4.1 Derivatives as Rates of Change

In this lesson, you will learn about the beauty and power of calculus, velocity and acceleration of an object moving, growth rate, average and marginal cost, and application problem-solving.

Lesson Objectives

Upon completion of the lesson 4.1, you will be able to:

• • Provide examples of the beauty and power of calculus as the mathematics of change.

  • Find average and instantaneous velocity and acceleration of an object moving horizontally or vertically and use the velocity to determine direction of motion.

  • Find average and instantaneous growth rate.

  • Find average and marginal cost.

  • Solve application problems involving average and instantaneous rates of change.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. YouTube video: Average Velocity

• 1. YouTube video: Instantaneous Velocity

  2. Note: Since instantaneous velocity is the limit of the average velocity (calculated from the position function/data) as Δt approaches 0, therefore, instantaneous velocity is the derivative of the position function.

• 1. YouTube video: Position, Velocity, Acceleration

• 1. YouTube video: Instantaneous Velocity, Definition of Derivative

  2. Note: This video finds the derivative by using the basic limit definition. However, you may use your derivative rules to find the derivative

• 1. YouTube video: Position, Velocity, Acceleration, and Speed - Essential Calculus Video 10

• 1. YouTube video: Average Rate of Change

• 1. YouTube video: Finding Instantaneous Rates of Change Using Def'n of Derivative

  2. Note: This video finds the derivative by using the basic limit definition. However, you should use your derivative rules to find the derivative.

• 1. Brightstorm videos: Economics: Marginal Cost and Revenue

  2. The Derivative as a Rate of Change (and Related Rates) [University of Houston] . View only pages 1 - 5.

  3. The derivatives as a rate of change (PDF) [UMass Lowell] .

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

• 1. Rates of Change [Wake Forest University] (average rate of change, instantaneous rate of change, practice problems with solutions provided)

  2. Rates of Change in Natural and Social Sciences [Furman University]

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. Velocity Problem [Paul's Online Math Notes] Do only problem #17.

  2. The Derivative as a Rate of Chang [University of Houston] (Problems with solutions provided)

Additional Resources

Below are additional resources that help reinforce the content for this module.

1. Chapter two - Rate of Change: The Derivative

   1. Read section 2.1 - pp. 99 - 100; section 2.3 - pp. 11 - 115, until the end of example 7;

   2. section 2.4 - pp. 120 - 122 (Interpretation of the Second Derivative as a rate of change);

   3. section 2.5 - pp. 125 - 128, until the end of example 3.

   4. Rectilinear Motion [Wiley]

   5. View only pages 1 - 8.