Home / Rates of Change
3.12 · Paper 2C
Gradients, chords and tangents, greatest and least values, and the link between distance, speed and acceleration.
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The rate of change is the gradient: change in $y$ ÷ change in $x$, with units 'per' the $x$ unit. On a curve the rate changes. The average rate between two points is the gradient of the chord; as the two points get closer the chord gets closer to the tangent — the curve looks like a straight line when you zoom in.
The gradient of a curve at a point is the gradient of the tangent there. At a maximum or minimum the tangent is horizontal, so the gradient is 0. This is how to find greatest and least values, e.g. the biggest area for a fixed length of fencing.
On a distance–time graph, gradient = speed. On a speed–time graph, gradient = acceleration and the area under the graph = distance travelled. For a curve, use a tangent to find the instantaneous speed or acceleration.
Estimate the gradient of $y = x^2$ at $x = 3$ by using a chord from $x = 3$ to $x = 3.01$.
At $x = 3$: $y = 9$. At $x = 3.01$: $y = 9.0601$.
Gradient of the chord $= \dfrac{9.0601 - 9}{0.01} = \dfrac{0.0601}{0.01} = 6.01$.
The gradient at $x = 3$ is approximately 6 (the pattern for $y = x^2$ is gradient $= 2x$).
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