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Unformatted text preview: Chapter 3 Average velocity, Average Rates of Change, and Secant Lines In this chapter, we extend the idea of slope of a straight line to a related concept for a curve. We will first encounter the idea of an average slope, (also denoted average rate of change, and average velocity in the example to come). That quantity is the slope of a secant line (a line that connects two points on a given curve). We will then consider how a process of refinement can lead us to the concept of the derivative, which is the slope of a tangent line (also called instantaneous rate of change, or instantaneous velocity in the following example). The connection between these two concepts will form a major theme in this course 1 . We begin with a motivating example in which such ideas can be developed naturally, the motion of a falling object. 3.1 Observations about falling objects The left panel of Figure 3.1 shows a set of three stroboscopic images combined (for visualization purposes) on a single graph. Each set of dots shows successive vertical positions of an object falling from a height of 20 meters over a 2 second time period. In the first image, at left, the location of the ball is given at times t = 0 , . 5 , 1 , 1 . 5, and 2 . 0 seconds, i.e. at intervals of Δ t = 0 . 5 seconds. (A strobe flashing five times, once every Δ t = 0 . 5 would produce this data.) We might wonder where the ball is located at times between these successive measurements. Did it vanish? Did it continue in a straight or a looped path? To find out what happened during the intervals between data points, we could increase the strobe frequency, and record measurements more often: for example, in the image at the center of the right panel in Figure 3.1 measurements were made for t = 0 , . 2 , . 4 , . . ., 2 . 0 seconds, i.e. at intervals Δ t = 0 . 2 seconds. An even closer set of points appears at right, where the time interval between strobe flashes was decreased to Δ t = 0 . 1 second. By determining the position of the ball at closer time points, we can determine the trajectory of the ball with greater accuracy. The idea of making measurements at finer and finer time increments is important in this example. We will return to it often in our goal of understanding rates of change of natural processes. In this chapter, we would like to establish some understanding of the idea of a velocity. Uniform motion is defined as motion in which a constant distance is covered in constant time intervals. For particles moving uniformly, velocity is constant, and is simply the distance travelled divided by 1 This connection will form the theme of Lab 3 in this Calculus course v.2005.1 - September 4, 2009 1 Math 102 Notes Chapter 3 0.0 3.0 0.0 20.0 0.0 6.0 0.0 20.0 Figure 3.1: Left panel: Three experiments with a falling object are shown here. The time interval between successive positions of the object is Δ t = 0 . 5 in the experiment shown on the left, Δ t = 0 . 2 in the middle set, and Δ t = 0 . 1 in the set at the right. Right panel: The positions of the ball1 in the set at the right....
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- Winter '09