Lesson 1.3: Exponential Functions AP Calculus Mrs. Mongold

Review of Exponent Rules ax ∙ ay = ax+y (ax)y = axy ax ∙ bx = (ab)x

Exponential Function Let a be a positive real number other than 1. Then f(x) = ax is the exponential function with base a. Domain of f(x) = ax is (-∞, ∞) Range of f(x) = ax is (0, ∞)

Quick Examples y = 2x y = 2-x

Exponential Growth y = Pax P = initial amount a = 1 + rate as a decimal x = # of times compounded F(x) = P(1+r)x

Example Find growth of $100 invested in 2009 at an interest rate of 5.5% in 2013 which was compounded annually. Find amount in 2020

Example Predict the world population in 2020 using an exponential model and the table Year Population (millions) Ratio 2005 4936 2006 5023 2007 5111 2008 5201 2009 5329 2010 5422

Exponential Decay The function y=kax k>0 and 0<a<1

Half-life example A certain radioactive substance has a half-life of 20 days. If there are 5 grams present initially when will there be only 1 gram remaining?

The number e e=2.718281828 We can define e to be the number that the function f(x) = (1+ 1/x)x approaches as x approaches infinity. y=ex and y=e-x are frequently used as models for exponential growth and decay.

Example y=Pert is used for interest compounded continually

Homework Pgs 24-25/1-6 all, 7-17 odd, 23, 25, 29, 30, 33