10
Introduction to the
Derivative
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10.4
Average Rate of Change
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Average Rate of Change of a Function
Numerically and Graphically
3
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):
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Example 1 – Standard and Poor’s 500
cont’d
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?
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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.
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Example 1(a) – Solution
cont’d
We can write the calculation this way:
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Example 1(a) – Solution
cont’d
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
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Example 1(a) – Solution
cont’d
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
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Example 1(a) – Solution
cont’d
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).
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Example 1(b) – Solution
cont’d
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.
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Example 1(b) – Solution
Figure 19(a)
cont’d
Figure 19(b)
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Example 1(b) – Solution
cont’d
We saw in part (a) that the average rate of change of S
over the interval [5, 7] is the ratio
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Example 1(b) – Solution
cont’d
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
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Example 1(b) – Solution
cont’d
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.
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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
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Average Rate of Change of a Function Numerically and Graphically
= Slope of line through points P and Q
(see figure).
Average rate of change = Slope of PQ
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.
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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.
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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
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Average Rate of Change of a Function
Using Algebraic Data
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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 2 + 144t + 150 dollars
(7.5 ≤ t ≤ 10.5)
where t is time in hours.
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Example 3 – Average Rate of Change from a Formula
cont’d
See Figure 23. t = 8 represents 8:00 AM.
G(t) = –8t 2 + 144t + 150
Figure 23
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Example 3 – Average Rate of Change from a Formula
cont’d
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)?
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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)2 + 144(9.5) + 150
= 796
G(8) = −8(8)2 + 144(8) + 150
= 790.
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Example 3 – Solution
cont’d
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.
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