Edexcel A Level Further Maths: Core Pure:复习笔记5.2.2 Mean Value of a Function

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

    • Note that the mean value  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  is a real number, the graph of   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.

5-2-2-mean-value-rectangle

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

  • If  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  over the interval [a, b]
    • kf(x) has mean value  over the interval [a, b]
    • -f(x) has mean value - over the interval [a, b]
  • If  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  be the function defined by .

a)  Find the exact mean value of  over the interval .5-2-2-edx-a-fm-we1a-soltn
b) Write down the exact mean value of each of the following functions over the interval :
(i)
(ii)
(iii)

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

 

 

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