Finding Gradients of Tangents (Edexcel IGCSE Maths A) : Revision Note

Finding Gradients of Tangents

  • The gradient of a graph at a point is equal to the gradient of the tangent to the curve at that point

    • A tangent is a line that touches a curve, but does not cross it

A quadratic with two tangents drawn on it. The gradient of the curve at the point x=1 will be equal to the gradient of the purple tangent. The gradient of the curve at th epoint x=6 will be equal to the gradient of the green tangent.

How do I estimate the gradient of a curve using a tangent?

  • To find an estimate for the gradient of a curve at a point:

    • Draw a tangent to the curve at the point

    • Find the gradient of the tangent using

      • Gradient = rise ÷ run

      • or difference in y ÷ difference in x

      • In the example below, the gradient of the tangent at x = 4 would be fraction numerator negative 2.5 over denominator 4 end fraction equals negative 0.625

      • Remember that the rise is negative if it is going down

      • This means the gradient of the curve at x = 4 is also -0.625

    A curve with a tangent drawn on at a point on the curve (4, 2.5). The rise of the tangent is 2.5 and the run is 4.
  • It is an estimate because the tangent has been drawn by eye and is not exact

    • To find the exact gradient we would need to use differentiation

What does the gradient represent?

  • The gradient represents the rate of change of y with x

    • I.e. For every increase in x by 1, how much does y increase?

  • Consider the quantities used for the axes to determine the meaning of the gradient

    • In a distance-time graph, the gradient is the rate of change of distance with time

      • This is the speed

    • In a speed-time graph, the gradient is the rate of change of speed with time

      • This is the acceleration

    • In a graph of volume against radius, e.g. as a balloon is inflated, the gradient is the rate of change of volume as the radius increases

Examiner Tips and Tricks

  • When drawing a tangent by hand:

    • Use a ruler

    • Draw the line as long as you can

  • When finding the gradient of the tangent:

    • Pick two points that are far away from one another

    • This will reduce the effect of any inaccuracy

Worked Example

The graph below shows y equals cube root of x for 0 less or equal than x less or equal than 1.

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

Graph of y=cube root of x between 0 and 1

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

Tangent to curve at x=0.5

Find suitable, easy to read coordinates as far apart as possible and draw a right-angled triangle between them

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

Finding gradient of tangent

Find the gradient by dividing the difference in (rise) by the difference in x

Gradient space equals fraction numerator 0.3 over denominator 0.5 end fraction equals 3 over 5 equals 0.6

The gradient of the tangent at x = 0.5 is equal to the gradient of the curve at = 0.5

Estimate of gradient = 0.6

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Amber

Author: Amber

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Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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