[playground] Test formula

$\color{blue}{ \LARGE \alpha}$	$\color{red}{ \LARGE \sin(\alpha)}$
$\color{blue}{ \LARGE 0 \text{ rad}; \ 0^\circ}$	$\color{blue}{ \LARGE 0}$
$\frac{\pi}{12} \text{ rad}; \ 15^\circ$	$\frac{\sqrt{6} - \sqrt{2}}{4}$
$\frac{\pi}{10} \text{ rad}; \ 18^\circ$	$\frac{\sqrt{5} - 1}{4}$
$\frac{\pi}{8} \text{ rad}; \ 22^\circ 30$	$\frac{\sqrt{2 -\sqrt{2}}}{2}$
$\frac{\pi}{6} \text{ rad}; \ 30^\circ$	$\frac{1}{2}$
$\frac{\pi}{5} \text{ rad}; \ 36^\circ$	$\frac{\sqrt{10-2 \sqrt{5}}}{4}$
$\frac{\pi}{4} \text{ rad}; \ 45^\circ$	$\frac{\sqrt{2}}{2}$
$\frac{3\pi}{10} \text{ rad}; \ 54^\circ$	$\frac{\sqrt{5}+1}{4}$
$\frac{\pi}{3} \text{ rad}; \ 60^\circ$	$\frac{\sqrt{3}}{2}$

Radici dell’equazione caratteristica	Integrale generale
$\Delta>0\;\rightarrow\; z_1\neq z_2$ (reali e distinte)	$y=c_1\cdot e^{z_1x}+c_2\cdot e^{z_2x+}$
$\Delta=0\;\rightarrow\; z_1=z_2$ (reali e coincidenti)	$y=e^{z_1x}(c_1+c_2x)$
$\Delta<0\;\rightarrow\; z_{1,2}=\alpha+\beta i$ (complesse e coniugate)	$y=e^{\alpha x}(c_1\cos{\beta x}+c_2\sin{\beta x})$

Problema 1	a
Problema 2	a
Quesito 1	Sia ABC un triangolo rettangolo in A. Sia O il centro del quadrato BCDE costruito sull’ipotenusa, dalla parte opposta al vertice A. Dimostrare che O è equidistante dalle rette AB e AC.
Quesito 2	Un dado truccato, con le facce numerate da 1 a 6, gode della proprietà di avere ciascuna faccia pari che si presenta con probabilità doppia rispetto a ciascuna faccia dispari. Calcolare le probabilità di ottenere, lanciando una volta il dado, rispettivamente: un numero primo un numero almeno pari a 3 un numero al più pari a 3
Quesito 3	Considerata la retta 𝑟 passante per i due punti 𝐴(1, −2, 0) e 𝐵(2, 3, −1), determinare l’equazione cartesiana della superficie sferica di centro 𝐶(1, −6, 7) e tangente a 𝑟.
Quesito 4	Tra tutti i parallelepipedi a base quadrata di volume 𝑉, stabilire se quello di area totale minima ha anche diagonale di lunghezza minima.
Quesito 5	Determinare l’equazione della retta tangente alla curva di equazione $\Large y=\sqrt{25-x^2}$ nel suo punto di ascissa 3, utilizzando due metodi diversi.
Quesito 6	Determinare i valori dei parametri reali a e b affinché: aaaa $\Large \lim_{x\to 0}\frac{\sin x -(ax^3+bx)}{x^3}=1$
Quesito 7	Si consideri la funzione: aaaa $\Large f(x)=\begin{cases}-1+\arctan x & \;x<0 \\ ax+b & \;x \ge 0 \end{cases}$ Determinare per quali valori dei parametri reali a, b la funzione è derivabile. Stabilire se esiste un intervallo di ℝ in cui la funzione 𝑓 soddisfa le ipotesi del teorema di Rolle. Motivare la risposta.
Quesito 8	Data la funzione $f_a(x)=x^5-5ax+a$ , definita nell’insieme dei numeri reali, stabilire per quali valori del parametro 𝑎 > 0 la funzione possiede tre zeri reali distinti.

$ \begin{align} &y=y’-6x^2=0 \\ &y’=6x^2\\ &\int y’\;dy=\int 6x^2\;dx\\ &y=6\int x^2\;dx\\ &y=6\cdot \dfrac{x^3}{3}+c\\ &y=2x^3+c \end{align} ” /> <img decoding=$

$\displaystyle \Large \lim_{x \to 1}\;\frac{x^2+x+1}{x^2-3x+3}=$

$\Large \frac{1^2+1+1}{1^2-3\cdot 1+3}=\frac{3}{1}=3$

$\vskip\smallskipamount\\</pre> \frac{0}{l}=0\;\;(l\neq 0)\\ \frac{l}{0}=+\infty\;\;(l\neq 0)\\$

$\begin{equation} \begin{cases} \displaystyle -\frac{b}{2a}=2 \\ \displaystyle -\frac{\Delta}{4a}=-1\\ \displaystyle -\frac{1+\Delta}{4a}=3 \end{cases} \end{equation}$

[qsm_link id=1]Click here[/qsm_link]

ANGOLI $\vspace{0.5cm} \LARGE 0\cdot l =0\\ \frac{0}{l}=0\;\;(l\neq 0)\\ \frac{l}{0}=+\infty\;\;(l\neq 0)\\$

$A\widehat{B}C$

$x+\frac{3}{4}x+\frac{3}{4}x=180^\circ\\ \vspace{0.5cm}\\ x+\frac{6}{4}x=180^\circ$

$x+\frac{3}{2}x=180^\circ\\ \frac{2x+3x}{2}=180^\circ$

$\frac{5}{2}x=180^\circ\\ x=180^\circ \cdot \frac{2}{5}=72^\circ \mbox { (angolo al vertice)}$

$a_m=\frac{\Delta v}{\Delta t}=\frac{(v-v_0)}{(t-t_0)}$

$a_m=\frac{(0,647-0)}{(2,16-0)}\simeq 0,300\; m/s^2$

$M'=\begin{vmatrix} 2 & 1 & -1 \\ 1 & 1 & -1 \\ 1 & 2 & -2 \end{vmatrix}=$

$=2\begin{vmatrix} 1 & -1 \\ 2 & -2 \end{vmatrix}-1\begin{vmatrix} 1 & -1 \\ 1 & -2 \end{vmatrix}-1\begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix}$

$=2\cdot(-2+2)-1\cdot(-2+1)+1\cdot(2-1) =$

$=2\cdot(0)-1\cdot(-1)+1\cdot(1) =1+1=2\neq 0\\$

Test Formula

$VOLUME_{totale}=Sb\cdot b_1-2\cdot \frac{Sb\cdot OB}{3}=\pi9^2\cdot 32-2\cdot \frac{\pi9^2\cdot 8}{3}=81\cdot 32\pi-2\cdot \frac{81\cdot 8\pi}{3}=2592\pi-432\pi=2160\pi\;\;\;cm^3$

$a=\bar{a}\pm\Delta{a}$

$b=\bar{b}\pm\Delta{b}$

$\begin{cases} \displaystyle x \leq 1 \displaystyle\\\frac{-2x+7+x}{-2x+2-x}\leq0\end{cases}$

$ \begin{eqnarray} \begin{cases} \displaystyle x \leq 1\\\mbox{}\displaystyle\\\frac{-2x+7+x}{-2x+2-x}\leq0\end{cases} \end{eqnarray} $

$ \Bigg \{ \begin{array}{rl} \frac{a}{b}=\frac{c}{d}\\ \frac{x}{y}=1\\ \end{array} $

RUFFINI

$\begin{array}{c|ccccc|c}& 1 & 0 & 0 & 0 & 0 & 1 \vspace{2}\\[2]\\-1 & -1 & +1 & -1 & +1 & -1\\\hline& 1 & -1 & +1 & -1 & +1 & 0\\\end{array}$

$\Large \begin{array}{c|ccc|c}& +1 & 0 & -7 & +6 \vspace{2}\\ \\+1 & & +1 & +1 & -6\\\hline& +1 & +1 & -6 & 0\\\end{array}$

$\begin{eqnarray} \Bigg \{ \begin{array}{rl} \frac{a}{b}=\frac{c}{d}\\ \frac{x}{y}=1\\ \end{array} \end{eqnarray}$













TestTest

$\LARGE \alpha$

$\color{blue}{ \LARGE \alpha}$	$\color{red}{ \LARGE \sin(\alpha)}$	$\color{red}{ \LARGE \cos(\alpha)}$	$\color{red}{ \LARGE \tan(\alpha)}$	$\color{red}{ \LARGE \cot(\alpha)}$
$\color{blue}{ \LARGE 0 \text{ rad}; \ 0^\circ}$	$\color{blue}{ \LARGE 0}$	$\color{blue}{ \LARGE 1}$	$\color{blue}{ \LARGE 0}$	$\color{blue}{ \LARGE \text{non definito}}$
$\frac{\pi}{12} \text{ rad}; \ 15^\circ$	$\frac{\sqrt{6} - \sqrt{2}}{4}$	$\frac{\sqrt{6} + \sqrt{2}}{4}$	$2 - \sqrt{3}$	$2 + \sqrt{3}$
$\frac{\pi}{10} \text{ rad}; \ 18^\circ$	$\frac{\sqrt{5} - 1}{4}$	$\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$	$\sqrt{1 - \frac{2\sqrt{5}}{5}}$	$\sqrt{5 +2\sqrt{5}}$
$\frac{\pi}{8} \text{ rad}; \ 22^\circ 30$	$\frac{\sqrt{2 -\sqrt{2}}}{2}$	$\frac{\sqrt{2 +\sqrt{2}}}{2}$	$\sqrt{2} - 1$	$\sqrt{2} + 1$
$\frac{\pi}{6} \text{ rad}; \ 30^\circ$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$	$\sqrt{3}$
$\frac{\pi}{5} \text{ rad}; \ 36^\circ$	$\frac{\sqrt{10-2 \sqrt{5}}}{4}$	$\frac{\sqrt{5}+1}{4}$	$\sqrt{5-2\sqrt{5}}$	$\frac{\sqrt{25+10\sqrt{5}}}{5}$
$\frac{\pi}{4} \text{ rad}; \ 45^\circ$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$	$1$
$\frac{3\pi}{10} \text{ rad}; \ 54^\circ$	$\frac{\sqrt{5}+1}{4}$	$\frac{\sqrt{10-2 \sqrt{5}}}{4}$	$\frac{\sqrt{25+10\sqrt{5}}}{5}$	$\sqrt{5-2\sqrt{5}}$
$\frac{\pi}{3} \text{ rad}; \ 60^\circ$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{\sqrt{3}}{3}$
$\frac{3\pi}{8} \text{ rad}; \ 67^\circ 30'$	$\frac{\sqrt{2+\sqrt{2}}}{2}$	$\frac{\sqrt{2-\sqrt{2}}}{2}$	$\sqrt{2}+1$	$\sqrt{2}-1$
$\frac{2\pi}{5} \text{ rad}; \ 72^\circ$	$\frac{\sqrt{10+2 \sqrt{5}}}{4}$	$\frac{\sqrt{5}-1}{4}$	$\sqrt{5+2\sqrt{5}}$	$\frac{\sqrt{25-10\sqrt{5}}}{5}$
$\frac{5\pi}{12} \text{ rad}; \ 75^\circ$	$\frac{\sqrt{6} + \sqrt{2}}{4}$	$\frac{\sqrt{6} - \sqrt{2}}{4}$	$2 + \sqrt{3}$	$2 - \sqrt{3}$
$\color{blue}{ \LARGE \frac{\pi}{2} \text{ rad}; \ 90^\circ}$	$\color{blue}{ \LARGE 1}$	$\color{blue}{ \LARGE 0}$	$\color{blue}{ \LARGE\text{non definito}}$	$\color{blue}{ \LARGE 0}$
$\frac{7\pi}{12} \text{ rad}; \ 105^\circ$	$\frac{\sqrt{6} + \sqrt{2}}{4}$	$\frac{\sqrt{2} - \sqrt{6}}{4}$	$-2 - \sqrt{3}$	$\sqrt{3}-2$
$\frac{3\pi}{5} \text{ rad}; \ 108^\circ$	$\frac{\sqrt{10+2 \sqrt{5}}}{4}$	$\frac{1-\sqrt{5}}{4}$	$-\sqrt{5+2\sqrt{5}}$	$-\frac{\sqrt{25-10\sqrt{5}}}{5}$
$\frac{5\pi}{8} \text{ rad}; \ 112^\circ 30'$	$\frac{\sqrt{2+\sqrt{2}}}{2}$	$-\frac{\sqrt{2-\sqrt{2}}}{2}$	$-\sqrt{2}-1$	$1-\sqrt{2}$
$\frac{2\pi}{3} \text{ rad}; \ 120^\circ$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$-\sqrt{3}$	$-\frac{\sqrt{3}}{3}$
$\frac{7\pi}{10} \text{ rad}; \ 126^\circ$	$\frac{\sqrt{5}+1}{4}$	$-\frac{\sqrt{10-2 \sqrt{5}}}{4}$	$-\frac{\sqrt{25+10\sqrt{5}}}{5}$	$-\sqrt{5-2\sqrt{5}}$
$\frac{3\pi}{4} \text{ rad}; \ 135^\circ$	$\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$-1$	$-1$
$\frac{4\pi}{5} \text{ rad}; \ 144^\circ$	$\frac{\sqrt{10-2 \sqrt{5}}}{4}$	$-\frac{\sqrt{5}+1}{4}$	$-\sqrt{5-2\sqrt{5}}$	$-\frac{\sqrt{25+10\sqrt{5}}}{5}$
$\frac{5\pi}{6} \text{ rad}; \ 150^\circ$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$	$-\sqrt{3}$
$\frac{7\pi}{8} \text{ rad}; \ 157^\circ 30'$	$\frac{\sqrt{2 -\sqrt{2}}}{2}$	$-\frac{\sqrt{2 +\sqrt{2}}}{2}$	$1-\sqrt{2}$	$-\sqrt{2} - 1$
$\frac{9\pi}{10} \text{ rad}; \ 162^\circ$	$\frac{\sqrt{5} - 1}{4}$	$-\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$	$-\sqrt{1 - \frac{2\sqrt{5}}{5}}$	$-\sqrt{5 +2\sqrt{5}}$
$\frac{11\pi}{12} \text{ rad}; \ 165^\circ$	$\frac{\sqrt{6} - \sqrt{2}}{4}$	$-\frac{\sqrt{6} + \sqrt{2}}{4}$	$\sqrt{3}-2$	$-2 - \sqrt{3}$
$\color{blue}{ \LARGE \pi \text{ rad}; \ 180^\circ}$	$\color{blue}{\LARGE 0}$	$\color{blue}{ \LARGE -1}$	$\color{blue}{\LARGE 0}$	$\color{blue}{\LARGE \text{non definito}}$
$\frac{13\pi}{12} \text{ rad}; \ 195^\circ$	$\frac{\sqrt{2} - \sqrt{6}}{4}$	$-\frac{\sqrt{6} + \sqrt{2}}{4}$	$2 - \sqrt{3}$	$2 + \sqrt{3}$
$\frac{11\pi}{10} \text{ rad}; \ 198^\circ$	$\frac{1-\sqrt{5}}{4}$	$-\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$	$\sqrt{1 - \frac{2\sqrt{5}}{5}}$	$\sqrt{5 +2\sqrt{5}}$
$\frac{9\pi}{8} \text{ rad}; \ 202^\circ 30'$	$-\frac{\sqrt{2-\sqrt{2}}}{2}$	$-\frac{\sqrt{2 +\sqrt{2}}}{2}$	$\sqrt{2} - 1$	$\sqrt{2} + 1$
$\frac{7\pi}{6} \text{ rad}; \ 210^\circ$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$	$\sqrt{3}$
$\frac{6\pi}{5} \text{ rad}; \ 216^\circ$	$-\frac{\sqrt{10-2 \sqrt{5}}}{4}$	$-\frac{\sqrt{5}+1}{4}$	$\sqrt{5-2\sqrt{5}}$	$\frac{\sqrt{25+10\sqrt{5}}}{5}$
$\frac{5\pi}{4} \text{ rad}; \ 225^\circ$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$1$	$1$
$\frac{13\pi}{10} \text{ rad}; \ 234^\circ$	$-\frac{\sqrt{5}+1}{4}$	$-\frac{\sqrt{10-2 \sqrt{5}}}{4}$	$\frac{\sqrt{25+10\sqrt{5}}}{5}$	$\sqrt{5-2\sqrt{5}}$
$\frac{4\pi}{3} \text{ rad}; \ 240^\circ$	$-\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$\sqrt{3}$	$\frac{\sqrt{3}}{3}$
$\frac{11\pi}{8} \text{ rad}; \ 247^\circ 30'$	$-\frac{\sqrt{2 +\sqrt{2}}}{2}$	$-\frac{\sqrt{2-\sqrt{2}}}{2}$	$\sqrt{2}+1$	$\sqrt{2}-1$
$\frac{7\pi}{5} \text{ rad}; \ 252^\circ$	$-\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$	$\frac{1-\sqrt{5}}{4}$	$\sqrt{5+2\sqrt{5}}$	$\frac{\sqrt{25-10\sqrt{5}}}{5}$
$\frac{17\pi}{12} \text{ rad}; \ 255^\circ$	$-\frac{\sqrt{6} + \sqrt{2}}{4}$	$\frac{\sqrt{2} - \sqrt{6}}{4}$	$2 + \sqrt{3}$	$2 - \sqrt{3}$
$\color{blue}{ \LARGE \frac{3\pi}{2} \text{ rad}; \ 270^\circ}$	$\color{blue}{ \LARGE -1}$	$\color{blue}{ \LARGE 0}$	$\color{blue}{ \LARGE \text{non definito}}$	$\color{blue}{ \LARGE 0}$
$\frac{19\pi}{12} \text{ rad}; \ 285^\circ$	$-\frac{\sqrt{6} + \sqrt{2}}{4}$	$\frac{\sqrt{6} - \sqrt{2}}{4}$	$-2 - \sqrt{3}$	$\sqrt{3}-2$
$\frac{8\pi}{5} \text{ rad}; \ 288^\circ$	$-\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$	$\frac{\sqrt{5}-1}{4}$	$-\sqrt{5+2\sqrt{5}}$	$-\frac{\sqrt{25-10\sqrt{5}}}{5}$
$\frac{13\pi}{8} \text{ rad}; \ 292^\circ 30'$	$-\frac{\sqrt{2 +\sqrt{2}}}{2}$	$\frac{\sqrt{2-\sqrt{2}}}{2}$	$-\sqrt{2}-1$	$1-\sqrt{2}$
$\frac{5\pi}{3} \text{ rad}; \ 300^\circ$	$-\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$-\sqrt{3}$	$-\frac{\sqrt{3}}{3}$
$\frac{17\pi}{10} \text{ rad}; \ 306^\circ$	$-\frac{\sqrt{5}+1}{4}$	$\frac{\sqrt{10-2 \sqrt{5}}}{4}$	$-\frac{\sqrt{25+10\sqrt{5}}}{5}$	$-\sqrt{5-2\sqrt{5}}$
$\frac{7\pi}{4} \text{ rad}; \ 315^\circ$	$-\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$-1$	$-1$
$\frac{9\pi}{5} \text{ rad}; \ 324^\circ$	$-\frac{\sqrt{10-2 \sqrt{5}}}{4}$	$\frac{\sqrt{5}+1}{4}$	$-\sqrt{5-2\sqrt{5}}$	$-\frac{\sqrt{25+10\sqrt{5}}}{5}$
$\frac{11\pi}{6} \text{ rad}; \ 330^\circ$	$-\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$	$-\sqrt{3}$
$\frac{15\pi}{8} \text{ rad}; \ 337^\circ 30'$	$-\frac{\sqrt{2-\sqrt{2}}}{2}$	$\frac{\sqrt{2 +\sqrt{2}}}{2}$	$1-\sqrt{2}$	$-\sqrt{2} - 1$
$\frac{19\pi}{10} \text{ rad}; \ 342^\circ$	$\frac{1-\sqrt{5}}{4}$	$\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$	$-\sqrt{1 - \frac{2\sqrt{5}}{5}}$	$-\sqrt{5 +2\sqrt{5}}$
$\frac{23\pi}{12} \text{ rad}; \ 345^\circ$	$\frac{\sqrt{2} - \sqrt{6}}{4}$	$\frac{\sqrt{6} + \sqrt{2}}{4}$	$\sqrt{3}-2$	$-2 - \sqrt{3}$
$\color{blue}{ \LARGE 2 \pi \text{ rad}; \ 360^\circ}$	$\color{blue}{ \LARGE 0}$	$\color{blue}{ \LARGE 1}$	$\color{blue}{ \LARGE 0}$	$\color{blue}{ \LARGE \text{non definito}}$