Guided Practice for 3.5: Related Rates

Overview

This section introduces a new kind of derivative problem called a related rates problem. In a related rates problem, we are given a relationship between two or more quantities and the rates of change of all but one of those quantities, and then we find the rate of change in the remaining quantity. For example, in an electrical circuit, voltage, current, and resistance are related by the formula V = IR. If we know the rate at which voltage and resistance are changing with respect to time, we might want to find the rate at which current is changing with respect to time. Therefore this is an application in particular of implicit differentiation to real-world scientific problems and problems in other areas.

Learning objectives

Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).

• Read a quantitative problem carefully and identify the independent and dependent variables in the problem as well as the constraints on the variables.
• Explain what a “related rates” problem is. (Hint: See the Overview above.)
• Identify when more than one variable in a problem depends on a common third variable, such as when multiple variables depend on time.
• In the context of a related rates problem, identify the rate of change you wish to find and use implicit differentiation/Chain Rule to take the derivative.

Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.

• Set up and solve related rates problems using calculus.

Learning resources

To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.

Textbook: In Active Calculus, read Section 3.5. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos at the MTH 201 YouTube playlist (http://bit.ly/GVSUCalculus).

• Quick review: Related Rates (2:15)
• Basic related rates example (6:38)
• Related rates example with trigonometry (8:11)

Here are some more OPTIONAL screencasts to watch for students who would like to view extra examples:

• Related rates using implicit differentiation (9:57)
• Related rates using cones (2:47)
• Related rates with triangles (14:03)

Activities

The following activity is to be done during and following your reading and viewing of the resources. Go to student.desmos.com and join using your name in the format Last, First. For example, I would enter my name as Ballif, Serge. Complete each part of the activity. Some of these problems will require you to work them out on paper before entering your answer. Practice producing high quality work so that your work is readable and meaningful. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline. Remember to use the Piazza discussion board to ask about any questions you have.