Calculus I

Natural numbers $\mathbb{N}$: $\mathbb{N}={1,2,3,\dots}$. Used for counting discrete objects. In $\mathbb{N}$ we can add and multiply, but not freely subtract ($3-5\notin\mathbb{N}$).

Integers $\mathbb{Z}$: $\mathbb{Z}={\dots,-2,-1,0,1,2,\dots}$. By introducing negatives, subtraction becomes always possible. Every integer has an opposite ($-n$).

Rational numbers $\mathbb{Q}$: $\mathbb{Q}={p/q : p\in\mathbb{Z}, q\in\mathbb{Z}\setminus{0}}$. Fractions allow division between any integers (except $0$). However $\mathbb{Q}$ has "gaps": there are points on the number line that do not correspond to any rational number.

Why $\mathbb{Q}$ is not enough: we prove $\sqrt{2} \notin \mathbb{Q}$ by contradiction. Suppose $\sqrt{2}=p/q$ with $p,q$ coprime integers. Then $p^2=2q^2$, so $p^2$ is even, hence $p$ is even. Write $p=2k$. Substituting: $4k^2=2q^2 \Rightarrow q^2=2k^2$, so $q^2$ is even, hence $q$ is even. But $p$ and $q$ cannot both be even because they were chosen coprime — contradiction. Therefore $\sqrt{2}$ is not rational.

Real numbers $\mathbb{R}$: to "fill the gaps" we introduce real numbers, which include all limits of convergent sequences of rationals (completion of $\mathbb{Q}$). $\mathbb{R}$ is a complete ordered field: every non-empty bounded above subset has a supremum in $\mathbb{R}$.

Complex numbers $\mathbb{C}$: $\mathbb{C}={a+bi : a,b\in\mathbb{R}, i^2=-1}$. They extend $\mathbb{R}$ by including solutions to $x^2+1=0$.

In summary: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$. Each step extends the possible operations and fills previous gaps.

Upper and lower bounds: let $A\subseteq\mathbb{R}$ be non-empty. $M$ is an upper bound of $A$ if $x\leq M$ for all $x\in A$. If $A$ admits upper bounds, it is bounded above. Similarly, $m$ is a lower bound if $x\geq m$ for all $x\in A$.

Supremum: the smallest upper bound, denoted $\sup A$. The supremum may or may not belong to $A$. For example, $\sup(0,1)=1$ but $1\notin(0,1)$, while $\sup[0,1]=1$ and $1\in[0,1]$. If $\sup A\in A$, then $\sup A$ is also the maximum of $A$.

Characterisation of $\sup$: $s=\sup A$ iff (i) $s$ is an upper bound ($x\leq s;\forall x\in A$); (ii) for every $\varepsilon>0$ there exists $x\in A$ with $x>s-\varepsilon$ (i.e., $s$ cannot be improved).

Infimum: the largest lower bound: $\inf A = \max{m : m\leq x;\forall x\in A}$. If $\inf A\in A$, then $\inf A$ is the minimum.

Completeness axiom (Dedekind): every non-empty subset of $\mathbb{R}$ bounded above has a $\sup$ in $\mathbb{R}$. This axiom distinguishes $\mathbb{R}$ from $\mathbb{Q}$: in $\mathbb{Q}$, the set ${q\in\mathbb{Q}: q^2<2}$ is bounded above but has no sup in $\mathbb{Q}$ (the sup would be $\sqrt{2}$, which is not rational).

Existence theorem for sup and inf: if $A$ is non-empty and bounded (both above and below), then $\sup A$ and $\inf A$ exist in $\mathbb{R}$.

A function $f: A \to B$ assigns to each $x \in A$ exactly one element $y = f(x) \in B$. $A$ is the domain, $B$ the codomain, and $f(A)={f(x):x\in A}\subseteq B$ the image (range).

Key properties:

• Injective (one-to-one): $f(x_1)=f(x_2) \Rightarrow x_1=x_2$. Different inputs give different outputs. Graphically: no horizontal line intersects the graph more than once (horizontal line test).
• Surjective (onto): $f(A)=B$, i.e., every $y\in B$ has at least one preimage. Equivalently: the equation $f(x)=y$ has a solution for every $y\in B$.
• Bijective: both injective and surjective. Then the inverse function $f^{-1}:B\to A$ exists with $f^{-1}(y)=x \iff f(x)=y$.
• Even: $f(-x)=f(x)$ for all $x$. Graph symmetric about the $y$-axis. Examples: $x^2$, $\cos x$, $|x|$.
• Odd: $f(-x)=-f(x)$ for all $x$. Graph symmetric about the origin. Examples: $x^3$, $\sin x$, $\tan x$.
• Monotone increasing: $x_1<x_2 \Rightarrow f(x_1)\leq f(x_2)$. Strictly increasing if $f(x_1)<f(x_2)$.
• Periodic with period $T>0$: $f(x+T)=f(x)$ for all $x$. The fundamental period is the smallest $T>0$ with this property.

Function composition: $(f \circ g)(x) = f(g(x))$. Composition is defined when the image of $g$ is contained in the domain of $f$. Composition is associative ($f\circ(g\circ h)=(f\circ g)\circ h$) but not commutative ($f\circ g \neq g\circ f$ in general).

Inverse function: if $f$ is bijective, $f^{-1}:B\to A$ is defined by $f^{-1}(y)=x\iff f(x)=y$. The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$. Properties: $f^{-1}(f(x))=x$ and $f(f^{-1}(y))=y$.

Exponential function $e^x$: domain $\mathbb{R}$, range $(0,+\infty)$, strictly increasing, $e^0=1$. Key property: $e^{x+y}=e^x e^y$. Growth hierarchy: $e^x$ grows faster than any power $x^\alpha$ as $x\to+\infty$ ($\lim_{x\to+\infty} x^\alpha/e^x=0$ for any $\alpha$). General exponential: $a^x = e^{x\ln a}$.

Natural logarithm $\ln x$: inverse of $e^x$. Domain $(0,+\infty)$, range $\mathbb{R}$, strictly increasing. Properties: $\ln(xy)=\ln x+\ln y$, $\ln(x/y)=\ln x-\ln y$, $\ln(x^\alpha)=\alpha\ln x$. Change of base: $\log_a x = \ln x / \ln a$.

Trigonometric functions: $\sin x$ and $\cos x$ are periodic with period $2\pi$, bounded in $[-1,1]$. Pythagorean identity: $\sin^2 x+\cos^2 x=1$. Addition formulas: $\sin(x\pm y)=\sin x\cos y\pm\cos x\sin y$, $\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y$. $\tan x = \sin x/\cos x$ is periodic with period $\pi$, defined for $x\neq\pi/2+k\pi$.

Inverse trigonometric functions: $\arcsin:[-1,1]\to[-\pi/2,\pi/2]$ (inverse of $\sin$ restricted to $[-\pi/2,\pi/2]$); $\arccos:[-1,1]\to[0,\pi]$; $\arctan:\mathbb{R}\to(-\pi/2,\pi/2)$.

Hyperbolic functions: $\sinh x = (e^x-e^{-x})/2$, $\cosh x = (e^x+e^{-x})/2$. $\cosh^2 x-\sinh^2 x=1$ (analogue of Pythagorean identity).