\begin{frame} \frametitle{Linear Approximation and Differentials} \begin{exampleblock}{} What is the linear approximation of $f(x)= \cos x$ at $0$?\\ Use it to approximate $\cos 0.01$. \pause\bigskip We have: \begin{talign} f(0) &= \cos 0 = \mpause[1]{1} \\ \mpause[2]{f'(x) }&\mpause[2]{= }\mpause[3]{-\sin x} & \mpause[4]{f'(0) = }\mpause[5]{0} \end{talign} \pause\pause\pause\pause\pause\pause Thus the linear approximation of $\cos x$ at $0$ is: \begin{talign} L(x) = \mpause[1]{1 + 0 (x-0)} \mpause[2]{= 1} \end{talign} \pause\pause\pause We use this to approximate $\cos 0.01$: \begin{talign} \cos 0.01 \approx \mpause[1]{L(0.01) = 1} \end{talign} \end{exampleblock} \pause\pause\bigskip Approximations for $\sin$ and $\cos$ are often applied in physics (e.g. optics). \end{frame}