Consider the following question:

A car travels $390$ meters in $30$ seconds.

What is the velocity of the car in ${\text{m s}}^{-1}$?

You probably want to answer

However, that gives us the average velocity. In actuality, the car might have been going faster than this for a few seconds, and slower than this for a few seconds, but it all balances out to give $390$ meters in $30$ seconds.

So, instead of asking "what is the velocity of the car?" we should really ask "what was the velocity of the car at a particular instant along its path?"

That is the question that differentiation can answer. Before we learn to do this, however, we'll learn how to estimate the answer.

A car is travelling in a straight line. Relative to a fixed origin, the position, in meters, of the car at time $t$ is modeled by the function

1. The displacement of the car after $4$ seconds.

2. Using an interval of width $0.1$ seconds, estimate the velocity of the car when $t=4.$

3. Explain how the estimate for the velocity of the car could be made more accurate.