Astone is tossed into the air from the ground level with an initial velocity 15 m/sec. it’s height in meters a t seconds is h(t) = 1.5t - 4.9t^2. compute the average velocity of the stone over the given time intervals, then use that information to calculate the instantaneous velocity of the stone at t = 1sec. show your work time intervals:

a) [1, 1.0]

b) [1, 1.01]

c) [1, 1.005]

d) [1, 1.001]

For the interval [1, 1.0] the average velocity is

For the interval [1, 1.01] the average velocity is

For the interval [1, 1.005] the average velocity is

For the interval [1, 1.001] the average velocity is

The instantaneous velocity at t = 1 is .

Explanation:

The average velocity over is

where

= change in position

= change in time (length of interval)

We know that the height in meters a t seconds is given

so the average velocity is

Consider the interval [1, 1.0]

For the interval [1, 1.01]

For the interval [1, 1.005]

For the interval [1, 1.001]

The instantaneous rate of change is the limit of the average rates of change. We estimate the instantaneous rate of change at by computing the average rate of change over smaller and smaller intervals

From the calculations above we can see when the time interval shrinks the average velocity tends to the value . This suggests that this value is a good candidate for the instantaneous velocity at t = 1.

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