find the average rate of change of the function ( f(x)=sqrt{x} ) from ( x_{1}=36 ) to ( x_{2}=144 ). the…

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)=\sqrt{x}), (x_1 = 36) and (x_2=144). For (x_1 = 36), (f(36)=\sqrt{36}=6). For (x_2 = 144), (f(144)=\sqrt{144}=12).

Step3: Substitute into the formula

Substitute (f(x_1) = 6), (f(x_2)=12), (x_1 = 36) and (x_2 = 144) into (\frac{f(x_2)-f(x_1)}{x_2 - x_1}). We get (\frac{12 - 6}{144-36}=\frac{6}{108}).

Step4: Simplify the fraction

Simplify (\frac{6}{108}) by dividing both the numerator and the denominator by 6. (\frac{6\div6}{108\div6}=\frac{1}{18}).

Answer:

(\frac{1}{18})