This article explores the concept of the **average rate of change over an interval**, aiming to **illuminate** this **mathematical** tool in a manner accessible to everyone.

## Defining Average Rate of Change Over an** Interval**

**Interval**

The **average rate of change** over an **interval** refers to the change in the value of a **function** between two **points** divided by the difference in the **independent variables** of these two points. In simpler terms, it measures how much the **output** (or **dependent variable**) changes per unit change in the **input** (or **independent variable**) over a specific **interval**.

Mathematically, it can be expressed as:

Average Rate of Change = [f(b) – f(a)] / (b – a)

where **f(b)** and **f(a)** are the function values at points **b** and **a**, respectively, and **b** and **a** are the endpoints of the **interval** on which the **rate of change** is being determined. This is essentially the slope of the **secant line** passing through the points (**a, f(a)**) and (**b, f(b)**) on the graph of the function.

Figure-1.

The **average rate of change** is fundamental in **calculus** and **underpins** more **complex** ideas, such as the **instantaneous rate of change** and the **derivative**.

**Properties**

Much like many **mathematical** concepts, the** average rate of change** has certain properties integral to its understanding and application. These properties are fundamental aspects of the **average rate of change behavior**. Here are some of them in detail:

**Linearity**

One of the key properties of the **average rate of change** is its **linearity**, which stems from the fact that it represents the slope of the **secant line** between two points on a function graph. This essentially means that if the function being considered is **linear** (i.e., it represents a straight line), the **average rate of change** over any interval is constant and equals the **slope** of the **line**.

**Dependence on Interval**

The **average rate of change** is dependent on the specific **interval** chosen. In other words, the average rate of change between two different pairs of points (i.e., different intervals) on the same function can be different. This is particularly evident in **non-linear functions**, where the average rate of change is not constant.

**Symmetry**

The **average rate of change** is **symmetric** in that reversing the **interval** will only change the sign of the rate. If the average rate of change from **‘a’** to **‘b’** is calculated to be** ‘r,’** then the average rate of change from** ‘b’** to** ‘a’** will be **‘-r.’**

**Interval Average vs. Instantaneous Change**

The **average rate of change** over an **interval** gives an overall view of the behavior of a **function** within that interval. It does not reflect **instantaneous changes** within the interval, which may differ greatly. This fundamental concept leads to the idea of a **derivative** in calculus, which represents the **instantaneous rate of change** at a point.

**Connection to Area Under Curve**

In the context of **integral calculus**, the **average rate of change** of a function over an interval is equal to the **average value** of its **derivative** over that interval. This is a consequence of the **fundamental theorem of calculus**.

**Exercise **

### Example 1

**Linear Function Example**

Given f**(x) = 3x + 2**. Find the **average rate of change** from **x = 1** to** x = 4**.

### Solution

Average Rate of Change = [f(4) – f(1)] / (4 – 1)

Average Rate of Change = [(3*4 + 2) – (3*1 + 2)] / (4 – 1)

Average Rate of Change = (14 – 5) / 3

Average Rate of Change = 3

This means that for every unit increase in **x**, the function increases by **3** units on average between **x = 1** and **x = 4**.

### Example 2

**Quadratic Function Example**

Suppose **f(x) = x²**. Find the **average rate of change** from** x = 2** to **x = 5**.

Figure-2.

### Solution

Average Rate of Change = [f(5) – f(2)] / (5 – 2)

Average Rate of Change = [(5²) – (2²)] / (5 – 2)

Average Rate of Change = (25 – 4) / 3

Average Rate of Change = 7

### Example 3

**Exponential Function Example**

Suppose **f(x) = 2ˣ**. Find the** average rate of change** from **x = 1** to **x = 3**.

Average Rate of Change = [f(3) – f(1)] / (3 – 1)

Average Rate of Change = [(2³) – (2^1)] / (3 – 1)

Average Rate of Change = (8 – 2) / 2

Average Rate of Change = 3

### Example 4

**Cubic Function Example**

Suppose **f(x) = x³**. Find the average rate of change from **x = 1** to **x = 2**.

Figure-3.

### Solution

Average Rate of Change = [f(2) – f(1)] / (2 – 1)

Average Rate of Change = [(2³) – (1³)] / (2 – 1)

Average Rate of Change = (8 – 1) / 1

Average Rate of Change = 7

### Example 5

**Square Root Function Example**

Suppose **f(x) = √x**. Find the **average rate of change** from** x = 4** to** x = 9**.

### Solution

Average Rate of Change = [f(9) – f(4)] / (9 – 4)

Average Rate of Change = [(√9) – (√4)] / (9 – 4)

Average Rate of Change = (3 – 2) / 5

Average Rate of Change = 0.2

### Example 6

**Inverse Function Example**

Suppose **f(x) = 1/x**. Find the average rate of change from **x = 1** to **x = 2**.

Figure-4.

### Solution

Average Rate of Change = [f(2) – f(1)] / (2 – 1)

Average Rate of Change = [(1/2) – (1/1)] / (2 – 1)

Average Rate of Change = (-0.5) / 1

Average Rate of Change = -0.5

### Example 7

**Absolute Value Function Example**

Suppose **f(x) = |x|**. Find the **average rate of change** from **x = -2** to** x = 2**.

### Solution

Average Rate of Change = [f(2) – f(-2)] / (2 – -2)

Average Rate of Change = [(2) – (2)] / (2 – -2)

Average Rate of Change = 0 / 4

Average Rate of Change = 0

### Example 8

**Trigonometric Function Example**

Suppose **f(x) = sin(x)**. Find the average rate of change from **x = π/6** to **x = π/3**. (Note that we use radians for x in trigonometric functions.)

### Solution

Average Rate of Change = [f(π/3) – f(π/6)] / (π/3 – π/6)

Average Rate of Change = [sin(π/3) – sin(π/6)] / (π/6)

Average Rate of Change = [(√3/2) – (1/2)] / (π/6)

Average Rate of Change = (√3 – 1) / (π/2)

Average Rate of Change ≈ 0.577

**Applications **

The **average rate of change over an interval** is widely applicable in various fields. Here are a few examples:

**Physics**

In **physics**, the **average rate of change** is commonly used in **kinematics**, the study of motion. For example, the **average velocity** of an object over a given time interval is the average rate of change of its position with respect to time during that interval. Similarly, the **average acceleration** is the average rate of change of velocity.

**Economics**

In **economics** and **finance**, the **average rate of change** can be used to understand changes in various metrics over time. For instance, it can be used to analyze the average growth rate of a company’s revenue or profits over several years. It could also be used to evaluate changes in** stock prices**, **GDP**, **unemployment rates**, etc.

**Biology**

In **population biology** and **ecology**, the **average rate of change** can be used to measure the growth rate of a population. This could be the rate of change of the number of individuals in a **population** or the change in the concentration of a substance in an** ecosystem**.

**Chemistry**

In **chemistry**, the rate of **reaction** is essentially an average **rate of change**—it represents the change in concentration of a **reactant** or **product** per unit of time.

**Environment Science**

In **environmental studies**, the **average rate of change** can be used to measure **pollution levels**, **temperature changes** (global warming), **deforestation rates**, and many more.

**Medical Science**

In **medical science**, it can measure the **rate of change** in a patient’s condition over time. This could be the change in **heart rate**, **blood sugar levels**, or tumor growth rate.

**Geography**

In **geography**, it’s used to assess changes in various parameters over time, such as the **erosion rate** of a **riverbank**, **glacier melting rates**, or **even urban sprawl rates**.

**Computer Science**

In **computer science**, the **average rate of change** can be used in algorithms to predict **future trends** based on **past data**.

These are just a few examples. The **average rate of change** is an essential mathematical tool that finds **wide-ranging** applications across virtually all fields of **science**, **technology**, and beyond.

*All images were created with GeoGebra and MATLAB.*