Question

given the function ( g(x)=-x^{2}+10x + 33 ), determine the average rate of change of the function over the interval ( 0leq xleq6 ).

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = g(x)\) over the interval \([a,b]\) is \(\frac{g(b)-g(a)}{b - a}\). Here \(a = 0\) and \(b=6\).

Step2: Calculate \(g(0)\)

Substitute \(x = 0\) into \(g(x)=-x^{2}+10x + 33\).
\(g(0)=-(0)^{2}+10(0)+33=33\)

Step3: Calculate \(g(6)\)

Substitute \(x = 6\) into \(g(x)=-x^{2}+10x + 33\).
\(g(6)=-(6)^{2}+10(6)+33=-36 + 60+33=57\)

Step4: Calculate the average rate of change

Use the formula \(\frac{g(6)-g(0)}{6 - 0}\).
\(\frac{57-33}{6}=\frac{24}{6}=4\)

Answer:

\(4\)