6.1: Best Linear Approximations
- Page ID
- 22671
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We say a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is linear if for every \(x, y \in \mathbb{R}\),
\[f(x+y)=f(x)+f(y)\]
and for every \(\alpha \in \mathbb{R}\) and \(x \in \mathbb{R}\),
\[f(\alpha x)=\alpha f(x).\]
Show that if \(f: \mathbb{R} \rightarrow \mathbb{R}\) is linear, then there exists \(m \in \mathbb{R}\) such that \(f(x)=m x\) for all \(x \in \mathbb{R}\).
Suppose \(D \in \mathbb{R}, f: D \rightarrow \mathbb{R},\) and \(a\) is an interior point of \(D\). We say \(f\) is differentiable at \(a\) if there exists a linear function \(d f_{a}: \mathbb{R} \rightarrow \mathbb{R}\) such that
\[\lim _{x \rightarrow a} \frac{f(x)-f(a)-d f_{a}(x-a)}{x-a}=0.\]
We call the function \(d f_{a}\) the best linear approximation to \(f\) at \(a,\) or the differential of \(f\) at \(a .\)
Suppose \(D \subset \mathbb{R}, f: D \rightarrow \mathbb{R},\) and \(a\) is an interior point of \(D .\) Then \(f\) is differentiable at \(a\) if and only if
\[\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\]
exists, in which case \(d f_{a}(x)=m x\) where
\[m=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}.\]
- Proof
-
Let \(m \in \mathbb{R}\) and let \(L: \mathbb{R} \rightarrow \mathbb{R}\) be the linear function \(L(x)=m x .\) Then
\[\begin{aligned} \frac{f(x)-f(a)-L(x-a)}{x-a} &=\frac{f(x)-f(a)-m(x-a)}{x-a} \\ &=\frac{f(x)-f(a)}{x-a}-m. \end{aligned}\]
Hence
\[\lim _{x \rightarrow a} \frac{f(x)-f(a)-L(x-a)}{x-a}=0\]
if and only if
\[\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}=m.\]
Q.E.D.