1.3 Exponential Functions Acadia National Park, Maine

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

We will be using the TI-89 calculator in this class. You may use either the TI-89 Titanium or the TI-84. TI-89 Titanium 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

If $100 is invested for 4 years at 5.5% interest, compounded annually, the ending amount is: On the TI-89 TI 84 or N-Spire: 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

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

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

Population growth can often be modeled with an exponential function: Ratio: World Population: million The world population in any year is about times the previous year. in 2010: 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: 40 days: t days: In Pre-Calc you solved this using logs. Today we are going to solve it graphically for practice.

Y= WINDOW GRAPH

WINDOW GRAPH Upper bound and lower bound are x-values. F5 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

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.

TblSet We can have the calculator construct a table to investigate how this function behaves as x gets much larger. tblStart …….1000 tbl……… ENTER TABLE Move to the y1 column and scroll down to watch the y value approach e. 

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: million To simplify, let represent 1880, represent 1890, etc. 2nd { 0,1,2,3,4,5,6,7,8,9 } STO alpha L 1 ENTER 2nd MATH 63 StatisticsRegressions 2 ExpReg alpha L 1 L 2 ENTER Done The calculator should return:, (Upper case L used for clarity.)

2nd MATH 68 StatisticsShowStat ENTER The calculator gives you an equation and constants: 2nd MATH 63 StatisticsRegressions 2 ExpReg alpha L 1 alpha L 2 ENTER Done The 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 WINDOW

Plot 1 ENTER WINDOW GRAPH

WINDOW GRAPH

What does this equation predict for the population in 1990? F3 Trace 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 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 = : or 