Question

given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval (4leq xleq6).

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, \(a = 4\) and \(b=6\).

Step2: Identify \(f(a)\) and \(f(b)\)

From the table, when \(x = 4\), \(f(4)=9\) (so \(f(a)=9\)) and when \(x = 6\), \(f(6)=81\) (so \(f(b)=81\)).

Step3: Substitute into the formula

Substitute \(f(a)=9\), \(f(b)=81\), \(a = 4\), and \(b = 6\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{81 - 9}{6-4}\).

Step4: Simplify the expression

First, calculate the numerator: \(81-9=72\). Then calculate the denominator: \(6 - 4=2\). So \(\frac{72}{2}=36\).

Answer:

\(36\)