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\begin{document}

\title{Integration workshop 2020  \\-- Single Variable Calculus --}
\author{Colin Clark}
\date{August 6, 2020}
\maketitle

\section{Problems: Taylor Series}
\begin{enumerate}
  \item   Determine which differentiability class each of the functions belong to:
    \begin{enumerate}[(a)]
      \item \(f(x) = \frac{1}{1-x}\) for (i) \(-1 < x < 1\), and (ii) \(x \in \mathbb{R}\).
      \item \(f(x) = x\, |x| \) for (i) \(1< x < \infty\), and (ii) \(x \in \mathbb{R}\).
      \item \(f(x) = x^{4/3}\) for (i) \(1< x < \infty\), and (ii) \(x \in \mathbb{R}\).
    \end{enumerate}
  \item Derive (or state) the first three non-zero terms of the Taylor Series centered at 0 for the following functions:
  \begin{enumerate}
    \item \(\ln(1 + x)\)
    \item \(\cos(t^2)\)
    \item \(\mathrm{sinc}(kx) := \begin{cases} \frac{\sin(kx)}{kx} & x \neq 0 \\ 1 & x = 0 \end{cases}\)
    \item \((1 + x)^\alpha\) for \(\alpha > 0\)
  \end{enumerate}
  % \item Compute the radius of convergence for
  % \item Estimate the number of terms
  % \item Express the
  \item Describe how you could use a Taylor polynomial to estimate the integral \(\int_{-1}^1 e^{-x^2}\, dx\) to 2 decimal places.
  \item Identify the domains on which the following functions are (real) analytic.
  \begin{enumerate}[(a)]
    \item \(f(x) = 1 - 3x + 4x^3\)
    \item \(f(x) = e^x\)
    \item \(f(x) = (1 + x)^\alpha\) for \(\alpha > 0\)
    \item \(f(x) = 1 / (1 + x^2) \)
    \item \(f(x) = 1/ (1-x) \)
    \item \(f(x) = |x|\)
    \item \(f(x) = e^{-1/(1-x^2)} \) if \(|x| < 1\) and \(f(x) = 0\) if \(|x| \geq 1\)
  \end{enumerate}
  \item What does the Taylor series for \(e^x\) centered at \(a = 0\) imply about the inequality \(e^x \geq 1 + x\).
  % \item What does the Taylor series for \(\tfrac{1}{1-x}\) centered at \(a = 0\) imply about the inequality \(\tfrac{1}{1-x} \geq 1 + x\).
\end{enumerate}


\section{Problems: Improper Integrals}
\begin{enumerate}
  \item
  \item Determine for which values a \(\alpha\) does  \(\displaystyle \int_1^\infty \frac{1}{x\, [\ln(x)]^\alpha}\, dx\) converge.
  \item Use the comparison test to determine whether the following integrals converge
  \begin{enumerate}
    \item \(\displaystyle \int_1^\infty \frac{\sin^2(x)}{x^2}\, dx \)
    \item \(\displaystyle \int_1^\infty \frac{1}{1 + \sqrt{x}} \, dx \)
    \item \(\displaystyle \int_{0}^\infty \frac{e^{-x}}{\sqrt{x}} \, dx \)
  \end{enumerate}
\end{enumerate}
\end{document}
