Applications of the Second Derivative (OCR A Level Maths: Pure)

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Applications of the Second Derivative

What applications of the second derivative do I need to know?

  • The second order derivative (or simply second derivative) is encountered at AS level
    • At AS level second derivatives are used to help determine the nature of a stationary point
  • At A level you need to be able to use the second derivative to determine if a function is convex or concave on a given interval
    • Where a function is convex its graph 'curves up' and its tangent lines lie below the graph
    • Where a function is concave its graph 'curves down' and its tangent lines lie above the graph

Appl 2nd Der Illustr 1, AS & A Level Maths revision notes

  • It's the sign of the second derivative (+ or -) that's important:
    • A function f(x) is convex on a given interval if and only if f''(x) ≥ 0 for every value of x in the interval
    • A function f(x) is concave on a given interval if and only if f''(x) ≤ 0 for every value of x in the interval

    Appl 2nd Der Illustr 2, AS & A Level Maths revision notes

Examiner Tip

  • To get full marks in an exam your working needs to include the condition for a function to be convex or concave.

Worked example

Appl 2nd Der Example, AS & A Level Maths revision notes

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.