$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Let $U\subseteq\mathbb{R}$, and $\vec{a}\in\text{int}(U)$. The linear approximation of a function $f$ with respect to $\vec{a}$ is a linear function, $$\begin{align*} f(\vec{x})\approx l_{\vec{a}}^f:=f(\vec{a})+Df(\vec{a})(\vec{x}-\vec{a}) \end{align*}$$

Concepts

This comes from the definition and alternate definition of derivatives. If $f$ is differentiable at a point $\vec{a}$, then $$\begin{align*} f(\vec{x})=f(\vec{a})+Df(\vec{a})(\vec{x}-\vec{a})+r(\vec{x}-\vec{a}), \end{align*}$$ where $r(\vec{x}-\vec{a})$ satisfies $\lim_{\vec{x}\to\vec{a}}\frac{r(\vec{x}-\vec{a})}{\vec{x}-\vec{a}}$.
If $\vec{x}-\vec{a}$ is small, as it is for a linear approximation near $\vec{a}$, then $r(\vec{x}-\vec{a})$ is really small, so $$\begin{align*} f(\ve{x})\approx f(\vec{x})+Df(\vec{x})(\vec{x}-\vec{a}) \end{align*}$$

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Let $U\subseteq\mathbb{R}$, and $\vec{a}\in\text{int}(U)$. The linear approximation of a function $f$ with respect to $\vec{a}$ is a linear function, $$\begin{align*} f(\vec{x})\approx l_{\vec{a}}^f:=f(\vec{a})+Df(\vec{a})(\vec{x}-\vec{a}) \end{align*}$$

Concepts

This comes from the definition and alternate definition of derivatives. If $f$ is differentiable at a point $\vec{a}$, then $$\begin{align*} f(\vec{x})=f(\vec{a})+Df(\vec{a})(\vec{x}-\vec{a})+r(\vec{x}-\vec{a}), \end{align*}$$ where $r(\vec{x}-\vec{a})$ satisfies $\lim_{\vec{x}\to\vec{a}}\frac{r(\vec{x}-\vec{a})}{\vec{x}-\vec{a}}$.
If $\vec{x}-\vec{a}$ is small, as it is for a linear approximation near $\vec{a}$, then $r(\vec{x}-\vec{a})$ is really small, so $$\begin{align*} f(\ve{x})\approx f(\vec{x})+Df(\vec{x})(\vec{x}-\vec{a}) \end{align*}$$

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definition
concepts
used in
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FullPage
definition
concepts
used in
hypothesis
results
Let $U\subseteq\mathbb{R}$, and $\vec{a}\in\text{int}(U)$. The linear approximation of a function $f$ with respect to $\vec{a}$ is a linear function, $$\begin{align*} f(\vec{x})\approx l_{\vec{a}}^f:=f(\vec{a})+Df(\vec{a})(\vec{x}-\vec{a}) \end{align*}$$

Concepts

This comes from the definition and alternate definition of derivatives. If $f$ is differentiable at a point $\vec{a}$, then $$\begin{align*} f(\vec{x})=f(\vec{a})+Df(\vec{a})(\vec{x}-\vec{a})+r(\vec{x}-\vec{a}), \end{align*}$$ where $r(\vec{x}-\vec{a})$ satisfies $\lim_{\vec{x}\to\vec{a}}\frac{r(\vec{x}-\vec{a})}{\vec{x}-\vec{a}}$.
If $\vec{x}-\vec{a}$ is small, as it is for a linear approximation near $\vec{a}$, then $r(\vec{x}-\vec{a})$ is really small, so $$\begin{align*} f(\ve{x})\approx f(\vec{x})+Df(\vec{x})(\vec{x}-\vec{a}) \end{align*}$$

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Let $U\subseteq\mathbb{R}$, and $\vec{a}\in\text{int}(U)$. The linear approximation of a function $f$ with respect to $\vec{a}$ is a linear function, $$\begin{align*} f(\vec{x})\approx l_{\vec{a}}^f:=f(\vec{a})+Df(\vec{a})(\vec{x}-\vec{a}) \end{align*}$$

Concepts

This comes from the definition and alternate definition of derivatives. If $f$ is differentiable at a point $\vec{a}$, then $$\begin{align*} f(\vec{x})=f(\vec{a})+Df(\vec{a})(\vec{x}-\vec{a})+r(\vec{x}-\vec{a}), \end{align*}$$ where $r(\vec{x}-\vec{a})$ satisfies $\lim_{\vec{x}\to\vec{a}}\frac{r(\vec{x}-\vec{a})}{\vec{x}-\vec{a}}$.
If $\vec{x}-\vec{a}$ is small, as it is for a linear approximation near $\vec{a}$, then $r(\vec{x}-\vec{a})$ is really small, so $$\begin{align*} f(\ve{x})\approx f(\vec{x})+Df(\vec{x})(\vec{x}-\vec{a}) \end{align*}$$

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definition
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used in
hypothesis
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