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Chapter 3 | Derivatives
sin t −sin0
sin t t
t
t −0 =
−0.1
0.998334166
−0.01
0.9999833333
−0.001 0.999999833
0.001
0.999999833
0.01
0.9999833333
0.1
0.998334166
Table 3.1 Average velocities using values of t approaching 0
From the table we see that the average velocity over the time interval [−0.1, 0] is 0.998334166, the average velocity over the time interval [−0.01, 0] is 0.9999833333, and so forth. Using this table of values, it appears that a good estimate is v (0) =1. By using Equation 3.5 , we can see that v (0) = s ′(0) = lim t →0 sin t −sin0 t −0 = lim t →0 sin t t =1. Thus, in fact, v (0) =1.
3.4 A rock is dropped from a height of 64 feet. Its height above ground at time t seconds later is given by s ( t ) =−16 t 2 +64, 0≤ t ≤2. Find its instantaneous velocity 1 second after it is dropped, using Equation 3.5 .
As we have seen throughout this section, the slope of a tangent line to a function and instantaneous velocity are related concepts. Each is calculated by computing a derivative and each measures the instantaneous rate of change of a function, or the rate of change of a function at any point along the function.
Definition The instantaneous rate of change of a function f ( x ) at a value a is its derivative f ′( a ).
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