6.1 Optimization Problems

In this lesson, you will learn to use calculus to solve optimization problems.

Lesson Objectives

Upon completion of the lesson 6, you will be able to:

• • Identify optimization problems from other types of problems.

  • Identify variables and constants for an optimization problem.

  • Identify what quantity is to be maximized or minimized and write it in terms of the variables in the problem.

  • Write the constraints of the optimization problem in terms of the variables in the problem.

  • Solve an optimization problem using the guidelines for solving optimization problems:

    • Read the problem carefully, identify variables for the problem, and sketch a picture of the situation, where appropriate.

    • Identify the objective function (the function to be optimized). Write it in terms of the variables of the problem.

    • Identify the constraints. Write them in terms of the variables of the problem.

    • Use the constraints to eliminate all but one independent variable of the objective function.

    • Find the domain for the single independent variable left in the objective function that is pertinent to the problem.

    • Use the methods of calculus to find the absolute maximum or minimum value of the objective function on the domain. Check endpoints if necessary.

Lesson Content

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

• 1. YouTube video: Calculus I - Lecture 20 - Optimization

  2. Note: This video is a little over one hour long. The parts of the video are given with the respective times so that you may view this video in parts rather than all at one time. - Applied Maximum & Minimum Problems [ about 4 min.: 0 - 3:40]

  3. - Optimization over a (finite) Closed Interval: Maximizing Area or Volume, Minimizing Cost [about 21.5 min.: 3:40 - 25:08]

  4. - Optimization over Other Intervals: Minimizing Materials or Distance [about 11 min.: 25:08 - 36:11]

  5. - An Economics Application: Cost, Revenue, Profit, & Marginal Analysis [about 9 min.: 36:11 - 45:33]

  6. - Some Exercises [about 17 min.: 45:33 - 1:02:23]

• 1. YouTube video: Calculus Optimization Problems

1. Applied Optimization Problems: Five Examples [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. Optimization [Paul's Online Math Notes]

1. More Optimization Problems [Paul's Online Math Notes]

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. Optimization Problems [Paul's Online Math Notes] (solutions provided)

  2. Some Max-Min Problems [University of Houston] (answers and solutions provided)

  3. Maximum/Minimum Problems [UC Davis] Note: Skip problem #14.

Additional Resources

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

• 1. YouTube video: Optimization with Calculus 3

• 1. Optimization [Phen Kim Ving]

  2. YouTube videos: Optimization Problems

  3. Problem 1:

• 1. Problem 2:

• 1. Problem 3: