Rates of Change

Average Rate of Change

The average rate of change of a function describes how the output of the function changes, on average, as the input changes over a specified interval. It is a fundamental concept in calculus, often used to model real-world phenomena such as velocity, profit, or cost over time.

• Definition: For a function f(x), the average rate of change from x = a to x = b is given by:

• This formula is identical to the slope formula for a straight line passing through the points (a, f(a)) and (b, f(b)).

• It can be used to find average velocity, average profit, average cost, and similar quantities.

Example 1: Find the average rate of change of f(x) = x^2 between x = 2 and x = 6.

• Average rate of change:

Example 2: The AI market grew from $134 billion in 2023 to $184 billion in 2024. Find the average rate of change (in billions of dollars per year) between 2023 and 2024.

• Average rate of change: billion dollars per year

• Interpretation: The AI market increased by $50$ billion per year on average during this period.

Example 3: A company's spending is modeled by (in billions of dollars), where represents 2018. Find the average rate of change between 2018 () and 2023 ().

• Average rate of change: billion dollars per year

• Interpretation: The company spent approximately billion more per year, on average, from 2018 to 2023.

Instantaneous Rate of Change

The instantaneous rate of change of a function at a specific point measures how the function is changing at that exact input value. In calculus, this is defined as the derivative of the function at that point.

• Definition: The instantaneous rate of change of f(x) at x = a is given by the limit:

• This is called the difference quotient and represents the slope of the tangent line to the graph of f(x) at x = a.

• It is used to find instantaneous velocity, marginal profit, and other rates at a specific moment.

Example 1: The position of an object moving in a straight line is given by , where is in seconds. Find the instantaneous velocity at .

• First, compute the derivative:

• At :

• Interpretation: At seconds, the object is moving at $28$ feet per second.

Marginal Analysis: Marginal Cost and Marginal Profit

Marginal cost and marginal profit refer to the instantaneous rate of change of cost or profit with respect to the number of units produced or sold. In business applications, the marginal value is interpreted as the cost or profit of producing or selling one additional unit.

• Definition: If is the profit from selling units, then the marginal profit at is .

Example: The total profit (in thousands of dollars) from selling TVs is , for . Find the marginal profit when $15$ TVs are sold.

• Compute the derivative:

• At , marginal profit is $5$ (thousand dollars per TV).

• Interpretation: The profit earned for selling the 16th TV is about (or $5$ thousand dollars).

Summary Table: Average vs. Instantaneous Rate of Change