1. 8.3 The number e p. 480 What is the Euler number? How is it defined? Do laws of exponents apply to “e” number? How do you use “e” on your calculator? When graphing base e, how do you know if you have growth or decay? What is the formula for continuously compounded interest?

2. The Natural base e Much of the history of mathematics is marked by the discovery of special types of numbers like counting numbers, zero, negative numbers, Л, and imaginary numbers.

3. Natural Base e Like Л and ‘i’, ‘e’ denotes a number. Called The Euler Number after Leonhard Euler (1707-1783) It can be defined by: e= 1 + 1 + 1 + 1 + 1 + 1 +… 0! 1! 2! 3! 4! 5! = 1 + 1 + ½ + 1/6 + 1/24 +1/120+... ≈ 2.718281828459….

4. The number e is irrational – its’ decimal representation does not terminate or follow a repeating pattern. The previous sequence of e can also be represented: As n gets larger (n→∞), (1+1/n) n gets closer and closer to 2.71828….. Which is the value of e.

5. Examples e 3 · e 4 = e 7 10e 3 = 5e 2 2e 3-2 = 2e (3e -4x ) 2 9e (-4x)2 9e -8x 9 e 8x

6. More Examples! 24e 8 = 8e 5 3e 3 (2e -5x ) -2 = 2 -2 e 10x = e 10x 4

7. Using a calculator Evaluate e 2 using a graphing calculator Locate the e x button you need to use the second button 7.389

8. Evaluate e -.06 with a calculator

9. Graphing f(x) = ae rx is a natural base exponential function If a>0 & r>0 it is a growth function If a>0 & r<0 it is a decay function

10. Graphing examples Graph y=e x Remember the rules for graphing exponential functions! The graph goes thru (0,a) and (1,e) (0,1) (1,2.7) y=0

11. Graphing cont. Graph y=e -x (0,1) (1,.368) y=0

12. Graphing Example Graph y=2e 0.75x State the Domain & Range Because a=2 is positive and r=0.75, the function is exponential growth. Plot (0,2)&(1,4.23) and draw the curve. (0,2) (1,4.23) y=0

13. Using e in real life. In 8.1 we learned the formula for compounding interest n times a year. In that equation, as n approaches infinity, the compound interest formula approaches the formula for continuously compounded interest: A = Pe rt

14. Continuously Compounded Interest A = Pe rt Shampoo Problems

15. Example of continuously compounded interest You deposit $1000.00 into an account that pays 8% annual interest compounded continuously. What is the balance after 1 year? P = 1000, r =.08, and t = 1 A=Pe rt = 1000e.08*1 ≈ $1083.29

16. What is the Euler number? Natural base e How is it defined? 2.718 - - it is an irrational number like pi Do laws of exponents apply to “e” number? Yes- - all of them. How do When graphing base e, how do you know if you have growth or decay? Growth rises on the right and decay rises on the left. What is the formula for continuously compounded interest? Pe rt

17. Assignment Page 483, 17-65 odd, 76