📚 FORMULARIO MATEMATICA - LICEO SCIENTIFICO

$\mathbb{N}$ = numeri naturali $\{0,1,2,3,...\}$

$\mathbb{Z}$ = numeri interi $\{...,-2,-1,0,1,2,...\}$

$\mathbb{Q}$ = numeri razionali $\frac{p}{q}$, $q \neq 0$

$\mathbb{R}$ = numeri reali (razionali + irrazionali)

$\mathbb{C}$ = numeri complessi $a+bi$

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

$(a,b) = \{x \in \mathbb{R} : a < x < b\}$ (aperto)

$[a,b] = \{x \in \mathbb{R} : a \leq x \leq b\}$ (chiuso)

$[a,b) = \{x \in \mathbb{R} : a \leq x < b\}$ (semiaperto)

$(a,+\infty) = \{x \in \mathbb{R} : x > a\}$

$(-\infty,b] = \{x \in \mathbb{R} : x \leq b\}$

$a^m \cdot a^n = a^{m+n}$

$\frac{a^m}{a^n} = a^{m-n}$

$(a^m)^n = a^{m \cdot n}$

$(a \cdot b)^n = a^n \cdot b^n$

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

$a^0 = 1$ (se $a \neq 0$)

$a^{-n} = \frac{1}{a^n}$

$\sqrt[n]{a^m} = a^{\frac{m}{n}}$

$\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$

$\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$

$\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \cdot m]{a}$

$\left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$

$\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$

$\frac{1}{\sqrt{a} + \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a-b}$

$\frac{1}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} + \sqrt{b}}{a-b}$

$(a+b)^2 = a^2 + 2ab + b^2$

$(a-b)^2 = a^2 - 2ab + b^2$

$(a+b+c)^2 = a^2+b^2+c^2+2ab+2ac+2bc$

$(a+b)(a-b) = a^2 - b^2$

$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

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

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$ (coefficiente binomiale)

$ax^2 + bx + c = 0$ (con $a \neq 0$)

$x_{1,2} = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$\Delta = b^2 - 4ac$ (discriminante)

Se $b=2b'$: $x_{1,2} = \frac{-b' \pm \sqrt{b'^2-ac}}{a}$

$\Delta > 0$: due soluzioni reali distinte

$\Delta = 0$: una soluzione doppia $x=-\frac{b}{2a}$

$\Delta < 0$: nessuna soluzione reale (2 complesse)

$x_1 + x_2 = -\frac{b}{a}$

$x_1 \cdot x_2 = \frac{c}{a}$

$y = ax^2 + bx + c$

Vertice: $V\left(-\frac{b}{2a}, -\frac{\Delta}{4a}\right)$

Asse: $x = -\frac{b}{2a}$

$a>0$: concavità verso l'alto

$a<0$: concavità verso il basso

$y = mx + q$

$m$ = coefficiente angolare (pendenza)

$q$ = ordinata all'origine

$ax + by + c = 0$

$m = -\frac{a}{b}$ (se $b \neq 0$)

$m = \tan\alpha$ (α = angolo con asse x)

$m = \frac{y_2-y_1}{x_2-x_1}$ (dati 2 punti)

$d(P_1,P_2) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

$M\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

$r \parallel s \Leftrightarrow m_r = m_s$

$r \perp s \Leftrightarrow m_r \cdot m_s = -1$

$d(P,r) = \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$

$(x-\alpha)^2 + (y-\beta)^2 = r^2$

Centro: $C(\alpha, \beta)$

Raggio: $r$

$x^2 + y^2 + ax + by + c = 0$

Centro: $C\left(-\frac{a}{2}, -\frac{b}{2}\right)$

Raggio: $r = \sqrt{\frac{a^2}{4} + \frac{b^2}{4} - c}$

$y = ax^2 + bx + c$ (se $a \neq 0$)

Vertice: $V\left(-\frac{b}{2a}, -\frac{\Delta}{4a}\right)$

Fuoco: $F\left(-\frac{b}{2a}, \frac{1-\Delta}{4a}\right)$

Direttrice: $y = -\frac{1+\Delta}{4a}$

Asse: $x = -\frac{b}{2a}$

$y = ax^2$

Vertice: $V(0,0)$

Fuoco: $F\left(0, \frac{1}{4a}\right)$

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (con $a > b > 0$)

$a$ = semiasse maggiore

$b$ = semiasse minore

$F_1(-c, 0)$, $F_2(c, 0)$ se $a>b$

$c = \sqrt{a^2 - b^2}$ (semidistanza focale)

$e = \frac{c}{a}$ con $0 < e < 1$

$A = \pi ab$

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

$F_1(-c, 0)$, $F_2(c, 0)$

$c = \sqrt{a^2 + b^2}$ (sempre $c > a$)

$y = \pm \frac{b}{a} x$

$e = \frac{c}{a}$ con $e > 1$

$a = b$: $x^2 - y^2 = a^2$

Asintoti: $y = \pm x$ (perpendicolari)

$xy = k$

$k > 0$: rami nel I e III quadrante

$k < 0$: rami nel II e IV quadrante

$180° = \pi$ rad

$\alpha_{\text{rad}} = \alpha_{\text{gradi}} \cdot \frac{\pi}{180}$

$\alpha_{\text{gradi}} = \alpha_{\text{rad}} \cdot \frac{180}{\pi}$

$\sin^2\alpha + \cos^2\alpha = 1$

$\tan\alpha = \frac{\sin\alpha}{\cos\alpha}$

$\cot\alpha = \frac{\cos\alpha}{\sin\alpha}$

$\tan\alpha \cdot \cot\alpha = 1$

$1 + \tan^2\alpha = \frac{1}{\cos^2\alpha}$

$1 + \cot^2\alpha = \frac{1}{\sin^2\alpha}$

$\sin 30° = \frac{1}{2}$; $\cos 30° = \frac{\sqrt{3}}{2}$

$\sin 45° = \frac{\sqrt{2}}{2}$; $\cos 45° = \frac{\sqrt{2}}{2}$

$\sin 60° = \frac{\sqrt{3}}{2}$; $\cos 60° = \frac{1}{2}$

$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$

$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$

$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$

$\sin(2\alpha) = 2\sin\alpha\cos\alpha$

$\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha$

$\cos(2\alpha) = 2\cos^2\alpha - 1$

$\cos(2\alpha) = 1 - 2\sin^2\alpha$

$\tan(2\alpha) = \frac{2\tan\alpha}{1-\tan^2\alpha}$

$\sin\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1-\cos\alpha}{2}}$

$\cos\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1+\cos\alpha}{2}}$

$\tan\left(\frac{\alpha}{2}\right) = \frac{\sin\alpha}{1+\cos\alpha} = \frac{1-\cos\alpha}{\sin\alpha}$

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

R = raggio circonferenza circoscritta

$a^2 = b^2 + c^2 - 2bc \cos A$

$b^2 = a^2 + c^2 - 2ac \cos B$

$c^2 = a^2 + b^2 - 2ab \cos C$

$A = \frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ac \sin B$

Formula di Erone: $A = \sqrt{p(p-a)(p-b)(p-c)}$

$p = \frac{a+b+c}{2}$ (semiperimetro)

$e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = 2.71828...$

$\log_a b = x \Leftrightarrow a^x = b$

Condizioni: $a > 0$, $a \neq 1$, $b > 0$

$\log_a (x \cdot y) = \log_a x + \log_a y$

$\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

$\log_a x^p = p \log_a x$

$\log_a \sqrt[n]{x} = \frac{1}{n} \log_a x$

$\log_a b = \frac{\log_c b}{\log_c a}$

$\log_a b = \frac{1}{\log_b a}$

$\log_a 1 = 0$

$\log_a a = 1$

$\log_a a^x = x$

$a^{\log_a x} = x$

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

$\lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2}$

$\lim_{x \to 0} \frac{\tan x}{x} = 1$

$\lim_{x \to 0} \frac{e^x-1}{x} = 1$

$\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$

$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$

$\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $1^{\infty}$, $0^0$, $\infty^0$

$f'(x_0) = \lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h}$

$D[k] = 0$ (costante)

$D[x^n] = nx^{n-1}$

$D[\sqrt{x}] = \frac{1}{2\sqrt{x}}$

$D\left[\frac{1}{x}\right] = -\frac{1}{x^2}$

$D[e^x] = e^x$

$D[a^x] = a^x \ln a$

$D[\ln x] = \frac{1}{x}$

$D[\log_a x] = \frac{1}{x \ln a}$

$D[\sin x] = \cos x$

$D[\cos x] = -\sin x$

$D[\tan x] = \frac{1}{\cos^2 x} = 1 + \tan^2 x$

$D[\cot x] = -\frac{1}{\sin^2 x}$

$D[\arcsin x] = \frac{1}{\sqrt{1-x^2}}$

$D[\arccos x] = -\frac{1}{\sqrt{1-x^2}}$

$D[\arctan x] = \frac{1}{1+x^2}$

$D[k f(x)] = k f'(x)$

$D[f(x) + g(x)] = f'(x) + g'(x)$

$D[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)$

$D\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

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

$\int k \, dx = kx + c$

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + c$ (se $n \neq -1$)

$\int \frac{1}{x} \, dx = \ln|x| + c$

$\int e^x \, dx = e^x + c$

$\int a^x \, dx = \frac{a^x}{\ln a} + c$

$\int \sin x \, dx = -\cos x + c$

$\int \cos x \, dx = \sin x + c$

$\int \frac{1}{\cos^2 x} \, dx = \tan x + c$

$\int \frac{1}{1+x^2} \, dx = \arctan x + c$

$\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + c$

$\int_a^b f(x) \, dx = F(b) - F(a) = [F(x)]_a^b$

dove $F'(x) = f(x)$

$\int f'(x)g(x) \, dx = f(x)g(x) - \int f(x)g'(x) \, dx$

$n! = 1 \cdot 2 \cdot 3 \cdot ... \cdot n$

$0! = 1$ per convenzione

$D_{n,k} = \frac{n!}{(n-k)!}$

$P_n = n!$

$C_{n,k} = \binom{n}{k} = \frac{n!}{k!(n-k)!}$

$P(E) = \frac{\text{casi favorevoli}}{\text{casi possibili}}$

$P(\overline{E}) = 1 - P(E)$

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$P(A \cap B) = P(A) \cdot P(B)$

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

$A_{\text{tot}} = 6l^2$

$V = l^3$

Diagonale: $d = l\sqrt{3}$

$A_{\text{tot}} = 2(ab + ac + bc)$

$V = a \cdot b \cdot c$

Diagonale: $d = \sqrt{a^2 + b^2 + c^2}$

$A_{\text{laterale}} = 2\pi r h$

$A_{\text{totale}} = 2\pi r(r + h)$

$V = \pi r^2 h$

$A_{\text{laterale}} = \pi r a$ (a=apotema)

$A_{\text{totale}} = \pi r(r + a)$

$V = \frac{\pi r^2 h}{3}$

$a = \sqrt{r^2 + h^2}$

$A = 4\pi r^2$

$V = \frac{4}{3}\pi r^3$

$A_{\text{laterale}} = \frac{p \cdot a}{2}$

$V = \frac{A_b \cdot h}{3}$

$i = \sqrt{-1}$

$i^2 = -1$

$i^3 = -i$

$i^4 = 1$

$z = a + bi$

$a = \text{Re}(z)$ (parte reale)

$b = \text{Im}(z)$ (parte immaginaria)

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

$\overline{z} = a - bi$

$z \cdot \overline{z} = a^2 + b^2 = |z|^2$

$z = \rho(\cos\theta + i\sin\theta)$

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

$\theta = \arg(z)$ (argomento)

$z^n = \rho^n(\cos(n\theta) + i\sin(n\theta))$

$\sum_{k=1}^n k = \frac{n(n+1)}{2}$

$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$

$\sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2$

$a_n = a_1 + (n-1)d$

$S_n = \frac{n(a_1+a_n)}{2}$

$a_n = a_1 \cdot q^{n-1}$

$S_n = a_1\frac{q^n-1}{q-1}$ (se $q \neq 1$)

$S_{\infty} = \frac{a_1}{1-q}$ (se $|q| < 1$)

$\pi = 3.14159265...$

$e = 2.71828182...$

$\phi = \frac{1+\sqrt{5}}{2} = 1.618...$ (sezione aurea)

$\sqrt{2} = 1.41421356...$

$\sqrt{3} = 1.73205080...$