Question

find the average rate of change of g(x) = -3\sqrt{x} over the interval \left5, 14\
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 \( g(x) \) over the interval \([a, b]\) is given by \(\frac{g(b) - g(a)}{b - a}\). Here, \( a = 5 \), \( b = 14 \), and \( g(x)=- 3\sqrt{x}\).

Step2: Calculate \( g(5) \) and \( g(14) \)

First, find \( g(5) \):
\( g(5)=-3\sqrt{5}\)
Then, find \( g(14) \):
\( g(14)=-3\sqrt{14}\)

Step3: Substitute into the average rate of change formula

The average rate of change \( r=\frac{g(14)-g(5)}{14 - 5}=\frac{-3\sqrt{14}-(-3\sqrt{5})}{9}=\frac{3\sqrt{5}-3\sqrt{14}}{9}=\frac{\sqrt{5}-\sqrt{14}}{3}\)

Now, calculate the numerical values:
\(\sqrt{5}\approx2.236\), \(\sqrt{14}\approx3.742\)
So, \(\frac{2.236 - 3.742}{3}=\frac{- 1.506}{3}\approx - 0.5\)

Answer:

\(-0.5\)