Problem 55 - College Algebra

Question

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}.\) $$f(x)=3 x \text { from } x_{1}=0 \text { to } x_{2}=5$$

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x)=3x\) from \(x_{1}=0\) to \(x_{2}=5\) is 3.

Step 1: Function Evaluation at \(x_{1}\)

First, evaluate the function \(f(x)=3x\) at \(x_{1}=0\). Substitute \(x_{1}=0\) into the function to get \(f(x_{1}) = 3*0 = 0\).

Step 2: Function Evaluation at \(x_{2}\)

Next, evaluate the function \(f(x)=3x\) at \(x_{2}=5\). Substitute \(x_{2}=5\) into the function to get \(f(x_{2})=3*5 = 15\).

Step 3: Compute the Average Rate of Change

With \(f(x_{1})\) and \(f(x_{2})\) calculated, the average rate of change of the function from \(x_{1}\) to \(x_{2}\) can be defined as \(\dfrac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}\). Substituting the given and calculated values, we have: \(\dfrac{15 - 0}{5 - 0} = 3\).