Question

find the average rate of change of f(x) = -12\sqrt{x + 17} over the interval \left-8, -6\
ight. 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 \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \( a=-8 \), \( b = - 6\), and \( f(x)=-12\sqrt{x + 17}\).

Step2: Calculate \( f(-8) \)

Substitute \( x=-8 \) into \( f(x) \):
\( f(-8)=-12\sqrt{-8 + 17}=-12\sqrt{9}=-12\times3=-36 \)

Step3: Calculate \( f(-6) \)

Substitute \( x = - 6\) into \( f(x) \):
\( f(-6)=-12\sqrt{-6+17}=-12\sqrt{11}\approx-12\times3.3166\approx - 39.7992\)

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\), substitute \( a=-8 \), \( b=-6 \), \( f(-8)=-36 \), and \( f(-6)\approx - 39.7992\):
\(\frac{f(-6)-f(-8)}{-6-(-8)}=\frac{-39.7992-(-36)}{-6 + 8}=\frac{-39.7992 + 36}{2}=\frac{-3.7992}{2}\approx-1.9\)

Answer:

\(-1.9\)