Copyright © Cengage Learning. All rights reserved. 10 Introduction to the Derivative

Copyright © Cengage Learning. All rights reserved Average Rate of Change

33 Average Rate of Change of a Function Numerically and Graphically

44 Example 1 – Standard and Poor’s 500 The following table lists the approximate value of Standard and Poor’s 500 stock market index (S&P) during the period 2000–2008 (t = 0 represents 2000):

55 Example 1 – Standard and Poor’s 500 a.What was the average rate of change in the S&P over the 2-year period 2005–2007 (the period 5  t  7 or [5, 7] in interval notation); over the 4-year period 2000–2004 (the period 0  t  4 or [0, 4]); and over the period [1, 5]? b.Graph the values shown in the table. How are the rates of change reflected in the graph? cont’d

66 Example 1(a) – Solution During the 2-year period [5, 7], the S&P changed as follows: Thus, the S&P increased by 300 points in 2 years, giving an average rate of change of 300/2 = 150 points per year.

77 Example 1(a) – Solution We can write the calculation this way: cont’d

88 Example 1(a) – Solution Interpreting the result: During the period [5, 7] (that is, 2005–2007), the S&P increased at an average rate of 150 points per year. Similarly, the average rate of change during the period [0, 4] was cont’d

99 Example 1(a) – Solution Interpreting the result: During the period [0, 4] the S&P decreased at an average rate of 87.5 points per year. Finally, during the period [1, 5], the average rate of change was cont’d

10 Example 1(a) – Solution Interpreting the result: During the period [1, 5] the average rate of change of the S&P was zero points per year (even though its value did fluctuate during that period). cont’d

11 Example 1(b) – Solution The rate of change of a quantity that changes linearly with time is measured by the slope of its graph. However, the S&P index does not change linearly with time. Figure 19 shows the data plotted two different ways: (a) as a bar chart and (b) as a piecewise linear graph. Bar charts are more commonly used in the media, but Figure 19(b) illustrates the changing index more clearly. cont’d

12 Example 1(b) – Solution Figure 19(a)Figure 19(b) cont’d

13 Example 1(b) – Solution We saw in part (a) that the average rate of change of S over the interval [5, 7] is the ratio cont’d

14 Example 1(b) – Solution Notice that this rate of change is also the slope of the line through P and Q shown in Figure 20, and we can estimate this slope directly from the graph as shown. Figure 20 cont’d

15 Example 1(b) – Solution Average Rate of Change as Slope: The average rate of change of the S&P over the interval [5, 7] is the slope of the line passing through the points on the graph where t = 5 and t = 7. Similarly, the average rates of change of the S&P over the intervals [0, 4] and [1, 5] are the slopes of the lines through pairs of corresponding points. cont’d

16 Average Rate of Change of a Function Numerically and Graphically Change and Average Rate of Change of f over [a, b]: Difference Quotient The change in f (x) over the interval [a, b] is Change in f =  f = Second value – First value = f (b) – f (a). The average rate of change of f (x) over the interval [a, b] is

17 Average Rate of Change of a Function Numerically and Graphically = Slope of line through points P and Q (see figure). We also call this average rate of change the difference quotient of f over the interval [a, b]. (It is the quotient of the differences f (b) – f (a) and b – a.) A line through two points of a graph like P and Q is called a secant line of the graph. Average rate of change = Slope of PQ

18 Average Rate of Change of a Function Numerically and Graphically Units The units of the change in f are the units of f (x). The units of the average rate of change of f are units of f (x) per unit of x. Quick Example If f (3) = –1 billion dollars, f (5) = 0.5 billion dollars, and x is measured in years, then the change and average rate of change of f over the interval [3, 5] are given by Change in f = f (5) – f (3) = 0.5 – (–1) = 1.5 billion dollars = 0.75 billion dollars/year.

19 Average Rate of Change of a Function Numerically and Graphically Alternative Formula: Average Rate of Change of f over [a, a + h] (Replace b above by a + h.) The average rate of change of f over the interval [a, a + h] is

20 Average Rate of Change of a Function Using Algebraic Data

21 Example 3 – Average Rate of Change from a Formula You are a commodities trader and you monitor the price of gold on the New York Spot Market very closely during an active morning. Suppose you find that the price of an ounce of gold can be approximated by the function G(t) = –8t t dollars (7.5 ≤ t ≤ 10.5) where t is time in hours.

22 Example 3 – Average Rate of Change from a Formula See Figure 23. t = 8 represents 8:00 AM. Figure 23 G(t) = – 8t t cont’d

23 Example 3 – Average Rate of Change from a Formula Looking at the graph, we can see that the price of gold rose rather rapidly at the beginning of the time period, but by t = 8.5 the rise had slowed, until the market faltered and the price began to fall more and more rapidly toward the end of the period. What was the average rate of change of the price of gold over the -hour period starting at 8:00 AM (the interval [8, 9.5] on the t-axis)? cont’d

24 Example 3 – Solution We have Average rate of change of G over [8, 9.5] From the formula for G(t), we find G(9.5) = −8(9.5) (9.5) = 796 G(8) = −8(8) (8) = 790.

25 Example 3 – Solution Thus, the average rate of change of G is given by In other words, the price of gold was increasing at an average rate of $4 per hour over the given -hour period. cont’d