Question

find the average rate of change of ( g(x) = 3sqrt{x + 17} ) over the interval ( -12, -7 ). 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 average rate of change of a function \( g(x) \) over the interval \([a, b]\) is given by \(\frac{g(b)-g(a)}{b - a}\). Here, \( a=-12 \) and \( b = - 7\).

Step2: Calculate \( g(-12) \)

Substitute \( x=-12 \) into \( g(x)=3\sqrt{x + 17}\):
\( g(-12)=3\sqrt{-12 + 17}=3\sqrt{5}\approx3\times2.236 = 6.708\)

Step3: Calculate \( g(-7) \)

Substitute \( x = - 7\) into \( g(x)=3\sqrt{x + 17}\):
\( g(-7)=3\sqrt{-7+17}=3\sqrt{10}\approx3\times3.162=9.486\)

Step4: Calculate the average rate of change

Using the formula \(\frac{g(b)-g(a)}{b - a}\), with \( a=-12\), \( b=-7\), \( g(a)\approx6.708\) and \( g(b)\approx9.486\):
\(\frac{g(-7)-g(-12)}{-7-(-12)}=\frac{9.486 - 6.708}{-7 + 12}=\frac{2.778}{5}=0.5556\approx0.6\) (rounded to the nearest tenth)

Answer:

\(0.6\)