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