A particle moves according to a law of motions=f(t),t0,

wheretis measured in seconds andsin feet.

f(t) =t³12t²+45t

(a) Find the velocity at timet.

v(t) =3t224t+45

(b) What is the velocity after4s?

v(4) =ft/s

(c) When is the particle at rest?t=s (smaller value)t=s (larger value)

(d) When is the particle moving in the positive direction? (Enter your answer in interval notation.)

t

(e) Find the total distance traveled during the first6s.

ft

(f) Find the acceleration at timet.

a(t) =

Find the acceleration after4s.

a(4) =ft/s²

(g) Graph the position, velocity, and acceleration functions for the first6s.

(h) When, for0t<,

is the particle speeding up? (Enter your answer in interval notation.)

When, for0t<,

is it slowing down? (Enter your answer in interval notation.)