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 • 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. 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…. • 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. Examples • · 7 •e 3 e 4 e = 3 •10e = 5e2 3-2 = •2e •2e -4x 2 •(3e ) (-4x)2 •9e •9e-8x • 9 e8x More Examples! • 8 24e = 5 8e • 3e3 -5x -2 •(2e ) = •2-2e10x= 10x •e 4 Using a calculator 2 • Evaluate e using a graphing calculator • Locate the ex button • you need to use the second button 7.389 Evaluate e-.06 with a calculator Graphing • f(x) = rx ae 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 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) Graphing cont. • Graph y=e-x (0,1) (1,.368) 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) 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 = rt Pe 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