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Warm up Graph

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**Objective: To use the exponential function y = ex**

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

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**Natural Base e Like and ‘i’, ‘e’ denotes a number.**

Called The Euler Number after Leonhard Euler ( ) It can be defined by: e= … 0! 1! 2! 3! 4! 5! = ½ + 1/6 + 1/24 + 1/ ≈ ….

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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 ….. Which is the value of e.

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**Using a calculator Evaluate e2 using a graphing calculator**

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

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**Graphing examples Graph y=ex**

Remember the rules for graphing exponential functions! The graph goes thru (0,1) and (1,e) (1,2.7) (0,1)

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

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**A = Pert 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 = Pert

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**Example of continuously compounded interest**

You deposit $ 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 ≈ $

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Practice An amount of $1, 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 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 t 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 t A = 12e (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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Journal: Write an exponential growth equation using the natural base with a horizontal asymptote of y=-2.

Journal: Write an exponential growth equation using the natural base with a horizontal asymptote of y=-2.

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