Problem 36 - College Algebra

Question

What is the average rate of change of a function?

Step-by-Step Solution

Answer

The average rate of change of a function over an interval [a, b] is given by the formula \(\frac{f(b) - f(a)}{b - a}\), which is the slope of the secant line over that interval.

Step 1: Understand the Concept

Know that the average rate of change of a function over an interval [a, b] is given by the formula \(\frac{f(b) - f(a)}{b - a}\). This formula represents the change in the output value of the function (the difference in the y-values or f-values) divided by the change in the input value (the difference in the x-values).

Step 2: Apply the Formula

Apply the formula to the specific function. For example, for a linear function \(f(x) = mx + c\), the average rate of change from \(x = a\) to \(x = b\) would be \(\frac{f(b) - f(a)}{b - a} = \frac{m*b + c - (m*a + c)}{b - a} = m\). Thus, for a linear function, the average rate of change over any interval is simply the slope of the function.

Step 3: Interpret the Result

The final result is a single number, which represents the average rate of change of the function over the interval from \(x = a\) to \(x = b\) . This is the slope of the secant line that passes through the points \((a, f(a))\) and \((b, f(b))\). Positive values indicate the function is increasing on the interval, negative values indicate it is decreasing.