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If a ball is thrown straight up into the air with an initial velocity of $80$ ft/s, its height in feet after $t$ seconds is given by $$y=80t−16t^2$$ Find the average velocity for the time period beginning when $t=1$ and lasting

(i) $0.5$ seconds

(ii) $0.1$ seconds

(iii) $0.01$ seconds

Finally, based on the above results, guess what the instantaneous velocity of the ball is when $t=1$.

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  • $\begingroup$ Can you share your thoughts on the problem, and explain what you've tried? For example, do you know the definition of average velocity, or at least have a rough idea of what it means? $\endgroup$ – user61527 Oct 25 '13 at 6:53
  • $\begingroup$ Average velocity = distance/time $\endgroup$ – dnova Oct 25 '13 at 6:57
  • $\begingroup$ Yes. So can you find the distance and time for each part? $\endgroup$ – user61527 Oct 25 '13 at 6:58
  • $\begingroup$ Well that's where I'm stuck since what I've tried is plugging in the given values into my equation $\endgroup$ – dnova Oct 25 '13 at 6:59
  • $\begingroup$ Can you show what you've tried, then? $\endgroup$ – user61527 Oct 25 '13 at 6:59
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The answer is $\frac{y(1+\delta)-y(1)}{\delta} = 16(3-\delta)$. Substitute the various values of $\delta$ above.

Since $\frac{dy(1)}{dt} = \lim_{\delta \downarrow 0} \frac{y(1+\delta)-y(1)}{\delta}$, you can estimate $\frac{dy(1)}{dt}$.

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