CHAPTER 1:PREREQUISITES FOR CALCULUS

SECTION 1.3:EXPONENTIAL FUNCTIONS

AP CALCULUS AB

Exponential GrowthExponential DecayApplicationsThe Number e…and why

Exponential functions model many growth patterns.

What you’ll learn about…

Exponential Function

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

( )

is the .

x

a

f x a

a

=

exponential function with base

The domain of f(x) = ax is (-, ) and the range is (0,).

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

Exponential Growth

If 1 the graph of looks like the graph

of 2 in Figure 1.22ax

a f

y=

Exponential Growth

If 0 1 the graph of looks like the graph

of 2 in Figure 1.22b.x

a f

y -=

Section 1.3 – Exponential Functions

Example: Graph the function State its domain and range.

3 2 3xf x

Section 1.3 – Exponential Functions

You try: Graph each function State its domain and range.1. 3 6 2. 2 4x xy y

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 aa

ba b

a a a

+

-

> >

× = × =

æö÷ç= =÷ç ÷çè ø

= =

b

a x

Rules for Exponents

Rules for Exponents If a > 0 and b > 0, then the following hold for all real numbers

x and y.Rule Example

1.

2.

3.

4.

5.

yxyx aaa 6642 9 333 yx

y

x

aa

a 34

7

xx

x

xyxyyx aaa 842 33

xxx abba

x

xx

b

a

b

a

xxx 632

2733

3

3

33xxx

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.

Note: Carbon-14 half-life is about 5730 years.

Half-life

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. In the following equation, n represents the half-life.

nx

ky

2

1

Section 1.3 – Exponential Functions

Example: Suppose the half-life of a certain radioactive substance is 12 days and that there are 8 grams present initially. When will there be only 1.5 grams of the substance remaining?

(Hint: Solve graphically)

Section 1.3 – Exponential Functions

You try: The half-life of a radioactive substance is 20 days. The number of grams present initially is 10 grams. Determine when 4 grams of the substance will remain.

Exponential Growth and Exponential Decay

The function , 0, is a model for

if 1, and a model for if 0 1.

xy k a k

a a

exponential growth

exponential decay

= × >

> < <

Zeros of Exponential Functions

To find the zeros of an exponential function using a graphing calculator (TI-83 or 84):

1. Enter the equation in y1.2. Graph in the appropriate window.3. Use the following keystrokes:

CALC (2nd TRACE)ZEROWhen it says “Left Bound?”, go just left of

the x-intercept and hit ENTER.When it says “Right Bound?”, go just right

of the x-intercept and hit ENTER.When it says “Guess?”, go to approximately

the x-intercept and hit ENTER.It will print out

ZEROx = ________ y = ________

The zero is the x-value.

Example Exponential Functions

( )Use a grapher to find the zero's of 4 3.xf x = -

( ) 4 3xf x = -

[-5, 5], [-10,10]

Section 1.3 – Exponential Functions

Example: Find the zeros of graphically.

7 1.25xf x

Section 1.3 – Exponential Functions

You try: Find the zeros of each functiongraphically.

1.

2. 4 1.25xf x

2 1.20xf x

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

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 P

r

-= =

= ×

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

Example The Number e

[0,100] by [0,120] in 10’s

Remember

Compounding Formulas:1. Simple Interest:

2. Compounded n times per year:

3. Compounded continuously:

trAtA 10

nt

n

rAtA

10

percent. on the form decimal theis where,0 rePtP rt