Exponential Functions - PowerPoint Presentation

Slide1

Chapter 1.3

Exponential FunctionsSlide2

The Exponential Function

DEFINITION:

Let

be a positive real number other than 1. The function

is the exponential function with base a.

2Slide3

The Exponential Function

The domain of an exponential function is

and the range

is

)If , the function values increase as x increases, while if , the function values decrease as x

increasesExponential change occurs when function values change by a set percentage for every unit of change in the input valueThat is,

For example, if

, then

, meaning that function values decrease by half each time x increases by 1.If , then , so the function values double for each unit of increase in

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Example 1: Graphing an Exponential Function

Graph the function

and state its domain and range.

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Example 1: Graphing an Exponential Function

Since the parent function

, with

, has domain

, then the domain of the given function is also

. We also have that

for all values of x. So multiplying by a positive constant C on both sides gives

and adding constant k gives

. This implies that the range is all values greater than the constant . So the given function has range . The graph is given on the next slide. 5Slide6

Example 1: Graphing an Exponential Function

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Example 2: Finding Zeros

Find the

zeros

of

. 

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Example 2: Finding Zeros

Later you will be able to solve this using logarithms, but for now you will learn two ways to solve equations on the TI89 calculator. This is important to know because, on the calculator portions of the AP test, you are not required to solve equations by hand. Indeed, you will sometimes end up with equations that cannot be solved by hand.

We will first solve the equation

by defining a function on the Home screen, then using the solve function from the Math button.

Then, we will graph the function and find the zero on the graph. 

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Example 2: Finding Zeros

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Rules for Exponents

Rules for Exponents

If

, the following hold for all real numbers x and y

These are the familiar rules for exponents, but applied in this case when the exponent is a variable. Since

an

exponential equation has domain

, the rules can

also be

applied to exponential equations.

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Example 3: Predicting United States Population

Use the data in the table below and an exponential model to predict the population of the United States in the year 2010.

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Year

Population (millions)Ratio2002

287.92003

290.4

2004

293.22005295.92006298.82007301.6YearPopulation (millions)Ratio2002287.9

2003290.4

2004

293.2

2005

295.9

2006

298.8

2007

301.6Slide12

Example 3: Predicting United States Population

Note that the rate of change from one year to the next is approximately the same. This is characteristic of exponential growth. The exponential growth follows the form

, where

a

is the constant ratio

and

is the initial function value (i.e.,

), and

t is the time in years. So using and , our model in this case is . We need the initial value of t to be zero, so for 2002 we let , for 2003 we let , and so forth. Since 2012 is 10 years after the initial time, million. 12Slide13

Exponential Decay

For the function,

, if

the graph of the function rises as

x increases; if

, then the graph falls as x increasesThe former is exponential growth and the latter exponential decay

We could also take

in every case so that exponential decay would follow the form

Exponential decay models the behavior of radioactive substances that decay over timeThe half-life of a radioactive substance is the time it takes for half of a given amount to decay 13Slide14

Example 4: Modeling Radioactive Decay

Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram of substance remaining?

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Example 4: Modeling Radioactive Decay

In this problem, the unit of time is one 20-day period, during which half the amount of substance decays. Since the exponent in both growth and decay models represents the number of unit time changes (whether it be one year or one 20-day period), then our exponent will be

. Hence, starting with 5 grams, after 20 days there will remain

grams. After a second 20-day period there will remain

grams. We see, then, that our decay model follows the form

, or alternatively,

. Solve this graphically for

t

if

gram. The solution is approximately 46.4 days.

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Example 4: Modeling Radioactive Decay

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Exponential Change

DEFINITIONS:

The function

is a model for

exponential growth if

.The function

is a model for

exponential decay

if . Alternatively, we can take so that the model for exponential decay becomes . 17Slide18

Example 5: Predicting U.S. Population

Use the population data in the table below to estimate the population for the year 2000. Compare the results with the actual 2000 population of approximately 281.4 million.

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Year

Population (millions)188050.21890

63.01900

76.2

1910

92.21920106.01930123.21940132.11950151.31960179.31970203.31980226.51990248.7Slide19

Example 5: Predicting U.S. Population

Using the exponential regression function on the TI89, the best-fit growth function is

(Note that

is in 10-year intervals;

is the first 10-year interval,

is the second 10-year interval, and so on.)

Extrapolating for the year 2000 gives

This is an overestimate of the actual population by about 26.8 million or about

. 19Slide20

The Number

e

The letter

e

is used to represent the so-called natural number will become very important later on in our studies and will show up continuously when you study differential equationsExponential growth and decay can be expressed as

respectively

We will see that we can define

e

asfor very large values of n (a positive integer) 20Slide21

Exercise 1.3

Online Exercise 1.3

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