1.3 Exponential Functions AP Calculus AB/BC 1.3 Exponential Functions

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.

We will be using the TI-84 Plus calculator in this class. You may use either the TI-84 Platinum or the TI-84 Plus. TI-84 The pictures in the lectures will usually illustrate the TI-84 Plus. Although the buttons on the Platinum Edition are different shapes and colors, they are in the same positions and have the same functions.

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-84: 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 (Note: I like to use b for base. The text uses a for base.)

Graph for in a [-5,5] by [-2,5] window: MODE Graph…………. FUNCTION Display Digits… FLOAT Angle………….. RADIAN 2ND QUIT Y= First, clear any previous functions. WINDOW

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

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

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

Population growth can often be modeled with an exponential function: World Population: 1986 4936 million 1987 5023 1988 5111 1989 5201 1990 5329 1991 5422 Ratio: The world population in any year is about 1.018 times the previous year. Nineteen years past 1991. in 2010: About 7.6 billion people.

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: Of course, you can solve this using logs. Today we are going to solve it graphically for practice.

Y= WINDOW GRAPH

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

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 [0,10] by [-5,10] window. Use “trace” to investigate the function.

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 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*

The TI-89 and TI-84 have the exponential growth and decay model built in as an exponential regression equation. The following instructions are for the TI-83 and TI-84. You will have to translate them to work on the TI-89. A regression equation starts with the points and finds the equation.

To simplify, let represent 1880, represent 1890, etc. 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 To simplify, let represent 1880, represent 1890, etc. 2nd { 0,1,2,3,4,5,6,7,8,9 2nd } STO 2nd L1 ENTER The calculator should return: STAT CALC 0:ExpReg ENTER

The calculator gives you an equation and constants: The calculator should return: STAT CALC 0:ExpReg ENTER The calculator gives you an equation and constants:

We can use the calculator to plot the new curve along with the original points: Y= y1=regeq(x) VARS 5:Statistics EQ 1:RegEQ Press to highlight Plot 1 then press ENTER WINDOW

WINDOW GRAPH

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.

What does this equation predict for the population in 1990? 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? 11 ENTER Enters an x-value of 11.

p To find the annual rate of growth: Since we used 10 year intervals with b=1.160626 : or p