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Pensemos en que se nos pide hallar la ecuación de la recta tangente a la curva, cuya ecuación es [math](x-2)^2+(y+3)^2=17[/math] , en el punto P(1;1).


Tratado desde el Algebra :

                                                    debemos saber que la curva a la que hace referencia el ejercicio es una circunferencia con centro en C(2;-3) y radio[math] 17^{1/2} [/math].Además necesitamos conocer que la tangente a una circunferencia en un punto de ella es perpendicular al radio. Por otra parte no podemos desconocer que dos rectas perpendiculares tienen pendientes inversas y opuestas.
Con todo esto es probable que podamos hallar la ecuación de la recta sostén del radio que une C con P. Por lo tanto  [math] y+3=-4(x-2) [/math] donde su pendiente es -4. Consecuentemente la ecuación de la recta sostén de la tangente pedida es [math](y-1)=(1/4).(x-1) [/math]. Tal vez lo quieran expresar como [math] y=1/4 x+3/4 [/math].


Tratado desde el Análisis Matemático:

                                                                            debemos saber que la derivada en un punto determina el valor de la pendiente de la recta tangente a la curva en ese punto. Por lo tanto tenemos que derivar correctamente.
Si tomamos la función podremos derivarla después de despejar y: por lo tanto [math] |y+3| =\sqrt{17-(x-2)^2} [/math] pero como P está en un cuadrante positivo de la función tomaremos la raíz cuadrada positiva, por consiguiente: [math] y=\sqrt{17-(x-2)^2}-3 [/math]. Su derivada es: [math] y'=\frac{1}{2.\sqrt{17-(x-2)^2}}.(-2).(x-2) [/math]

Tal que en el punto P(1,1) la derivada es [math] y_{(1)}=\frac{(-2)(-1)}{2\sqrt{16}}=\frac{2}{8}=\frac{1}{4} [/math]

Por lo tanto si la derivada en P es [math] \frac{1}{4} [/math], la pendiente de la recta que pasa por P y resulta tangente a la curva es [math] \frac{1}{4} [/math].En consecuencia [math] (y-1)=\frac{1}{4}.(x-1) [/math]

Pero si sabemos derivada de funciones implícitas la función [math] (x-2)^2+(y+3)^2=17 [/math] se puede derivar del siguiente modo:

[math] \frac{\partial _y}{\partial _x}=-\frac{2.(x-2)}{2.(y+3)} [/math] que en el punto P(1;1) resulta [math] \frac{\partial _y}{\partial _x}=-\frac{2.(-1)}{2.4}=\frac{2}{8}=\frac{1}{4} [/math]

La idea de mostrar estas posibilidades es hacer comprensible al conocimiento de las herramientas que la matemática ofrece para una misma situación, todo depende de lo que sepamos. Esto ratifica aquello de que la matemática es un modelo teórico de un alcance ilimitado dado que, frente a una situación siempre habrá un modelo al cual recurrir o en su defecto se podrá generar el modelo necesario basado, por supuesto, en conocimientos elementales y objetivos claros de lo que necesitamos resolver.
Los estudiantes deben comprender que el aprendizaje  es un proceso lento y que acompañado con esfuerzo y dedicación nos otorgará, a medida que se recorre el camino, todo lo que necesitaremos para nuestra vida profesional.

Lo de hoy fue solo un ejemplo.

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