Derivatives as Rates of Change

Introduction

The derivative is a fundamental concept in calculus that measures the rate of change of a function. It is widely used to analyze how quantities change over time or with respect to other variables. In this section, we explore the interpretation of derivatives as rates of change in physical and business contexts, including velocity, acceleration, and marginal cost.

Velocity and Instantaneous Rate of Change

Average and Instantaneous Velocity

Velocity describes how the position of an object changes over time. The average velocity over a time interval is the slope of the secant line connecting two points on the position curve, while the instantaneous velocity is the slope of the tangent line at a specific point, given by the derivative of the position function.

• Average Velocity:

• Instantaneous Velocity:

• Interpretation: The average velocity is the change in position divided by the change in time, while the instantaneous velocity is the derivative of the position function at a given time.

Velocity, Speed, and Acceleration

For an object moving along a line with position :

• Velocity:

• Speed:

• Acceleration:

Applications: One-Dimensional Motion

Airline Travel Example

The position function models the distance traveled by an airliner over time. The graph below shows the position as a function of time for a round trip from Seattle to Minneapolis.

• Average Velocity (First 1.5 hours): The slope of the secant line from to gives the average velocity during this interval.

• Average Velocity (Return Trip): The slope of the secant line from to gives the average velocity during the return trip.

• Velocity is Zero: The velocity is zero when the airliner is at rest, corresponding to a horizontal tangent on the position graph (e.g., at the maximum height).

• Negative Velocity: When the airliner returns to Seattle, the slope of the tangent line is negative, indicating motion in the opposite direction.

Business Applications: Marginal Cost

Average and Marginal Cost

In manufacturing, the cost function gives the total cost to produce items. The average cost is the total cost divided by the number of items, and the marginal cost is the derivative of the cost function, representing the approximate cost to produce one additional item.

• Average Cost:

• Marginal Cost:

• Interpretation: Marginal cost estimates the cost of producing one more item after items have already been produced.

Graphical Interpretation of Marginal and Average Cost

The slope of the tangent line to the cost curve at gives the marginal cost, while the slope of the secant line between and gives the average cost of producing one additional item.

Summary Table: Velocity, Speed, Acceleration, and Cost Functions

Examples and Applications

Example: Airline Travel

Given the position function for an airliner, calculate:

• Average velocity during the first 1.5 hours

• Average velocity during the return trip (7.5 to 8.5 hours)

• Times when velocity is zero (airliner at rest)

• Velocity at noon (), explaining why it is negative

Example: Marginal Cost in Manufacturing

Suppose the cost function is given. The average cost decreases as more items are produced, and the marginal cost decreases linearly if the cost function is linear. Marginal cost is used to estimate the cost of producing one additional item.

Conclusion

Derivatives provide powerful tools for analyzing rates of change in both physical and business contexts. Understanding the distinction between average and instantaneous rates, as well as their graphical interpretations, is essential for solving real-world problems in calculus.