Speed is not merely distance divided by time. Intuitively we mean the distance travelled in a particular time divided by the time taken to travel that distance. The calculated speed will be the average speed over that time interval, but the speed may in fact vary considerably and give a very poor estimate of the speed at any one time. By reducing the size of the size interval we can obtain a more accurate estimate of the speed at any one time, but we will always be finding an average speed over a certain time interval.
For example the speed at the point x above may be estimated as
or
The second is the more accurate. In general smaller intervals gives more accurate estimates. The most accurate estimate can be found by drawing a tangent at the point x and finding the gradient of the tangent, equal to
for the graph below. The gradient is called the instantaneous speed speed. It is the speed at an instant, not over a period of time, however small.
