Question

Find the average value of \({\rm{f}}\)on\(\left( {{\rm{0,8}}} \right)\).

Short answer

\(\frac{9}{8}\)

Step-by-step solution

The graph of the function\(f\).

Average value of a function is represented by this graph,

The average value of a function is represented by this graph is

\({{\rm{f}}_{{\rm{axe}}}}{\rm{ = }}\frac{{\rm{1}}}{{{\rm{b - a}}}}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{f(}}{{\rm{x}}_{\rm{i}}}{\rm{)\Delta x}}} \)

Here \({\rm{\Delta x = }}\frac{{{\rm{b - a}}}}{{\rm{n}}}\) and \(n\)is number of terms.

\(\begin{aligned}{c}{{\rm{f}}_{{\rm{axe}}}}\frac{{\rm{1}}}{{{\rm{b - a}}}}\sum\limits_{{\rm{i &= 1}}}^{\rm{n}} {{\rm{f(}}{{\rm{x}}_{\rm{i}}}{\rm{)\Delta x}}} \\{\rm{ &= }}\frac{{\rm{1}}}{{{\rm{8 - 0}}}}\left( {{\rm{\Delta x}}\left( \begin{aligned}{l}{\rm{f}}\left( {{\rm{0}}{\rm{.5}}} \right){\rm{ + f}}\left( {{\rm{1}}{\rm{.5}}} \right){\rm{ + f}}\left( {{\rm{2}}{\rm{.5}}} \right){\rm{ + f}}\left( {{\rm{3}}{\rm{.5}}} \right)\\{\rm{ + f}}\left( {{\rm{4}}{\rm{.5}}} \right){\rm{ + f}}\left( {{\rm{5}}{\rm{.5}}} \right){\rm{ + f}}\left( {{\rm{6}}{\rm{.5}}} \right){\rm{ + f}}\left( {{\rm{7}}{\rm{.5}}} \right)\end{aligned} \right)} \right)\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{8}}}{\rm{.9 &= }}\frac{{\rm{9}}}{{\rm{8}}}\end{aligned}\)