Find the average rate of change of the function from x₁ to x₂. f(x) = 3x from x₁ = 0 to x₂ = 5

Verified step by step guidance

1. Identify the function given: \(f(x) = 3x\).
2. Note the interval over which to find the average rate of change: from \(x_1 = 0\) to \(x_2 = 5\).
3. Recall the formula for the average rate of change of a function \(f(x)\) over the interval \([x_1, x_2]\): \[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]
4. Calculate \(f(x_1)\) and \(f(x_2)\) by substituting the values into the function: \[f(0) = 3 \times 0\] \[f(5) = 3 \times 5\]
5. Substitute these values into the average rate of change formula and simplify the expression (without calculating the final numeric value).

Key Concepts

Average Rate of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input. It is calculated as the difference in function values divided by the difference in input values, similar to the slope of a secant line between two points on the graph.

Evaluating Functions at Given Points

To find the average rate of change, you must first evaluate the function at the specified input values. This involves substituting the given x-values into the function expression to find the corresponding y-values or outputs.

Linear Functions and Their Properties

A linear function like f(x) = 3x has a constant rate of change, meaning the average rate of change between any two points is the same as the slope of the line. Understanding this helps simplify calculations and interpret results quickly.