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Rates of Change & Related Rates

Emily Davis

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

  1. Introduction to Derivatives
  2. Meaning of a Derivative
  3. Worked Examples
  4. Practice Questions
  5. Glossary
  6. Summary and Key Takeaways

Introduction to Derivatives

Key Concept

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.
Consider a simple example: y=3xy = 3x
  • The derivative, or the rate of change, is given by: dydx=3\frac{dy}{dx} = 3

This means for every 1 unit increase in xx, yy increases by 3 units. In this case, the rate of change is constant (3) at every point on the graph of yy against xx.

For a more complex example, consider: v=13t3v = \frac{1}{3}t^3
  • The derivative, or the rate of change, is: dvdt=t2\frac{dv}{dt} = t^2

This means vv changes at a rate of t2t^2. Here, the rate of change depends on tt.

  • At t=2t = 2, the rate of change is: 22=42^2 = 4
  • At t=5t = 5, the rate of change is: 52=255^2 = 25

The rate of change at a particular point is the ...

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