Question

find the average rate of change of ( k(x) = \frac{-12}{x - 19} ) over the interval ( 8, 11 ). 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 \( k(x) \) over the interval \([a, b]\) is given by \(\frac{k(b) - k(a)}{b - a}\). Here, \( a = 8 \), \( b = 11 \), and \( k(x)=\frac{-12}{x - 19}\).

Step2: Calculate \( k(8) \)

Substitute \( x = 8 \) into \( k(x) \):
\( k(8)=\frac{-12}{8 - 19}=\frac{-12}{-11}=\frac{12}{11} \)

Step3: Calculate \( k(11) \)

Substitute \( x = 11 \) into \( k(x) \):
\( k(11)=\frac{-12}{11 - 19}=\frac{-12}{-8}=\frac{3}{2} \)

Step4: Calculate the average rate of change

Using the formula \(\frac{k(11)-k(8)}{11 - 8}\), substitute the values of \( k(11) \) and \( k(8) \):
First, find \( k(11)-k(8)=\frac{3}{2}-\frac{12}{11} \). Find a common denominator, which is \( 22 \):
\(\frac{3}{2}=\frac{3\times11}{2\times11}=\frac{33}{22}\) and \(\frac{12}{11}=\frac{12\times2}{11\times2}=\frac{24}{22}\)
So, \(\frac{33}{22}-\frac{24}{22}=\frac{33 - 24}{22}=\frac{9}{22}\)
Then, divide by \( 11 - 8 = 3 \):
\(\frac{\frac{9}{22}}{3}=\frac{9}{22}\times\frac{1}{3}=\frac{3}{22}\approx0.1\) (rounded to the nearest tenth)

Answer:

\(\frac{3}{22}\) (or approximately \( 0.1 \))