Question

The height of a ball thrown in to the air is given by the formula

y= s(t) = 16t^2 + 50t + 2 where s(t) is in feet and t in seconds

A. Find the average velocity of the object over the interval [1,2] and include units.
B. Find a simplified expression that gives the average rate of change of s(t) on the interval [1, 1 + h]. Your answer will be an expression involving h.

Answer 1

A. To find the average velocity of the object over the interval [1,2], we need to find the change in position (or height) over the change in time.

s(2) = 16(2)^2 + 50(2) + 2 = 146
s(1) = 16(1)^2 + 50(1) + 2 = 68

The change in position over the interval [1,2] is 146 - 68 = 78 feet.

The change in time is 2 - 1 = 1 second.

Therefore, the average velocity of the object over the interval [1,2] is 78 feet per second.

B. The average rate of change of s(t) on the interval [1, 1 + h] is given by:

[s(1+h) - s(1)] / h

Substituting in the formula for s(t), we get:

[s(1+h) - s(1)] / h = [(16(1+h)^2 + 50(1+h) + 2) - (16(1)^2 + 50(1) + 2)] / h

Simplifying the expression, we get:

[s(1+h) - s(1)] / h = (32h + 16) / h = 32 + 16/h

Therefore, the average rate of change of s(t) on the interval [1, 1 + h] is 32 + 16/h.