Question

find the average rate of change of the function f(X)=3-sinx over the given interval (0,90)​

Answer 1

$\large\maltese \underbrace{ \frak{Concept :-} }$

Average Rate of Change over any interval (a,b) is calculated by the following formula :-

$\\ \leadsto\underline{ \boxed{ \sf average = \frac{f(b) - f(a)}{b - a} }} \bigstar$

Let's find f(b) and f(a) individually and then substitute them into the formula!

$\large\maltese \underbrace{ \frak{Solution :-} }$

$\\ \quad \dag\underline{ \sf calculating \: f(a), a = 0 : - }$

$\\ \quad \rightarrow\sf f(x) = 3 - sinx$

$\\ \quad\rightarrow \sf f(0) = 3 - sin(0)$

$\\ \quad \rightarrow\sf f(0) = 3 - 1$

$\\ \quad \rightarrow \boxed{ \sf f(0) = 2} \bigstar$

$\\ \quad \dag\underline{ \sf calculating \: f(b), b = 90 : - }$

Note:- 90 can be written as π/2

$\\ \quad \rightarrow \sf f \bigg(\dfrac{ \pi}{2} \bigg) = 3 - sin \bigg( \frac{\pi}{2} \bigg)$

$\\ \quad \rightarrow \sf f \bigg( \frac{\pi}{2} \bigg) = 3 - 1$

$\\ \quad \rightarrow \sf f \bigg( \frac{\pi}{2} \bigg) = 3 - 1$

$\\ \quad \rightarrow \boxed{ \sf f \bigg( \frac{\pi}{2} \bigg) = 2} \bigstar$

$\\ \quad \dag\underline{ \sf substituting \: these \: values \: in \: the \: formula : - }$

$\\ \quad \rightarrow \sf average = \frac{f \bigg( \dfrac{\pi}{2} \bigg) - f \bigg( 0\bigg)}{b - a}$

$\\ \quad \rightarrow \sf average = \frac{2 - 3}{ \dfrac{\pi}{2} - 0 }$

$\\ \quad \rightarrow \sf average = \frac{ - 1}{ \dfrac{\pi}{2} }$

$\\ \quad \therefore\boxed{ \boxed{\sf average = \frac{ - 2}{\pi}}}\bigstar$

Answer 2

$\large\underline{\sf{Solution-}}$

Given function is

$\rm :\longmapsto\:f(x) =3- sinx \: \: on \: \: \bigg[0, \: \dfrac{\pi}{2} \bigg]$

So,

$\rm :\longmapsto\:f(0) = 3-sin0 = 3$

and

$\rm :\longmapsto\:f\bigg[\dfrac{\pi}{2} \bigg] =3- sin\bigg[\dfrac{\pi}{2} \bigg] = 3-1=2$

We know that,

Average rate of change in function f(x) on the interval [ a, b ] is

$\red{ \boxed{ \sf{ \:Average \: rate \: of \: change \: = \: \frac{f(b) - f(a)}{b - a} }}}$

So,

$\rm :\longmapsto\: \:Average \: rate \: of \: change \: = \: \dfrac{f\bigg[\dfrac{\pi}{2} \bigg] - f(0)}{\dfrac{\pi}{2} - 0}$

$\rm :\longmapsto\: \:Average \: rate \: of \: change \: = \: \dfrac{2 - 3}{\dfrac{\pi}{2} }$

$\rm :\longmapsto\: \:Average \: rate \: of \: change \: = \: \dfrac{-1}{\dfrac{\pi}{2}}$

$\bf\implies \:\: \:Average \: rate \: of \: change \: = \: \dfrac{-2}{\pi}$

Additional Information :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cox & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}$