Exercises for the Velocity Module

• 1:
  Compute the average velocity of the canteloupe over the time interval 2.0 <  t  < 2.5 and over the interval 0.5 <  t  < 1.5.
  

• 2:
  What is the average velocity of the canteloupe over the interval [0.99,1]? ... over the interval [1,1.01]? Use the following table to get average velocities over correspondingly shorter intervals:

  t(secs)	0.999	1	1.001
  x(feet)	84.026984	84.000000	83.972984

  What value (to two decimal places, say) should we assign to the instantaneous velocity of the canteloupe at time t = 1 sec?

• 3 :
  
  • a.) Even if you did the above problems by hand, why not use Maple this time? Approximate the instantaneous velocity of the canteloupe at time t=2 sec by first defining the function

     f:= t -> 95 + 5*t - 16*t^2; 

    and then calculating average velocities over short intervals which begin or end at t=2:

           (f(2.01) - f(2))/ (2.01-2); 

           (f(2)-f(1.99))/(2-1.99);  

    and so forth. What value would you assign for the instantaneous velocity at t=2?
  • b.) You can get fancier: Let's define the function

          vav := t -> (f(t+h)-f(t))/h;  

    This will give the average velocity over the time interval [t,t+h] once we have assigned a value to h. Do part (a) again using vav. Then use vav to find the instantaneous velocity at several (at least three) other values of t.
  • c.) It's even better if you graph it. Assign a (small) value to h, and then

            plot({f(t),vav(t)},t=0..2.6)   

    Do this several times with decreasing values of h (but only turn in \it one plot). What do the graphs of all the 'vav' s have in common? How do they differ? How is the graph of f(t) related to that of vav(t) . In particular how does f behave when vav is positive? negative? big? small? -- Is this behavior precise or only approximate?

• 4:
  
  • a.) Use Maple's solve command to determine precisely at what time the canteloupe hits the ground.
  • b.) How would you determine the highest height reached by the canteloupe? At what time does this occur?

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