\begin{frame} \frametitle{Linear Approximation and Differentials} \begin{center} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=1cm] \coordinate (l1) at (1.75,0.75); \coordinate (l2) at (2.25,1.25); \coordinate (l3) at (1.75,1.25); \coordinate (l4) at (2.25,0.75); \diagram{-.5}{4}{-.5}{4}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cgreen,ultra thick] plot[smooth,domain=-0:3.3,samples=200] function{2+(x-1)**2 -x}; \tangent{2cm}{2cm}{2+pow(\x-1,2) -\x}{2} \end{scope} \mpause[1]{ \begin{scope}[xshift=-15cm,yshift=-9cm,xscale=12,yscale=12] \coordinate (r1) at (1.75,0.75); \coordinate (r2) at (2.25,1.25); \coordinate (r3) at (1.75,1.25); \coordinate (r4) at (2.25,0.75); \begin{scope}[dashed,cblue] \draw (l1) rectangle (l2); \draw (r1) rectangle (r2); \draw (l1) -- (r1); \draw (l2) -- (r2); \draw (l3) -- (r3); \draw (l4) -- (r4); \end{scope} \clip (r1) rectangle (r2); \diagram{-.5}{4}{-.5}{4}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cgreen,ultra thick] plot[smooth,domain=-0:3.3,samples=200] function{2+(x-1)**2 -x}; \tangent{2cm}{2cm}{2+pow(\x-1,2) -\x}{2} \end{scope} \end{scope} } \end{tikzpicture} } \end{center} \pause\pause \begin{block}{} A curve is very close to its tangent close to the point of tangency (touching). \end{block} We can use this for approximating values of the function\ldots \vspace{10cm} \end{frame}