On the left you'll find a copy of skeletal notes I used in MA 231 for both Summer 2014 and Summer 2015. The lesson topic is the Lagrange Multiplier method, used for constrained optimization of multi-variate functions. This lesson is taught during the first week of the five week course. On the right, I describe what it is I am trying to achieve, and I discuss what I think did and did not work with these notes.
MA 231
Section 7.4: Lagrange Multipliers and Constrained Optimization
Lesson Objectives:
To understand the concept of Lagrange Multiplier method
To learn the steps of the Lagrange Multiplier method
To apply the steps of the LM method to a problem
Ex 1) The manager of a department store wants to build a 600 sq. ft rectangular enclosure on the store's parking lot to display some equipment. Three sides of the enclosure will be built of redwood fencing at a cost of $ 14 per running foot. The fourth side will be built of cement blocks, at a cost of $ 28 per running foot. Find the dimensions that will minimize the total cost of the building materials.
Ex 1 is an optimization problem, and we solved it using the method of finding min/max. Now we'll learn about Lagrange Multiplier method. We do this by presenting a problem:
Let f(x, y) and g(x, y) be functions of 2 variables. Find values of x and y that maximize (or minimize) the objective function f(x, y) and also satisfy the constraint equation g(x, y) = 0.
To solve this problem, we'll let F(x, y, L) be an auxiliary function, defined as
F(x, y, L) =
where L is and it always the constraint function.
Theorem:
Ex 2)
In summary, LM steps
Identify the objective function and set it equal to f(x, y).
Set
Find partial derivative of f with respect to x, partial derivative of f with respect to y, partial derivative of f with respect to L. Set all equal to 0.
Solve
Set
Then you have all the means to find the remaining values.
Ex 3) The US postal rules require that the length plus the girth of a package cannot exceed 84 inches. Find the dimensions of the rectangular package of greatest volume that can be mailed. Note: 84 = length + girth.
Ex 4)
In the workshop FIT: Learning Styles, I learned that to maximize the amount of students who are learning in your classroom you must be willing to teach from their learning style. I created skeletal notes for my lectures in hopes of doing just that.
The Lesson Objectives are meant to guide those students who like to know where we are headed in the lesson. (global)
I provide some notes already written for the students who like to listen to the instructor speak. (auditory)
In addition, I leave some notes blank for those who need notes to learn. (visual)
For this particular lesson, we have specific steps that students can use to guide them through real examples. (sequential)
The last two examples are for the students to work on their own and in their groups, respectively. (reflective and active)
The first example is a real-life example which helps students see how the Lagrange Multiplier method is applicable outside the classroom. (sensing)
While it isn't stated in these skeletal notes, this lesson is similar to, and thus connects to, a prior lesson on finding extrema for multi-variate functions. (intuitive)
What did work:
The students who used the skeletal notes participated more in class.
These students asked questions that required more thought (rather than the typical clarification questions students ask because they're lost from taking notes).
What did not work:
Not all students used the skeletal notes, although they are available on the course website.
These students would ask clarification questions, which slowed the lecture for the students using the skeletal notes.
I think in the future I may require students to use these notes because I've only seen positive things and heard positive feedback from students who are using the skeletal notes.