94 CHAPTER 2 Functions and Their Graphs In Words The symbol Δ is the Greek capital letter delta and is read “change in.” DEFINITION Average Rate of Change Suppose the domain of a function y f x( ) = is an interval I. If a and b a b , , ≠ are in I, the average rate of change of f from a to b is defined as y x f b f a b a a b Average rate of change ( ) ( ) = Δ Δ = − − ≠ (1) Finding the Average Rate of Change Find the average rate of change of ( ) = f x x3 :2 (a) From 1 to 3 (b) From 1 to 5 (c) From 1 to 7 EXAMPLE 7 Solution (a) The average rate of change of ( ) = f x x3 2 from 1 to 3 is y x f f 3 1 3 1 27 3 3 1 24 2 12 ( ) ( ) Δ Δ = − − = − − = = (b) The average rate of change of ( ) = f x x3 2 from 1 to 5 is y x f f 5 1 5 1 75 3 5 1 72 4 18 ( ) ( ) Δ Δ = − − = − − = = (c) The average rate of change of ( ) = f x x3 2 from 1 to 7 is y x f f 7 1 7 1 147 3 7 1 144 6 24 ( ) ( ) Δ Δ = − − = − − = = NOTE The difference quotient of a function f can be used to find the average rate of change between the points ( ) ( ) x f x , and ( ) ( ) + + x h f x h , . j Figure 34 f x x3 2 ( ) = x y 40 80 160 120 6 4 2 (3, 27) (5, 75) Average rate of change 5 12 Average rate of change 5 18 Average rate of change 5 24 (7, 147) (0, 0) (1, 3) Figure 35 Secant line y x b a (b, f (b )) Secant line (a, f (a )) f (b ) 2 f (a ) b 2 a y 5 f (x ) See Figure 34 for a graph of ( ) = f x x3 .2 The function f is increasing on the interval [ )∞ 0, . The fact that the average rate of change is positive for any ≠ x x x x , , , 1 2 1 2 in the interval [ ] 1, 7 indicates that f is increasing on ≤ ≤ x 1 7. Now Work PROBLEM 65 The Secant Line The average rate of change of a function has an important geometric interpretation. Look at the graph of ( ) = y f x in Figure 35. Two points are labeled on the graph: ( ) ( ) a f a , and ( ) ( ) b f b , . The line containing these two points is called a secant line ; its slope is m f b f a b a sec ( ) ( ) = − − The symbol Δy in equation (1) is the “change in y, ” and Δx is the “change in x. ” The average rate of change of f is the change in y divided by the change in x.
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