Simple Derivative Question?

Question

s(t) = 2/3t^3

Find the average velocity of the particle over the time interval [1, 1 + h]

a) 2/3 + 2/3h + 2h^3

b) 2/3 + 2h + 2/3h^3

c) 2/3 + 2h + 2/3h^2

d) 2 + 2/3h + 2/3h^2

e) 2 + 2/3h + 2h^3^3

f) 2 +2h + 2/3h^2

Answer 1

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

Take it one at a time.

s(1+h) = 2/3(1+h)^3 = 2/3(1 +3h + 3h^2 +h^3)

= 2/3 + 2h + 2h^2 + 2/3h^3

s(1) = 2/3(1)^3 = 2/3

1+h - 1 = h

Then we have

(2/3 + 2h + 2h^2 + 2/3h^3 - 2/3) / h

= (2h + 2h^2 + 2/3h^3)/h

= 2 + 2h + 2/3h^2