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\title{Calculus and Elements of Linear Algebra I}
\author{Homework 5}
\date{Due on Moodle, Monday, October 12, 2020}
\maketitle

\begin{enumerate}

\item Use implicit differentiation to find an equation of the tangent
line to the graph of the given equation at the given point.
\begin{enumerate}
\item $x^2 \, y = x+2$ at point $(2,1)$
\item $x^{\tfrac23} + y^{\tfrac23} = 5$ at point $(8,1)$
\end{enumerate}

\item An airplane flying horizontally at a height of $5\,$km with a
speed of $300\,$m/s passes directly above an observer.  What is the
rate of increase of distance to the observer $30\,$s later?

\item A balloon is filled at a rate of $100\pi \,\mathrm{cm}^3$ per
second.  At what rate is the radius of the balloon increasing when the
radius is $10\,$cm?

\item Show that
\[
  \frac{\mathrm d \arcsin x}{\mathrm d x}
  = \frac{1}{\sqrt{1-x^2}} \,.
\]
(The function $y=\arcsin x$ is the inverse function of $x=\sin y$.)

\item Find all critical points (points where $f'(x)=0$) for the
following functions, and characterize whether they correspond to a
local minimum, a local maximum, or neither.
\begin{enumerate}
\item $f(x) = x^3 -3x +3$
\item $f(x) = x^3 +3x +3$
\item $g(t) = \cos(\omega t)$ with $\omega \neq 0$
\end{enumerate}


\end{enumerate}

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