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{6 x + 5} )

Explanation:

Step1: Recall the formula for average rate of change

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

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

• For \(x_1 = 6\):

Substitute \(x = 6\) into \(f(x)=\sqrt{6x + 5}\). Then \(f(6)=\sqrt{6\times6+5}=\sqrt{36 + 5}=\sqrt{41}\approx6.40\).

• For \(x_2 = 7\):

Substitute \(x = 7\) into \(f(x)=\sqrt{6x + 5}\). Then \(f(7)=\sqrt{6\times7+5}=\sqrt{42+5}=\sqrt{47}\approx6.86\).

Step3: Calculate the average rate of change

Substitute \(f(x_1)\) and \(f(x_2)\) into the average - rate - of - change formula.
\(\frac{f(7)-f(6)}{7 - 6}=\frac{\sqrt{47}-\sqrt{41}}{1}\approx6.86-6.40 = 0.46\)

Answer:

\(0.46\)