Question

find the average rate of change of ( g(x)=\frac{1}{-x} ) over the interval ( -3,-2 ). write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = g(x)\) over the interval \([a,b]\) is \(\frac{g(b)-g(a)}{b - a}\). Here, \(a=-3\), \(b = - 2\), and \(g(x)=\frac{1}{-x}\).

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

When \(x=-3\), \(g(-3)=\frac{1}{-(-3)}=\frac{1}{3}\). When \(x = - 2\), \(g(-2)=\frac{1}{-(-2)}=\frac{1}{2}\).

Step3: Substitute into the formula

\(\frac{g(-2)-g(-3)}{-2-(-3)}=\frac{\frac{1}{2}-\frac{1}{3}}{-2 + 3}\). First, simplify the numerator: \(\frac{1}{2}-\frac{1}{3}=\frac{3-2}{6}=\frac{1}{6}\). The denominator \(-2+3 = 1\).

Answer:

\(\frac{1}{6}\approx0.2\)