13.1 Introduction

This chapter is an introduction to Calculus. Calculus is a branch of mathematics that mainly deals with the study of change in the value of a function as the points in the domain change. First, we give an intuitive idea of the derivative (without actually defining it). Then we give a naive definition of limit and study some algebra of limits. Then we come back to a definition of derivative and study some algebra of derivatives. We also obtain derivatives of certain standard functions.

What is Calculus?

Calculus is essentially the mathematics of change, and limits and derivatives are its two foundational pillars. Here is a breakdown of what they mean, how they work, and examples for each.

Limits: The Concept of “Approaching”

A limit tells us what value a function is approaching as the input variable (\(x\)) gets closer and closer to a specific number. It doesn’t care what the function actually is at that exact point-only what it looks like it’s going to be.
We write it as:
\(
\lim _{x \rightarrow a} f(x)=L
\)
(Read as: “The limit of \(f(x)\) as \(x\) approaches \(a\) equals \(L\).”)

Why do we need limits?

Sometimes, plugging a number directly into a function gives us something undefined, like \(\frac{0}{0}\). Limits allow us to sneak up on that point and see what the behavior is.

Example of a Limit:
Consider the function:
\(
f(x)=\frac{x^2-4}{x-2}
\)
If you try to plug in \(x=2\) directly, you get:
\(
f(2)=\frac{2^2-4}{2-2}=\frac{0}{0} \quad(\text { Undefined })
\)
However, we can find the limit as \(x\) approaches 2 by simplifying the algebra first (factoring the top):
\(
f(x)=\frac{(x-2)(x+2)}{x-2}
\)
As long as \(x \neq 2\), we can cancel out (\(x-2\)):
\(
f(x)=x+2
\)
Now, let’s look at what happens as \(x\) gets super close to 2 :
If \(x=1.99\), then \(f(1.99)=1.99+2=3.99\)
If \(x=2.01\), then \(f(2.01)=2.01+2=4.01\)
As \(x\) approaches 2, the value of the function safely approaches 4. Therefore:
\(
\lim _{x \rightarrow 2} \frac{x^2-4}{x-2}=4
\)

Derivatives: The Instantaneous Rate of Change

A derivative measures how quickly a function is changing at any given single point. Graphically, it is the slope of the tangent line to a curve at a specific point. If you drive a car 120 miles in 2 hours, your average speed is 60 mph . But if you look at your speedometer at exactly 1 hour and 15 minutes, it might say 67 mph . That exact speedometer reading is the derivative of your position function at that specific moment.
We define the derivative of a function \(f(x)\) using limits:
\(
f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}
\)

Example of a Derivative:
Let’s find the derivative of \(f(x)=x^2\) using standard derivative rules (the Power Rule dictates that the derivative of \(x^n\) is \(n \cdot x^{n-1}\)).
The derivative of \(f(x)=x^2\) is:
\(
f^{\prime}(x)=2 x
\)
What does this mean?
This new formula (\(2 x\)) tells us the slope of the curve \(x^2\) at any point \(x\).
At \(x=1\), the slope of the curve is \(2(1)=2\).
At \(x=3\), the slope of the curve is \(2(3)=6\).
At \(x=-2\), the slope of the curve is \(2(-2)=-4\).

How They Connect?

You cannot have derivatives without limits. To find the slope at a single instantaneous point, you take two points on a curve, calculate the slope between them (a secant line), and then use a limit to bring those two points so close together that the distance between them (h) approaches zero. This turns the secant line into a tangent line, giving you the derivative.