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\begin{document}

\title{Calculus and Elements of Linear Algebra I}
\author{Homework 4}
\date{Due on Moodle, Monday, October 5, 2020}
\maketitle

\begin{enumerate}

\item Consider the polynomial
\[
  p(x) = -2 + 9 \, x - 6 \, x^2 + x^3 \,.
\]
\begin{enumerate}
\item Find the roots of $p$.

\emph{Hint:} One of the roots is $x=2$.

\item Compute $p'(x)$ and find its roots.

\item Compute $p''(x)$ and find its roots.

\item Do you see a pattern?  Explain.
\end{enumerate}

\item Compute the derivatives of the following functions.
\begin{enumerate}
\item $\displaystyle f(x) = \frac{x}{a + x^2}$ where $a$ is a
constant
\item $\displaystyle g(t) = \cos(\omega t + \phi)$ where $\omega$ and
$\phi$ are constants
\item $\displaystyle h(s) = \sin (s^3)$
\item $\displaystyle j(s) = (\sin s)^3$
\item $\displaystyle k(x) = \ln(x^a + x^{-a})$ where $a\neq 0$ is a constant

\emph{Note:} You can use without further discussion that $(\ln x)' = 1/x$.
\item $\displaystyle \ell(x) = \ln(a^x + a^{-x})$ where $a>0$ is a
constant
\item $\displaystyle u(x) = \exp(bx)$ where $b$ is a constant
\item $\displaystyle v(x) = x^2 \, \exp(x)$
\item $\displaystyle w(x) = \exp(-x^2)$
\item $\displaystyle z(x) = x^x$
\end{enumerate}

\item Use the definition of the derivative,
\[
  f(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}h \,,
\]
to show that the function $f(x) = \lvert x \rvert$ is not
differentiable at $x=0$.

\item Suppose that $f \colon (a,b) \to \R$ is differentiable at
$x \in (a,b)$ with $f'(x) \neq 0$ and that $f$ is invertible in some
neighborhood of $x$.
\begin{enumerate}
\item Show that the inverse function $f^{-1}$ is differentiable in
some neighborhood of $y=f(x)$ with
\[
  (f^{-1})'(y) = \frac1{f'(x)} \,.
\]
\item Give a geometrical explanation of why this is true.
\end{enumerate}


\end{enumerate}

\end{document}
