Question

given the function ( f(x)=-x^{2}+x + 8 ), determine the average rate of change of the function over the interval ( 0leq xleq7 ).

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 0\), \(b=7\).

Step2: Calculate \(f(0)\)

Substitute \(x = 0\) into \(f(x)=-x^{2}+x + 8\).
\(f(0)=-(0)^{2}+0 + 8=8\)

Step3: Calculate \(f(7)\)

Substitute \(x = 7\) into \(f(x)=-x^{2}+x + 8\).
\(f(7)=-(7)^{2}+7 + 8=-49 + 7+8=-34\)

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\), substitute \(a = 0\), \(b = 7\), \(f(0)=8\), \(f(7)=-34\).
\(\frac{f(7)-f(0)}{7-0}=\frac{-34 - 8}{7}=\frac{-42}{7}=-6\)

Answer:

\(-6\)