Question

1. find the average rate of change of ( g ( x ) = \frac { x ^ { 2 } } { 2 } ) over the interval ( - 2,4 ).
2. write your answer as an integer, fraction, or decimal rounded to the nearest tenth.

simplify any fractions.

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \(y = g(x)\) over the interval \([a,b]\) is given by \(\frac{g(b)-g(a)}{b - a}\). Here, \(a=-2\), \(b = 4\), and \(g(x)=\frac{-2}{x}\).

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

When \(x=-2\), \(g(-2)=\frac{-2}{-2}=1\). When \(x = 4\), \(g(4)=\frac{-2}{4}=-\frac{1}{2}\).

Step3: Substitute into the formula

\(\frac{g(4)-g(-2)}{4-(-2)}=\frac{-\frac{1}{2}-1}{4 + 2}=\frac{-\frac{1}{2}-\frac{2}{2}}{6}=\frac{-\frac{3}{2}}{6}\).

Step4: Simplify the fraction

\(\frac{-\frac{3}{2}}{6}=-\frac{3}{2}\times\frac{1}{6}=-\frac{1}{4}=-0.25\).

Answer:

\(-0.3\) (rounded to the nearest tenth)