4.3 Related Rates

In this lesson, you will learn to solve related rates problems.

Lesson Objectives

Upon completion of the lesson 4.3, you will be able to:

• • Identify the rates that are given in the problem and identify the rate that is to be determined.

• Differentiate between a constant in the problem and a quantity that change over time.

• Write one or more equations that express the basic relationships among the variables in the problem.

• Solve related rated problems.

Lesson Content

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

• 1. Application of the derivative: Lesson 27- Related rates [Internet Archive]

  2. The module includes defining the problem, example problems, and homework with answers.

  3. Note 1: For one part of the baseball problem, if you submit the wrong answer, the program may freeze. If that happens to you, close the program and click on the link again to restart.

  4. Note 2: The third problem (ship problem) uses inverse cotangent function in its solution. We have not learned the derivative of the inverse cotangent function. Therefore, you could use the tangent function. In that given problem:

• 1. and then simplify to get . When then the ship is 1 km from the lighthouse, solve for the hypotenuse of the right triangle with legs 1 km and 0.1 km. hypotenuse

• 1. . Then, you will be able to find cos θ to substitute that value into the related rates equation. Also, substitute the value of x=1 km, dx/dt=- 20 km/hr. You will get the same answer that the lesson gets (1.980 radians/hr).

  2. YouTube videos: Related Rates Related Rates

  3. #1 - Problem Using Implicit Differentiation:

• 1. Related Rates #6 - Rate at Which the Circumference of a Circle is Changing:

• 1. Related Rates #8 - Cars Traveling from an Intersection - Rate of Change in Perimeter:

• 1. Rates of Change & Related Rates [University of Houston]

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. Related Rates [Paul's Online Math Notes]

  2. Related Rates [Pheng Kim Ving]

  3. Related Rates [UMass Lowell]

1. Related Rates

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. Related Rates [Paul's Online Math Notes] Don't do problem #1.

  2. 5.4 Related Rates [Stefan Waner]

  3. Rates of Change per Unit Time [University of Houston]

Additional Resources

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

• 1. Related Rates [Math with Mr. Leckie]

  2. Read only the guidelines for solving related-rate problems.

  3. YouTube videos: Related Rates #5

  4. Involving Baseball:

• 1. Using Cones:

• 1. YouTube video: Related Rates

  2. #7 Ladder Sliding Down Wall, Finding Rate of Change of Area Under Ladder:

• 1. Lecture 12: Related Rates [MIT OpenCourseWare]

  2. Read to the end of example 2 on p. 3. Also watch the following videos for the lecture.

  3. YouTube video: Calculus I - Lecture 12

  4. Note: Start watching the first video from 45:30 since the topic of related rates starts at that point.

• 1. YouTube video: Calculus I - Lecture 13

  2. Note: Stop watching the second video at 18:44 when the professor finishes his second related rated problem and questions that students have about it.

• 1. YouTube video: Related Rates