Rolle's Theorem

Question

The height of a ball t seconds after it is thrown upward from a height of 5 feet and with an initial velocity of 64 feet per second is f (t) = -16t2 + 64t + 5.

 

Verify that f(1) = f(3). In ft.
f (1) =
f (3) =

According to Rolle's Theorem, what must be the velocity at some time in the interval (1, 3)? In ft/sec

Find that time in seconds. 

Answer 1

f(t) = -16t² + 64t + 5

 

   f(t) is continuous on [1,3] and is differentiable on (1,3).

 

   f(1) = f(3) = 53

 

So, according to Rolle's Theorem, f'(c) = 0 for at least one value, c, in the interval (1,3).

 

      f'(c) = -32c + 64 = 0

 

                             c = 2

 

f'(2) = 0 and 2 is in the interval (1,3).