The position function of an object moving along a straight line is given by s = f(t). The average velocity of the object over the time interval [a, b] is the average rate of change of f over [a, b]; its (instantaneous) velocity at t = a is the rate of change of f at a.

A ball is thrown straight up with an initial velocity of 112 ft/sec, so that its height (in feet) after t sec is given by s = f(t)=112t−16t^2.
(a) What is the average velocity of the ball over the following time intervals?
[3,4] ft/sec
[3,3.5] ft/sec
[3,3.1] ft/sec

To find the average velocity of the ball over a given time interval, we need to calculate the displacement of the ball during that interval and divide it by the length of the interval.

(a) Average velocity over [3, 4]:
To find the displacement, we subtract the initial position from the final position.
Initial position (at t=3 seconds): s(3) = 112(3) - 16(3)^2 = 336 - 144 = 192 feet
Final position (at t=4 seconds): s(4) = 112(4) - 16(4)^2 = 448 - 256 = 192 feet
Displacement = Final position - Initial position = 192 - 192 = 0 feet

The length of the interval [3, 4] is 4 - 3 = 1 second.

Average velocity = Displacement / Length of interval = 0 / 1 = 0 ft/sec

Therefore, the average velocity of the ball over the interval [3, 4] is 0 ft/sec.

(b) Average velocity over [3, 3.5]:
Initial position (at t=3 seconds): s(3) = 192 feet
Final position (at t=3.5 seconds): s(3.5) = 112(3.5) - 16(3.5)^2 = 392 - 196 = 196 feet
Displacement = Final position - Initial position = 196 - 192 = 4 feet

The length of the interval [3, 3.5] is 3.5 - 3 = 0.5 seconds.

Average velocity = Displacement / Length of interval = 4 / 0.5 = 8 ft/sec

Therefore, the average velocity of the ball over the interval [3, 3.5] is 8 ft/sec.

(c) Average velocity over [3, 3.1]:
Initial position (at t=3 seconds): s(3) = 192 feet
Final position (at t=3.1 seconds): s(3.1) = 112(3.1) - 16(3.1)^2 = 347.2 - 152.88 = 194.32 feet
Displacement = Final position - Initial position = 194.32 - 192 = 2.32 feet

The length of the interval [3, 3.1] is 3.1 - 3 = 0.1 seconds.

Average velocity = Displacement / Length of interval = 2.32 / 0.1 = 23.2 ft/sec

Therefore, the average velocity of the ball over the interval [3, 3.1] is 23.2 ft/sec.