05-diff-integration.tex

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  pdftitle={PSET 5: Functions of several variables and optimization with several variables},
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\title{PSET 5: Functions of several variables and optimization with
several variables}
\author{}
\date{\vspace{-2.5em}}

\begin{document}
\maketitle

\textbf{Note: all homework uploads should be as a PDF \emph{and} have
the questions identified}. We'll be giving zero credit for submissions
that don't follow this protocol as it adds considerable time to grading.
Thank you!

\begin{itemize}
\item
  Name
\item
  How long did this problem set take you?
\item
  How difficult was this problem set? very easy 1 2 3 4 5 very
  challenging
\end{itemize}

\section{Find first partial
derivatives}\label{find-first-partial-derivatives}

Find all of the first partial derivatives of each function.\footnote{Inspired
  by Grimmer HW6.3}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(f(x,y) = 3x + 2y^4\)

  ~ ~\\
  \strut \\
\item
  \(f(x,y) = \frac{1}{3}x^5 - 3x^3y^2 + 3xy^4\)

  ~ ~\\
  \strut \\
\item
  \(g(x,y) = xe^{4y}\)

  ~ ~\\
  \strut \\
\item
  \(k(x,y) = \frac{x+y}{x-y}\)

  ~\\
\item
  \(h(x,y,z) = x^3 e^{yz}\)

  ~\\
  \newpage
\end{enumerate}

\section{Find the gradient}\label{find-the-gradient}

Find the gradient \(\nabla f\) of the following functions and evaluate
them at the given points.\footnote{Inspired by Grimmer HW6.4}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(f(x,y) = \sqrt{x^2 + y^2}\), \quad \((x,y) = (3,4)\)

  ~\\
  \strut \\
\item
  \(f(x,y,z) = (x+z)e^{x-y}\), \quad \((x,y,z) = (1,1,1)\)

  ~\\
  \strut \\
\end{enumerate}

\section{Find the Hessian}\label{find-the-hessian}

Find the Hessian \(H\) for the following functions.\footnote{Inspired by
  Grimmer HW7.3}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(g(x,y) = x^4 - 3x^2 y^3\)

  \strut \\
  \strut ~\\
  \strut ~ ~\\
  \strut ~
\item
  \(f(x,y,z) = xyz - x^2\)

\begin{verbatim}
 \     
 \  
 \     
 \
\end{verbatim}
\end{enumerate}

\section{Find the critical points}\label{find-the-critical-points}

Find the local minimum values, local maximum values, and saddle point(s)
of the function. Remember the process we discussed in class: Calculate
the gradient, set it equal to zero to solve the system of equations,
calculate the Hessian, and assess the Hessian at critical values. Be
sure to show your work on each of these steps.\footnote{Inspired by
  Grimmer HW7.4}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(f(x,y) = x^4 + y^4 - 4xy + 2\)

  ~ ~ ~\\
\item
  \(k(x,y) = (1+xy)(x+y)\)

  ~\\
  \newpage
\end{enumerate}

\section{Definite integrals}\label{definite-integrals}

Solve the following definite integrals using the antiderivative
method.\footnote{Inspired by Gill 5.10 and Grimmer HW4.1}

For all these problems, the basic approach to compute the definite
integral of \(f(x)\) from \(a\) to \(b\) is by using the formula
\(F(b) - F(a)\), where \(F(x)\) is the \textbf{antiderivative} of \(f\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(\int_6^8 x^3 \,dx\)

  ~\\
\item
  \(\int_{-1}^{0} (3x^2 -1) \,dx\)

  ~\\
\item
  \(\int_2^4 e^y \,dy\)

  ~\\
\item
  \(\int_3^3 \sqrt{x^5 + 2} \,dx\)

  ~\\
\end{enumerate}

\section{Indefinite integrals}\label{indefinite-integrals}

Calculate the following indefinite integrals:\footnote{Inspired by Gill
  5.13 and 5.14}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(\int (x^2-x^{-\frac{1}{2}}) \,dx\)

  ~\\
\item
  \(\int 360t^6 \,dt\)
\item
  \(\int 2x\log(x^2) \,dx\)
\end{enumerate}

\section{Determining convergence}\label{determining-convergence}

Determine whether each integral is convergent or divergent. Evaluate
those that are convergent.\footnote{Inspired by Grimmer HW 4.3}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(\int_1^{\infty} \left(\frac{1}{3x}\right)^2 \,dx\)

\begin{verbatim}
 \     \  \     \ 
\end{verbatim}
\item
  \(\int_0^{\infty} e^{-x} \,dx\)

  \strut \\
  \strut ~ ~ ~ ~
\item
  \(\int_{-\infty}^0 x^3 \,dx\)
\item
  \(\int_{0}^1 \int_{2}^{3} x^2y^3 \,\, dxdy\)
\item
  \(\int_{0}^2 \int_0^{\sqrt{1-x^2}} 2x^3y \, \, dy dx\)
\end{enumerate}

\newpage

\section{Applied integration}\label{applied-integration}

A group of three unidentified first-year graduate students at the
University of Chicago are worn out after a week of math camp. Wanting to
unwind, the students agree to not talk about math and decide to chat
over some casual drinks at Medici.

After five shots of tequila each, two pitchers of beer, a bottle of
wine, and a large Chicago-style pizza, the three students have had
enough fun and decide to start the trip back home.

\begin{itemize}
\tightlist
\item
  Student \(A\) gets on a bike and starts pedaling away at a velocity of
  \(v_A(t) = 2t^4 + t\), where \(t\) represents minutes. However, the
  student crashes into the side of an Uber and ends the journey after
  only 2 minutes.
\item
  Student \(B\) has no bike, so starts running at a velocity of
  \(v_B(t) = 4\sqrt{t}\). Sadly, after only 4 minutes, the student's
  legs give out and the student decides to sing a song, instead.
\item
  Student \(C\) can't even stand up, so has no choice but to slowly
  crawl at a velocity of \(v_C(t) = 2e^{-t}\). Student \(C\) steadily
  plods along for 20 minutes before falling asleep on the sidewalk.
\end{itemize}

Generally, if an object moves along a straight line with position
function \(s(t)\), then its velocity is \(v(t) = s'(t)\). The
Fundamental Theorem of Calculus then tells us that

Without using a calculator, use this formula to find the distance
traveled by Students \(A\), \(B\), and \(C\). (Assume, however
unrealistic in may be, that all three students traveled in a straight
line.) Who traveled the farthest? The least far?\footnote{Inspired by
  Grimmer HW4.2}

\section{AI and Resources statement}\label{ai-and-resources-statement}

\begin{itemize}
\tightlist
\item
  Please list (in detail) all resources you used for this assignment. If
  you worked with people, list them here as well. It is not enough to
  say that you used a resource for help, you need to be specific on the
  link and \emph{how} it was helpful. W/R/T gen AI tools (including GPT,
  etc. ) you cannot use them to do work on your behalf -- you cannot put
  in any of the questions, etc. You can ask for help on logic / sample
  problems. If you do use GPT or other AI tools, you need to provide a
  link to your chat transcript. Any suspected academic integrity
  violations will be immediately reported.
\end{itemize}

\end{document}