Question

find the average rate of change of ( f(x) = 7sqrt{x + 6} ) over the interval (-2, 0). write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall average rate of change formula

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, \( a=-2 \), \( b = 0 \), and \( f(x)=7\sqrt{x + 6}\).

Step2: Calculate \( f(-2) \)

Substitute \( x=-2 \) into \( f(x) \):
\( f(-2)=7\sqrt{-2 + 6}=7\sqrt{4}=7\times2 = 14 \).

Step3: Calculate \( f(0) \)

Substitute \( x = 0 \) into \( f(x) \):
\( f(0)=7\sqrt{0 + 6}=7\sqrt{6}\approx7\times2.45 = 17.15 \) (rounded to two decimal places for calculation).

Step4: Apply the average rate of change formula

Using \(\frac{f(0)-f(-2)}{0 - (-2)}=\frac{7\sqrt{6}-14}{2}\).
First, compute \( 7\sqrt{6}\approx17.15 \), so \( 17.15 - 14 = 3.15 \). Then, \(\frac{3.15}{2}=1.575\approx1.6\) (rounded to the nearest tenth).
(Alternatively, using exact values: \(\frac{7\sqrt{6}-14}{2}=\frac{7(\sqrt{6}-2)}{2}\approx\frac{7(2.449 - 2)}{2}=\frac{7\times0.449}{2}=\frac{3.143}{2}=1.5715\approx1.6\))

Answer:

\( 1.6 \)