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Chapter 2 | Limits
2.1 Estimate the slope of the tangent line (rate of change) to f ( x ) = x 2 at x =1 by finding slopes of secant lines through (1, 1) and the point ⎛ ⎝ 5 4 , 25 16 ⎞ ⎠ on the graph of f ( x ) = x 2 .
We continue our investigation by exploring a related question. Keeping in mind that velocity may be thought of as the rate of change of position, suppose that we have a function, s ( t ), that gives the position of an object along a coordinate axis at any given time t . Can we use these same ideas to create a reasonable definition of the instantaneous velocity at a given time t = a ? We start by approximating the instantaneous velocity with an average velocity. First, recall that the speed of an object traveling at a constant rate is the ratio of the distance traveled to the length of time it has traveled. We define the average velocity of an object over a time period to be the change in its position divided by the length of the time period. Definition Let s ( t ) be the position of an object moving along a coordinate axis at time t . The average velocity of the object over a time interval [ a , t ] where a < t (or [ t , a ] if t < a ) is (2.2) v ave = s ( t )− s ( a ) t − a . As t is chosen closer to a , the average velocity becomes closer to the instantaneous velocity. Note that finding the average velocity of a position function over a time interval is essentially the same as finding the slope of a secant line to a function. Furthermore, to find the slope of a tangent line at a point a , we let the x -values approach a in the slope of the secant line. Similarly, to find the instantaneous velocity at time a , we let the t -values approach a in the average velocity. This process of letting x or t approach a in an expression is called taking a limit . Thus, we may define the instantaneous velocity as follows. Definition For a position function s ( t ), the instantaneous velocity at a time t = a is the value that the average velocities approach on intervals of the form [ a , t ] and [ t , a ] as the values of t become closer to a , provided such a value exists.
Example 2.2 illustrates this concept of limits and average velocity. Example 2.2 Finding Average Velocity
A rock is dropped from a height of 64 ft. It is determined that its height (in feet) above ground t seconds later (for 0≤ t ≤2) is given by s ( t ) =−16 t 2 +64. Find the average velocity of the rock over each of the given time intervals. Use this information to guess the instantaneous velocity of the rock at time t =0.5. a. ⎡ ⎣ 0.49, 0.5 ⎤ ⎦
b. ⎡
⎣ 0.5, 0.51 ⎤ ⎦
Solution Substitute the data into the formula for the definition of average velocity. a. v ave = s (0.5)− s (0.49) 0.5−0.49 =−15.84
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