Question

what is the average rate of change of ( f(x) ) from ( x_1 = 6 ) to ( x_2 = 7 )? please write your answer rounded to the nearest hundredth.

( f(x)=sqrt{3 x + 9} )

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) from \(x_1\) to \(x_2\) is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). Here, \(x_1 = 6\), \(x_2=7\).

Step2: Calculate \(f(x_1)\) and \(f(x_2)\)

• For \(x_1 = 6\):

\(f(6)=\sqrt{3\times6 + 9}=\sqrt{18 + 9}=\sqrt{27}=3\sqrt{3}\approx3\times1.732 = 5.196\)

• For \(x_2 = 7\):

\(f(7)=\sqrt{3\times7+9}=\sqrt{21 + 9}=\sqrt{30}\approx5.477\)

Step3: Substitute into the average - rate - of - change formula

\(\frac{f(7)-f(6)}{7 - 6}=\frac{\sqrt{30}-\sqrt{27}}{1}\approx5.477-5.196 = 0.281\approx0.28\)

Answer:

\(0.28\)