AQA A Level Maths: Pure复习笔记7.4.1 Applications of the Second Derivative

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

7.4.1-Appl-2nd-Der-Illustr-1

  • 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

    7.4.1-Appl-2nd-Der-Illustr-2

Exam 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

7.4.1-Appl-2nd-Der-Example

 

 

 

 

 

 

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