4: Aproximación Lineal de una Función Escalar de Varias Variables
- Page ID
- 51020
This page is a draft and is under active development.
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Recordemos que, bajo ciertas condiciones de derivabilidad, una función real puede aproximarse por una recta cuando se producen cambios pequeños (\(\varepsilon\)) en la variable independiente (aproximación por la recta tangente, figura 5),
\[f(x+\varepsilon) \approx f(x) + \varepsilon f'(x)\]
La función \(f(x) = x^2\) puede aproximarse, en puntos cerca de \(x=1\) por la recta: \[f(1+\varepsilon) \approx f(1) +f'(1)\varepsilon = 1+2\varepsilon\]
Del mismo modo, para una función \(f: \mathbb{R}^n \rightarrow \mathbb{R}\) que depende de varias variables, tendremos: \[f(x_1+\varepsilon_1,x_2+\varepsilon_2,...,x_n+\varepsilon_n) \approx f(x_1,x_2,...,x_n) + \sum_{i=1}^n \varepsilon_i \dfrac{\partial f}{\partial x_i} (x_1,x_2,...,x_n)\] Esta expresión puede escribirse de forma más compacta usando la definición de gradiente (Definición 4) y llamando \(\boldsymbol{\varepsilon}=(\varepsilon_1,\varepsilon_2,...,\varepsilon_n)\), vector fila donde se agrupan los pequeños cambios que se producen en cada una de las variables independientes: \[f(\textbf{x}+\boldsymbol{\varepsilon}) \approx f(\textbf{x}) + \boldsymbol{\varepsilon} \nabla f(\textbf{x})\]
En este caso, si tenemos \(n=2\), aproximamos la función cerca de un punto, por su plano tangente.
La función \(f(x,y) = x^2+xy+y^2\), en un entorno de \((1,0)\), tomará valores cercanos a la aproximación de primer orden.
Para calcularla necesitamos \(f(1,0)\) y \(\nabla f(1,0)\) que valen
\(\bullet \quad f(1,0)=1\)
\(\bullet \quad \nabla f(1,0) = (2x+y,x+2y)\left|_{x=1,y=0}\right. = (2,1)\)
y se emplean para determinar el plano tangente en el punto \((1,0,f(1,0))\), cuya ecuación es \(2x+y=~2\). El vector normal al plano es \(\nabla f(1,0)\).
