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8.3 The number e

8.3 The number e. Mrs. Spitz Algebra 2 Spring 2007. Objectives:. Use the number e as the base of exponential functions. Use the natural base e in real-life situations such as finding the air pressure on Mount Everest. Assignment. Worksheet 8.3A. The Natural base e.

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8.3 The number e

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  1. 8.3 The number e Mrs. Spitz Algebra 2 Spring 2007

  2. Objectives: • Use the number e as the base of exponential functions. • Use the natural base e in real-life situations such as finding the air pressure on Mount Everest.

  3. Assignment • Worksheet 8.3A

  4. 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.

  5. 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….

  6. 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.

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

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

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

  10. Evaluate e-.06witha calculator

  11. 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

  12. 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)

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

  14. 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)

  15. Using e in real life. • 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

  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

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