UTPA STEM/CBI Courses/Business Math/Business Calculus/Optimization Applications

Course Title: Business Calculus

Lecture Topic: Optimization Applications (textbook: Mathematics with Applications in the Management, Natural, and Social Sciences, Tenth Edition by Margaret L. Lial; Thomas W. Hungerford; John P. Holcomb Jr.)

Instructor: Mau Nam Nguyen

Institution: University of Texas-Pan American

Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
  • Convert real world optimization problems into mathematics problems
  • Find the solutions of the optimization problems

• Sub Objectives- The objectives will require that students be able to:
  • Understand what is given in a particular problem and what is required to use a particular method to solve
  • Decide which variable is to be maximized or minimized
  • Use suitable derivative tests to solve the optimization problem

• Difficulties- Students may have difficulty:
  • Converting various real world optimization problems into solvable, mathematical optimization problems.
  • Using appropriate derivative tests to solve the problems.

• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
  • Solving optimization problems by finding the optimal solutions and optimal values of the problems. The term optimization refers to the study of problems in which one seeks to minimize or maximize a function by choosing the values of variables from an allowed set. Optimization theory has a variety of applications ranging from science, engineering, economics, and industry. Since economics' primary goal is to maximize profits and minimize costs, optimization models and methods play a particularly important role in successful economics.

Model of Knowledge

• Concept Map
  • Critical Numbers
  • Local Extrema
  • Absolute Extrema
  • Optimal Solutions
  • Optimal Values

• Content Priorities
  • Enduring Understanding
    • Solving optimizations problems in which cost functions are differentiable and constraints are closed intervals
    • Solving optimization problems in which the cost functions are differentiable and constraints are of other types
    • Analyzing the real world problems to reduce them to optimizations problems involving cost functions of one variable
  • Important to Do and Know
    • Extreme-Value Theorem: If a function f is continuous on a closed interval [a,b] then f has both an absolute maximum and an absolute minimum on the interval. Each of these values occurs either at an endpoint of the interval or at a critical number of f.
    • Critical- Point Theorem: Suppose that a function f is continuous on an interval I and that f has exactly one critical number in the interval I, say, x=c. If f has a local maximum at x=c, then this local maximum is the absolute maximum of f on the interval I. If f has a local minimum at x=c, then this local minimum is the absolute minimum of f on the interval I.
    • Construct a table containing the constraint, the derivative, and the monotonicity of the cost function to identify absolute extrema.
  • Worth Being Familiar with
    • Solving optimization problems in which the cost functions are continuous but not necessarily differentiable.

Assessment of Learning

• Formative Assessment
  • In Class (groups)
    • The instructor will introduce optimization problems. Students will work in groups in each of the following steps:
      • Step 1: Read the problem carefully. Make sure you understand what is given and what is asked for.
      • Step 2: If possible, sketch a diagram and label the various parts.
      • Step 3: Decide which variable is to be maximized or minimized. Express that variable as a function of one other variable. Be sure to determine the domain of this function.
      • Step 4: Find the critical numbers for the function in Step 3.
      • Step 5: If the domain is a closed interval, evaluate the function at the endpoints and at each critical number to see which yields the absolute maximum or minimum. If the domain is a (not necessarily closed) interval in which there is exactly one critical number, apply the critical-point theorem, if possible, to find an absolute extremum.
    • Students from each group will report their progress to the whole class.
  • Homework (individual)
    • Assign homework after the lecture
    • Assign the online practice problems on WebWork.
• Summative Assessment
  • Assign a quiz focusing on optimization applications
  • Assign problems involving optimization applications in the test for the chapter

Legacy Cycle

OBJECTIVE

By the next class period, students will be able to:

• Convert real world optimization problems in mathematical language
• Solve the optimization problems

The objectives will require that students be able to:

• Understand important tools to solve optimization problems from the previous classes
• Use five steps of solving applied problems mentioned previously

THE CHALLENGE

You have a friend who has 1200 m of fencing. He wants to enclose a rectangular field bordering a river, with no fencing needed along the river. Please help him determine dimensions of the field that give the maximum area.

GENERATE IDEAS

• Interaction between the instructor and students:
  • Give some examples of different dimensions yielding different areas
  • Determine what to maximize
  • Review the formula for computing the area of a rectangle: A=lw

• Group activity:
  • Use the given information to express the area as a function of one variable
  • Determine the constraints
  • Solve the problem

MULTIPLE PERSPECTIVES

• One group of students will present their solution.
• The instructor will ask other groups for comments and critique.
• The instructor will give comments to and present the complete solution.
• Continue with more examples.

RESEARCH & REVISE

• The instructor will summarize and present the general method of solving optimization problems.

TEST YOUR METTLE

• Students are required to work on a similar problem in class.
• Students will work on a quiz and a test related to this lecture.

GO PUBLIC

• Students in each group will propose an optimization problem and their strategy to solve the problem.
• Use different types of homework problems that address various situations to reinforce what students have learned.

Pre-Lesson Quiz

1. Write a formula to compute the area of a rectangle given its length l and width w.
2. A farmer has 1200 m of fencing. He wants to enclose a rectangular field bordering a river, with no fencing needed along the river. Compute the areas of the garden in the following cases:
   1. The width is 100 m.
   2. The width is 300 m.
   3. The width is 600 m.
3. In the setting of Problem 2, let x represent the width of the garden.
   1. Write an expression for the length of the field.
   2. Find a function A(x) that represents the area of the garden.
   3. What is the domain of the area function A(x)?

Test Your Mettle Quiz

1. What is the maximum product of two numbers whose sum is 100?
2. A rectangular field is to be enclosed with 2000 m of fencing. Find the dimensions of the field that maximize its area.
3. Find the dimensions of a rectangle with area 5000 square meters whose perimeter is minimal.
4. A farmer with 1000 ft. of fencing wants to enclose a rectangular field and then divide it into five pens with fencing parallel to one side of the field. That is the largest possible area of the field.
5. A rectangular field is to be enclosed with a fence. One side of the field is against an existing fence, so no fence is needed on that side. If the material for the fence costs $2 per foot for the two ends and $4 per foot for the side parallel to the existing fence, find the dimensions of the field of largest area that can be enclosed for $1000.