Presentation on theme: "Warm up Graph. Lesson 11-3 The Number e Objective: To use the exponential function y = e x."— Presentation transcript:
Warm up Graph
Lesson 11-3 The Number e Objective: To use the exponential function y = e x
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/ ≈ ….
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.
Using a calculator Evaluate e 2 using a graphing calculator Locate the e x button you need to use the second button 7.389
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)
Graphing cont. Graph y=e -x (0,1) (1,.368)
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
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=Pe rt = 1000e.08*1 ≈ $
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
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 A = 3.58