A particle moves along a number line such that its position s at any time t, t(greaterthan or equal to), is given by s(t)=2t^3-15t^2+24t+1.

A) Find the average velocity over the time interval [1,2].

B) Find the instantaneous velocity at t=2

C) When is the particle at rest?

It has been a while since we have been on this topic and i have forgotten how to set any of these up. Any help would be wonderful =)

Sure! I can help you with these questions. Let's break them down one by one.

A) To find the average velocity over the time interval [1,2], we need to calculate the change in position and divide it by the change in time. The formula for average velocity is:

Average Velocity = change in position / change in time

In this case, the change in position is given by the difference of the position function at t=2 and t=1, and the change in time is 2-1.

So, let's calculate the average velocity:
Change in position = s(2) - s(1) = (2*(2^3) - 15*(2^2) + 24*2 + 1) - (2*(1^3) - 15*(1^2) + 24*1 + 1)

Evaluate the expression inside the parentheses to get the change in position. Then divide it by (2-1) to find the average velocity.

B) To find the instantaneous velocity at t=2, we need to find the derivative of the position function with respect to time, and then substitute t=2 into the derivative.

First, let's find the derivative of the position function s(t):
s'(t) = d/dt (2t^3 - 15t^2 + 24t + 1)

To do that, we differentiate each term separately using the power rule and sum rule of derivatives. Then substitute t=2 into the derivative to find the instantaneous velocity.

C) To find when the particle is at rest, we need to find the values of t where the instantaneous velocity is zero. This corresponds to the times when the particle is not moving or changing its position.

For this, we can set the derivative s'(t) = 0, and solve for t. The values of t that satisfy this equation will be the times when the particle is at rest.

Now, let's go through the steps for each of these questions.