1. 4.3 Use Functions Involving e p. 244 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, π, and imaginary numbers.

3. Natural Base e Like π 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….

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. Use a calculator to evaluate the expression. a.e4e4 b.e –0.09 ExpressionKeystrokesDisplay 54.59815003 0.9139311853 [ ] e x 4 [ ]exex 0.09

9. Simplify the expression. 4. (10 e –4x ) 3 = 10 3 ( e –4x ) 3 = 1000 e 12x Use a calculator to evaluate5.e 3/4. 2.117 e 3/4 = = 1000 e –12x

10. 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

11. 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

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

13. 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

14. Graph the function. State the domain and range. SOLUTION The domain is all real numbers, and the range is y > 1. b.y = e –0.75(x – 2) + 1 a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit. e –0.75x

15. Using e in real life. In 4.1 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

16. Continuously Compounded Interest A = Pe rt “Shampoo” Problems

17. 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

18. A = Pe rt SOLUTION FINANCE: You deposit $2500 in an account that pays 5% annual interest compounded continuously. Find the balance after each amount of time? Use the formula for continuously compounded interest. Write formula. Substitute 2500 for P, 0.05 for r, and 2 for t. ≈2762.9 The balance at the end of 2 years is $2762.90. ANSWER a. 2 years =2500e 0.10 =2500e (0.05 2) =2500 1.105

19. 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

20. 4.3 Assignment Page 247, 3-48 every third problem