Definitions

Calculus is the study of functions $f(x)$ over the real numbers $\mathbb{R}$: \[ f: \mathbb{R} \to \mathbb{R}. \] The function $f$ takes as input some number, usually called $x$ and gives as output another number $f(x)=y$. You are familiar with many functions and have used them in many problems.

In this chapter we will learn about different operations that can be performed on functions. It worth understanding these operations because of the numerous applications which they have.

Differential calculus is all about derivatives:

• $f'(x)$: the derivative of $f(x)$ is the rate of change of $f$ at $x$.

The derivative is also a function of the form

  \[
     f': \mathbb{R} \to \mathbb{R},
  \]
  The output of $f'(x)$ represents the //slope// of 
  a line parallel (tangent) to $f$ at the point $(x,f(x))$.

Integral calculus is all about integration:

• $\int_a^b f(x)\:dx$: the integral of $f(x)$ from $x=a$ to $x=b$

corresponds to the area under $f(x)$ between $a$ and $b$:

  \[
      A(a,b) = \int_a^b f(x) \: dx.
  \]
  The $\int$ sign is a mnemonic for //sum//.
  The integral is the "sum" of $f(x)$ over that interval. 
* $F(x)=\int f(x)\:dx$: the anti-derivative of the function $f(x)$ 
  contains the information about the area under the curve for 
  //all// limits of integration.
  The area under $f(x)$ between $a$ and $b$ is computed as the
  difference between $F(b)$ and $F(a)$:
  \[
     A(a,b) = \int_a^b f(x)\;dx = F(b)-F(a).
  \]
  

Functions are usually defined for continuous inputs $x\in \mathbb{R}$, but there are also functions which are defined only for natural numbers $n \in \mathbb{N}$. Sequences are the discrete analogue functions.

• $a_n$: sequence of numbers $\{ a_0, a_1, a_2, a_3, a_4, \ldots \}$.

You can think about each sequence as a function

  \[
     a: \mathbb{N} \to \mathbb{R},
  \]
  where the input $n$ is an integer (index into the sequence) and
  the output is $a_n$ which could be any number.

NOINDENT The integral of a sequence is called a series.

• $\sum$: sum.

The summation sign is the short way to express

  the sum of several objects:
  \[
    a_3 + a_4 + a_5 + a_6 + a_7 
    \equiv \sum_{3 \leq i \leq 7} a_i 
    \equiv \sum_{i=3}^{7} a_i.
  \]
  Note that summations could go up to infinity.
* $\sum a_i$: the series corresponds to the running total of a sequence until $n$:
  \[
     S_n = \sum_{i=1}^{n} a_i  = a_1 + a_2 + \cdots + a_{n-1} + a_n.
  \]
* $f(x)=\sum_{i=0}^\infty a_i x^i$: a //power series// is a series
  which contains powers of some variable $x$.
  Power series give us a way to express any function $f(x)$ as
  an infinitely long polynomial. 
  For example, the power series of $\sin(x)$ is
  \[
    \sin(x) 
       = x - \frac{x^3}{3!}  + \frac{x^5}{5!} 
          - \frac{x^7}{7!} + \frac{x^9}{9!}+ \ldots.
  \]

Don't worry if you don't understand all the notions and the new notation in the above paragraphs. I just wanted to present all the calculus actors in the first scene. We will talk about each of them in more detail in the following sections.

Actually, we have not mentioned the main actor yet: the limit. In calculus, we do a lot of limit arguments in which we take some positive number $\epsilon>0$ and we make it progressively smaller and smaller:

• $\displaystyle\lim_{\epsilon \to 0}$: the mathematically rigorous

way of saying that the number $\epsilon$ becomes smaller and smaller. We can also take limits to infinity, that is, we imagine some number $N$ and we make that number bigger and bigger:

• $\displaystyle\lim_{N \to \infty}$: the mathematical

way of saying that the number $N$ will get larger and larger.

Indeed, it wouldn't be wrong to say that calculus is the study of the infinitely small and the infinitely many. Working with infinitely small quantities an infinitely large numbers can be tricky business but it is extremely important that you become comfortable with the concept of a limit which is the rigorous way of talking about infinity. Before we learn about derivatives, integrals and series we will spend some time learning about limits.