Question

what is the average rate of change of ( f(x) ) from ( x_1=-2 ) to ( x_2 = 3 )? please write your answer rounded to the nearest hundredth.

( f(x)=sqrt{2x + 9} )

Explanation:

Step1: Calculate \(f(x_1)\) and \(f(x_2)\)

For \(x_1=-2\), \(f(-2)=\sqrt{2\times(-2)+9}=\sqrt{-4 + 9}=\sqrt{5}\approx2.24\)
For \(x_2 = 3\), \(f(3)=\sqrt{2\times3+9}=\sqrt{6 + 9}=\sqrt{15}\approx3.87\)

Step2: Use the average rate of change formula

The average rate of change formula is \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\)
Substitute \(x_1=-2,x_2 = 3,f(x_1)\approx2.24,f(x_2)\approx3.87\) into the formula:
\(\frac{3.87-2.24}{3-(-2)}=\frac{1.63}{5}=0.326\approx0.33\)

Answer:

\(0.33\)