6.1: Mejores aproximaciones lineales
- Page ID
- 108869
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Decimos que una función\(f: \mathbb{R} \rightarrow \mathbb{R}\) es lineal si por cada\(x, y \in \mathbb{R}\),
\[f(x+y)=f(x)+f(y)\]
y para todos\(\alpha \in \mathbb{R}\) y\(x \in \mathbb{R}\),
\[f(\alpha x)=\alpha f(x).\]
Mostrar que si\(f: \mathbb{R} \rightarrow \mathbb{R}\) es lineal, entonces existe\(m \in \mathbb{R}\) tal que\(f(x)=m x\) para todos\(x \in \mathbb{R}\).
Supongamos\(D \in \mathbb{R}, f: D \rightarrow \mathbb{R},\) y\(a\) es un punto interior de\(D\). Decimos que\(f\) es diferenciable en\(a\) si existe una función lineal\(d f_{a}: \mathbb{R} \rightarrow \mathbb{R}\) tal que
\[\lim _{x \rightarrow a} \frac{f(x)-f(a)-d f_{a}(x-a)}{x-a}=0.\]
Llamamos a\(d f_{a}\) la función la mejor aproximación lineal a\(f\) at\(a,\) o el diferencial de\(f\) at\(a .\)
Supongamos\(D \subset \mathbb{R}, f: D \rightarrow \mathbb{R},\) y\(a\) es un punto interior de\(D .\) Entonces\(f\) es diferenciable en\(a\) si y solo si
\[\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\]
existe, en cuyo caso\(d f_{a}(x)=m x\) donde
\[m=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}.\]
- Prueba
-
Dejar\(m \in \mathbb{R}\) y dejar\(L: \mathbb{R} \rightarrow \mathbb{R}\) ser la función lineal\(L(x)=m x .\) Entonces
\[\begin{aligned} \frac{f(x)-f(a)-L(x-a)}{x-a} &=\frac{f(x)-f(a)-m(x-a)}{x-a} \\ &=\frac{f(x)-f(a)}{x-a}-m. \end{aligned}\]
De ahí
\[\lim _{x \rightarrow a} \frac{f(x)-f(a)-L(x-a)}{x-a}=0\]
si y solo si
\[\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}=m.\]
Q.E.D.