The position of an object moving vertically along a line is given by the function s (t) = - 16t^2 + 128t. Find the average velocity of the object over the following intervals.

a. [1, 4]

b. [1, 3]

c. [1, 2]

d. [1, 1 + h], where h > 0 is a real number

a) 48

b) 64

c) 80

d) 96-16h

Step-by-step explanation:

a) s (1) = 112 and s (4) = 256

average velocity on [1,4] = (256-112) / (4-1) = 48

b) s (1) = 112 and s (3) = 240

average velocity on [1,3] = (240-112) / (3-1) = 64

c) s (1) = 112 and s (2) = 192

average velocity on [1,2] = (192-112) / (2-1) = 80

the next one's tricky to type. watch the parentheses carefully:

d) s (1) = 112 and s (1+h) = - 16 (1+h) ^2 + 128 (1+h)

average velocity on [1,1+h] =

(s (1+h) - s (1)) / ((1+h) - 1) = (-16 (1+h) ^2 + 128 (1+h) - (112)) / h

= (-16 (1+2h+h^2) + 128+128h - 112) / h

= (-16 - 32h - 16h^2 + 16 + 128h) / h

= (96h - 16 h^2) / h

= 96 - 16h