Average Rate of Change Calculator

How to Calculate Average Rate of Change

The average rate of change is a fundamental concept in calculus that measures how much a function changes, on average, over a given interval. It's essentially the slope of the secant line between two points on the graph of a function.

Formula

The average rate of change of a function f(x) over the interval [a, b] is given by:

$$\text{Average Rate of Change}=\frac{f(b)-f(a)}{b-a}$$

Where:

• f(x) is the given function
• a is the starting point of the interval
• b is the ending point of the interval
• f(a) is the function value at point a
• f(b) is the function value at point b

Calculation Steps

1. Identify the function f(x) and the interval [a, b].
2. Calculate f(a) by substituting x with a in the function.
3. Calculate f(b) by substituting x with b in the function.
4. Subtract f(a) from f(b) to get the change in y-values.
5. Subtract a from b to get the change in x-values.
6. Divide the change in y-values by the change in x-values.

Example

Let's calculate the average rate of change for f(x) = x² over the interval [1, 4]:

1. f(x) = x², a = 1, b = 4
2. f(a) = f(1) = 1² = 1
3. f(b) = f(4) = 4² = 16
4. f(b) - f(a) = 16 - 1 = 15
5. b - a = 4 - 1 = 3
6. Average Rate of Change = (16 - 1) / (4 - 1) = 15 / 3 = 5

Therefore, the average rate of change of f(x) = x² over the interval [1, 4] is 5.

Visual Representation

This diagram illustrates the average rate of change for f(x) = x² over the interval [1, 4]. The blue curve represents the function, and the red line shows the secant line between the points (1, 1) and (4, 16). The slope of this red line is the average rate of change.