Question

1. what is the average rate of change of $g(x)=x^{2}$ on the interval $3 \leq x \leq 9$?

(1) 8
(2) 12
(3) 15
(4) 18

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = g(x)\) on the interval \([a,b]\) is \(\frac{g(b)-g(a)}{b - a}\). Here, \(a = 3\), \(b=9\), and \(g(x)=x^{2}\).

Step2: Calculate \(g(a)\) and \(g(b)\)

When \(x = 3\), \(g(3)=3^{2}=9\). When \(x = 9\), \(g(9)=9^{2}=81\).

Step3: Substitute into the formula

Substitute \(g(3) = 9\), \(g(9)=81\), \(a = 3\), and \(b = 9\) into \(\frac{g(b)-g(a)}{b - a}\). We get \(\frac{81 - 9}{9-3}=\frac{72}{6}\).

Step4: Simplify the expression

\(\frac{72}{6}=12\).

Answer:

(2) 12