Moderate -0.8 This is a straightforward application of the mean value formula requiring a single standard integration (power rule with x^{-1/2}) and division by the interval length. The integration is routine and the question involves no problem-solving or conceptual challenges beyond recalling the mean value definition.
7 The function f is defined by
$$f ( x ) = \frac { 1 } { \sqrt { x } } \quad 4 \leq x \leq 7$$
Find the mean value of f over the interval \(4 \leq x \leq 7\)
Give your answer in exact form.
Integrates \(\dfrac{1}{\sqrt{x}}\) to form \(ax^{\frac{1}{2}}\) where \(a\) is non-zero, substitutes 7 and 4 and subtracts; PI by 1.291… or 0.430…
\(= \dfrac{2}{3}(\sqrt{7}-2)\)
A1
Obtains \(\dfrac{2}{3}(\sqrt{7}-2)\); ignore an approximated answer
## Question 7:
| $\dfrac{1}{7-4}\displaystyle\int_4^7 \dfrac{1}{\sqrt{x}}\,dx$ | M1 | Writes correct expression for mean value; PI by 0.430… |
| $= \dfrac{1}{3}\left[2x^{\frac{1}{2}}\right]_4^7$ | M1 | Integrates $\dfrac{1}{\sqrt{x}}$ to form $ax^{\frac{1}{2}}$ where $a$ is non-zero, substitutes 7 and 4 and subtracts; PI by 1.291… or 0.430… |
| $= \dfrac{2}{3}(\sqrt{7}-2)$ | A1 | Obtains $\dfrac{2}{3}(\sqrt{7}-2)$; ignore an approximated answer |
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7 The function f is defined by
$$f ( x ) = \frac { 1 } { \sqrt { x } } \quad 4 \leq x \leq 7$$
Find the mean value of f over the interval $4 \leq x \leq 7$
Give your answer in exact form.\\
\hfill \mbox{\textit{AQA Further AS Paper 1 2024 Q7 [3]}}