2 Objective: To use the exponential function y = ex Lesson 11-3 The Number eObjective: To use the exponential function y = ex
3 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/≈ ….
4 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.
5 Using a calculator Evaluate e2 using a graphing calculator 7.389Evaluate e2 using a graphing calculatorLocate the ex buttonyou need to use the second button
6 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)
8 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
9 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 = 1A=Pert = 1000e.08*1 ≈ $
10 PracticeAn 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)= $1,331.57A = 1240e(.0285)(2.5)= $1,331.57
11 Exponential DecayAn 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 tA = 12e (10,000)A = 12e-1.21A = 3.58