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Mathematics LibreTexts

6.1: Best Linear Approximations

  • Page ID
    22671
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    Definition

    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 \(x \in \mathbb{R}\),

    \[f(\alpha x)=\alpha f(x).\]

    Exercise \(\PageIndex{1}\)

    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 \(x \in \mathbb{R}\).

    Definition

    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 \(a .\)

    Proposition \(\PageIndex{1}\)

    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.

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