Question

How do I find the average rate of change of f from 0 to
`(7\pi )/(2)`?

Answer 1

`\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill`

`f(x)= \sin\left( (x)/(2)\right) \qquad \begin{cases} x_1=0\\ x_2=(7\pi )/(2) \end{cases}\implies \cfrac{f\left( (7\pi )/(2) \right)-f(0)}{(7\pi )/(2) - 0}`

`\cfrac{ ~~ \sin\left( ( ~~ (7\pi )/(2) ~~ )/(2) \right)-\sin\left( (0)/(2) \right) ~~ }{(7\pi )/(2)}\implies \cfrac{ ~~ \sin\left( ~~ (7\pi )/(4) ~~ \right)-\sin(0) ~~ }{(7\pi )/(2)} \\\\\\ \cfrac{-(1)/(√(2))~~ - ~~0}{(7\pi )/(2)}\implies -\cfrac{1}{√(2)}\cdot \cfrac{2}{7\pi }\implies \boxed{-\cfrac{√(2)}{7\pi }}`