1.3 Exponential Functions

Interest

• If $100 is invested for 4 years at 5.5% interest compounded annually, find the ending amount.

This is an example of an exponential function:

f(x) = k●ax

Graph for in a [-5,5] by [-2,5] window:xy a 2, 3, 5a

Where is ?2 3 5x x x

0,x

Where is ?2 3 5x x x

,0x

Where is ?2 3 5x x x

0x

Population growth can often be modeled with an exponential function:

Ratio:

5023 4936 1.0176 5111 5023 1.0175

1.01761.02461.0175

World Population:

1986 4936 million1987 50231988 51111989 52011990 53291991 5422

The world population in any year is about 1.018 times the previous year.

in 2010: 195422 1.018P 7609.7

About 7.6 billion people.

Nineteen years past 1991.

Radioactive decay can also be modeled with an exponential function:

Suppose you start with 5 grams of a radioactive substance that has a half-life of 20 days. When will there be only one gram left?

After 20 days:1 5

52 2

40 days:2 51

542

t days:201

52

t

y

In Pre-Calc and in AB Cal you solved this using logs. Today we are going to solve it graphically for practice.

Many real-life phenomena can be modeled by an exponential function with base , where .e 2.718281828e

e can be approximated by: 11

xf x

x

As , x f x e

Graph:y=(1+1/x)^x in a[-10,10] by [-5,10]window.

Use “trace” to investigate the function.

y = ex and y = e–x are used as forms of exponential growth and decay.

Interest Compounded Continuosly:y = Pert

P is the principle investmentr is the interest rate (decimal)t is the time (years)