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Understanding Rates of Change

Every time you fill a differently shaped glass with water from a tap, the water level rises at a different speed. The relationship between the changing depth of the water and time is a rate of change, which measures how much one variable alters in response to another. On a standard graph, the rate of change is equal to the gradient.

For a straight line, the steepness is constant, meaning the rate of change is uniform.
For a non-linear graph (a curve), the steepness varies at every coordinate, meaning the rate of change is always shifting.
The units for a rate of change are always the units of the $y$ -axis divided by the units of the $x$ -axis (for example, $m/s$ for a distance-time graph, or $c m^{3} /$ for a volume-time graph).

The Gradient Formula

Whether you are finding an exact average or estimating an instantaneous rate, you must use the gradient formula:

m = \frac{y_2 - y_1}{x_2 - x_1}

Where:

$m$ = gradient (the rate of change)
$(x_1, y_1)$ = the coordinates of the first point
$(x_2, y_2)$ = the coordinates of the second point

Average Rate of Change (Chords)

The average rate of change looks at how a quantity alters over a specific interval. To find this geometrically on a curve, you draw a chord — a straight line segment that connects the start point and the end point of your interval. The gradient of this chord gives you the average rate of change. You only ever use the first and last points of the timeframe; ignoring any fluctuations that happen in between.

Worked Example: Average Rate of Change

A particle's distance from a starting point is given by the equation $d = 2t^2 + 1$ , where $d$ is distance in metres and $t$ is time in seconds. Calculate the average rate of change of distance (average speed) between $t = 1$ and .

Step 1: Calculate the coordinates for the start and end of the interval.

At $t = 1$ : $d = 2(1)^2 + 1 = 3$ . The first point is $(1,$ .

Step 2: Substitute the coordinates into the gradient formula for a chord.

$m = \frac{19 - 3}{3 - 1}$

Step 3: Calculate the numerator and denominator.

$m = \frac{16}{2}$

Step 4: Find the final answer and include units.

Average rate of change = $8\,m/s$

Instantaneous Rate of Change (Tangents)

The instantaneous rate of change is the precise rate at one specific, frozen moment in time. To find this on a curve, you must draw a tangent. A tangent is a straight line that rests against the outside of the curve, touching it at exactly one point without crossing through it. Because you must draw the tangent 'by eye' with a ruler, calculating its gradient provides an estimate of the instantaneous rate of change.

Worked Example: Instantaneous Rate of Change

A volume-time graph shows water emptying from a tank. The curve passes through the point $(4, 20)$ . A tangent is drawn touching the curve at $t = 4$ . The tangent straight line also passes through the coordinates $(0, 36)$ and $(8, 4)$ . Estimate the instantaneous flow rate at seconds.

Step 1: Identify two points on the tangent line that are far apart.

The points given on the tangent are $(0, 36)$ and $(8, 4)$ .

Step 2: Substitute these coordinates into the gradient formula.

$m = \frac{4 - 36}{8 - 0}$

Step 3: Calculate the numerator and denominator.

$m = \frac{-32}{8}$

Step 4: Calculate the final estimate and state the meaning of the sign.

Instantaneous rate of change = $-4$ units of volume per second.
The negative sign indicates that the rate is decreasing (the tank is emptying).

Students often calculate gradients upside down by dividing the horizontal change by the vertical change. Always ensure your calculation is change in y divided by change in x.

When drawing a tangent in an exam, draw the line as long as possible across the graph grid; this minimises reading errors and makes your gradient estimate much more accurate.

If a question asks you to calculate an 'average rate' from a table, only use the start and end values of the requested interval — do not mathematically average the intermediate readings.

Always check if a tangent or chord is sloping downwards. If it is, your final gradient calculation must be a negative number.

A measure of how much one quantity (the dependent variable) changes in relation to another (the independent variable).

A mathematical measure of the steepness of a line, calculated by dividing the change in the vertical axis by the change in the horizontal axis.

A graph that forms a curve rather than a straight line, indicating that its rate of change is constantly varying.

The measure of how much a quantity changes over a specific interval or timeframe, represented by the gradient of a chord.

A straight line segment that joins two distinct points on a curve.

The estimated rate of change at one specific instant or value, represented by the gradient of a tangent.

A straight line that touches a curve at exactly one point without crossing through it.