Guided Practice for 3.4: Applied Optimization

Overview

In this section we come to one of the most significant applications of the derivative. Given a function that models something in real life – the temperature of a control rod, the size of a population, the value of a stock portfolio, etc. – we want to find the “best” values of that function: the lowest temperature, the highest population size, the most lucrative portfolio value. We do this by finding the absolute extreme values of a function using the tools we developed in Sections 3.1 and 3.3. So there is not much new math in this section but a lot of applying what we know to new problems that matter to important groups of people.

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.
• In the context of an optimization problem, identify the quantity to be optimized.
• Set up an equation to relate the quantity to be optimized to the independent variables.
• Use a constraint in the problem to reduce the equation from above down to one variable.
• Use calculus to solve a properly set up optimization problem.

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 an optimization problem.
• Provide full and careful justification using calculus techniques that you have found the optimum value.

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 3.4. 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: Applied optimization (2:16)
• Fencing optimization (7:09)
• Optimization with trigonometry (13:25)

Here are some more OPTIONAL screencasts to watch—just some randomly chosen, yet potentially valuable extra examples of applied optimization problems:

• Optimizing the area of a rectangle inscribed in a parabola (7:06)
• Optimizing the surface area of a cylinder (8:08) <– Audio is terrible but it does feature a Led Zeppelin clip, so it has that going for it
• Maximizing profit (7:07)

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.