Question

what is the average rate of change of the function ( f(x)=sqrt{100 - x^{2}} ) on the interval (-8,0)?

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) on the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \(a=-8\), \(b = 0\), and \(f(x)=\sqrt{100 - x^{2}}\).

Step2: Calculate \(f(-8)\) and \(f(0)\)

• For \(x=-8\), \(f(-8)=\sqrt{100-(-8)^{2}}=\sqrt{100 - 64}=\sqrt{36}=6\).
• For \(x = 0\), \(f(0)=\sqrt{100-0^{2}}=\sqrt{100}=10\).

Step3: Substitute into the average - rate - of - change formula

\(\frac{f(0)-f(-8)}{0-(-8)}=\frac{10 - 6}{0 + 8}\).
Simplify the fraction: \(\frac{4}{8}=\frac{1}{2}\).

Answer:

\(\frac{1}{2}\)