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\frametitle{Linear Approximation and Differentials}

Why approximate values of a function using a tangent?
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\item might be easy to compute $f(a)$ and $f'(a)$,
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\item but difficult to compute values $f(x)$ with $x$ near $a$
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We use the tangent line at $(a,f(a))$ to approximate $f(x)$ when $x$ is close to $a$.
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The tangent at $(a,f(a))$ is:
\begin{talign}
L(x) = f(a) + f'(a)\cdot (x-a)
\end{talign}
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This function is called \emph{linearization} of $f$ at $a$.
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When $x$ is close to $a$, we approximate $f(x)$ by:
\begin{talign}
f(x) \approx f(a) + f'(a)\cdot (x-a)
\end{talign}
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This is called
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\item \emph{linear approximation} of $f$ at $a$, or
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\item \emph{tangent line approximation} of $f$ at $a$.
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