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Rates of Change
Lesson 1.2
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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Assignment
Lesson 1.2 Page 15 Exercises
3, 5, 7, 9, 11, 12, 13, 15, 21