Rates of Change & Related Rates

Emily Davis
7 min read
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Study Guide Overview
This study guide covers the meaning of a derivative in context, focusing on its interpretation as a rate of change. It explains how to determine the units of the derivative, interpret it in exam questions, and distinguish between a function representing an amount versus a rate. The guide includes worked examples involving real-world scenarios like depth of water and volume change, along with practice questions and a glossary of key terms like derivative, instantaneous rate of change, and slope of the tangent.
Study Notes on Derivatives
Table of Contents
- Introduction to Derivatives
- Meaning of a Derivative
- Worked Examples
- Practice Questions
- Glossary
- Summary and Key Takeaways
Introduction to Derivatives
Derivatives are a fundamental concept in calculus that measure how a function changes as its input changes. They are essential for understanding rates of change and have numerous applications in various fields such as physics, engineering, and economics.
Meaning of a Derivative
What does the derivative mean?
- The derivative of a function represents the rate of change of that function.
- The rate of change describes how the dependent variable changes as the independent variable changes.
- The derivative, or the rate of change, is given by:
This means for every 1 unit increase in , increases by 3 units. In this case, the rate of change is constant (3) at every point on the graph of against .
- The derivative, or the rate of change, is:
This means changes at a rate of . Here, the rate of change depends on .
- At , the rate of change is:
- At , the rate of change is:
The rate of change at a particular point is the instantaneous rate of change.
The derivative (rate of change) at a point is equal to the slope of the tangent at that point.
Units for the rate of change
- The units for the rate of change are derived from the units of the dependent and independent variables.
Interpreting rate of change in exam questions
- Read the scenario description carefully.
- Determine if the function describes an amount or a rate of change.
If unsure whether a function is a rate or an amount, checking the units can provide helpful clues.
Worked Examples
Example 1: Depth of Water in a Harbor
The depth of the water in a harbor, measured in feet, is modeled by the function: where represents the number of hours after midnight.
(a) Maximum depth of water
Question: State the maximum depth of the water in the harbor according to the model.
Answer: The maximum value of is 1. Thus, the maximum depth is:
(b) Rate of change at 6 am
Question: Find the rate at which the depth of the water in the harbor is changing at 6 am. State appropriate units for your answer.
Answer: The rate of change is given by :
At 6 am ():
(c) Interpretation of given derivatives
Question: Explain the meaning of and in the context of the model.
Answer:
- means that at noon, the depth of water is decreasing at a rate of 2.261 feet per hour.
- means that at noon, the rate at which the depth is decreasing is itself decreasing by 0.322 feet per hour per hour.
Example 2: Volume of Water in a Container
The rate of change of the volume of water in a container is modeled by the function , where is measured in minutes.
(a) Interpretation of
Question: Explain the meaning of in the context of the model.
Answer: models the rate of change of volume. Thus, at minutes (6 seconds), the volume of water is increasing at a rate of 2 gallons per minute.
(b) Interpretation of being positive and negative
Question: Explain what it means if is positive and is negative.
Answer: The volume of water is increasing, but the rate of increase is slowing down.
(c) Units of
Question: State the units for the quantity found by calculating .
Answer: Integrating the rate of change of volume gives the total change in volume. Thus, the units of will be gallons.
Practice Questions
Practice Question
Question 1: Find the derivative of .
Practice Question
Question 2: If , what is the rate of change of at ?
Practice Question
Question 3: The rate of change of the temperature in a room is given by $$ h(t) = 3 \sin(t) ). What is the maximum rate of change?
Glossary
- Derivative: The rate of change of a function with respect to its independent variable.
- Rate of Change: How a quantity changes with respect to another quantity.
- Instantaneous Rate of Change: The rate of change at a specific point.
- Slope of the Tangent: The slope of the line that just touches a curve at a given point, representing the derivative at that point.
Summary and Key Takeaways
- The derivative of a function measures its rate of change.
- The units of the rate of change depend on the units of the dependent and independent variables.
- Interpreting a rate of change requires understanding whether the function describes an amount or a rate.
- The slope of the tangent to a curve at a point gives the instantaneous rate of change at that point.
- Practice interpreting and calculating derivatives to gain proficiency.
Key Takeaways
- Understand that the derivative represents how a function changes as its input changes.
- Be able to calculate derivatives and interpret their meanings in real-world contexts.
- Recognize the units associated with rates of change and how they relate to the problem context.
- Use derivative concepts to solve practical problems, such as finding rates of change and interpreting graphs.
By mastering these concepts, you'll be well-prepared to understand and apply derivatives in various contexts, both in exams and real-world scenarios.

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Question 1 of 9
What does the derivative of a function, , represent? ๐ค
The area under the curve of
The rate of change of with respect to
The inverse of
The y-intercept of