User:Jorgemontes

=Límites=

1) $$\lim_{x\to 0}\frac {senx}{2x}$$
 * $$\frac {1}{2} \lim_{x\to 0}\frac {senx}{x}$$
 * $$\frac {1}{2}.1$$
 * $$\frac {1}{2}$$ by:Jorge Montes Jorgemontes

2)$$\lim_{x\to -1}\frac {x^3-4x^2+x+6}{x+1}$$
 * $$\lim_{x\to -1}\frac {(x+1)(x^2-5x+6)}{x+1}$$
 * $$\lim_{x\to -1}\ {x^2-5x+6}$$
 * $$\ {(-1)^2-5(-1)+6}$$
 * $$\ {1+5+6}$$
 * $$\ {12}$$                                      by:Jorge Montes Jorgemontes

3)$$\lim_{x\to 3}\frac {x^4-18x^2+81}{(x-3)^2}$$
 * $$\lim_{x\to 3}\frac {(x-3)(x^3+3x^2-9x-27)}{(x-3)^2}$$
 * $$\lim_{x\to 3}\frac {x^3+3x^2-9x-27}{x-3}$$
 * $$\lim_{x\to 3}\frac {(x-3)(x^2+6x+9)}{x-3}$$
 * $$\lim_{x\to 3}\ {x^2+6x+9}$$
 * $$\ {(3)^2+6(3)+9}$$
 * $$\ {9+18+9}$$
 * $$\ {36}$$                                       by:Jorge Montes Jorgemontes

4)$$\lim_{h\to 0}\frac {(x+h)^2-x^2}{h}$$
 * $$\lim_{h\to 0}\frac {x^2+2hx+h^2-x^2}{h}$$
 * $$\lim_{h\to 0}\frac {2hx+h^2}{h}$$
 * $$\lim_{h\to 0}\frac {(h)(2x+h)}{h}$$
 * $$\lim_{h\to 0}\ {(2x+h)}$$
 * $$\ {(2x)}$$                                     by:Jorge Montes Jorgemontes

5) $$\lim_{x \to 0} \frac {tan^2x}{senx}$$
 * $$\lim_{x \to 0} \frac {(sen^2x/cos^2x)}{senx}$$
 * $$\lim_{x \to 0} \frac {2senx cosx}{(senx)(cos^2x-sen^2x)}$$
 * $$2\lim_{x \to 0} \frac {cosx}{(cos^2x-sen^2x)}$$
 * $$2\lim_{x \to 0} \frac {cosx}{cos^2x-(1-cos^2x)}$$
 * $$2\lim_{x \to 0} \frac {cosx}{2cos^2x-1}$$
 * $$\frac $$
 * $$\frac {(2)(1)}{(2)(1)-1}$$
 * $$\ 2$$                                           by:Jorge Montes Jorgemontes

6)$$\frac {1}{\sqrt [2]{3x}}$$
 * $$\lim_{h\to 0}\frac {\frac {1}{\sqrt [2]{3x+3h}}-\frac {1}{\sqrt [2]{3x}}}{h}$$
 * $$\lim_{h\to 0}\frac {\frac {\sqrt {3x}-\sqrt {3x+3h}}{(\sqrt {3x+3h})(\sqrt {3x})}}{h}$$
 * $$\lim_{h\to 0}\frac {\sqrt {3x}-\sqrt {3x+3h}}{(h)((\sqrt {3x+3h})(\sqrt {3x}))}$$
 * $$\lim_{h\to 0}\frac {\sqrt {3x}-\sqrt {3x+3h}}{(h)((\sqrt {3x+3h})(\sqrt {3x}))}$$
 * $$\lim_{h\to 0}\frac {\sqrt {3x}-\sqrt {3x+3h}}{(\sqrt {h^2})((\sqrt {3x+3h})(\sqrt {3x}))}$$
 * $$\lim_{h\to 0}\frac {\sqrt {3x}-\sqrt {3x+3h}}{(\sqrt {(h^2)(3x^2+9hx)})}$$
 * $$\lim_{h\to 0}\frac {\sqrt {3x}-\sqrt {3x+3h}}{\sqrt {(3x^2h^2+9h^3x)}}$$
 * $$\lim_{h\to 0}\frac {\sqrt {3x}-\sqrt {3x+3h}}{\sqrt {3x^2h^2+9h^3x}}$$
 * $$\lim_{h\to 0}\frac {\sqrt {3x}-\sqrt {3x+3h}}{\sqrt {(3h^2x)(3x+3h)}}$$

=Derivadas=

1)$$\lim_{h\to 0}\frac {(2+h)^2-4}{h}$$
 * $$\lim_{h\to 0}\frac {4+4h+h^2-4}{h}$$
 * $$\lim_{h\to 0}\frac {4h+h^2}{h}$$
 * $$\lim_{h\to 0}\frac {h(4+h)}{h}$$
 * $$\lim_{h\to 0}\ {4+h}$$
 * $$\ {4+0}$$
 * $$\ {4}$$                           by:Jorge Montes

Jorgemontes

2)Derivar:$$\ F(x)= \frac {(x+1)^2}{x-1}$$
 * $$\lim_{h\to 0}\frac {\frac {(((x+h)+1)^2)}{(x+h)-1}-\frac {(x+1)^2}{x-1}}{h}$$
 * $$\lim_{h\to 0}\frac {\frac {hx^2+h^2x-3h-2hx}{(x+h-1)(x-1)}}{h}$$
 * $$\lim_{h\to 0}\frac {(h)(x^2+hx-3-2x)}{(h)((x+h-1)(x-1))}$$
 * $$\lim_{h\to 0}\frac {x^2+hx-3-2x}{(x+h-1)(x-1)}$$
 * $$\frac {x^2-2x-3}{(x-1)^2}$$ by: jorge montes Jorgemontes

3)Hallar la derivada de:
 * y=x
 * $$\ y'=1$$
 * $$\ lim_{x\to 0}\frac {f(x+h)-F(x)}{h}$$
 * $$\ lim_{x\to 0}\frac {x+h-x}{h}$$
 * $$\ lim_{x\to 0}\frac {h}{h}$$
 * $$\ 1$$

by: jorge montes and Zorraidorsito Jorgemontes

4)Derive:$$\frac {6}{x^2+1}$$
 * $$\lim_{h\to 0}\frac {\frac {6}{(x+h)^2+1}-\frac {6}{x^2+1}}{h}$$
 * $$\lim_{h\to 0}\frac {\frac {6x^2+6-6x^2-12hx-6h^2-6}{((x+h)^2+1)(x^2+1)}}{h}$$
 * $$\lim_{h\to 0}\frac {\frac {-6h^2-12hx}{((x+h)^2+1)(x^2+1)}}{h}$$
 * $$\lim_{h\to 0}\frac {-6h^2-12hx}{(h)(((x+h)^2+1)(x^2+1))}$$
 * $$\lim_{h\to 0}\frac {(h)(-6h-12x)}{(h)(((x+h)^2+1)(x^2+1))}$$
 * $$\lim_{h\to 0}\frac {-6h-12x}{((x+h)^2+1)(x^2+1)}$$
 * $$\frac {-6(0)-12x}{((x+(0))^2+1)(x^2+1)}$$
 * $$\frac {-12x}{(x^2+1)^2}$$ by: jorge Montes Jorgemontes

5)Derive por implícita: $$\ {y^2-x^2=1}$$
 * $$\ {\frac {d}{dx}y^2-\frac {d}{dx}x^2=\frac {d}{dx}1}$$
 * $$\ {\frac {d}{dx}2y-\frac {d}{dx}2x=\frac {d}{dx}0}$$
 * $$\ {\frac {d}{dx}2y-\frac {d}{dx}2x=0}$$
 * $$\ {\frac {d}{dx}(2y-2x)=0}$$
 * $$\ {\frac {d}{dx}(2y)=2x}$$
 * $$\ {\frac {d}{dx}=\frac {2x}{2y}}$$
 * $$\ {\frac {d}{dx}=\frac {x}{y}}$$

6)calcular $$\ f^,(0),f^,(\frac {1}{2}),f^,(1),f^,(-10)$$
 * $$\ {f(x)=2+x-x^2}$$
 * $$\ {f^,(x)=1-2x}$$


 * $$\ {f^,(0)=1}$$
 * $$\ {f^,(\frac {1}{2})=0}$$
 * $$\ {f^,(1)=-1}$$
 * $$\ {f^,(-10)=-19}$$

=Recta Tangente=

Encuentre la ecuación de la recta tangente en el punto propuesto

1)$$\ {x^3y+y^3x=30}$$

=Parcial=

1)Se da una tabla de valores para f, g, f´ y g´


 * / x  /  f(x)  /  g(x)  /  f´(x)  /  g´(x)  /
 * /  1   /   3    /   2    /   4     /   6     /
 * /  2   /   1    /   8    /   5     /   7     /
 * /  3   /   7    /   2    /   7     /   9     /


 * Si h(x) = g(f(x)), el valor de h´(1) es igual a:
 * a)4    b)5     c)36     d)24     e)30


 * R/

2)

3)Un rectángulo tiene dos vértices sobre el eje x y otros dos sobre la parabola $$\ {y=12-x^2}$$ ¿Cuáles son las dimensiones del rectángulo de este tipo con área máxima?
 * $$\ {A = (2x)(y)}$$
 * $$\ {A = (2x)(12-x^2)}$$
 * $$\ {A = 24x-2x^3}$$
 * $$\ {A^, = 24-6x^2}$$
 * $$\ {0 = 24-6x^2}$$
 * $$\ {24 = 6x^2}$$
 * $$\ {\frac {24}{6} = x^2}$$
 * $$\ {4 = x^2}$$
 * $$\ {\sqrt {4} = x}$$
 * $$\ {2 = x}$$
 * $$\ {y=12-x^2}$$
 * $$\ {y=12-(2)^2}$$
 * $$\ {y=12-4}$$
 * $$\ {y=8}$$
 * $$\ {A = (2(2))(8)}$$
 * $$\ {A = (4)(8)}$$
 * $$\ {A = 32}$$

--Jorgemontes 13:07, 19 Jul 2004 (UTC)

4)