Gradient of a Curve (Tangents)

Grade 7

Why curves are different

A straight line has the same gradient everywhere. A curve doesn't — the steepness changes at every point. To find the gradient at one specific point, you need to find the gradient of the line that just touches the curve at that point. That line is called the tangent.

How to draw a tangent

Place your ruler at the point on the curve you're interested in. Rotate it until the line it makes has the same slope as the curve at that point — meaning the curve curves away equally on both sides of your ruler. Then draw the line.

Once you've drawn the tangent, it's a straight line — so you can calculate its gradient using the normal rise-over-run method.

Use two points that are as far apart as possible when calculating the gradient of your tangent. This reduces the effect of small drawing inaccuracies — a small angle error matters less over a longer distance.

Worked example — instantaneous speed

EXAMPLE

A distance-time graph shows a curve. At t = 4 seconds, a tangent is drawn. It passes through the points (2, 10) and (6, 50). What is the instantaneous speed at t = 4?

Gradient of tangent = rise ÷ run = (50 − 10) ÷ (6 − 2) = 40 ÷ 4 =10 m/s.

Estimating — not exact

Drawing a tangent by hand gives an estimate of the gradient, not an exact value. Examiners know this — a mark scheme for a tangent question will typically accept a range of answers. Do your best to draw it carefully, and always show your two chosen points and the gradient calculation clearly.

Try this one

The diagram shows a distance-time graph for two runners, A and B, in a race.

(a) Find the speed of runner A. [2]

(b) Find the speed of runner B. [1]

Nova's hint:

This question uses the gradient of a distance-time graph to find speed — speed = distance ÷ time.

Ask yourself: 'What are the start and end coordinates for each line, and how do I use these to find gradient?'

Then subtract the two distances at t = 40 to find the difference.

Try to Solve it with Nova?