DP IB Maths: AA SL

Revision Notes

5.2.6 Derivatives & Graphs

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Derivatives & Graphs

How are derivatives and graphs connected?

  • If the graph of a functionspace y equals f left parenthesis x right parenthesis is known, or can be sketched, then it is also possible to sketch the graphs of the derivativesspace y equals f apostrophe left parenthesis x right parenthesis andspace y equals f apostrophe apostrophe left parenthesis x right parenthesis
  • The key properties of a graph include
    • thebold space bold italic y-axis intercept
    • thebold space bold italic x-axis intercepts – the roots of the function; wherespace f left parenthesis x right parenthesis equals 0
    • stationary points; wherespace f apostrophe left parenthesis x right parenthesis equals 0
      • turning points – (local) minimum and maximum points
      • (horizontal) points of inflection
    • (non-stationary,space straight f apostrophe left parenthesis x right parenthesis not equal to 0) points of inflection
    • asymptotesvertical and horizontal
    • intervals where the graph is increasing and decreasing
    • intervals where the graph is concave down and concave up
  • Not all graphs have all of these properties and not all can be determined without knowing the expression of the function
  • However questions will provide enough information to sketch
    • the shape of the graph
    • some of the key properties such as roots or turning points

How do I sketch the graph of y = f'(x) from the graph of y = f(x)?

  • The graph ofspace y equals f apostrophe left parenthesis x right parenthesis will have its
    • space x-axis intercepts at thespace x-coordinates of the stationary points ofspace y equals f left parenthesis x right parenthesis
    • turning points at thespace x-coordinates of the  points of inflection ofspace y equals f left parenthesis x right parenthesis
  • For intervals wherespace y equals f left parenthesis x right parenthesis is concave up,space y equals f apostrophe left parenthesis x right parenthesis will be increasing
  • For intervals wherespace y equals f left parenthesis x right parenthesis is concave down ,space y equals f apostrophe left parenthesis x right parenthesis will be decreasing
  • For intervals wherespace y equals f left parenthesis x right parenthesis is increasing,space y equals f apostrophe left parenthesis x right parenthesis will be positive
  • For intervals wherespace y equals f left parenthesis x right parenthesis is decreasing,space y equals f apostrophe left parenthesis x right parenthesis will be negative

How do I sketch the graph of y = f''(x) from the graph of y = f(x)?

  • First sketch the graph ofspace y equals f apostrophe left parenthesis x right parenthesis fromspace y equals f left parenthesis x right parenthesis, as per the above process
  • Then, using the same process, sketch the graph ofspace y equals f apostrophe apostrophe left parenthesis x right parenthesis from the graph ofspace y equals f apostrophe left parenthesis x right parenthesis
  • There are a couple of things you can deduce about the graph ofspace y equals f apostrophe apostrophe left parenthesis x right parenthesis directly from the graph ofspace y equals f left parenthesis x right parenthesis
    • The graph of space y equals f apostrophe apostrophe left parenthesis x right parenthesis will have its x-axis intercepts at the x-coordinates of the points of inflection of space y equals f left parenthesis x right parenthesis
    • For intervals wherespace y equals f left parenthesis x right parenthesis is concave up,space y equals f apostrophe apostrophe left parenthesis x right parenthesis will be positive
    • For intervals wherespace y equals f left parenthesis x right parenthesis is concave down,space y equals f apostrophe apostrophe left parenthesis x right parenthesis will be negative

5-2-6-ib-sl-aa-only-y-dy-d2y

Is it possible to sketch the graph of y = f(x) from the graph of a derivative?

  • It is possible to sketch a graph ofspace y equals f left parenthesis x right parenthesis by considering the reverse of the above
    • For intervals wherespace y equals f apostrophe left parenthesis x right parenthesis is positive,space y equals f left parenthesis x right parenthesis will be increasing but is not necessarily positive
    • For intervals wherespace y equals f apostrophe left parenthesis x right parenthesis is negative,space y equals f left parenthesis x right parenthesis will be decreasing but is not necessarily negative
    • Roots ofspace y equals f apostrophe left parenthesis x right parenthesis give thespace x-coordinates of the stationary points ofspace y equals f left parenthesis x right parenthesis
  • There are some properties of the graph ofspace y equals f left parenthesis x right parenthesis that cannot be determined from the graph ofspace y equals f apostrophe left parenthesis x right parenthesis
    • thebold space bold italic y-axis intercept
    • the intervals for whichspace y equals f left parenthesis x right parenthesis is positive and negative
    • the roots ofspace y equals f left parenthesis x right parenthesis
  • Unless a specific point the curve passes through is known, the constant of integration cannot be determined
    • the exact location of the curve will remain unknown
    • but it will still be possible to sketch its shape
  • If starting from the graph of the second derivative,space y equals f apostrophe apostrophe left parenthesis x right parenthesis, it is easier to sketch the graph ofspace y equals f apostrophe left parenthesis x right parenthesis first, then sketchspace y equals f left parenthesis x right parenthesis

Worked example

The graph ofspace y equals f left parenthesis x right parenthesis is shown in the diagram below.

qwwD1Cx~_5-2-6-ib-sl-aa-only-we-quest

On separate diagrams sketch the graphs ofspace y equals f apostrophe left parenthesis x right parenthesis andspace y equals f apostrophe apostrophe left parenthesis x right parenthesis, labelling any roots and turning points.

5-2-6-ib-sl-aa-only-we-soltn

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Paul

Author: Paul

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.