Rates of Change Lesson 1.2

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.

Rate of Change Given function y = 3x + 5 • • • • x y 5 1 8 2 11 3 14 4 5 1 8 2 11 3 14 4 17 Given function y = 3x + 5 • 6 2 • • •

Rate of Change Try calculating for different pairs of (x, y) points 5 1 8 2 11 3 14 4 17 Try calculating for different pairs of (x, y) points You should discover that the rate of change is constant

You may need to specify the beginning x value and the increment Rate of Change Consider the function Enter into Y= screen of calculator View tables on calculator ( Y) You may need to specify the beginning x value and the increment

Rate of Change As before, determine the rate of change for different sets of ordered pairs x sqrt(x) 0.00 1 1.00 2 1.41 3 1.73 4 2.00 5 2.24 6 2.45 7 2.65

Rate of Change View Geogebra demo which demonstrates results of the formula below.

Rate of Change (NOT a constant) You should find that the rate of change is changing – NOT a constant. Contrast to the first function y = 3x + 5

Function Defined by a Table Year 1982 1984 1986 1988 1990 1992 1994 CD sales 5.8 53 150 287 408 662 LP sales 244 205 125 72 12 2.3 1.9 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

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

Increasing, Decreasing Functions A decreasing function An increasing function

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

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

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)

Assignment Lesson 1.2 Page 15 Exercises 3, 5, 7, 9, 11, 12, 13, 15, 21