1.3 Exponential Functions

Slide 1- 2

Exponential Function

Let be a positive real number other than 1. The function

( ) is the .

x

a

f x aa

=exponential function with base

The domain of ( ) is ( , ) and the range is (0, ).

Compound interest investment and population growth are examples of exponential growth.

xf x a= - ¥ ¥ ¥

Slide 1- 3

Rules for Exponents

( )

( ) ( )

If 0 and 0, the following hold for all real numbers and .

1. 4.

2. 5.

3.

xx y x y x x

xx xx y

y x

y xx y xy

a b x y

a a a a b ab

a a aaba b

a a a

+

-

> >× = × =

æö÷ç= =÷ç ÷çè ø= =

Slide 1- 4

Half-life

Exponential functions can also model phenomena that produce decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation.

Slide 1- 5

Exponential Growth and Exponential Decay

The function , 0, is a model for if 1, and a model for if 0 1.

xy k a ka a

exponential growthexponential decay

= × >> < <

Ex: The half-life of a certain radioactive substance is 14 days. There are 6.6 g present initially.

A. Express the amount of substance remaining as a function of time t.

B. When will there be 1g remaining?

Slide 1- 7

The Number e

Many natural, physical and economic phenomena are best modeled by an exponential function whose base is the famous number , which is 2.718281828 to nine decimal places.

We can define to be the numbe

e

e ( ) 1r that the function 1

approaches as approaches infinity.

x

f xx

x

æ ö÷ç= + ÷ç ÷çè ø

Slide 1- 8

The Number e

The exponential functions and are frequently used as models of exponential growth or decay.

Interest compounded continuously uses the model , where is the initial investment, is t

x x

r t

y e y e

y P e Pr

-= =

= ×he interest rate as a decimal and is the time in years.t

Do example 5 on page 25

Hwk: p. 26, #1-17 e.o.o. 19,21,25,33,39,41-46