Calculating Average Rate of Change: Formula and Example

Average Rate of Change

The average rate of change is a concept in mathematics that measures the average rate at which a quantity changes over a given interval

The average rate of change is a concept in mathematics that measures the average rate at which a quantity changes over a given interval. It is commonly used to analyze the rate of change of a function over a specific interval.

To calculate the average rate of change, you need to find the difference in the values of the quantity being measured between the endpoints of the interval, and divide it by the difference in the input values (usually denoted as x-values) of the endpoints.

The formula for average rate of change can be expressed as:

Average rate of change = (change in y)/(change in x) = (y2 – y1)/(x2 – x1)

where (x1, y1) and (x2, y2) represent the coordinates of the two points on the function, located at the endpoints of the interval.

Let’s work through an example to better understand how to calculate the average rate of change.

Example:
Given the function f(x) = 2x + 3, find the average rate of change over the interval [1, 5].

Solution:
To calculate the average rate of change, we need to find the difference in the y-values (change in y) and the difference in the x-values (change in x) between the endpoints of the interval.

(x1, y1) = (1, f(1)) = (1, 2(1) + 3) = (1, 5)
(x2, y2) = (5, f(5)) = (5, 2(5) + 3) = (5, 13)

Now, we can calculate the average rate of change using the formula:

Average rate of change = (y2 – y1)/(x2 – x1) = (13 – 5)/(5 – 1) = 8/4 = 2

Therefore, the average rate of change of the function f(x) = 2x + 3 over the interval [1, 5] is 2.

This means that on average, the function increases by 2 units for every 1 unit increase in the x-value within the interval [1, 5].

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