Average and Instantaneous Rates of Change

From a changing graph to a rate

A gradient compares a vertical change with a horizontal change: change in $y$ divided by change in $x$. On a straight line this ratio is the same everywhere. A curve has no single gradient for its whole length: it may be steep in one place and shallow in another. We therefore need to distinguish the rate over an interval from the rate at one point.

Average rate: join two points

Mark two points $(x_1,y_1)$ and $(x_2,y_2)$ on the curve and join them with a straight line called a chord. Its gradient is the average rate of change between those points:

$\text{average rate of change}=\frac{y_2-y_1}{x_2-x_1}.$

For example, on $y=x^2-1$, the points at $x=2$ and $x=3$ are and . The chord’s gradient is . Across that interval, increases by an average of 5 for each increase of 1 in . This does say that the curve’s gradient is 5 at every point between them; the chord replaces a varying rate with one constant rate for the interval. Nor is this an arithmetic mean of the two -values.

Instantaneous rate: match the curve at a point

To find the rate at a particular point, draw a tangent that matches the curve’s direction there. The tangent’s gradient is the instantaneous rate of change. On a printed graph, draw the tangent carefully, choose two well-spaced points on that line, and divide its vertical change by its horizontal change. Because the line is drawn by eye, the result is usually an estimate.

For $y=x^2-1$ at $(2,3)$, the tangent shown has gradient 4. That is different from the chord’s average gradient of 5 between and . The connection becomes clearer if the chord’s second point moves towards : using gives a chord gradient of 4.5, and using gives 4.1. The gradients approach 4 as the interval shrinks. Algebraically, a second point at gives a chord gradient of , which approaches 4 as approaches zero. The tangent captures that limiting steepness without requiring a chord of zero width.

Interpreting the answer

Always read the axes before interpreting a gradient. If $x$ is time in seconds and $y$ is distance in metres, a chord’s gradient is average speed over the time interval, while a tangent’s gradient estimates speed at one instant; both have units of metres per second. If the graph instead shows temperature in °C against time in minutes, a negative tangent gradient means the temperature is falling at that moment, in °C per minute. The same chord-and-tangent method works from coordinates, an equation or a graph: identify the interval or point, find the relevant gradient, then explain its sign and units in context.