Mathematics, 06.10.2019 22:40 Anavail
Determine if the mean value theorem for integrals applies to the function f of x equals the square root of x minus 1 on the interval [1, 5]. if so, find the x-coordinates of the point(s) guaranteed to exist by the theorem.
the lesson didn't cover how to do this, ! you in advance.
Answers: 1
![\bf f(x)=\sqrt{x-1}\qquad \begin{array}{ccllll} [1&,&5]\\ a&&b \end{array}\\\\ -----------------------------\\\\ \cfrac{dy}{dx}=\cfrac{1}{2}(x-1)^{-\frac{1}{2}}\implies \cfrac{dy}{dx}=\cfrac{1}{2\sqrt{x-1}} \\\\\\ \textit{now, the mean value theorem says} \\\\\\ f'(c)=\cfrac{f(b)-f(a)}{b-a}\implies \cfrac{1}{2\sqrt{c-1}}=\cfrac{f(5)-f(1)}{5-1} \\\\\\ \cfrac{1}{2\sqrt{c-1}}=\cfrac{2-0}{4}\implies \cfrac{1}{2\sqrt{c-1}}=\cfrac{1}{2} \\\\\\ 1=\sqrt{c-1}\implies 1^2+1=c\implies \boxed{2=c}](/tpl/images/0295/2500/71ec8.png)
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