$$A = bh,\ \text{donde } b \text{ es la base y } h \text{ es la altura}$$

$$A = \frac{1}{2}(bh),\ \text{donde } b \text{ es la base y } h \text{ es la altura}$$

$$A = \frac{1}{2}(a+b)h,\ \text{donde } a \text{ y } b \text{ son los lados paralelos y } h \text{ la altura}$$

$$A = \pi r^{2},\ \text{donde } r \text{ es el radio}$$

$$C = 2\pi r,\ \text{donde } r \text{ es el radio}$$

$$V = lwh,\ \text{donde } l \text{ es la longitud, } w \text{ el ancho y } h \text{ la altura}$$

$$V = \pi r^{2}h,\ \text{donde } r \text{ es el radio y } h \text{ la altura}$$

$$V = Ah,\ \text{donde } A \text{ es el área de la sección transversal y } h \text{ la altura}$$

$$A = 2\pi rh,\ \text{donde } r \text{ es el radio y } h \text{ la altura}$$

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

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

$$\text{Las soluciones de } ax^{2} + bx + c = 0 \text{ son}$$

$$x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a},\ a \neq 0$$

$$u_n = u_1 + (n-1)d$$

$$S_n = \dfrac{n}{2}\bigl(2u_1 + (n-1)d\bigr);\quad S_n = \dfrac{n}{2}(u_1 + u_n)$$

$$u_n = u_1 r^{n-1}$$

$$S_n = \dfrac{u_1(r^n - 1)}{r - 1} = \dfrac{u_1(1 - r^n)}{1 - r},\ r \neq 1$$

$$FV = PV \times \left(1 + \dfrac{r}{100k}\right)^{kn}$$

$$\text{FV: valor futuro,\quad PV: valor presente (actual)}$$

$$\text{n: número de años,\quad k: número de periodos de composición al año}$$

$$\text{r$$

$$a^x = b \iff x = \log_a b,\ \text{donde } a > 0,\ b > 0,\ a \neq 1$$

$$\varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right| \times 100$$

$$\log_a xy = \log_a x + \log_a y$$

$$\log_a \dfrac{x}{y} = \log_a x - \log_a y$$

$$\log_a x^m = m\log_a x$$

$$\text{para } a,\ x,\ y > 0$$

$$S_\infty = \dfrac{u_1}{1 - r},\ |r| < 1$$

$$z = a + b\mathrm{i}$$

$$\Delta = b^2 - 4ac$$

$$z = r(\cos\theta + \mathrm{i}\operatorname{sen}\theta) = r\mathrm{e}^{\mathrm{i}\theta} = r\operatorname{cis}\theta$$

$$\boldsymbol{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \implies \det\boldsymbol{A} = |\boldsymbol{A}| = ad - bc$$

$$\boldsymbol{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \implies \boldsymbol{A}^{-1} = \dfrac{1}{\det\boldsymbol{A}}\begin{pmatrix} d & -b \ -c & a \end{pmatrix},\ ad \neq bc$$

$$\boldsymbol{M}^n = \boldsymbol{P}\boldsymbol{D}^n\boldsymbol{P}^{-1},\ \text{donde } \boldsymbol{P} \text{ es la matriz de vectores propios y } \boldsymbol{D} \text{ la matriz diagonal de valores propios}$$

$$y = mx + c;\quad ax + by + d = 0;\quad y - y_1 = m(x - x_1)$$

$$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

$$f(x) = ax^2 + bx + c \implies \text{el eje de simetría es } x = -\dfrac{b}{2a}$$

$$f(x) = \dfrac{L}{1 + C\mathrm{e}^{-kx}},\ L,\ k,\ C > 0$$

$$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}$$

$$\left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2},\ \dfrac{z_1 + z_2}{2}\right)$$

$$V = \dfrac{1}{3}Ah,\ \text{donde } A \text{ es el área de la base y } h \text{ la altura}$$

$$V = \dfrac{1}{3}\pi r^2 h,\ \text{donde } r \text{ es el radio y } h \text{ la altura}$$

$$A = \pi r l,\ \text{donde } r \text{ es el radio y } l \text{ la generatriz}$$

$$V = \dfrac{4}{3}\pi r^3,\ \text{donde } r \text{ es el radio}$$

$$A = 4\pi r^2,\ \text{donde } r \text{ es el radio}$$

$$\dfrac{a}{\operatorname{sen} A} = \dfrac{b}{\operatorname{sen} B} = \dfrac{c}{\operatorname{sen} C}$$

$$c^2 = a^2 + b^2 - 2ab\cos C;\quad \cos C = \dfrac{a^2 + b^2 - c^2}{2ab}$$

$$A = \dfrac{1}{2}ab\operatorname{sen} C$$

$$l = \dfrac{\theta}{360} \times 2\pi r,\ \text{donde } \theta \text{ es el ángulo en grados y } r \text{ el radio}$$

$$A = \dfrac{\theta}{360} \times \pi r^2,\ \text{donde } \theta \text{ es el ángulo en grados y } r \text{ el radio}$$

$$l = r\theta,\ \text{donde } r \text{ es el radio, } \theta \text{ el ángulo en radianes}$$

$$A = \dfrac{1}{2}r^2\theta,\ \text{donde } r \text{ es el radio, } \theta \text{ el ángulo en radianes}$$

$$\cos^2\theta + \operatorname{sen}^2\theta = 1$$

$$\tan\theta = \dfrac{\operatorname{sen}\theta}{\cos\theta}$$

$$\begin{pmatrix} \cos 2\theta & \operatorname{sen} 2\theta \ \operatorname{sen} 2\theta & -\cos 2\theta \end{pmatrix},\ \text{simetría respecto a la recta } y = (\tan\theta)x$$

$$\begin{pmatrix} k & 0 \ 0 & 1 \end{pmatrix},\ \text{estiramiento horizontal (paralelo al eje } x\text{), factor de escala } k$$

$$\begin{pmatrix} 1 & 0 \ 0 & k \end{pmatrix},\ \text{estiramiento vertical (paralelo al eje } y\text{), factor de escala } k$$

$$\begin{pmatrix} k & 0 \ 0 & k \end{pmatrix},\ \text{homotecia con factor de escala } k \text{ y centro } (0,,0)$$

$$\begin{pmatrix} \cos\theta & -\operatorname{sen}\theta \ \operatorname{sen}\theta & \cos\theta \end{pmatrix},\ \text{rotación antihoraria de ángulo } \theta \text{ alrededor del origen } (\theta > 0)$$

$$\begin{pmatrix} \cos\theta & \operatorname{sen}\theta \ -\operatorname{sen}\theta & \cos\theta \end{pmatrix},\ \text{rotación horaria de ángulo } \theta \text{ alrededor del origen } (\theta > 0)$$

$$|\boldsymbol{v}| = \sqrt{v_1^{,2} + v_2^{,2} + v_3^{,2}},\ \text{donde } \boldsymbol{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix}$$

$$\boldsymbol{r} = \boldsymbol{a} + \lambda\boldsymbol{b}$$

$$x = x_0 + \lambda l,\quad y = y_0 + \lambda m,\quad z = z_0 + \lambda n$$

$$\boldsymbol{v} \cdot \boldsymbol{w} = v_1 w_1 + v_2 w_2 + v_3 w_3,\ \text{donde } \boldsymbol{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix},\ \boldsymbol{w} = \begin{pmatrix} w_1 \ w_2 \ w_3 \end{pmatrix}$$

$$\boldsymbol{v} \cdot \boldsymbol{w} = |\boldsymbol{v}|,|\boldsymbol{w}|\cos\theta,\ \text{donde } \theta \text{ es el ángulo que forman } \boldsymbol{v} \text{ y } \boldsymbol{w}$$

$$\cos\theta = \dfrac{v_1 w_1 + v_2 w_2 + v_3 w_3}{|\boldsymbol{v}|,|\boldsymbol{w}|}$$

$$\boldsymbol{v} \times \boldsymbol{w} = \begin{pmatrix} v_2 w_3 - v_3 w_2 \ v_3 w_1 - v_1 w_3 \ v_1 w_2 - v_2 w_1 \end{pmatrix},\ \text{donde } \boldsymbol{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix},\ \boldsymbol{w} = \begin{pmatrix} w_1 \ w_2 \ w_3 \end{pmatrix}$$

$$|\boldsymbol{v} \times \boldsymbol{w}| = |\boldsymbol{v}|,|\boldsymbol{w}|\operatorname{sen}\theta,\ \text{donde } \theta \text{ es el ángulo que forman } \boldsymbol{v} \text{ y } \boldsymbol{w}$$

$$A = |\boldsymbol{v} \times \boldsymbol{w}|,\ \text{donde } \boldsymbol{v} \text{ y } \boldsymbol{w} \text{ constituyen dos lados adyacentes del paralelogramo}$$

$$\mathrm{RIC} = Q_3 - Q_1$$

$$\bar{x} = \dfrac{\sum_{i=1}^{k} f_i x_i}{n},\ \text{donde } n = \sum_{i=1}^{k} f_i$$

$$\mathrm{P}(A) = \dfrac{n(A)}{n(U)}$$

$$\mathrm{P}(A) + \mathrm{P}(A') = 1$$

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

$$\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B)$$

$$\mathrm{P}(A \mid B) = \dfrac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)}$$

$$\mathrm{P}(A \cap B) = \mathrm{P}(A),\mathrm{P}(B)$$

$$\mathrm{E}(X) = \sum x,\mathrm{P}(X = x)$$

$$X \sim \mathrm{B}(n,, p)$$

$$\mathrm{E}(X) = np$$

$$\operatorname{Var}(X) = np(1 - p)$$

$$\mathrm{E}(aX + b) = a\mathrm{E}(X) + b$$

$$\operatorname{Var}(aX + b) = a^2\operatorname{Var}(X)$$

$$\mathrm{E}(a_1 X_1 \pm a_2 X_2 \pm \dots \pm a_n X_n) = a_1\mathrm{E}(X_1) \pm a_2\mathrm{E}(X_2) \pm \dots \pm a_n\mathrm{E}(X_n)$$

$$\operatorname{Var}(a_1 X_1 \pm a_2 X_2 \pm \dots \pm a_n X_n) = a_1^2\operatorname{Var}(X_1) + a_2^2\operatorname{Var}(X_2) + \dots + a_n^2\operatorname{Var}(X_n)$$

$$s_{n-1}^{,2} = \dfrac{n}{n-1}s_n^{,2}$$

$$X \sim \operatorname{Po}(m)$$

$$\mathrm{E}(X) = m$$

$$\operatorname{Var}(X) = m$$

$$\boldsymbol{T}^n \boldsymbol{s}_0 = \boldsymbol{s}_n,\ \text{donde } \boldsymbol{s}_0 \text{ es el estado inicial}$$

$$f(x) = x^n \implies f'(x) = nx^{n-1}$$

$$\int x^n,\mathrm{d}x = \dfrac{x^{n+1}}{n+1} + C,\ n \neq -1$$

$$A = \int_a^b y,\mathrm{d}x$$

$$\int_a^b y,\mathrm{d}x \approx \dfrac{1}{2}h\bigl((y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1})\bigr),\ \text{donde } h = \dfrac{b - a}{n}$$

$$f(x) = \operatorname{sen} x \implies f'(x) = \cos x$$

$$f(x) = \cos x \implies f'(x) = -\operatorname{sen} x$$

$$f(x) = \tan x \implies f'(x) = \dfrac{1}{\cos^2 x}$$

$$f(x) = \mathrm{e}^x \implies f'(x) = \mathrm{e}^x$$

$$f(x) = \ln x \implies f'(x) = \dfrac{1}{x}$$

$$y = g(u),\ \text{donde } u = f(x) \implies \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\mathrm{d}y}{\mathrm{d}u} \times \dfrac{\mathrm{d}u}{\mathrm{d}x}$$

$$y = uv \implies \dfrac{\mathrm{d}y}{\mathrm{d}x} = u\dfrac{\mathrm{d}v}{\mathrm{d}x} + v\dfrac{\mathrm{d}u}{\mathrm{d}x}$$

$$y = \dfrac{u}{v} \implies \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{v\dfrac{\mathrm{d}u}{\mathrm{d}x} - u\dfrac{\mathrm{d}v}{\mathrm{d}x}}{v^2}$$

$$\int \dfrac{1}{x},\mathrm{d}x = \ln|x| + C$$

$$\int \operatorname{sen} x,\mathrm{d}x = -\cos x + C$$

$$\int \cos x,\mathrm{d}x = \operatorname{sen} x + C$$

$$\int \dfrac{1}{\cos^2 x},\mathrm{d}x = \tan x + C$$

$$\int \mathrm{e}^x,\mathrm{d}x = \mathrm{e}^x + C$$

$$A = \int_a^b |y|,\mathrm{d}x \ \text{o}\ A = \int_a^b |x|,\mathrm{d}y$$

$$V = \int_a^b \pi y^2,\mathrm{d}x \ \text{o}\ V = \int_a^b \pi x^2,\mathrm{d}y$$

$$a = \dfrac{\mathrm{d}v}{\mathrm{d}t} = \dfrac{\mathrm{d}^2 s}{\mathrm{d}t^2} = v\dfrac{\mathrm{d}v}{\mathrm{d}s}$$

$$\text{distancia} = \int_{t_1}^{t_2} |v(t)|,\mathrm{d}t$$

$$\text{desplazamiento} = \int_{t_1}^{t_2} v(t),\mathrm{d}t$$

$$y_{n+1} = y_n + h \times f(x_n,, y_n);\quad x_{n+1} = x_n + h,\ \text{donde } h \text{ es una constante (‘paso’)}$$

$$x_{n+1} = x_n + h \times f_1(x_n,, y_n,, t_n)$$

$$y_{n+1} = y_n + h \times f_2(x_n,, y_n,, t_n)$$

$$t_{n+1} = t_n + h,\ \text{donde } h \text{ es una constante (‘paso’)}$$

$$\boldsymbol{x} = A\mathrm{e}^{\lambda_1 t}\boldsymbol{p}_1 + B\mathrm{e}^{\lambda_2 t}\boldsymbol{p}_2$$