AS June 2024 Paper 1 Q7
7 The function f is defined by
\[\mathrm{f}(x) = \frac{1}{\sqrt{x}} \qquad 4 \leqslant x \leqslant 7\]Find the mean value of f over the interval \(4 \leqslant x \leqslant 7\)
Give your answer in exact form. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Writes a correct expression for the mean value, eg \(\dfrac{1}{7 - 4}\displaystyle\int_4^7 \frac{1}{\sqrt{x}}\,\mathrm{d}x\) PI by 0.430… | M1 | 1.1a |
| Integrates \(\dfrac{1}{\sqrt{x}}\) to an expression of the form \(ax^{\frac{1}{2}}\) where \(a\) is non-zero and substitutes 7 and 4 and subtracts. PI by 1.291… or 0.430… Note: \(\dfrac{1}{4}\left(\dfrac{1}{\sqrt{4}} + \dfrac{1}{\sqrt{5}} + \dfrac{1}{\sqrt{6}} + \dfrac{1}{\sqrt{7}}\right) = 0.433\) is M0 | M1 | 1.1a |
| Obtains \(\dfrac{2}{3}\left(\sqrt{7} - 2\right)\) Ignore an approximated answer. | A1 | 1.1b |
| (3 marks) |
Typical solution
The mean of f
\[\begin{aligned} &= \frac{1}{7 - 4}\int_4^7 x^{-\frac{1}{2}}\,\mathrm{d}x \\ &= \frac{1}{3}\left[2x^{\frac{1}{2}}\right]_4^7 \\ &= \frac{2}{3}\left(\sqrt{7} - 2\right)\end{aligned}\]