Project: Maximizing profit, minimizing cost: A hands-on approach to optimization

Context

Math is a science that is present in all fields of life. Be it at home, at school or at work, it is indispensable for our daily lives. Within the mathematical universe, a topic of great importance and applicability is the study of maximum and minimum points of functions that have applications in several branches such as economics, physics, engineering, among others.

The maximum and minimum points of a function play a crucial role in the optimization of processes, in economics, for example, these points are used to maximize profits or minimize costs, in engineering they are essential to find the points of greatest or least tension in a structure. Learning about these concepts is fundamental to understanding how things work around us.

In the modern world, technology plays an increasingly relevant role in our lives. Smartphone apps, games, music and video streaming platforms, among others, all run on complex algorithms that often use mathematical principles to optimize their operations. In this sense, the study of maximums and minimums is a powerful tool for modeling and solving complex problems.

Introduction

The analysis of functions and the study of their behavior are essential parts of mathematics. Among the aspects that we can analyze in a function, the maximum and minimum points occupy a prominent place. Almost all of us have already come across such concepts, even if we have not noticed it. They are present in everyday situations, such as finding the largest or smallest quantity that can be obtained from a certain good or resource, or the longest or shortest distance that something can travel, for example.

Maximum and minimum points are points where the function reaches its highest or lowest values, respectively, in a given interval. In the graph of a function, these points are easily visible as the peaks and valleys of the curve. The importance of this concept in mathematics is vast and its applications are countless.

The study of maximum and minimum points involves an understanding of other key concepts in mathematics, such as quadratic functions and their graph representation, the derivative of a function, and the calculation of local extrema. We will study these concepts in detail and learn to apply them in varied contexts to solve complex problems.

For a more in-depth study of the concepts, the following sources are recommended:

1. Book: "Functions", by authors Iezzi, Gelson and Hazzan, Samuel. Editora Atual.
2. Website: Khan Academy, section "Quadratic and Polynomial Functions". Available at: https://pt.khanacademy.org/math/algebra/quadratics
3. Video: Canal Matemática Rio, video "Quadratic Function - Maximums and Minimums". Available at: https://www.youtube.com/watch?v=Y7lJbiJw9Cw

Practical Activity

Activity title: "Maximizing profit, minimizing cost: A hands-on approach to optimization"

Project objective

This project's main objective is to provide students with a practical experience related to the topic of "Maximums and Minimums" of quadratic functions. They will work in the context of a fictional company and, through the analysis of the business, will have to apply the concepts learned to optimize the company's profits.

The activity will be carried out by groups of 3 to 5 students and should take about fifteen hours divided into two weeks.

Detailed project description

The project is based on the simulation of a small manufacturing company that manufactures three different types of products. Each product has a different production cost and generates different profits. The goal of the simulation is to maximize the company's profit while minimizing costs. To do this, students will have to investigate and calculate the maximum and minimum points of the representative functions of profit and cost, respectively, and then define a production strategy that optimizes profits.

In addition, students will have the challenge of understanding the theoretical concepts behind market functioning and the Financial Mathematics involved. It will be necessary to research and understand key economic concepts, such as supply and demand, fixed and variable costs, and equilibrium price.

In this context, students will also be working on their socio-emotional skills, such as time management, communication and problem solving, which are fundamental for the success and efficiency of teamwork.

Necessary materials

1. Calculation and graphics software such as GeoGebra, Excel or Google Sheets.
2. Internet access for research.
3. Presentation material (PowerPoint, for example).

Detailed step-by-step for carrying out the activity

1. Separate the students into groups of 3-5 participants.
2. Each group must choose the three products that the fictional company will produce.
3. Research and understand concepts of Financial Mathematics and economics that will be necessary to simulate the company's scenarios.
4. Establish the functions that represent the cost of production and the profit for each product. (These functions must be quadratic functions)
5. With the help of the software, calculate the minimum point of the cost functions and the maximum point of the profit functions.
6. Based on the results, establish the production and sales strategy for the products that maximize the company's profit.
7. Prepare a PowerPoint presentation explaining the process taken, showing the graphs obtained and justifying the decisions made.

Project deliverables

1. Written report, containing the following topics: Introduction to the activity, theoretical explanation of the concepts used (based on the theory learned in class and on additional research), detailed description of the activity carried out (with details of the methodology used and presentation of the results obtained) and conclusions drawn from the activity .
2. PowerPoint presentation showing the problem-solving process, the graphs obtained and the final strategy to maximize the company's profit.
3. Appendices containing the calculations performed and the graphs generated.

The written part of the work is essential so that students can organize their ideas, present the concepts learned and their applications, and present and interpret the results obtained. The report should follow the standard structure (Introduction, Development, Conclusions and Bibliography), and should be written with care and attention, as it is an important part of the project evaluation.

The development of the report and the presentation will require students to be able to synthesize and communicate complex ideas in a clear and concise way. In addition, they will need to demonstrate research skills by seeking additional information to support their findings.