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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\text{r$$

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

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

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

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

$$\log_a x = \dfrac{\log_b x}{\log_b a}$$

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

$$(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \dots + \binom{n}{r}a^{n-r}b^r + \dots + b^n$$

$$\binom{n}{r} = {}^{n}C_r = \dfrac{n!}{r!,(n-r)!}$$

$${}^{n}C_r = \dfrac{n!}{r!,(n-r)!}$$

$${}^{n}P_r = \dfrac{n!}{(n-r)!}$$

$$(a+b)^n = a^n\left(1 + n\left(\dfrac{b}{a}\right) + \dfrac{n(n-1)}{2!}\left(\dfrac{b}{a}\right)^2 + \dots\right)$$

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

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

$$\bigl[r(\cos\theta + \mathrm{i}\operatorname{sen}\theta)\bigr]^n = r^n(\cos n\theta + \mathrm{i}\operatorname{sen} n\theta) = r^n\mathrm{e}^{\mathrm{i}n\theta} = r^n\operatorname{cis} n\theta$$

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

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

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

$$ax^2 + bx + c = 0 \implies x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a},\ a \neq 0$$

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

$$a^x = \mathrm{e}^{x\ln a};\quad \log_a a^x = x = a^{\log_a x},\ \text{donde } a,\ x > 0,\ a \neq 1$$

$$\text{La suma es } \dfrac{-a_{n-1}}{a_n}$$

$$\text{el producto es } \dfrac{(-1)^n a_0}{a_n}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\operatorname{sen} 2\theta = 2\operatorname{sen}\theta\cos\theta$$

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

$$\sec\theta = \dfrac{1}{\cos\theta}$$

$$\operatorname{cosec}\theta = \dfrac{1}{\operatorname{sen}\theta}$$

$$1 + \tan^2\theta = \sec^2\theta$$

$$1 + \cot^2\theta = \operatorname{cosec}^2\theta$$

$$\operatorname{sen}(A \pm B) = \operatorname{sen} A\cos B \pm \cos A\operatorname{sen} B$$

$$\cos(A \pm B) = \cos A\cos B \mp \operatorname{sen} A\operatorname{sen} B$$

$$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$$

$$\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}$$

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

$$\boldsymbol{v} \cdot \boldsymbol{w} = v_1 w_1 + v_2 w_2 + v_3 w_3$$

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

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

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

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

$$\dfrac{x - x_0}{l} = \dfrac{y - y_0}{m} = \dfrac{z - z_0}{n}$$

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

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

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

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

$$\boldsymbol{r} \cdot \boldsymbol{n} = \boldsymbol{a} \cdot \boldsymbol{n}$$

$$ax + by + cz = d$$

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

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

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

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

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

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

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

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

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

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

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

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

$$z = \dfrac{x - \mu}{\sigma}$$

$$\mathrm{P}(B \mid A) = \dfrac{\mathrm{P}(B),\mathrm{P}(A \mid B)}{\mathrm{P}(B),\mathrm{P}(A \mid B) + \mathrm{P}(B'),\mathrm{P}(A \mid B')}$$

$$\mathrm{P}(B_i \mid A) = \dfrac{\mathrm{P}(B_i),\mathrm{P}(A \mid B_i)}{\mathrm{P}(B_1),\mathrm{P}(A \mid B_1) + \mathrm{P}(B_2),\mathrm{P}(A \mid B_2) + \mathrm{P}(B_3),\mathrm{P}(A \mid B_3)}$$

$$\sigma^2 = \dfrac{\sum_{i=1}^{k} f_i (x_i - \mu)^2}{n} = \dfrac{\sum_{i=1}^{k} f_i x_i^2}{n} - \mu^2$$

$$\sigma = \sqrt{\dfrac{\sum_{i=1}^{k} f_i (x_i - \mu)^2}{n}}$$

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

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

$$\mathrm{E}(X) = \mu = \int_{-\infty}^{\infty} x f(x),\mathrm{d}x$$

$$\operatorname{Var}(X) = \mathrm{E}\bigl[(X - \mu)^2\bigr] = \mathrm{E}(X^2) - \bigl[\mathrm{E}(X)\bigr]^2$$

$$\operatorname{Var}(X) = \sum (x - \mu)^2 \mathrm{P}(X = x) = \sum x^2 \mathrm{P}(X = x) - \mu^2$$

$$\operatorname{Var}(X) = \int_{-\infty}^{\infty} (x - \mu)^2 f(x),\mathrm{d}x = \int_{-\infty}^{\infty} x^2 f(x),\mathrm{d}x - \mu^2$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$y = f(x) \implies \dfrac{\mathrm{d}y}{\mathrm{d}x} = f'(x) = \lim_{h \to 0}\left(\dfrac{f(x+h) - f(x)}{h}\right)$$

$$f(x) = \tan x \implies f'(x) = \sec^2 x$$

$$f(x) = \sec x \implies f'(x) = \sec x \tan x$$

$$f(x) = \operatorname{cosec} x \implies f'(x) = -\operatorname{cosec} x \cot x$$

$$f(x) = \cot x \implies f'(x) = -\operatorname{cosec}^2 x$$

$$f(x) = a^x \implies f'(x) = a^x (\ln a)$$

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

$$f(x) = \arcsin x \implies f'(x) = \dfrac{1}{\sqrt{1 - x^2}}$$

$$f(x) = \arccos x \implies f'(x) = -\dfrac{1}{\sqrt{1 - x^2}}$$

$$f(x) = \arctan x \implies f'(x) = \dfrac{1}{1 + x^2}$$