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Warm up Graph

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Lesson 11-3 The Number e Objective: To use the exponential function y = e x

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Natural Base e Like 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….

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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.

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Using a calculator Evaluate e 2 using a graphing calculator Locate the e x button you need to use the second button 7.389

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Graphing examples Graph y=e x Remember the rules for graphing exponential functions! The graph goes thru (0,1) and (1,e) (0,1) (1,2.7)

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

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Using e in real life. 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 = Pe rt

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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=Pe rt = 1000e.08*1 ≈ $1083.29

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Practice An amount of $1,240.00 is deposited in a bank paying an annual interest rate of 2.85 %, compounded continuously. Find the balance after 2½ years. A = 1240e (.0285)(2.5) = $1,331.57

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Exponential Decay An artifact originally had 12 grams of carbon- 14 present. The decay model A = 12e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in this artifact after 10,000 years? A = 12e -0.000121t A = 12e -0.000121(10,000) A = 12e -1.21 A = 3.58

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Sources myteacherpages.com/webpages/rrowe, 2/22/14.

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