Answer
(a) $$v_{ave1}=48.$$
(b) $$v_{ave2}=64.$$
(c) $$v_{ave3}=80;$$
(d)
$$v_{ave}=96-16h;$$
Work Step by Step
By definition, the average velocity is the change in position over the time interval, divided by the length of that interval:
(a) $$v_{ave1}=\frac{s(4)-s(1)}{4-1}=\frac{-16\cdot4^2+128\cdot4-(-16\cdot1^2+128\cdot1)}{3}=\frac{144}{3}=48.$$
(b) $$v_{ave2}=\frac{s(3)-s(1)}{3-1}=\frac{-16\cdot4^2+128\cdot4-(-16\cdot1^2+128\cdot1)}{2}=\frac{128}{2}=64.$$
(c) $$v_{ave3}=\frac{s(2)-s(1)}{2-1}=\frac{-16\cdot2^2+128\cdot2-(-16\cdot1^2+128\cdot1)}{1}=80;$$
(d)
$$v_{ave}=\frac{s(1+h)-s(1)}{1+h-1}=\frac{-16\cdot(1+h)^2+128\cdot(1+h)-(-16\cdot1^2+128\cdot1)}{2}=\frac{-16-2\cdot16h-16h^2+128+128h+16-128}{h}=\frac{96h-16h^2}{h}=96-16h;$$
