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Unformatted text preview: (3/19/08) Section 2.2 Average rates of change Overview: This section is background for the definition of the derivative in the next section. The examples of average velocity and other average rates of change considered here illustrate the types of mathematical models that we will use later with derivatives, and the derivative will be defined as a limit of average rates of change. Topics: Average velocity Other average rates of change Estimating instantaneous velocities with average velocity Average velocity Imagine that a pilot is flying a small airplane toward the west from an airport. As a mathematical model of her flight, we suppose that the plane is s ( t ) = t 3 + 30 t + 100 miles from the airport t hours after noon (Figure 1). We begin by calculating the planes average velocity during a particular period of time. t 1 2 3 4 5 s (miles) 100 200 300 400 s = t 3 + 30 t + 100 (hours) FIGURE 1 Example 1 What is the planes average velocity from t = 1 to t = 5? Solution The planes average velocity is equal to the distance it travels divided by the time taken. Since s (1) = 131 and s (5) = 375, its average velocity for 1 t 5 is Distance traveled Time taken = [ s (5)- s (1)] miles (5- 1) hours = (375- 131) miles 4 hours = 244 miles 4 hours = 61 miles hour . square (1) Notice that the average velocity (1) is the slope of the line in Figure 2 through the points at t = 1 and t = 5 on the graph of the planes distance from the airport. The rise is the distance the plane travels, and the run the time it takes to go that distance. The line is called a secant line because it passes through two points on the curve. The terminology secant line comes from the Latin word secare , meaning to cut. It is used because secant lines generally cut across curves. 78 Section 2.2, Average rates of change p. 79 (3/19/08) t 1 2 3 4 5 s (miles) 131 375 s = t 3 + 30 t + 100 (hours) 4 hours 244 miles FIGURE 2 bracketleftbigg Average velocity bracketrightbigg = Rise Run = 244 miles 4 hours = 61 miles hour Any average velocity can be interpreted as the slope of a secant line: Definition 1 If an object is at s = s ( t ) on an s-axis at time t , then its average velocity in the positive s-direction between times a and b is its change in position, s ( b )- s ( a ) , divided by the change in time, b- a . This ratio is the slope of the secant line through the points at t = a and t = b on the graph of s ( t ) (Figure 3): [Average velocity from time a to time b ] = Change in position Change in time = s ( b )- s ( a ) b- a ( a negationslash = b ) . (2) t s s = s ( t ) b- a s ( b )- s ( a ) a b s ( a ) s ( b ) FIGURE 3 bracketleftbigg Average velocity bracketrightbigg = Rise Run = s ( b )- s ( a ) b- a Other average rates of change Average velocity is the average rate of change of distance with respect to time. Consequently, Definition 1 is a special case of the following general definition of average rate of change....
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