Finding Gradients of Tangents

The gradient of a graph at any point is equal to the gradient of the tangent to the curve at that point
Remember that a tangent is a line that just touches a curve (and doesn’t cross it)

GoNL Notes fig4, downloadable IGCSE & GCSE Maths revision notes

To find an estimate for the gradient:
- Draw a tangent to the curve
- Find the gradient of the tangent using Gradient = RISE ÷ RUN
- - In the example above, the gradient at x = 4 would be $\frac{- 2.5}{4} = - 0.625$
It is an estimate because the tangent has been drawn by eye and is not exact

In a y-x graph, the gradient represents the rate of change of y against x
This has many practical applications, for example;
- in a distance-time graph, the gradient (rate of change of distance against time) is the speed
- in a speed-time graph, the gradient (rate of change of speed against time) is the acceleration

This is particularly useful when working with Speed-Time and Distance-Time graphs if they are curves and not straight lines

The graph below shows $y = \sqrt[3]{x}$ for $0 \leq x \leq 1 .$

estimating-areas-and-gradients-of-graphs-worked-example-1

Find an estimate of the gradient of the curve at the point where $x = 0.5 .$

Draw a tangent to the curve at the point where x = 0.5.

estimating-areas-and-gradients-of-graphs-worked-example-1-image-2

Find suitable, easy to read coordinates and draw a right-angled triangle between them.

Find the difference in the y coordinates (rise) and the difference in the x coordinates (run).

estimating-areas-and-gradients-of-graphs-worked-example-1-image-3

Divide the difference in y (rise) by the difference in x.

$Gradient = \frac{rise}{run} = \frac{0.3}{0.5} = \frac{3}{5}$

Estimate of gradient = 0.6

Finding Gradients of Tangents (AQA GCSE Maths)