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.
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
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.