1. Differentiation(average speed)

Determine the average speed given that the distance travelled, $\displaystyle s$, in metres, relative to time, $\displaystyle t$, in seconds is $\displaystyle s(t)=2t^2 + 1$, between $\displaystyle t=2$ and $\displaystyle t=4$

Do I first do the first derivative of $\displaystyle s(t)=2t^2 + 1$ to get the equation for the speed at a point and then substitute in $\displaystyle t=2$ and $\displaystyle t=4$, and then use those values returned to get the gradient between the 2 points?

2. Originally Posted by MarcoMP

Determine the average speed given that the distance travelled, $\displaystyle s$, in metres, relative to time, $\displaystyle t$, in seconds is $\displaystyle s(t)=2t^2 + 1$, between $\displaystyle t=2$ and $\displaystyle t=4$

Do I first do the first derivative of $\displaystyle s(t)=2t^2 + 1$ to get the equation for the speed at a point and then substitute in $\displaystyle t=2$ and $\displaystyle t=4$, and then use those values returned to get the gradient between the 2 points?
Avarage speed $\displaystyle = \frac{s(4) - s(2)}{4 - 2}$.

3. Originally Posted by mr fantastic
Avarage speed $\displaystyle = \frac{s(4) - s(2)}{4 - 2}$.
So I take it I don't have to do the average speed between the 2 points while using the instantaneous speed?

4. You would only need to use differetiation if you were given the model for distance as

$\displaystyle \frac{d}{dt}(d(t)) = s(t)$

then substitiute in those values as the previous poster suggested.

Originally Posted by MarcoMP
So I take it I don't have to do the average speed between the 2 points while using the instantaneous speed?
I agree as the average between 2 points is never instantaneous.

5. Originally Posted by MarcoMP
So I take it I don't have to do the average speed between the 2 points while using the instantaneous speed?
If you had to do that I would have said so.