Rates of Change

Lesson 1.2

2

Which Is Best?

Which roller coaster would you rather ride?

Why?

Today we will look at a mathematical explanation for

why one is preferable to another.

3

Rate of Change

Given function y = 3x + 5

•• •

•

x y

0 5

1 8

2 11

3 14

4 17

Change in y = y = 66

Change in x = = 2x

2

change in y 6Average rate of change = 3change in x 2

yx

4

Rate of Change

Try calculating for differentpairs of (x, y) points

You should discover that the rate of change is constant

x y

0 5

1 8

2 11

3 14

4 17

1 2

1 2

change in yAverage rate of change = change in x

yx

y yx x

5

Rate of Change

Consider the function Enter into Y= screen of calculator

View tables on calculator ( Y)

( )f x x

You may need to specify the

beginning x value and the increment

6

Rate of Change

As before, determine therate of change fordifferent sets of orderedpairs

x sqrt(x)

0 0.00

1 1.00

2 1.41

3 1.73

4 2.00

5 2.24

6 2.45

7 2.65

1 2 1 2

1 2 1 2

change in yAverage rate of change = change in x( ) ( )

yx

y y f x f xx x x x

7

Rate of Change

View spreadsheet which demonstrates results of the formula below.

1 2 1 2

1 2 1 2

change in yAverage rate of change = change in x

( ) ( )

yx

y y f x f xx x x x

8

Rate of Change (NOT a constant)

You should find that the rate of change is changing – NOT a constant.

Contrast to thefirst functiony = 3x + 5 ( )f x x

9

Function Defined by a Table

Consider the two functions defined by the table The independent variable is the year.

Predict whether or not the rate of change is constant

Determine the average rate of change for various pairs of (year, sales) values

Year 1982 1984 1986 1988 1990 1992 1994

CD sales 0 5.8 53 150 287 408 662

LP sales 244 205 125 72 12 2.3 1.9

10

Increasing, Decreasing Functions

Note that for the CD sales, the rates of change were always positive

For the LP sales, the rates of change were always negative

An increasing function

A decreasing function

11

Increasing, Decreasing Functions

0

100

200

300

400

500

600

700

1980 1985 1990 1995

CD sales

LP sales

An increasing function

A decreasing function

12

Increasing, Decreasing Functions

Given Q = f ( t ) A function, f is an increasing function if the

values of f increase as t increases The average rate of change > 0

A function, f is an decreasing function if the values of f decrease as t increases The average rate of change < 0

13

Using TI to Find Rate Of Change

Define a function f(x)3*x + 5 -> f(x)

We want to define the function

and assign it to a function

Use the STO> key( ) ( )f a f ba b

14

Using TI to Find Rate Of Change

Now call the function difquo( a, b ) using two different x values for a and b

For rate of change of a different function, redefine f(x)

15

Assignment

Lesson 1.2 Page 15 Exercises

3, 5, 7, 9, 11, 12, 13, 15, 21