The rollercoaster ride at The Wizarding World of Harry Potter theme park leaves
at time t = 0. The velocity of the rollercoaster is recorded for selected values over
the interval 0 ≤ t ≤12 seconds. Selected values are shown in the table below.
t (seconds) 0 2 4 6 8 10 12
v (t) (meters/sec) 3 4 7 10 14 10 30

show that there is some c, 0

Question

The rollercoaster ride at The Wizarding World of Harry Potter theme park leaves
at time t = 0. The velocity of the rollercoaster is recorded for selected values over
the interval 0 ≤ t ≤12 seconds. Selected values are shown in the table below.
t (seconds)           0 2 4 6 8 10 12
v (t) (meters/sec) 3 4 7 10 14 10 30

show that there is some c, 0<c<12 such that a(c) = 0 (justify)
a(t) is the acceleration of the roller coaster
Please help me.

anonymous · Answer

because the velocity function of the ride is constant and does not appear to have any irregularities, we may say that its derivative, or acceleration function is differentiable. (keep in mind this is not a complete list of things that a function needs to have/do to be derivative) If you can find 2 points in the acceleration of the ride that are the same, then you can use the MVT ( http://en.wikipedia.org/wiki/Mean_value_theorem) to justify that there is a point on the velocity function (a max or min or terrace point) that has a derivative of 0 (whish would mean that a(c)=0)