1. 8.3 The number e p. 480

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. 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

15. Homework P. 483 (17-73) odd