Thi Toán

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\firstpageheader{MATH 2410}{Midterm Exam 1}{Fall Semester}
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\begin{center}
  {\LARGE\bfseries Midterm Examination 1}\\[4pt]
  {\large MATH 2410: Multivariable Calculus and Linear Algebra}\\[4pt]
  Time allowed: 90 minutes \quad | \quad \numpoints\ points total
\end{center}

\vspace{0.6em}
\makebox[\linewidth]{Name: \enspace\hrulefill\hspace{2em} Student ID: \enspace\hrulefill}

\vspace{1.2em}
\noindent\fbox{\parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{%
\textbf{Honor statement.} I affirm that I will not give or receive any
unauthorized help on this exam, and that all work will be my own.\\[1.4em]
Signature: \enspace\hrulefill\hspace{2em} Date: \enspace\makebox[1.4in]{\hrulefill}}}

\vspace{1.2em}
\textbf{Instructions.} Closed book, no calculators. Show all work; correct
answers without justification receive little credit. If you need more room,
use the back of the page and indicate clearly where your work continues.

\vspace{1em}
\begin{center}
  \gradetable[h][questions]
\end{center}

\vspace{1em}
\begin{questions}

\question[10]
Let $f(x, y) = x^2 y + e^{xy}$.
\begin{parts}
  \part[5] Compute the partial derivatives $f_x$ and $f_y$.
  \part[5] Find the equation of the tangent plane to $z = f(x,y)$ at the
  point $(1, 0, 1)$.
\end{parts}

\question[10]
Let $A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$.
\begin{parts}
  \part[4] Find the eigenvalues of $A$.
  \part[4] Find an eigenvector for each eigenvalue.
  \part[2] Is $A$ diagonalizable? Justify your answer.
\end{parts}

\question[10]
Evaluate $\displaystyle\int_0^{1}\!\!\int_0^{x} 2xy \,dy\,dx$, then sketch
the region of integration and rewrite the integral with the order of
integration reversed.

\question[10]
Determine whether the series
$\displaystyle\sum_{n=1}^{\infty} \frac{n}{n^3 + 1}$ converges or diverges.
Name any convergence test you use and verify its hypotheses.

\end{questions}

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