Question

determine the average rate of change of f(x) over the interval -2, 3.

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, \(a=-2\) and \(b = 3\).

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

From the table, when \(x=-2\), \(f(-2)=-3\) (so \(f(a)=f(-2)=-3\)), and when \(x = 3\), \(f(3)=12\) (so \(f(b)=f(3)=12\)).

Step3: Substitute into the formula

Substitute \(a=-2\), \(b = 3\), \(f(a)=-3\), and \(f(b)=12\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{12-(-3)}{3-(-2)}\).
Simplify the numerator: \(12-(-3)=12 + 3=15\).
Simplify the denominator: \(3-(-2)=3 + 2=5\).
So, \(\frac{15}{5}=3\).

Answer:

\(3\)