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6.6 The Natural Base, e

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Natural Base, e The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.

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Natural Exponential Function The exponential function with base e, f(x)=e x, is called the natural exponential function. And e is called the natural base. You calculator has a FANCY e x button you can use!!!

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Evaluate f(x)=e x to the nearest thousandth for each value of x below: x = 2 x = -1/3 x = 0.25

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Simplify the expression. a.e2e2 e5e5 = e = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e (5(5 ) c.e –3x 2 = 5252 ( ) 2 = 25 e –6x = 25 e6xe6x Exponential Properties

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Simplify the expression. 1.e7e7 e4e4 e e8e8 4e5e5 12 e2e e –3 6e56e5 6e3e3 Try This! 4. ( 10 e –4x ) e 12x Use a calculator to evaluate

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The Graph of the Exponential Function

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Graph the function. State the domain and range. a.y = 3e 0.25x SOLUTION Because a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plug in 0 and 1. The domain is all real numbers, and the range is y > 0. Graph natural base functions x y 0 1

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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 Graph natural base functions

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Graph the function. State the domain and range. y = 2e 0.5x domain: all real numbers, range: y>0 Try This!

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Graph the function. State the domain and range. 1 2 f(x) = e -x +1 domain: all real numbers, range: y>1 Try This!

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Continuous Compounding Formula If P dollars are invested at an interest rate, r, that is compounding continuously, then the amount, A, of the investment at time t is given by: A = Pe rt

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An investment of $1000 earns an annual interest rate of 7.6% Compare the final amounts after 8 years for interest compounded quarterly and for interest compounded continuously. Compounded Quarterly Continuously

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Find the value: Of $500 after 4 years invested at an annual interest rate of 9% compounded continuously. $716.66

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Homework Green Book!! Page 142 # 1-25odd #27 and 28

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