1. ## Integral

The position of to particles on the x-axis are given

S1 = 3t^3 - 12t^2 + 18t + 5 and
S2 = -3^3 + 9t^2 - 12t , where "t" is measured in seconds.

When do the particles have the same speed?

I cannot figure out how to derivate/integrate these.. Someone help?

2. You can use the power rule to differentiate the expression:

$\displaystyle \frac{d}{dx} (x^n)=nx^{n-1}$

The derivatives of $\displaystyle S_1$ and $\displaystyle S_2$ are:

$\displaystyle S_1\ '(t)=9t^2-24t+18$

$\displaystyle S_2\ '(t)=-9t^2+18t-12$

And since they represent the instantaneous velocity of the particle, you are looking for when $\displaystyle S_1\ '(t)=S_2\ '(t)$

Solving for t, I get complex answers... so that would mean the particles never travel at the same speed.

In fact, knowing that the gradient of a function is its rate of change, you can graph and observe the original two functions $\displaystyle S_1(t)$ and $\displaystyle S_2(t)$ and notice that they are cubics which are a reflection of one another. In $\displaystyle S_1$, the gradient is always positive. In $\displaystyle S_2$, the gradient is always negative. This shows that their gradients and hence their rates of change can never be equal.

3. S2 = -t^3 - 24t + 18

But how do i figure their speed from this?

4. In that case, applying the power rule, you get

$\displaystyle \frac{d}{dx}(-t^3-24t+18)=-3t^2-24$

Equating $\displaystyle S_1\ '(t)$ and $\displaystyle S_2 \ '(t)$:

$\displaystyle -3t^2-24=9t^2-24t+18$

Once again I get complex numbers for t, meaning the particles don't ever have the same speed, but I'm in a bit of a hurry at the moment, so you'll have to check over that.