IGCSE Maths / Calculus
Gradient of a curve by drawing a tangent
Edexcel International GCSE Maths (4MA1), chapter 28: Calculus, specification point 3.3d. Free lesson.
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A curve has a different gradient at different points, so to estimate its steepness at one point, draw a tangent there and find its gradient. On a distance-time graph, that gradient gives the object's speed at that instant.
What a tangent tells you
A straight line has the same steepness all along it, but a curve does not. To find the curve's steepness at a particular point, draw a tangent: a straight line that touches the curve at that point and goes in the same direction as the curve there. Its gradient estimates the curve's gradient at that point. A tangent touches the curve at one point without cutting through it. The answer is an estimate, so a range of answers may be accepted. A careful tangent helps keep your estimate within that range.
How to estimate a gradient
Once you have drawn the tangent, calculate its gradient using change in y divided by change in x. Build a large triangle on the tangent so that small drawing errors have less effect on your measurements.
Tangent or chord?
A chord joins two points on the curve. It gives the average gradient over the stretch between those points, not the steepness at one point. The wording tells you which line to use. For a question asking for the gradient at t = 8, use a tangent at t = 8. For a question asking for the average between t = 4 and t = 12, use a chord between those times.
Speed from a distance-time graph
The distance-time graph for a car pulling away from rest curves upwards, showing that its speed changes. To estimate its speed at 8 seconds, use a tangent at that time. A large triangle on the tangent runs from 4 seconds to 12 seconds and rises from 0 metres to 32 metres. The gradient of a distance-time graph is speed, so the units are metres per second, or m/s.
Finding a gradient on y = x²
The same method works when a curve has no real-world context. For y = x² at x = 3, draw a tangent at that point and use a large triangle. The tangent runs from x = 1 to x = 5, rising from y = −3 to y = 21. There are no units when the axes have no units.
Check the direction of the calculation
The vertical change goes on top. Reversing the changes can turn a steep positive gradient into a small one, even if you have read the coordinates correctly. For a line that climbs 32 while moving 8 across, 32 ÷ 8 = 4, not 8 ÷ 32 = 0.25. A steeply rising line should have a large positive gradient.