Unit 8: Applied Mathematics
Section A: Mathematical Modeling Basics
Welcome
Welcome to Section A: Mathematical Modeling Basics! In this section, you’ll explore the principles of mathematical modeling, a powerful tool for representing real-world phenomena using mathematical equations and concepts. Understanding modeling will allow you to analyze and solve complex problems in science, engineering, economics, and other fields.
Imagine
Imagine you’re an environmental scientist modeling the impact of pollution on a river ecosystem. Mathematical modeling allows you to simulate different scenarios, predict outcomes, and develop strategies to protect and restore the environment.
Context
You’ve previously studied algebra, calculus, and other mathematical concepts. Now, we’ll apply these concepts to create mathematical models, helping you represent and analyze real-world systems and phenomena.
Overview
This section covers the basics of mathematical modeling, linear models and their applications, non-linear modeling in biology and economics, building and solving basic differential equation models, and using modeling software in science and engineering projects. You’ll learn to apply these concepts to create and analyze models that represent complex systems and scenarios.
Objectives
- Understand the principles of mathematical modeling and its significance in representing real-world phenomena.
- Learn and apply linear modeling techniques to represent and analyze systems with linear relationships.
- Explore non-linear modeling, representing more complex relationships in biology, economics, and other fields.
- Build and solve basic differential equation models, representing dynamic systems and changes over time.
- Use modeling software to create and analyze models in science and engineering, applying mathematical concepts to practical scenarios.
Preparatory Guidance
Definitions and Pronunciations
- Mathematical Model: A representation of a real-world system or phenomenon using mathematical equations and concepts.
- Linear Model: A mathematical model representing a system with a linear relationship between variables, often expressed as
.
Verbal Reading of Equations
- For linear models, say “a linear model represents a system with a straight-line relationship between variables, expressed as y equals m x plus b.”
- For differential equations in modeling, describe the process as “using calculus to represent and analyze dynamic systems that change over time.”
Problem-Solving Strategies
- Understand the principles of mathematical modeling and its significance in representing real-world phenomena.
- Apply linear modeling techniques to represent and analyze systems with linear relationships, solving problems in various fields.
- Explore non-linear modeling, representing more complex relationships in biology, economics, and other fields.
- Build and solve basic differential equation models, representing dynamic systems and changes over time.
- Use modeling software to create and analyze models in science and engineering, applying mathematical concepts to practical scenarios.
Considerations
How can understanding mathematical modeling help you analyze and solve complex problems in various fields? Why are models important for representing real-world systems and phenomena? In what ways can you apply mathematical modeling to improve decision-making and problem-solving in your chosen field?