Respuesta :
Answer:
Average rate of change is 1.
Step-by-step explanation:
The average rate of change of function [tex]f[/tex] over the intervals [tex]a\leqslant x\leqslant b[/tex] is given by the formula,
[tex]A\left(x\right)=\dfrac{f\left(b\right)-f\left(a\right)}{b-a}[/tex].
Rewriting the formula according to given data,
[tex]A\left(t\right)=\dfrac{g\left(b\right)-g\left(a\right)}{b-a}[/tex].
Given the interval as [tex]-4\leqslant t\leqslant 5[/tex]. Hence value of a and b is a = -4 and b = 5.
Now calculate [tex]g\left(b\right)[/tex] and [tex]g\left(a\right)[/tex] as follows.
Calculation for [tex]g\left(a\right)[/tex],
[tex]g\left(t\right)=-\left(t-1\right)^2+5[/tex]
Replace t as a,
[tex]g\left(a\right)=-\left(a-1\right)^2+5[/tex]
Substituting the value,
[tex]g\left(-4\right)=-\left(-4-1\right)^2+5[/tex]
[tex]g\left(-4\right)=-\left(-5\right)^2+5[/tex]
[tex]g\left(-4\right)=-25+5[/tex]
[tex]g\left(-4\right)=-20[/tex]
Calculation for [tex]g\left(b\right)[/tex],
[tex]g\left(b\right)=-\left(b-1\right)^2+5[/tex]
Substituting the value,
[tex]g\left(5\right)=-\left(5-1\right)^2+5[/tex]
[tex]g\left(5\right)=-\left(4\right)^2+5[/tex]
[tex]g\left(5\right)=-16+5[/tex]
[tex]g\left(5\right)=-11[/tex]
Now substituting the value,
[tex]A\left(t\right)=\dfrac{-11-\left(-20\right)}{5-\left(-4\right)}[/tex].
Simplifying,
[tex]A\left(t\right)=\dfrac{-11+20}{5+4}[/tex].
[tex]A\left(t\right)=\dfrac{9}{9}[/tex].
[tex]A\left(t\right)=1[/tex].
Hence average rate of change of the given function [tex]g\left(t\right)=-\left(t-1\right)^2+5[/tex] over the given interval [tex]-4\leqslant t\leqslant 5[/tex] is 1.
