Mathematical Models: Building Functions

Definition

Mathematical modeling is the process of translating a real-world scenario into a functional relationship, where one quantity (the dependent variable) is expressed in terms of another (the independent variable) based on physical, geometric, or economic constraints.

Why It Matters

Building functions is the act of translating a physical problem into a mathematical language; without this skill, we cannot apply the power of calculus and algebra to solve the real-world challenges of engineering and economics.

Core Concepts

• Variable Identification: Distinguishing between what is known, what is unknown, and which variable drives the other.
• Constraint Integration: Using established formulas (e.g., Geometry: $V = lwh$; Physics: $d = rt$; Economics: $R = xp$) to link variables.
  • How to read: “The volume equals length times width times height; the distance equals rate times time; and the revenue equals price times quantity.”
  • Meaning / when to use: These are the building blocks for single-variable reduction — substitute known relationships to express one dependent quantity in terms of one independent variable.
• Single-Variable Reduction: Often, a problem starts with multiple variables. Modeling involves using substitution to express everything in terms of one independent variable.
• Domain Sanity Check: Ensuring the resulting function only considers inputs that make sense in the real-world context (e.g., length must be positive).

Connected Concepts

• The Mathematical Modeling Process
• Empirical Models, Data Fitting
• Applied Optimization Strategy
• Linear Programming
• Quadratic Optimization
• Trade-offs
• Fundamental Theorem of calculus
• Leverage
• Map-Territory Relation
• Symmetry
• Bottlenecks