Part III — Functions and Mathematical Modeling

Chapters 51–90¶

The mental shift: In Parts I and II you worked with numbers — isolated values, sequences, arithmetic structures. In Part III, you start working with relationships between numbers. A function is a machine that transforms input into output, deterministically, with structure you can analyze, compose, invert, and fit to data. This is where mathematics becomes a modeling language.

1. What This Part Covers¶

Part III builds the concept of a function from the ground up, then expands it into a full toolkit for modeling real-world phenomena.

The arc:

• Chapters 51–57: What functions are, how to represent them, how to work with them in Python

• Chapters 58–64: The major function families — linear, quadratic, polynomial, rational, piecewise, step, sigmoid

• Chapters 65–70: How functions transform, shift, scale, and belong to parameterized families

• Chapters 71–78: Modeling — fitting, error, iteration, recursion, dynamical systems

• Chapters 79–90: Projects — 11 complete applied systems built from the concepts above

2. Why This Part Matters¶

Every model in machine learning, every simulation in science, every formula in engineering is a function. Understanding functions at a deep level — not just “f(x) = ...” but what it means to map, compose, invert, and fit — is the foundation for everything that follows.

Without this Part:

• Linear regression is a formula you copy. With it: it’s a parameterized function you minimize.

• Neural activation functions are black boxes. With it: they’re sigmoid/ReLU with understood behavior.

• Gradient descent is a recipe. With it: it’s an iterative function application on an error landscape.

Functions are the grammar of quantitative modeling.

3. Prerequisites from Prior Parts¶

Before starting Part III, you need:

• From Part I: Mathematical notation (ch009), abstraction (ch003), the idea of a model (ch017–018)

• From Part II: Real numbers and their properties (ch026), exponential and logarithmic behavior (ch041–045), floating-point precision (ch038), the concept of growth rates (ch046–048)

• Python: You must be comfortable writing functions, loops, and using NumPy arrays. Plotting with matplotlib is assumed.

4. Part Map — What Connects to What¶

ch051 What is a Function?
    └── ch052 Functions as Programs
        └── ch053 Domain and Range
            ├── ch054 Function Composition
            └── ch055 Inverse Functions
                └── ch056 Visualizing Functions
                    └── ch057 Graphing Functions with Python

ch058 Linear Functions ──────────────────────────────────────┐
ch059 Quadratic Functions ───────────────────────────────────┤
ch060 Polynomial Functions ──────────────────────────────────┤
ch061 Rational Functions ────────────────────────────────────┤ Function
ch062 Piecewise Functions ───────────────────────────────────┤ Families
ch063 Step Functions ────────────────────────────────────────┤
ch064 Sigmoid Functions ─────────────────────────────────────┤
ch065 Activation Functions in ML ────────────────────────────┘

ch066 Function Transformations
    ├── ch067 Scaling and Shifting
    ├── ch068 Reflection and Rotation
    ├── ch069 Parameter Sensitivity
    └── ch070 Function Families

ch071 Modeling Real Phenomena
    ├── ch072 Fitting Simple Models
    ├── ch073 Error and Residuals
    ├── ch074 Iterative Computation
    ├── ch075 Recursion and Mathematical Functions
    ├── ch076 Discrete Dynamical Systems
    ├── ch077 Chaos and Sensitivity
    ├── ch078 Simulation vs Analytical Solutions
    └── ch079 Numerical Experiments

ch080–090: Projects (11 applied systems)

5. Motivating Problem — You Cannot Solve This Yet¶

Here is a problem you will be able to fully solve by the end of Part III.

The problem: A disease is spreading through a city. You have daily case count data for 60 days. You want to:

1. Fit a mathematical model to the data

2. Identify when the peak occurred

3. Predict what happens after day 60

4. Estimate how sensitive the forecast is to the model parameters

Run the cell below. You will see the data. You do not yet have the tools to model it.

6. What You Will Build in This Part¶

By Chapter 90, you will have implemented:

Begin with ch051 — What is a Function?