Question

find the average rate of change of ( f(x) = \frac{-7}{x - 9} ) over the interval ( 0, 3 ). 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, \( a = 0 \), \( b = 3 \), and \( f(x)=\frac{-7}{x - 9}\).

Step2: Calculate \( f(0) \)

Substitute \( x = 0 \) into \( f(x) \):
\( f(0)=\frac{-7}{0 - 9}=\frac{-7}{-9}=\frac{7}{9} \)

Step3: Calculate \( f(3) \)

Substitute \( x = 3 \) into \( f(x) \):
\( f(3)=\frac{-7}{3 - 9}=\frac{-7}{-6}=\frac{7}{6} \)

Step4: Calculate the average rate of change

Using the formula \(\frac{f(3)-f(0)}{3 - 0}\), substitute the values of \( f(3) \) and \( f(0) \):
\[

$$\begin{align*} \frac{f(3)-f(0)}{3 - 0}&=\frac{\frac{7}{6}-\frac{7}{9}}{3}\\ &=\frac{\frac{21 - 14}{18}}{3}\\ &=\frac{\frac{7}{18}}{3}\\ &=\frac{7}{18}\times\frac{1}{3}\\ &=\frac{7}{54}\approx0.1 \end{align*}$$

\]

Answer:

\(\frac{7}{54}\) (or approximately \(0.1\))