1. 1.3 Exponential Functions
Acadia National Park, Maine
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2008

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

3. We will be using the TI-89 calculator in this class.
You may use either the TI-89 Titanium or the older TI-89.
TI-89
The pictures in the lectures will usually illustrate the older TI-89.
Although the buttons on the Titanium Edition are different shapes and colors, they are in the same positions and have the same functions.
TI-89 Titanium

4. This is an example of an exponential function:
If $100 is invested for 4 years at 5.5% interest, compounded annually, the ending amount is:
On the TI-89:
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:
exponent
base

5. Graph for in a [-5,5] by [-2,5] window:
MODE
Graph…….
FUNCTION
Display Digits…
FLOAT 6
Angle…….
RADIAN
ENTER Y=
WINDOW

6. Graph for in a [-5,5] by [-2,5] window:

7. Graph for in a [-5,5] by [-2,5] window:
Where is ?
Where is ?
Where is ?

8. Graph for in a [-5,5] by [-2,5] window:
Where is ?
What is the domain?
What is the range?
Where is ?
Where is ?

9. Population growth can often be modeled with an exponential function:
World Population:
million
Ratio:
The world population in any year is about times the previous year.
Nineteen years past 1991.
in 2010:
About 7.6 billion people.

10. 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:
40 days:
t days:
In Pre-Calc you solved this using logs. Today we are going to solve it graphically for practice.

11. Y=
WINDOW
GRAPH

12. WINDOW GRAPH F5 5 46 days Upper bound and lower bound are x-values.
Math 5
Intersection
Use the arrow keys to select a first curve, second curve, lower bound and upper bound, and press ENTER each time.
46 days

13. e can be approximated by:
Many real-life phenomena can be modeled by an exponential function with base , where
e can be approximated by:
Graph:
y=(1+1/x)^x in a
[-10,10] by [-5,10]
window.
Use “trace” to investigate the function.

14. We can have the calculator construct a table to investigate how this function behaves as x gets much larger.
TblSet
tblStart …….1000
ENTER
tbl………..1000
ENTER
ENTER
TABLE
* Stop here, or continue to include regression functions on the TI-89.
Move to the y1 column and scroll down to watch the y value approach e. p*

15. 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.

16. , To simplify, let represent 1880, represent 1890, etc. 50.2 million
U.S. Population:
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
50.2 million
63.0
76.0
92.0
105.7
122.8
131.7
151.3
179.3
203.3
2nd {
0,1,2,3,4,5,6,7,8,9
2nd }
STO
alpha
L 1
(Upper case L used for clarity.)
ENTER 6 3 2
alpha
L 1 ,
alpha
L 2
ENTER
2nd
MATH
The calculator should return:
ExpReg
Done
Statistics
Regressions

17. 6 3 2
alpha
L 1 ,
alpha
L 2
ENTER
2nd
MATH
The calculator should return:
ExpReg
Done
Statistics
Regressions 6 8
ENTER
2nd
MATH
Statistics
ShowStat
The calculator gives you an equation and constants:

18. We can use the calculator to plot the new curve along with the original points:Y=
y1=regeq(x) x )
regeq
2nd
VAR-LINK
Plot 1
ENTER
ENTER
WINDOW

19. Plot 1
ENTER
ENTER
WINDOW
GRAPH

20. WINDOW
GRAPH

21. What does this equation predict for the population in 1990?
Trace
This lets us see values for the distinct points.
This lets us trace along the line. 11
ENTER
Enters an x-value of 11.
Moves to the line.

22. What does this equation predict for the population in 1990?
In 1990, the population was predicted to be million.
This is an over estimate of 33 million, or 13%. Why might this be? 11
ENTER
Enters an x-value of 11.

23. p To find the annual rate of growth:
Since we used 10 year intervals with b= : or p