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GCSEMathsAQARevision Notes2. AlgebraEstimating Gradients & Areas under GraphsFinding Gradients of Tangents

Finding Gradients of Tangents (AQA GCSE Maths)

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Daniel I

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Daniel I

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Finding Gradients of Tangents

What is the gradient of a graph?

  • 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

How do I estimate the gradient under a graph?

  • To find an estimate for the gradient:
    • Draw a tangent to the curve
    • Find the gradient of the tangent using Gradient = RISE ÷ RUN

    cie-igcse-2-15

      • In the example above, the gradient at x = 4 would be fraction numerator negative 2.5 over denominator 4 end fraction equals negative 0.625
  • It is an estimate because the tangent has been drawn by eye and is not exact

What does the gradient represent?

  • 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

Examiner Tip

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

Worked example

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

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

Find an estimate of the gradient of the curve at the point where x space equals space 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 space equals space rise over run space equals fraction numerator space 0.3 over denominator 0.5 end fraction equals fraction numerator space 3 over denominator 5 end fraction

Estimate of gradient = 0.6

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    Daniel I

    Author: Daniel I

    Expertise: Maths

    Daniel has taught maths for over 10 years in a variety of settings, covering GCSE, IGCSE, A-level and IB. The more he taught maths, the more he appreciated its beauty. He loves breaking tricky topics down into a way they can be easily understood by students, and creating resources that help to do this.