Problem 59 - College Algebra

Question

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}.\) $$f(x)=\sqrt{x} \text { from } x_{1}=4 \text { to } x_{2}=9$$

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x)=\sqrt{x}\) from \(x_{1}=4\) to \(x_{2}=9\) is \(\frac{1}{5}\).

Step 1: Identify the given points

We have two points given, \(x_{1}=4\) and \(x_{2}=9\), and we know the function \(f(x)=\sqrt{x}\). First, determine the function values at these points. This gives us \(f(x_{1})=f(4)=\sqrt{4}=2\) and \(f(x_{2})=f(9)=\sqrt{9}=3\).

Step 2: Plug the values into the formula

Next, plug these values into the formula for average rate of change, \(\frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}\). Substitution gives us \(\frac{3 - 2}{9 - 4}\).

Step 3: Simplify the expression

When you simplify the expression \(\frac{3 - 2}{9 - 4}\), you get \(\frac{1}{5}\). Therefore, the average rate of change of the function from \(x_{1}=4\) to \(x_{2}=9\) is \(\frac{1}{5}\).