Mean Value of a Function (Edexcel A Level Further Maths: Core Pure)

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Mean Value of a Function

What is the mean value of a function?

  • The mean value of a function may be thought of as the ‘average’ value of a function over a given interval
  • For a function f(x), the mean value  of the function over the interval [a, b] is given by

straight f with bar on top equals fraction numerator 1 over denominator b minus a end fraction integral subscript a superscript b straight f left parenthesis x right parenthesis d x

    • Note that the mean value straight f with bar on top is simply a real number – it is not a function
    • The mean value depends on the interval chosen – if the interval [a, b] changes, then the mean value may change as well
  • Because straight f with bar on top is a real number, the graph of  y equals straight f with bar on top  is a horizontal line
    • This gives a geometrical interpretation of the mean value of a function over a given interval
    • If A is the area bounded by the curve y = f(x), the x-axis and the lines x = a and x = b, then the rectangle with its base on the interval [a, b] and with height  also has area A
      • i.e. left parenthesis b minus a right parenthesis straight f with bar on top equals integral subscript a superscript b straight f left parenthesis x right parenthesis d x

5-2-2-mean-value-rectangle

What are the properties of the mean value of a function?

  • If straight f with bar on top is the mean value of a function f(x) over the interval [a, b], and k is a real constant, then:
    • f(x) + k has mean value straight f with bar on top plus k over the interval [a, b]
    • kf(x) has mean value k straight f with bar on top over the interval [a, b]
    • -f(x) has mean value negative straight f with bar on top over the interval [a, b]
  • If straight f with bar on top equals 0 then the area that is above the x-axis and under the curve is equal to the area that is below the x-axis and above the curve

Worked example

Let straight f be the function defined by straight f left parenthesis x right parenthesis equals fraction numerator 1 over denominator x plus 1 end fraction.

a)

Find the exact mean value of straight f over the interval left square bracket 0 comma 1 right square bracket.5-2-2-edx-a-fm-we1a-soltn

b)
Write down the exact mean value of each of the following functions over the interval left square bracket 0 comma 1 right square bracket:
(i)
straight f left parenthesis x right parenthesis plus 3
(ii)
negative straight f left parenthesis x right parenthesis
(iii)
6 straight f left parenthesis x right parenthesis

5-2-2-edx-a-fm-we1b-soltn

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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.