Chapter 3 | Derivatives
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Figure 3.8 The slope of the secant line is the average velocity over the interval [ a , t ]. The slope of the tangent line is the instantaneous velocity.
We can use Equation 3.5 to calculate the instantaneous velocity, or we can estimate the velocity of a moving object by using a table of values. We can then confirm the estimate by using Equation 3.7 . Example 3.7 Estimating Velocity A lead weight on a spring is oscillating up and down. Its position at time t with respect to a fixed horizontal line is given by s ( t ) = sin t ( Figure 3.9 ). Use a table of values to estimate v (0). Check the estimate by using Equation 3.5 .
Figure 3.9 A lead weight suspended from a spring in vertical oscillatory motion.
Solution We can estimate the instantaneous velocity at t =0 by computing a table of average velocities using values of t approaching 0, as shown in Table 3.1 .
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