average velocity question for calc! please help?

Question

A particle is traveling along a straight line, and its distance from the original is given by the equation:

s(t)=t^3 +12t^2 +36t + 4
where t is greater than or equal to 0.

a. What is the average velocity of the particle on the interval [3, 6] ?

b. Find c in [3, 6] so the velocity at c is equal to the average velocity you found in the pervious question.

c.When is the particle's acceleration positive? negative?

d. When is the particle speeding up? slowing down?

CAN YOU PLEASE PROVIDE WORK so i understand how to do it myself next time?! thanksss a bunch!

im sorry. s(t)=t^3 - 12t^2 + 36t + 4

i made a mistake before! sorrry!

Answer 1

position = s(t) = t^3 +12 t^2 +36 t + 4
velocity = s'(t) = 3 t^2 + 24 t + 36

(a) Integrate the velocity from 3 to 6, then divide by 3 to get the average velocity.

(b) Set that average velocity equal to s'(t) and solve for t.

acceleration = s''(t) = 6 t + 24

(c) Set acceleration equal to zero and solve for t. The acceleration is positive for times greater than that value and negative for times less than that value.

(d) The particle is speeding up when the acceleration is positive and slowing down when it is negative.

Answer 2

a. s(t) = t^3 - 12^t + 36^t + 4
getting its derivative,
v(t) = 3t^2 - 24t + 36

v(3) = 3(3)^2 - 24(3) + 36
= 27 - 72 + 36
= -9

v(6) = 3(6)^2 - 24(6) + 36
= 0

v(av) = [v(6) - v(3)]/2
= (0 - (-9))/2
= 4.5

b. v(t) = 3t^2 - 24t + 36
4.5 = 3t^2 - 24t + 36
1.5 = t^2 - 8t +12
0 = t^2 - 8t + 10.5
t = {8 +/- sq.rt.(64 - 4(10.5))}/2
= (8 +/- 4.69)/2
= 6,345 s or -1.655

but since t must be greater than or equal to zero,
t = 6.345 s

c. getting the derivative of the velocity equation,
a(t) = 6t - 24
setting a(t) = 0
0 = 6t - 24
t = 4

at t greater than 4 s, the particle's acceleration is positive
at t less than 4 s, the particle' acceleration is negative

d. v(t) = 3t^2 - 24t + 36
= t^2 - 8t + 12
0 = t^2 - 8t + 12
t = [8 +/- sq.rt. (64 - 4(12))]/2
= ( 8 +/- 4)/2
= 2 s or 6 s

the particle is speeding up at t = [0,2) U (6, positive infinity)
the particle is slowing down at t = (2,6)