\section{Equations and Figure} \frame{\sectionpage} \begin{frame}{Ordinary Differential Equations} \uncover<+->{\begin{equation*} \dv{x} y(x) + \frac{1}{CR} y(x) = 0 \end{equation*}} \uncover<+->{\begin{equation} \dv[2]{x} y(x) + \gamma \dv{x} y(x) + \omega_0^2 y(x) = f(x) \end{equation}} \end{frame} \begin{frame}{} \uncover<+>{\begin{equation*} \dv[2]{x} y(x) + \gamma \dv{x} y(x) + \omega_0^2 y(x) = f(x) \end{equation*}} \uncover<+>{\[ \Downarrow \] \begin{equation*} \colch{\dv[2]{x} + \gamma \dv{x} + \omega_0^2} y(x) = f(x) \end{equation*}} \uncover<+>{\[ \Downarrow \] \begin{equation*} y(x) = \frac{f(x)}{\dv[2]{x} + \gamma \dv{x} + \omega_0^2} \end{equation*}} \end{frame} \begin{frame}{Imagem} \centering \includegraphics[width = 0.8\textwidth]{images/coke.jpg}\\ \footnotesize \textcolor{yellow}{Figure:} Some words about the figure here \end{frame} \begin{frame}{See how is cool the fourier serie} \uncover<+>{\begin{equation*} \mathcal{F}[f](\xi) = \hat{f}(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x) e^{-i x \xi} \dd{x} \end{equation*}} \uncover<+>{\begin{equation*} \mathcal{F}^{-1}[\hat{f}](x) = f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \hat{f}(\xi) e^{i x \xi} \dd{\xi} \end{equation*}} \end{frame} \begin{frame}{Quality Control} \uncover<+>{\begin{equation*} \widehat{(f + \alpha g)}(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \prnt{f(x) + \alpha g(x)} e^{-i x \xi} \dd{x} \end{equation*}} \uncover<+>{\[ \Downarrow \] \begin{equation*} \widehat{(f + \alpha g)}(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x) e^{-i x \xi} \dd{x} + \frac{\alpha}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} g(x) e^{-i x \xi} \dd{x} \end{equation*}} \uncover<+>{\[ \Downarrow \] \begin{equation*} \widehat{(f + \alpha g)}(\xi) = \hat{f}(\xi) + \alpha \hat{g}(\xi) \end{equation*}} \end{frame} \begin{frame}{Quality Control} \uncover<+>{\begin{equation*} \widehat{f'}(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f'(x) e^{-i x \xi} \dd{x} \end{equation*}} \uncover<+>{\[ \Downarrow \] \begin{equation*} \widehat{f'}(\xi) = \eval{\frac{f(x) e^{-ix\xi}}{\sqrt{2\pi}}}^{+\infty}_{-\infty} + i\xi \cdot \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x) e^{-i x \xi} \dd{x} \end{equation*}} \uncover<+>{\[ \Downarrow \] \begin{equation*} \widehat{f'}(\xi) = i\xi \widehat{f}(\xi) \end{equation*}} \end{frame} \begin{frame}{Quality Control} \centering \textcolor{green2}{\huge{The inverse does work}} \normalsize{for appropriate functions} \tiny{and, sometimes, the Fourier Transform of a function is not in the same set as the original function, but let's forget about this since we do not know a decent theory of integration} \end{frame}