\begin{frame} \frametitle{Linear Approximation and Differentials} The method of linear approximation with differentials: \begin{talign} f'(x) = \frac{dy}{dx} \end{talign} \pause We view $dx$ and $dy$ as variables, then: \begin{talign} dy = f'(x)\; dx \end{talign} \pause So $dy$ depends on the value of $x$ and $dx$. \medskip\pause \begin{minipage}{.43\textwidth} \begin{center} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=1cm] \def\diabordery{.7cm} \diagram{-0.5}{4}{-.5}{4}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cgreen,ultra thick] plot[smooth,domain=-.5:3,samples=200] function{(x-1)**2}; \tangent{1.5cm}{2.1cm}{pow(\x-1,2)}{1.5} \node[include=cred] (a) at (1.5,.25) {}; \draw[gray] (a) -- node [at end,below,black] {$x$} +(0,-.5); \draw[gray] (3,4) -- (3,-.25); \draw[gray,decorate,decoration={brace,amplitude=5pt,mirror,raise=2pt}] (1.5,-.5) -- node[below,black,yshift=-2mm] {$\Delta x = dx$} (3,-.5); \draw[cred,decorate,decoration={brace,amplitude=5pt,mirror,raise=10pt}] (3,.25) -- node[right,black,xshift=6mm] {$dy$} (3,1.75); \draw[cgreen,decorate,decoration={brace,amplitude=5pt,mirror,raise=4pt}] (3,.25) -- node[right,black,xshift=4mm] {$\Delta y$} (3,4); \end{scope} \end{tikzpicture} } \end{center} \end{minipage} \begin{minipage}{.56\textwidth} \begin{itemize} \pause \item $x =$ point of linearization \pause\smallskip \item $\Delta x = dx$ is the distance from $x$\\ \pause\smallskip \item $dy =$ change of $y$ of tangent \pause\smallskip \item $\Delta y = $ change of $y$ of curve~$f$ \end{itemize} \pause As formulas: \begin{itemize} \pause \item $dy = f'(x)\; dx$ \pause \item $\Delta y = f(x +\Delta x) - f(x)$ \end{itemize} \end{minipage} \end{frame}