# Instantaneous rates of change - Higher

When a relationship between two variables is defined by a curve it means that the gradient, or rate of change is always varying.

An average speed for a journey can be found from a distance-time graph by working out the gradient of the line between the two points of interest. This is like using a car's milometer and clock to take readings at the start and end of a journey.

However, they do not tell you the actual speed at a particular moment. You need a speedometer for that. On a graph, this can be estimated by drawing a to the curve and calculating its gradient.

### Example

This distance-time graph shows the first ten seconds of motion for a car.

The average speed over the 10 seconds = gradient of the line from (0, 0) to (10, 200) = .

To find an estimate of the speed after 6.5 seconds, draw the tangent to the curve at 6.5.

A velocity-time graph shows the velocity of a moving object on the vertical axis and time on the horizontal axis.

The gradient of a velocity time graph represents acceleration, which is the rate of change of velocity. If the velocity-time graph is curved, the acceleration can be found by calculating the gradient of a tangent to the curve.

### Example

The velocity of a sledge as it slides down a hill is shown in the graph.

Find the acceleration of the sledge when t = 6s.

Draw a tangent to the curve at the point where t = 6s and draw two lines to form a right angle triangle. The acceleration is equal to the gradient of the tangent which is

Notice, that after about 10 seconds, the gradients are negative meaning the sledge is slowing down or decelerating.