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4.3 Use Functions Involving e

4.3 Use Functions Involving e

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4.3 Use Functions Involving e

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  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….

  4. Examples • e3· e4 = • e7 • (3e-4x)2 • 9e(-4x)2 • 9e-8x • 9 e8x • 10e3= 5e2 • 2e3-2 = • 2e

  5. More Examples! • (2e-5x)-2= • 2-2e10x= • e10x 4 • 24e8= 8e5 • 3e3

  6. Using a calculator 7.389 • Evaluate e2 using a graphing calculator • Locate the ex button • you need to use the second button

  7. [ ] ex [ ] ex a. e4 4 b. e –0.09 0.09 Use a calculator to evaluate the expression. Expression Keystrokes Display 54.59815003 0.9139311853

  8. (10 ) 4. e –4x 103 3 = 1000 e –12x = = . 5. Use a calculator to evaluate e 3/4 ) ( e –4x 3 e 3/4 2.117 = 1000 e12x Simplify the expression.

  9. Graphing • f(x) = aerxis 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=ex • Remember the rules for graphing exponential functions! • The graph goes thru (0,a) and (1,e) (1,2.7) (0,1) y=0

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

  12. Graphing Example Graph y=2e0.75x 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. (1,4.23) (0,2) y=0

  13. + 1 b. y e –0.75(x – 2) = 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 Graph the function. State the domain and range. SOLUTION The domain is all real numbers, and the range is y > 1.

  14. 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 = Pert

  15. Continuously Compounded Interest A = Pert “Shampoo” Problems

  16. 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=Pert = 1000e.08*1 ≈ $1083.29

  17. A = Pert = 2500 e (0.05 •2) = 2500 e0.10 •1.105 = 2500 ANSWER The balance at the end of 2 years is $2762.90. FINANCE:You deposit $2500 in an account that pays 5% annual interest compounded continuously. Find the balance after each amount of time? a. 2 years SOLUTION Use the formula for continuously compounded interest. Write formula. Substitute 2500 for P, 0.05 for r, and 2 for t. ≈2762.9

  18. 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? Pert

  19. 4.3 Assignment Page 247, 3-48 every third problem