How to find The Average Rate of Change Over an Interval

Author: Noreen Niazi
Last Updated on: January 12, 2024

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How can you define the average rate of change over an interval? It’s clear from the name that the average rate of change is the average change in one quantity concerning another quantity.

Whether studying basic math or solving calculus problems, you often need to find the average rate of change. Is it a difficult topic? Do not worry. Here we discuss three easy to find the average rate of change over an interval.

What is the Average Rate of Change Over an Interval?

The average rate of change is the change in the value of the function at two points divided by the difference between two endpoints of intervals. Whether you are calculating your average speed or acceleration, the average rate of change helps you find it.

The average rate of change has its application in diverse fields such as mathematics, science, engineering, and economics, among others. Furthermore, it is also useful in analyzing data trends, making predictions, and identifying patterns[1].

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Importance of Average Rate of Change Over an Interval in Calculus

The average rate of change over an interval is an essential concept in calculus. It helps to calculate many different quantities. Here are some main things that you can find with an average rate of change.

Determine the slope of a curve at a particular point. The slope of a curve is a measure of the steepness of the curve, and it is essential in understanding the behavior of a function.
It also helps in finding the derivative of a function. Derivatives is essential in solving many calculus problems.
The average rate of change over an interval is also essential in understanding the relationship between two variables.
It helps to identify patterns, make predictions, and analyze data trends.
In economics, for example, the concept is used to measure the rate at which the demand or supply of a product changes over time.
The information is useful in making production, pricing, and marketing decisions.

Formula for average rate of change over an interval

You are familiar with the term average rate of change over an interval. Let’s talk about its formula.

Let $$f(x)$$ be the function defined over the interval $$[a,b]$$. Let $$<f(x)>$$ denote the average rate of change. Then its formula is The formula for calculating the average rate of change over an interval is as follows:

$$<f(x)>=\frac{f(b)-f(a)}{b-a}$$

Where $$f(a)$$ and $$f(b)$$ are the values of a function at two points a and b in an interval.

To calculate the average rate of change over an interval, you need to find the difference between the values of the function at two points and divide the difference by the length of the interval. The result is the average rate of change.

How to find average rate of change over an interval?

Let’s take an example to understand how to find the average rate of change over an interval.

Example:

Find the average rate of change of a function $$f(x) = 2x + 1$$ between the points $$x = 1$$ and $$x = 5$$.

Now, to find the average rate of change over an interval, we use the formula you would use the formula:

$$<f(x)>=\frac{f(b)-f(a)}{b-a}$$

Here

$$f(x)=2x+1$$
$$a=1$$ and $$b=5$$

$$<f(x) = \frac{f(5) – f(1))}{(5 – 1}$$

$$ <f(x)>= \frac{(2(5) + 1) – (2(1) + 1))}{(5 – 1)}$$

$$<f(x)> = \frac{(11 – 3)}{4}$$

$$<f(x)> = 2$$

Therefore, the average rate of change of the function $$f(x)=2x + 1$$ between the points $$x=1$$ and $$x=5$$ is $$2$$.

Real-Life Examples of the Average Rate of Change Over an Interval

The idea of average rate of change over an interval has multiple applications in real life.

A simple instance is the estimation of an object’s average velocity. Let us say a car covers 100 miles in 2 hours. 50 miles per hour is the average rate of change of the car’s position over 2 hours.
For instance, calculating the average rate of change in a country’s population over time. For instance, if within 10 years the population of a country has increased from 100 million to that of 57.
The idea is used in economics to determine the average rate of change of a product’s price over some time. For instance, if the price of a product goes up from $10 to $ 5 in five years, then on average the increase is about one dollar per year.[2].

How to Interpret the Results of the Average Rate of Change Over an Interval

Obtaining average rate of change over an interval is fundamental because it sheds light on how a function or variable behaves.

A positive average rate of change means that the function or variable is rising over this interval.
A negative one indicates it is decreasing.
Zer0 average rate of change means that the function or variable is constant within a given interval.

The measure of the average rate of change also gives valuable information about how a function or variable is performing. Larger values indicate quick changes in the function or variable, while smaller ones show slow changes.

Calculating the Average Rate of Change Over an Interval Using a Graph

One can compute the average rate of change over any interval through a graph. To do this, it is necessary to plot the function on a graph and identify two points determining the interval. After identifying these points, you can apply the formula to compute the average rate of change.

for instance, take the function $$f(x) = x^2-3x+2$$. The graph of the function is a parabola that is directed upwards. Suppose you wanted to compute this function’s average rate of change at intervals $$x=1$$ and $$x=3$$. First, you would graph the function and determine which two points to plot.

Once you have identified the points, you can use the formula to calculate the average rate of change as follows:

$$\text{average rate of change} = \frac{(f(3) – f(1))}{(3 – 1)}$$

$$ =\frac{ ((3)^2 – 3(3) + 2) – ((1)^2 – 3(1) + 2))}{(3 – 1)}$$

$$ = \frac{(2 – (-2))}{(2)}$$

$$ = 2$$

Therefore, the average rate of change of the function $$f(x)= x^2 – 3x + 2$$ between the points $$x=1$$ and $$x=3$$ is $$2$$.

Calculating the Average Rate of Change Over an Interval Using a Table

Another way to calculate the average rate of change over an interval is by using a table.

To do this, you need to create a table that lists the values of the function at the two points that define the interval. Once you have created the table, you can use the formula to calculate the average rate of change[3].

Example 1: Average Rate of Change of a Linear Function

Consider the linear function

$f (x) = 2 x + 1$

Let’s find the average rate of change over the interval $[1, 3].$

\begin{array}{cc} x & f(x) \\ 1 & 3 \\ 3 & 7​ \end{array}

$<f(x)> = \frac{Δ y}{Δ x} = \frac{f (3) - f (1)}{3 -1}$

Substitute the values

$<f(x)>$

=(7−3)/(3−1)

=4/2

So, the average rate of change of the function over the interval

$[1, 3]$

is 2.

Example 2: Average Rate of Change of a Quadratic Function

Consider the quadratic function g

$(x) = x^{2} - 2 x$

. Let’s find the average rate of change over the interval

$[- 2, 1].$

\begin{array}{cc} x & g (x) \\ - 2 & 8 \\ 1 & - 1​ \end{array}

$<g(x)> = \frac{Δ y}{Δ x} = \frac{g (1) -g (- 2)}{1 - (- 2)}$

Substitute the values:

$<g(x)> = \frac{(- 1) - 8}{1 + 2} = \frac{- 9}{3} = - 3$

So, the average rate of change of the function over the interval $[- 2, 1]$ is $- 3$ .

Common Mistakes to Avoid While Calculating the Average Rate of Change Over an Interval

When calculating the average rate of change over an interval, some common mistakes should be avoided.
Using the wrong formula: One of these errors is using the wrong formula. Make sure you use the appropriate formula for computing the average rate of change over an interval.
Using the wrong values for points: Using the wrong values for points that characterize this interval is common. Make sure you use the right values for the points to achieve accurate results.
Also, do not confuse the average rate of change with instantaneous rates. Instantaneous changes denote rate of change for a function at any given point, while average rates refer to the rate of change over any interval.

Applications of the Average Rate of Change Over an Interval in Different Fields

The average rate of change over an interval is a concept that has several applications in different fields. In physics, the concept calculates the velocity and acceleration of moving objects.
In economics, it is used to measure the rate of change of the demand or supply of a product. In biology, it measures the growth rate of cells or organisms.
In engineering, the concept is used to design and optimize systems, such as the flow of fluids or the rate of heat transfer. In finance, it calculates the rate of return on investments or the growth rate of a company’s revenue. In computer science, it is used to analyze algorithms and optimize their performance.

Conclusion

The average rate of change over an interval is a basic calculus principle with numerous real-life applications. It is used to ascertain the gradient of a curve at some point; it recognizes trends, suggests predictions; and makes observations on data patterns. To find the average rate of change over an interval, you should follow the formula and avoid those common mistakes like wrong formula or values. The idea can be used in various fields, including physics, economics biology engineering finance, and computer science.

Practice Questions

Here are ten practice questions to help you find the average rate of change over an interval:

Linear Function: Consider the function $f (x) = 3 x - 2$ . Find the average rate of change over the interval $[1, 4]$ .
Quadratic Function: Given the function $g (x) = x^{2} + 2 x - 1$ determine the average rate of change over the interval $[- 2, 3]$ .
Trigonometric Function: For the function $ℎ (x) = \sin (x)$ , calculate the average rate of change over the interval $[0, π]$ .
Exponential Growth: If $p (t) = 100 \cdot 1.0 5^{t}$ represents exponential growth, find the average rate of change over the interval $[2, 5]$ .
Logarithmic Function: Given $q (x) = \log_{2} (x)$ , calculate the average rate of change over the interval $[1, 8]$ .
Square Root Function: Determine the average rate of change of the function $r (x) = \sqrt{4 x - 3}$ over the interval.
Rational Function: For the function $s (x) = \frac{2 x}{x - 1}$ , find the average rate of change over the interval $[2, 5]$ .
Absolute Value Function: Consider the function $a (x) = ∣ x - 3 ∣$ . Calculate the average rate of change over the interval $[1, 5]$ .
Trigonometric Expression: If .