Number Sets and Functions

Number Sets — from N to R

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}

\sqrt{2} \notin \mathbb{Q}

Supremum, Infimum, and Completeness of R

\sup A = \min\{M : x \leq M\;\forall x\in A\}

A \text{ bounded} \Rightarrow \exists\,\sup A,\,\inf A \in \mathbb{R}

Real Functions — Definition and Properties

f: A\to B,\quad x\mapsto f(x)

(f\circ g)(x) = f(g(x))

f \text{ invertible} \iff f \text{ bijective}

Elementary Functions

e^{x+y}=e^x e^y,\quad \ln(xy)=\ln x+\ln y

\sin^2 x+\cos^2 x=1

\sin(x\pm y)=\sin x\cos y\pm\cos x\sin y

Complex Numbers

Algebraic form and operations

|z|=\sqrt{a^2+b^2}

\overline{z}=a-bi

Gauss plane and trigonometric form

z = r(\cos\theta + i\sin\theta) = r e^{i\theta}

e^{i\theta} = \cos\theta + i\sin\theta

Product, quotient and power in exponential form

(r e^{i\theta})^n = r^n e^{in\theta}

N-th roots of unity

z_k = r^{1/n} e^{i(\theta+2\pi k)/n},\; k=0,\dots,n-1

Complex exponential and applications

e^{x+iy}=e^x(\cos y+i\sin y)

Induction & Combinatorics

Principle of induction

\sum_{i=1}^n i = n(n+1)/2

P(1)\land[P(k)\Rightarrow P(k+1)]\Rightarrow\forall n\,P(n)

Binomial coefficients and Newton formula

{n \choose k} = n!/(k!(n-k)!)

(a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k

Sums and progressions

\sum_{k=1}^n a_k = n(a_1+a_n)/2

\sum_{k=1}^n a_1 q^{k-1} = a_1(1-q^n)/(1-q)

Limits of Sequences and Functions

Sequences — Definition and Convergence

\lim_{n\to\infty}a_n=L\iff\forall\varepsilon>0\;\exists N:\;n>N\Rightarrow|a_n-L|<\varepsilon

\lim_{n\to\infty}(1+1/n)^n=e

Function Limits — Definition and Properties

\lim_{x\to 0}\frac{\sin x}{x}=1,\quad\lim_{x\to 0}\frac{e^x-1}{x}=1,\quad\lim_{x\to 0}\frac{\ln(1+x)}{x}=1

Indeterminate Forms and Techniques

\text{Hierarchy: }\ln x\ll x^\alpha\ll e^x

\lim_{x\to0}\sin(\alpha x)/(\beta x)=\alpha/\beta

\lim_{x\to+\infty}x^n/e^x=0

Limits at Infinity and Asymptotes

m=\lim_{x\to\pm\infty}f(x)/x,\quad q=\lim_{x\to\pm\infty}[f(x)-mx]

\text{Vertical asymptote: }\lim_{x\to x_0}|f(x)|=+\infty

Derivatives and Differential Calculus Theorems

Definition of the Derivative

f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}

f\text{ differentiable at }x_0\Rightarrow f\text{ continuous}

Differentiation Rules

(fg)'=f'g+fg',\quad(f/g)'=(f'g-fg')/g^2

[f(g(x))]'=f'(g(x))\cdot g'(x)

(e^x)'=e^x,\quad(\ln x)'=1/x

Rolle's, Lagrange's, and Cauchy's Theorems

\exists c\in(a,b):\;f'(c)=\frac{f(b)-f(a)}{b-a}

f'\geq0\iff f\text{ non-decreasing}

L'Hôpital's Rule

\lim\frac{f}{g}\stackrel{0/0}{=}\lim\frac{f'}{g'}

Taylor and Maclaurin Series

f(x)=\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k+o((x-x_0)^n)

e^x=\sum x^k/k!,\quad\sin x=x-x^3/6+o(x^3)

Taylor Series

Power series and radius of convergence

R = 1/\lim_{n\to\infty}\sqrt[n]{|a_n|}

R = \lim_{n\to\infty}|a_n/a_{n+1}|

Taylor and MacLaurin series

f(x)=\sum_{k=0}^\infty\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k

R_n(x)=f^{(n+1)}(\xi)(x-x_0)^{n+1}/(n+1)!

Notable expansions

e^x=\sum_{k=0}^\infty x^k/k!

\sin x = \sum_{k=0}^\infty(-1)^k x^{2k+1}/(2k+1)!

\cos x = \sum_{k=0}^\infty(-1)^k x^{2k}/(2k)!

Applications to limits

\lim_{x\to0}\frac{\sin x - x}{x^3} = -\frac{1}{6}

f(x) = \sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k + o((x-x_0)^n)

Complete Function Analysis (Curve Sketching)

2. Domain and Symmetries

\text{Dom}(f) = \{x\in\mathbb{R} : f(x)\text{ is defined}\}

f \text{ even} \iff f(-x)=f(x)\;\forall x\in\text{Dom}

f \text{ odd} \iff f(-x)=-f(x)\;\forall x\in\text{Dom}

4. Axis Intercepts and Sign

f(0) = y\text{-intercept}

f(x) = 0 \Rightarrow x\text{-intercepts (zeros)}

\text{Sign: study }\text{sgn}(\text{num}) \cdot \text{sgn}(\text{den})

Limits at Domain Boundaries and Asymptotes

\text{Vert. asymp. at }x_0\iff\lim_{x\to x_0}|f(x)|=+\infty

\text{Horiz. asymp. }y=L\iff\lim_{x\to\pm\infty}f(x)=L

m=\lim_{x\to\infty}\frac{f(x)}{x},\quad q=\lim_{x\to\infty}[f(x)-mx]

First Derivative — Monotonicity, Maxima, Minima

f'(x_0)=0,\;f'\text{ changes }+\to-\Rightarrow x_0\text{ local max}

f'(x_0)=0,\;f'\text{ changes }-\to+\Rightarrow x_0\text{ local min}

f'(x_0)=0,\;f''(x_0)>0\Rightarrow\text{min};\quad f''(x_0)<0\Rightarrow\text{max}

Second Derivative — Concavity and Inflection Points

f''(x)>0\Rightarrow f\text{ convex (}\cup\text{)}

f''(x)<0\Rightarrow f\text{ concave (}\cap\text{)}

f''(x_0)=0\text{ with sign change}\Rightarrow x_0\text{ inflection point}

9. Summary Table and Graph

\text{Table: } x,\; f'(x),\; f(x),\; f''(x)

\text{Graph: asymptotes \rightarrow notable points \rightarrow curve}

Integrals — Indefinite and Definite

Indefinite Integral — Antiderivative

\int x^n\,dx=x^{n+1}/(n+1)+C,\quad\int1/x\,dx=\ln|x|+C

\int e^x\,dx=e^x+C,\quad\int\sin x\,dx=-\cos x+C

Integration Methods

\int u\,dv=uv-\int v\,du

\int f(g(x))g'(x)\,dx=\int f(t)\,dt

Definite Integral and Fundamental Theorem

\int_a^b f(x)\,dx=F(b)-F(a)

d/dx\int_a^x f(t)\,dt=f(x)

Improper Integrals

\int_1^{+\infty}x^{-\alpha}dx=1/(\alpha-1)\;(\alpha>1)

\int_0^{+\infty}e^{-x}dx=1

Numerical Series

Definition and Convergence

\sum_{n=0}^\infty q^n=1/(1-q)\;(|q|<1)

\sum 1/n^\alpha:\text{conv. if }\alpha>1,\text{ div. if }\alpha\leq1

Convergence Tests

L=\lim|a_{n+1}/a_n|:\;L<1\Rightarrow\text{conv.},\;L>1\Rightarrow\text{div.}

\sum(-1)^n b_n\text{ conv. if }b_n\searrow0

Ordinary Differential Equations (ODEs)

First-Order Separable ODEs

y'=ky\Rightarrow y=y_0e^{k(x-x_0)}

Second-Order Linear ODEs with Constant Coefficients

a\lambda^2+b\lambda+c=0

\Delta<0:\;y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)

First-Order Linear ODEs

General form and integrating factor

\mu(x)=e^{\int a(x)\,dx}

y = \frac{1}{\mu}\int \mu b\,dx + \frac{C}{\mu}

Cauchy problem

y(x_0)=y_0 \Rightarrow \exists!\,y\text{ solution}

y'+a(x)y=b(x),\;y(x_0)=y_0

Applications

L\,di/dt + Ri = V

i(t) = V/R(1-e^{-Rt/L})

Formulario di Analisi 1