Question

find the average rate of change of the function ( f(x)=sqrt{x} ) from ( x_{1}=49 ) to ( x_{2}=121 ).
the average rate of change is ( square ). (simplify your answer.)

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

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

Given \(f(x)=\sqrt{x}\), \(x_1 = 49\) and \(x_2=121\).
\(f(x_1)=\sqrt{49}=7\), \(f(x_2)=\sqrt{121} = 11\).

Step3: Substitute into the formula

\(\frac{f(x_2)-f(x_1)}{x_2 - x_1}=\frac{11 - 7}{121-49}\).
Simplify the denominator: \(121 - 49=72\).
So, \(\frac{11 - 7}{121-49}=\frac{4}{72}\).
Simplify the fraction: \(\frac{4\div4}{72\div4}=\frac{1}{18}\).

Answer:

\(\frac{1}{18}\)