 2000(e) (0.003*5) ≈ 2030.226 ≈ 2030.23 He has approx. $2030.23 in his account after five years.

24. Example 2 - Solution b.How many years will it take to double his initial investment? (Use the graphing calculator to approximate to the nearest year.) Initial investment is $2000 Double is $4000

25. Example 3 - Solution