1.3 Exponential Functions

Mt. St. Helens, Washington StateGreg Kelly, Hanford High School, Richland, Washington

Although some of today’s lecture is from the book, some of it is not. You must take notes to be successful in calculus.

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

4100 123.881.055

On the TI-89: 100 1.055 ^ 4 ENTER

At the end of each year, interest is paid on the amount in the account and added back into the account, so the amount of increase gets larger each year.

This is an example of an exponential function:

xf ax exponent

base

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

MODE Graph……. FUNCTION

Display Digits… FLOAT 6

Angle……. RADIAN ENTER

Y= 1 2 ^y x x

2 3^y x x

3 5 ^y x x

WINDOW

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

Y= 1 2 ^y x x

2 3^y x x

3 5 ^y x x

WINDOW

GRAPH

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

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

What is the domain?

,

What is the range?

0,

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 you solved this using logs. Today we are going to solve it graphically for practice.

Y= 1 1y x

2 5 (1/ 2) ^ ( / 20)y x x

WINDOW

GRAPH

WINDOW

GRAPH

Upper bound and lower bound arex-values.

F5Math

5Intersection

Use the arrow keys to select a first curve, second curve, lower bound and upper bound, and press ENTER each time.

46 days

The TI-89 has the exponential growth and decay model built in as an exponential regression equation.

A regression equation starts with the points and finds the equation.

U.S. Population:

1880189019001910192019301940195019601970

50.2 million 63.0 76.0 92.0105.7122.8131.7151.3179.3203.3

To simplify, let represent 1880, represent 1890, etc.

0x 1x

0,1,2,3,4,5,6,7,8,9 L1 ENTER

2nd { 0,1,2,3,4,5,6,7,8,9 2nd }

STO alpha L 1

ENTER

50.2,63,76,92,105.7,122.8,131.7,151.3,179.3,203.3 L2

ExpReg L1, L2 ENTER

2nd MATH 6 3

Statistics Regressions

2

ExpReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

(Upper case L used for clarity.)

50.2,63,76,92,105.7,122.8,131.7,151.3,179.3,203.3 L2

ShowStat ENTER

2nd MATH 6 8

Statistics ShowStat

ENTER

The calculator gives you an equation and constants:

xy a b 55.054258

1.160626

a

b

ExpReg L1, L2 ENTER

2nd MATH 6 3

Statistics Regressions

2

ExpReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

We can use the calculator to plot the new curve along with the original points:

Y= y1=regeq(x)

2nd VAR-LINK regeq

x )

Plot 1 ENTER

ENTER

WINDOW

Plot 1 ENTER

ENTER

WINDOW

GRAPH

WINDOW

GRAPH

What does this equation predict for the population in 1990?

F3Trace

This lets us see values for the distinct points.

Moves to the line.

This lets us trace along the line.

11 ENTER Enters an x-value of 11.

What does this equation predict for the population in 1990?

11 ENTER Enters an x-value of 11.

In 1990, the population was predicted to be 283.4 million.

This is an over estimate of 33 million, or 13%. Why might this be?

To find the annual rate of growth:

Since we used 10 year intervals with b=1.160626 :

10 1.160631 r

10 1.16063 1r

0.015r or 1.5%

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.

TblSet

We can have the calculator construct a table to investigate how this function behaves as x gets much larger.

tblStart …….1000

tbl………..1000

ENTER

ENTER ENTER

TABLE

Move to the y1 column and scroll down to watch the y value approach e.