Okay, you've learned about functions as ways to map inputs to outputs and about limits as the concept of approaching a value. How do these tie into the practical world of machine learning? It turns out they are fundamental building blocks.

Functions as Models

At its core, many machine learning models can be thought of as functions. Consider a simple task: predicting a house's price based on its size (square footage).

Input: House size (e.g., 1500 sq ft) - let's call this $x$ .
Output: Predicted price (e.g., $300,000) - let's call this$ y$.
Model: The mechanism that takes the input $x$ and produces the output $y$ .

This model essentially is a function $f$ . We feed it an input $x$ , and it gives us an output $y$ , so $y = f(x)$ .

For instance, a very simple linear model might try to capture this relationship using the familiar equation:

y = mx + b

Here, $x$ is the house size, $y$ is the predicted price, and $m$ and $b$ are parameters the model needs to learn from data. The function representing this model is $f(x) = mx + b$ . More complex models, like neural networks, are just more elaborate, multi-layered functions, but the principle remains the same: they map inputs to outputs.

Limits as the Foundation for Learning

So, functions represent our models. Where do limits come in? Limits are the theoretical bedrock upon which derivatives are built. We haven't discussed derivatives in detail yet (that's coming in the next chapter!), but here's the connection:

Derivatives measure the instantaneous rate of change of a function. Think about our house price model. We want to find the best values for $m$ and $b$ that make the model's predictions most accurate. "Accuracy" is often measured by how wrong the model is, using something called a cost function (more on this later). We want to minimize this cost or error.

How do we minimize the error? We need to know how changing $m$ or $b$ slightly will affect the error. Does increasing $m$ make the error go up or down? By how much? Derivatives answer exactly these questions. They tell us the slope or gradient of the cost function with respect to the parameters $m$ and $b$ .

And how are derivatives calculated? They are defined using limits! Specifically, a derivative is found by looking at the change in a function's output divided by the change in its input, as that input change approaches zero. This "approaching zero" part is precisely where the concept of a limit is essential.

A diagram showing how functions represent models, limits form the basis for derivatives, and derivatives guide the optimization process used to train models.

In essence:

Functions provide the structure for our machine learning models (mapping features to predictions).
Limits provide the mathematical foundation needed to precisely define how a function changes.
This definition of change (the derivative) allows us to systematically adjust model parameters to improve performance (optimization).

Without understanding functions, we can't represent models. Without understanding the concept of limits, we can't grasp derivatives. And without derivatives, we can't effectively train many machine learning models. This chapter laid the groundwork with functions and limits; next, we'll build upon this to understand the crucial concept of derivatives.