A Curved Line

Curved slope

Imagine we started the car from a stationary position, hit the accelerator hard, and held it down. Clearly, the starting speed is zero because we’re not moving initially.

Now imagine we’re pressing the accelerator so hard that the car doesn’t increase its speed at a constant rate. Instead, it keeps increasing exponentially. This means it is not adding 10 mph every minute. Instead, it is adding speed at a rate that itself goes up the longer we keep that accelerator pressed.

For this example, let’s imagine the speed is measured every minute, as shown in this table:

Time (mins)	Speed (mph)
0	0
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64

If we look closely, we can see that we’ve chosen to have the speed as the square of the time in minutes. That is, the speed at time $2$ is $2^{2}=4$, and at time $3$ is $3^2=9$, and time $4$ is $4^2=16$, and so on.

Speed calculation

The expression for this is easy to write:

$$s=t^2$$

Yes, we know this is a very contrived example of car speed, but it will illustrate how we might do calculus.

Let’s visualize this so we can get a feel for how the speed changes with time.