User:Margebtk

ENCUENTRE EL POLINOMIO DE GRADO 2 CUYA GRÁFICA PASA A TRAVÉS DE LOS PUNTOS DADOS:

2. (1,14), (2,22), (3,32)


 * $$\ a_0+a_1+a_2 = 14$$
 * $$\ a_0+2a_1+4a_2 = 22$$
 * $$\ a_0+3a_1+9a_2 = 32$$

$$\begin{pmatrix} 1 & 1 & 1 & 14 \\ 1 & 2 & 4 & 22 \\ 1 & 3 & 9 & 32\end{pmatrix}$$ $$\approx$$ $$f_2-f_1 \quad y\quad f_3-f_1$$

$$\begin{pmatrix} 1 & 1 & 1 & 14 \\ 0 & 1 & 3 & 8 \\ 0 & 2 & 8 & 18\end{pmatrix}$$ $$\approx$$ $$f_3-2f_2\quad y\quad f_1-f_2$$

$$\begin{pmatrix} 1 & 0 & -2 & 6 \\ 0 & 1 & 3 & 8 \\ 0 & 0 & 2 & 2\end{pmatrix}$$ $$\approx$$ $$\frac {1}{2}f_3$$

$$\begin{pmatrix} 1 & 0 & -2 & 6 \\ 0 & 1 & 3 & 8 \\ 0 & 0 & 1 & 1\end{pmatrix}$$ $$\approx$$ $$f_1+2f_3\quad y\quad f_2-3f_3$$

$$\begin{pmatrix} 1 & 0 & 0 & 8 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 1\end{pmatrix}$$


 * $$\ a_0+0+0 = 8$$
 * $$\ a_1+0 = 5$$
 * $$\ a_2 = 1$$


 * $$\ y = x^2+5x+8$$

4. (1,8) (3,26) (5,60)


 * $$\ a_0+a_1+a_2 = 8$$
 * $$\ a_0+3a_1+9a_2 = 26$$
 * $$\ a_0+5a_1+25a_2 = 60$$

$$\begin{pmatrix} 1 & 1 & 1 & 8 \\ 1 & 3 & 9 & 26 \\ 1 & 5 & 25 & 60\end{pmatrix}$$ $$\approx$$ $$f_2-f_1\quad y\quad f_3-f_1$$

$$\begin{pmatrix} 1 & 1 & 1 & 8 \\ 0 & 2 & 8 & 18 \\ 0 & 4 & 24 & 52\end{pmatrix}$$ $$\approx$$ $$\frac {1}{3}f_3\quad y\quad \frac {1}{2}f_2$$

$$\begin{pmatrix} 1 & 1 & 1 & 8 \\ 0 & 1 & 4 & 9 \\ 0 & 1 & 6 & 13\end{pmatrix}$$ $$\approx$$ $$f_3-f_2$$

$$\begin{pmatrix} 1 & 1 & 1 & 8 \\ 0 & 1 & 4 & 9 \\ 0 & 0 & 2 & 4\end{pmatrix}$$ $$\approx$$ $$\frac {1}{2}f_3\quad y\quad f_1-f_2$$

$$\begin{pmatrix} 1 & 0 & -3 & -1 \\ 0 & 1 & 4 & 9 \\ 0 & 0 & 1 & 2\end{pmatrix}$$ $$\approx$$ $$f_1+3f_3\quad y\quad f_2-4f_3$$

$$\begin{pmatrix} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2\end{pmatrix}$$


 * $$\ a_0 =5 $$
 * $$\ a_1 = 1$$
 * $$\ a_2 = 2$$


 * $$\ y = 2x^2+x+5$$

cuando x=2 


 * $$\ y = (2)2^2+2+5$$
 * $$\ y = 5+2+2(4)$$
 * $$\ y = 2+8+5$$
 * $$\ y = 15$$

6. (1,-3), (2,-1), (3,9), (4,33)


 * $$\ a_0+a_1+a_2+a_3 = -3$$
 * $$\ a_0+2a_1+4a_2+8a_3 = -1$$
 * $$\ a_0+3a_1+9a_2+27a_3 = 9$$
 * $$\ a_0+4a_1+16a_2+64a_3 = -3$$

$$\begin{pmatrix} 1 & 1 & 1 & 1 & -3 \\ 1 & 2 & 4 & 8 & -1 \\ 1 & 3 & 9 & 27 & 9 \\ 1 & 4 & 16 & 64 & 33\end{pmatrix}$$ $$\approx$$ $$f_2-f_1 \quad y\quad f_3-f_1 \quad y\quad f_4 - f_1$$

$$\begin{pmatrix} 1 & 1 & 1 & 1 & -3 \\ 0 & 1 & 3 & 7 & 2 \\ 0 & 2 & 8 & 26 & 12 \\ 0 & 3 & 15 & 63 & 36\end{pmatrix}$$ $$\approx$$ $$\frac{1}{2}f_3 \quad y\quad \frac {1}{3}f_4$$

$$\begin{pmatrix} 1 & 1 & 1 & 1 & -3 \\ 0 & 1 & 3 & 7 & 2 \\ 0 & 1 & 4 & 13 & 6 \\ 0 & 1 & 5 & 21 & 12\end{pmatrix}$$ $$\approx$$ $$f_1-f_2 \quad y\quad f_3-f_2 \quad y\quad f_4 - f_2$$

$$\begin{pmatrix} 1 & 0 & -2 & -6 & -5 \\ 0 & 1 & 3 & 7 & 2 \\ 0 & 0 & 1 & 6 & 4 \\ 0 & 0 & 2 & 14 & 10\end{pmatrix}$$ $$\approx$$ $$\frac{1}{2}f_4$$

$$\begin{pmatrix} 1 & 0 & -2 & -6 & -5 \\ 0 & 1 & 3 & 7 & 2 \\ 0 & 0 & 1 & 6 & 4 \\ 0 & 0 & 1 & 7 & 5\end{pmatrix}$$ $$\approx$$ $$f_1+2f_3 \quad y\quad f_2-3f_3 \quad y\quad f_4 - f_3$$

$$\begin{pmatrix} 1 & 0 & 0 & 6 & 3 \\ 0 & 1 & 0 & -11 & -10 \\ 0 & 0 & 1 & 6 & 4 \\ 0 & 0 & 0 & 1 & 1\end{pmatrix}$$ $$\approx$$ $$f_1-6f_4 \quad y\quad f_2+11f_4 \quad y\quad f_3 - 6f_4$$

$$\begin{pmatrix} 1 & 0 & 0 & 0 & -3 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & -2 \\ 0 & 0 & 0 & 1 & 1\end{pmatrix}$$


 * $$\ a_0 =-3 $$
 * $$\ a_1 =1 $$
 * $$\ a_2 =-2 $$
 * $$\ a_3 =1 $$


 * $$\ y = 2x^2+x+5$$

8.


 * $$\ union B: I_1+ I_2= I_3$$
 * $$\ union D: I_3= I_1+I_3$$


 * $$\ I_1+ I_2- I_3= 0$$
 * $$\ I_1+ 2I_3)= 9$$
 * $$\ I_1- 3(I_2)= 9- 17 $$
 * $$\ I_1- 3I_2 = -8 $$

$$\begin{pmatrix} 1 & 1 & -1 & 0 \\ 1 & 0 & 2 & 9 \\ 1 & -3 & 0 & -8\end{pmatrix}$$ $$\approx$$ $$f_2-f_1 \quad y\quad f_3-f_1 $$

$$\begin{pmatrix} 1 & 1 & -1 & 0 \\ 0 & -1 & 3 & 9 \\ 0 & -4 & 1 & -8\end{pmatrix}$$ $$\approx$$ $$f_1+f_2 \quad y\quad f_3-4f_2 $$

$$\begin{pmatrix} 1 & 0 & 2 & 9 \\ 0 & -1 & 3 & 9 \\ 0 & 0 & -11 & -44\end{pmatrix}$$ $$\approx$$ $$-f_2\quad y\quad -\frac {1}{11} $$

$$\begin{pmatrix} 1 & 0 & 2 & 9 \\ 0 & 1 & -3 & -9 \\ 0 & 0 & 1 & 4\end{pmatrix}$$ $$\approx$$ $$f_1-2f_3 \quad y\quad f_2+3f_3 $$

$$\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4\end{pmatrix}$$


 * $$\ I_1 = 1$$
 * $$\ I_2 = 3$$
 * $$\ I_3 = 4 $$

10.


 * $$\ union B: I_1+ I_2= I_3$$
 * $$\ union D: I_3= I_1+I_2$$


 * $$\ I_1+ I_2- I_3= 0$$
 * $$\ 2I_1+ 2I_3= 4$$
 * $$\ I_2+ 2I_3 + 3I_2= 2 $$


 * $$\ I_1+ I_2- I_3= 0$$
 * $$\ 2I_1+ 2I_3= 4$$
 * $$\ 4I_2+ 2I_3 = 2 $$

$$\begin{pmatrix} 1 & 1 & -1 & 0 \\ 2 & 0 & 2 & 4 \\ 0 & 4 & 2 & 2\end{pmatrix}$$ $$\approx$$ $$\ f_2-2f_1 \quad y\quad \frac {1}{2}f_3 $$

$$\begin{pmatrix} 1 & 1 & -1 & 0 \\ 0 & -2 & 4 & 4 \\ 0 & 2 & 1 & 1\end{pmatrix}$$ $$\approx$$ $$\ -\frac {1}{2}f_2 $$

$$\begin{pmatrix} 1 & 1 & -1 & 0 \\ 0 & 1 & -2 & -2 \\ 0 & 2 & 1 & 1\end{pmatrix}$$ $$\approx$$ $$\ f_1-2f_2 \quad y\quad f_3-2f_2 $$

$$\begin{pmatrix} 1 & 0 & 1 & 2 \\ 0 & 1 & -2 & -2 \\ 0 & 0 & 5 & 5\end{pmatrix}$$ $$\approx$$ $$\ \frac{1}{5}f_3 $$

$$\begin{pmatrix} 1 & 0 & 1 & 2 \\ 0 & 1 & -2 & -2 \\ 0 & 0 & 1 & 1\end{pmatrix}$$ $$\approx$$ $$\ f_2+2f_3 $$

$$\begin{pmatrix} 1 & 0 & 1 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1\end{pmatrix}$$


 * $$\ I_1 = 1$$
 * $$\ I_2 = 0$$
 * $$\ I_3 = 1 $$

16.
 * $$\ A. 100 + 200 = x_1 + x_4$$
 * $$\ B. x_1 + x_2 = 100 + 50$$
 * $$\ C. 50 + 50 = x_3 + x_2$$
 * $$\ D. x_3 + x_4 = 100 + 50 $$


 * $$\ A. x_1 + x_4 = 300$$
 * $$\ B. x_1 + x_2 = 150$$
 * $$\ C. x_2 + x_3 = 100$$
 * $$\ D. x_3 + x_4 = 150 $$

$$\begin{pmatrix} 1 & 0 & 0 & 1 & 300 \\ 1 & 1 & 0 & 0 & 250 \\ 0 & 1 & 1 & 0 & 100 \\ 0 & 0 & 1 & 1 & 150\end{pmatrix}$$ $$\approx$$ $$\ f_2-f_1 $$

$$\begin{pmatrix} 1 & 0 & 0 & 1 & 300 \\ 0 & 1 & 0 & -1 & -50 \\ 0 & 1 & 1 & 0 & 100 \\ 0 & 0 & 1 & 1 & 150\end{pmatrix}$$ $$\approx$$ $$\ f_3-f_2 $$

$$\begin{pmatrix} 1 & 0 & 0 & 1 & 300 \\ 0 & 1 & 0 & -1 & -50 \\ 0 & 0 & 1 & 1 & 150 \\ 0 & 0 & 1 & 1 & 150\end{pmatrix}$$ $$\approx$$ $$\ f_4-f_3 $$

$$\begin{pmatrix} 1 & 0 & 0 & 1 & 300 \\ 0 & 1 & 0 & -1 & -50 \\ 0 & 0 & 1 & 1 & 150 \\ 0 & 0 & 0 & 0 & 0\end{pmatrix}$$


 * $$\ x_1 + x_4 = 300$$
 * $$\ x_2 - x_4 = -50$$
 * $$\ x_3 + x_4 = 150$$


 * $$\ x_1 = 300 - x_4$$
 * $$\ x_2 = x_4 - 50$$
 * $$\ x_3 = 150 - x_4$$

Flujo AB


 * $$\ x_1 = 300 - x_4$$
 * $$\ x_4 = 300 - x_1$$


 * $$\ 300 - x_1 \ge 0$$
 * $$\ 300 \ge x_1$$

La afluencia mínima es de 300 entre A y B

17.


 * $$\ A. x_4 + 100 = x_1$$
 * $$\ B. x_1 = 200 + x_2$$
 * $$\ C. 150 + x_2 = x_3$$
 * $$\ D. x_3 = 50 + x_4$$


 * $$\ x_1 - x_4 = 100$$
 * $$\ x_1 - x_2 = 200$$
 * $$\ x_3 - x_2 = 150$$
 * $$\ x_3 - x_4 = 50$$

$$\begin{pmatrix} 1 & 0 & 0 & -1 & 100 \\ 1 & -1 & 0 & 0 & 200 \\ 0 & -1 & 1 & 0 & 150 \\ 0 & 0 & 1 & -1 & 50\end{pmatrix}$$ $$\approx$$ $$\ f_2-f_1 $$

$$\begin{pmatrix} 1 & 0 & 0 & -1 & 100 \\ 0 & -1 & 0 & 1 & 100 \\ 0 & -1 & 1 & 0 & 150 \\ 0 & 0 & 1 & -1 & 50\end{pmatrix}$$ $$\approx$$ $$\ -f_2 $$

$$\begin{pmatrix} 1 & 0 & 0 & -1 & 100 \\ 0 & 1 & 0 & -1 & -100 \\ 0 & -1 & 1 & 0 & 150 \\ 0 & 0 & 1 & -1 & 50\end{pmatrix}$$ $$\approx$$ $$\ f_3 + f_2 $$

$$\begin{pmatrix} 1 & 0 & 0 & -1 & 100 \\ 0 & 1 & 0 & -1 & -100 \\ 0 & 0 & 1 & -1 & 50 \\ 0 & 0 & 1 & -1 & 50\end{pmatrix}$$ $$\approx$$ $$\ f_4 - f_3 $$

$$\begin{pmatrix} 1 & 0 & 0 & -1 & 100 \\ 0 & 1 & 0 & -1 & -100 \\ 0 & 0 & 1 & -1 & 50 \\ 0 & 0 & 0 & 0 & 0\end{pmatrix}$$


 * $$\ x_1 - x_4 = 100$$
 * $$\ x_2 - x_4 = -100$$
 * $$\ x_3 - x_4 = 50$$


 * $$\ - x_4 = -100 - x_2$$
 * $$\ x_4 = 100 + x_2$$
 * $$\ x_2 + 100 \ge 0 $$
 * $$\ x_2 \ge -100 $$


 * No es posible porque no puede haber una vía negativa

Ejercicio de la matriz de entradas y salidas

a) La cantidad de electricidad que se consume en la producción del valor de $1 de acero


 * $$\ 1 / 0.25 $$
 * $$\ 4$$

b) La cantidad de acero que se consume en la producción del valor de $1 en la industria automotriz


 * $$\ 1 / 0.40 $$
 * $$\ 2.5 $$

c) El mayor consumo de carbón

Es 0.3 que equivale a la electricidad

d) El mayor consumo de electricidad

Es 0.25 que equivale al acero

e) La industria de la que depende más la industria automotriz

Es 0.10 que equivale al acero y 0.10 que equivale a la química

7.

$$ A = \begin{pmatrix} 0.20 & 0.40 \\ 0.50 & 0.10 \end{pmatrix}$$ $$ X = \begin{pmatrix} 8 \\ 10 \end{pmatrix}$$


 * $$ X = AX + D$$
 * $$ X - AX = D$$

$$ D = \begin{pmatrix} 8 \\ 10 \end{pmatrix} - \begin{pmatrix} 0.20 & 0.40 \\ 0.50 & 0.10 \end{pmatrix}\begin{pmatrix} 8 \\ 10 \end{pmatrix}$$

$$ D = \begin{pmatrix} 8 \\ 10 \end{pmatrix} - \begin{pmatrix} 9.6 \\ 5 \end{pmatrix}$$

$$ D = \begin{pmatrix} 2.4 \\ 5 \end{pmatrix}$$

Es la cantidad disponible de cada industria para el sector público

8.

$$ A = \begin{pmatrix} 0.10 & 0.20 & 0.30 \\ 0 & 0.10 & 0.40 \\ 0.50 & 0.40 & 0.20 \end{pmatrix}$$ $$ X = \begin{pmatrix} 10 \\ 10 \\ 20\end{pmatrix}$$


 * $$ X = AX + D$$
 * $$ X - AX = D$$

$$ D = \begin{pmatrix} 10 \\ 10 \\ 20\end{pmatrix} - \begin{pmatrix} 0.10 & 0.20 & 0.30 \\ 0 & 0.10 & 0.40 \\ 0.50 & 0.40 & 0.20 \end{pmatrix}\begin{pmatrix} 10 \\ 10 \\ 20\end{pmatrix}$$

$$ D = \begin{pmatrix} 10 \\ 10 \\ 20\end{pmatrix} - \begin{pmatrix} 9 \\ 9 \\ 13\end{pmatrix}$$

$$ D = \begin{pmatrix} 1 \\ 1 \\ 7\end{pmatrix}$$

Es la cantidad disponible de cada industria para el sector público