Question

find the average rate of change of the function ( f(x)=x^{2}+2x ) from ( x_{1}=1 ) to ( x_{2}=3 ).
the average rate of change is ( square ). (simplify your answer.)

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 \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).

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

Given \(f(x)=x^{2}+2x\), when \(x_1 = 1\), \(f(1)=1^{2}+2\times1=1 + 2=3\).
When \(x_2 = 3\), \(f(3)=3^{2}+2\times3=9+6 = 15\).

Step3: Calculate the average rate of change

Substitute \(f(x_1)=3\), \(f(x_2)=15\), \(x_1 = 1\), and \(x_2 = 3\) into the formula \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).
We get \(\frac{15 - 3}{3-1}=\frac{12}{2}\).

Answer:

\(6\)