average and instaneous velocity calculus

Question

s(t)=-16t^(2)+100

Write an expression for the average velocity of the ball on the time interval [2, 2 + h], where h is an arbitrary time in seconds. (An integral answer is not appropriate here and you do not need to simplify the expression.)

Using your work from 3(b)i, write a symbolic expression involving a limit that, if evaluated, would tell us the instantaneous velocity at time 2 second.

. Using proper notation and your work above, evaluate your symbolic expression (using algebra!) and write one sentence to summarize your results about the movement of the ball. Use another sheet of paper if you need more room to show your work. Show *all* your steps and use the proper notation from class.

Answer 1

The equation used to find average velocity over an interval is (f(b)-f(a))/(b-a).

Plugging in 2 seconds for t yields: -16(2)²+100=36=f(a)

Plugging in 2+h seconds for t yields: -16(h+2)²+100

-16(h²+4h+4)+100

-16h²-64h-64+100=-16h²-64h+36=f(b)

We now have f(a), f(b), a and b. Substituting these into the average velocity formula gives:

(-16h²-64h+36)-(36) which cleans up to -16h²-64h.

(h+2)-(2)

The second part is quite simple now that we have this expression. Let's call it f(t).

Lim f(t)=-16t²-64t

t-->2 t-2

Finally, we can determine some things about the velocity of the ball at some given time by looking at our expression from above:

-16t²-64t=-16t(t+4) and through factoring (-16t=0) (t+4=0) we can see that the ball will have no velocity at t=0 and t=-4.