# Calculus I

A core and fundamental course

## What is Calculus and used for?

The formal study of calculus started from the 17th century by well-known scientists and mathematicians Isaac Newton and Gottfried Leibniz. Calculus is a mathematical discipline that is primarily concerned with functions, limits, derivatives, integrals and infinite series. The emphasis is understanding concepts. The most fundamental concept in calculus is ** limit**.

### The Precise Definition of a Limit

Let $ f(x) $ be a function defined on some open interval that contains $ a $, except possibly at $ a $ itself. Then we say that **the limit of** $ f(x) $ **as** $ a $ **approaches** $ a $ **is** $ L $, and we write
$$ \underset{x\rightarrow a}\lim f(x)=L $$
if for every number $ ϵ > 0 $ there is a number $ δ > 0 $ such that
$$ \text{if}\quad 0<|x-a|<\delta\quad\text{then}\quad|f(x)-L|<\epsilon $$

### Continuity

Let $ f $ be a function and let $ a\in\mathbb{R} $. The function $ f $ is said to be **continuous at the point** $ a $ if the following three things are true:

- $ f (a) $ is defined (that is, $ a $ is in the domain of $ f $.)
- The limit $ \underset{x\rightarrow a}\lim f(x) $ exists.
- $ \underset{x\rightarrow a}\lim f(x)=f(a) $

### Derivative

The **derivative of a function** $ f $ **at a number** $ a $, denoted by $ f'(a) $, is
$$ f'(a)=\underset{h\rightarrow 0}\lim\frac{f(a+h)-f(a)}{h} $$
if this limit exists.

### MIT Mathematics

Calculus is the study of how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models.

Calculus is an exciting subject, justly considered to be one of the greatest achievements of the human intellect. I hope you will discover that it is not only useful but also intrinsically beautiful.

— James Stewart

### Reference Textbook

*Calculus*, James Stewart, The 8th, metric edition, Cengage Learning.