Question

find the average rate of change of the function ( f(x)=x^{2}+3x ) from ( x_{1}=1 ) to ( x_{2}=2 ).
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: Calculate \(f(x_1)\) and \(f(x_2)\)

Given \(f(x)=x^{2}+3x\), when \(x_1 = 1\), \(f(1)=1^{2}+3\times1=1 + 3=4\).
When \(x_2=2\), \(f(2)=2^{2}+3\times2=4 + 6=10\).

Step3: Substitute into the formula

Substitute \(f(x_1) = 4\), \(f(x_2)=10\), \(x_1 = 1\), \(x_2 = 2\) into \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).
We get \(\frac{10 - 4}{2-1}=\frac{6}{1}=6\).

Answer:

\(6\)