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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? 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. 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…. 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 Use a calculator to evaluate the expression. Expression Keystrokes a. e4 b. e –0.09 [ex] [ex ] Display 54.59815003 4 0.09 0.9139311853 Simplify the expression. 3 3 –4x 3 –4x ) 4. (10e ) = 10 ( e = 1000 e –12x 1000 = e12x 5. Use a calculator to evaluate e 3/4. e 3/4 = 2.117 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) y=0 Graphing cont. • Graph y=e-x (0,1) y=0 (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) y=0 Graph the function. State the domain and range. b. y = e –0.75(x – 2) + 1 SOLUTION a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = e –0.75x right 2 units and up 1 unit. The domain is all real numbers, and the range is y > 1. 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 = rt Pe Continuously Compounded Interest A= rt Pe “Shampoo” Problems 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 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. A = Pert Write formula. = 2500e (0.05 •2) = 2500 e0.10 = 2500 •1.105 ≈2762.9 Substitute 2500 for P, 0.05 for r, and 2 for t. ANSWER The balance at the end of 2 years is $2762.90. • 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 4.3 Assignment Page 247, 3-48 every third problem