Question

given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval ( 4 leq x leq 5 ).

Explanation:

Step1: Recall the formula for average rate of change

The formula for 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 \(b=5\).

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

From the table, when \(x = 4\), \(f(4)=6\) (so \(f(a)=6\)) and when \(x = 5\), \(f(5)=10\) (so \(f(b)=10\)).

Step3: Substitute into the formula

Substitute \(a = 4\), \(b = 5\), \(f(a)=6\), and \(f(b)=10\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{10 - 6}{5-4}\).

Step4: Simplify the expression

\(\frac{10 - 6}{5-4}=\frac{4}{1}\).

Answer:

\(4\)