244 CHAPTER 2 Graphs and Functions EXAMPLE 8 Interpreting Slope as Average Rate of Change In 2013, Amazon spent $6565 million on research and development. In 2017, Amazon spent $22,620 million on research and development. Assume a linear relationship, and find the average rate of change in the amount of money spent on R&D per year. Graph as a line segment, and interpret the result. SOLUTION To use the slope formula, we need two ordered pairs. Here, If x = 2013, then y = 6565, and if x = 2017, then y = 22,620. This gives the two ordered pairs 12013, 65652 and 12017, 22,6202. (Here y is in millions of dollars.) Average rate of change = 22,620 - 6565 2017 - 2013 = 16,055 4 ≈4014 The graph in Figure 41 confirms that the line through the ordered pairs rises from left to right and therefore has positive slope. Thus, the annual amount of money spent by Amazon on R&D increased by an average of about of $4014 million each year from 2013 to 2017. S Now Try Exercise 69. 2013 2014 2015 2016 2017 0 5,000 10,000 15,000 20,000 25,000 x y Year Dollars (in millions) Amazon Spending on R&D (2013, 6565) (2017, 22,620) Data from www.bloomberg.com Figure 41 Average rate of change of ƒ1x2 on the interval [a, b] is given by the slope of the secant line. m= ƒ1b2 −ƒ1a2 b −a Secant line y = f(x) a b f(a) (a, f(a)) f(b) (b, f(b)) x y Figure 42 EXAMPLE 9 Finding Average Rate of Change for a Nonlinear Function Figure 43 shows approximate annual federal spending for health care (in billions of dollars) from 1970 to 2017. Here y represents billions of dollars, and t represents number of years after 1970. Find and interpret the average rate of change in spending from 2000 to 2017. SOLUTION The year 2000 is 30 yr after 1970, and the year 2017 is 47 yr after 1970. The average rate of change from 2000 to 2017 is the slope of the line joining the two points shown in the figure. m= 3612.0 - 1329.4 47 - 30 = 2282.6 17 ≈134.3 Average rate of change Therefore, federal health care spending increased by an average of approximately $134.3 billion each year from 2000 to 2017. S Now Try Exercise 77. Year 10 20 30 40 50 5000 4000 3000 (30, 1329.4) (47, 3612.0) 2000 Dollars (in billions) 1000 t 0 Data from Kaiser Family Foundation analysis of National Health Expenditure (NHE) data. y Federal Health Care Spending Figure 43 The definition of average rate of change applies to every function, not just to linear functions. For linear functions, the average rate of change is the slope of the graph of the function, but for nonlinear functions, it is the slope of the line joining the two points 1a, ƒ1a22 and 1b, ƒ1b22. The line through these two points is a secant line. See Figure 42.
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