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\pgre{}{Due Thursday, Week 10}

I wrote a practice GRE!  It's a bit shorter (i.e. 40 problems in place of the normal 66) than a normal GRE exam, but it should still be a good warmup.  Try it out!  Give yourself about 2 hours or so to take this test, as it's a bit less than two-thirds the length of a normal GRE exam.



\begin{enumerate}
\item[]\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\item Consider the sequence
\begin{align*}
\{\alpha_n\}_{n=1}^\infty = \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac24, \frac34, \frac15, \frac25, \frac35, \frac45, \ldots
\end{align*}

Which of the following sets of numbers is the collection of all values $\beta$ for which a subsequence of $\{ \alpha_n \}^\infty_{n=1}$ converges to $\beta$? 

\begin{multicols}{3}
\begin{enumerate}
\item $\mathbb{Q}$.
\item $\mathbb{Q} \cap [0,1]$.
\item $\mathbb{R}$.
\item $[0,1]$.
\item $\emptyset$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Suppose that a function $f: \mathbb{C} \to \mathbb{C}$ has a complex derivative at every point in $\mathbb{C}$.  Which of the following must be true?
\begin{enumerate}
\item[I.] $f$ can be written as a power series that converges on all of $\mathbb{C}$. 
\item[II.] The integral of $f$ along any closed path is 0.
\item[III.] $f$ is a surjection from $\mathbb{C}$ to $\mathbb{C}$.
\end{enumerate}

\begin{multicols}{3}
\begin{enumerate}
\item I only.
\item II only.
\item II and III only.
\item I and II only.
\item I, II and III.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Suppose that $f(x)$ has Taylor series $\displaystyle \sum_{n=0}^\infty a_nx^n$, and that $g(x) = (f(x))^2$.  What is $g'''(0)$?



\begin{multicols}{3}
\begin{enumerate}
\item $2(a_0 + a_1 + a_2)$.
\item $6(a_0a_3 + a_1a_2)$
\item $3(a_1a_2 + a_0a_3)$.
\item $12(a_0a_3 + a_1a_2)$
\item Not enough information to tell.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}




\item Suppose that $G$ is a group and $g \in G$ is an element such that the set $\{g^{10}, g^6, g^{15}\}$ generates all of $G$.  Which of the following numbers are \textbf{not} possible orders of $G$?



\begin{multicols}{3}
\begin{enumerate}
\item 1.
\item 7.
\item 12.
\item 30.
\item They are all possible orders for $G$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item An element $r$ in a ring $R$ is called \textbf{irreducible} if we cannot write $r$ as the product of two non-unit elements. A ring $R$ is called a \textbf{unique factorization domain} if we can write every element of $r$ uniquely as some product of irreducible elements.

Which of the following are unique factorization domains?

\begin{enumerate}
\item[I.] $\mathbb{Z}[\sqrt{2}] = \{a + b\sqrt{2} ~|~ a, b \in \mathbb{Z}\}$.
\item[II.]  $\mathbb{R}[x]$, the collection of all real-valued polynomials with coefficients in $\mathbb{R}$.
\item[III.] $\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5} ~|~ a, b \in \mathbb{Z}\}$.
\end{enumerate}


\begin{multicols}{3}
\begin{enumerate}
\item I only.
\item II only.
\item II and III only.
\item I and II only.
\item I, II and III.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Fix any real number $a$.  How many solutions $(v,w,x,y,z)$ are there to the following set of linear equations?
\begin{alignat*}{1}
av+aw &= 1\\
aw+ax &= 1\\
ax+ay &= 1\\
ay+az &= 1\\
aw+az &= 1\\
\end{alignat*}



\begin{enumerate}
\item No solutions exist.
\item Exactly one solution exists.
\item Exactly two solutions exist.
\item Infinitely many solutions exist.
\item There is not enough information to answer this problem.
\end{enumerate}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\item Let $S$ be the surface given by the level set $x^2 + y^3 + z^4 = 3$.  Which of these planes are tangent to $S$ at the point $(1,1,1)$?



\begin{enumerate}
\item $x=z$.
\item $\frac{x}{2} + \frac{y}{3} + \frac{z}{4} = \frac{13}{12}$.
\item $2x+3y+4z = 3$.
\item $9 = x+y+z$.
\item None of these are the tangent plane.
\end{enumerate}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Let $g: \mathbb{R}^4 \to \mathbb{R}^2$ be defined by the equation $(a,b,c,d) = (a^b, c^d)$, and $h: \mathbb{R}^2 \to \mathbb{R}^4$ be defined by the equation $h(x,y) = (x,y,xy, x)$.  

Find the derivative of $(g \circ h)(x,y)$ with respect to $x$, evaluated at $(x,y) = (1,1)$.

\begin{multicols}{3}
\begin{enumerate}
\item (0,e).
\item (1,1).
\item $(e,1)$.
\item $(1,e^2)$.
\item Undefined.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}






\item Which of the following Fourier series corresponds to the \textbf{sawtooth wave} $s(x)$ drawn below?

\begin{align*}
s(x) =\left\{ \begin{array}{cc} x, & x \in [-\pi, \pi),\\ s(x \pm 2\pi), & \forall x \in \mathbb{R}. \end{array} \right.  \\
\end{align*}
\begin{center}
\includegraphics[width=3in]{math1d_wk4_sawtooth.pdf}
\end{center}

\begin{multicols}{3}
\begin{enumerate}
\item $\displaystyle \sum_{n=1}^\infty \frac{\pi (-1)^{n+1} \cdot\cos(nx)}{n}.$
\item $\displaystyle \sum_{n=1}^\infty \frac{\pi\sin(nx)}{2^n}.$
\item $\displaystyle \sum_{n=1}^\infty \frac{\sin(nx)x^n}{n!}.$
\item $\displaystyle \sum_{n=1}^\infty \frac{\pi (-1)^{n+1} \cdot\sin(nx)}{2n}.$
\item $\displaystyle \sum_{n=1}^\infty \frac{\sin(nx) + \cos(nx)}{n^2}.$
\end{enumerate}
\end{multicols}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\item Suppose that $\langle F, +, \cdot\rangle$ is a field.  Which of the following properties \textbf{must not} be true?





\begin{enumerate}
\item $F$ is finite.
\item $\forall x \in F, \exists y \in F$ with $x \cdot y = 1+1$.
\item $\exists x, y \in F$ with $xyx^{-1}y^{-1} \neq 1$.
\item $\mathbb{Q}$ is a subset of $F$.
\item If $G$ is a sobgroup of $\langle F^\times, \cdot\rangle$, then $G$ is cyclic.
\end{enumerate}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}




\item Suppose that $f(x)$ is a function such that $f'(x) = \cos(2x)e^{3x}$, and $f(0) = 0$.  What is $f(x)$?

\begin{enumerate}
\item $\displaystyle \frac{2}{9} \sin(2x)e^{3x} + \frac{2}{9}\cos(2x)e^{3x} - \frac{2}{9}$.
\item $\displaystyle\frac{1}{36}\sin(2x)e^{3x}$.
\item $\displaystyle\frac{3}{5} \sin(2x)e^{3x} + \frac{2}{5}\cos(2x)e^{3x} - \frac{3}{5}$.
\item $\displaystyle\frac{3}{13} \cos(2x)e^{3x} + \frac{2}{13}\sin(2x)e^{3x} - \frac{3}{13}$.
\item $\displaystyle\frac{5}{3} \sin(2x)e^{3x} + \frac{10}{9}\cos(2x)e^{3x} - \frac{5}{3}$.
\end{enumerate}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item What is the maximum value of $f(x,y) = 4x-3y$ given the constraint $x^2 + y^2 = 4$?

\begin{multicols}{5}
\begin{enumerate}
\item 1.
\item 2.
\item 6.
\item 8.
\item 10.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}
\item Look at the function $f: \mathbb{R} \to \mathbb{R}$ defined by
\begin{align*}
f(x) = \left\{ \begin{array}{cc} 0 & x \notin \mathbb{Q}\\ q, & x = \frac{p}{q}, p,q \in \mathbb{Z}, gcd(p,q) = 1\end{array} \right.
\end{align*}

Let $C$ denote the collection of all points at which $f(x)$ is continuous.  Which of the following properties hold for $C$?

\begin{multicols}{3}
\begin{enumerate}
\item[I.] $C$ is open.
\item[II.]  $C$ is closed.
\item[III.]  $C$ is compact.
\end{enumerate}
\end{multicols}
\begin{multicols}{3}
\begin{enumerate}
\item I only.
\item II only.
\item I and II only.
\item  II and III only.
\item I, II and III.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}



\item Find the integral 
\begin{align*}
\int_0^{1} \frac{x^{1/2}}{1+x^{1/3}}dx.
\end{align*}

\begin{enumerate}
\item $\displaystyle \frac12$.
\item  $\displaystyle 5\left( \frac15 + \frac13 - \frac{\pi}{4} \right)$
\item $\pi/4 - \frac{1}{5} + \frac{1}{7}$
\item $\displaystyle 6\left( \frac17 - \frac15 + \frac13 - 1 + \frac{\pi}{4} \right)$
\item Does not exist.
\end{enumerate}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}



\item Consider the process of rolling a fair $n$-sided die.  Let $X$ be the random variable corresponding to the face that appears when such a die is rolled.  What is the standard deviation of $X$?

\begin{multicols}{3}
\begin{enumerate}
\item $n$.
\item $\displaystyle \sqrt{\frac{n^2 - 1}{12}}$.
\item $\displaystyle\binom{n}{2}$.
\item $\displaystyle\sqrt{\frac{n+1}{2}}$.
\item $\displaystyle\frac{6}{\sqrt{(n+1)(2n+1)}}$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}
\item Suppose that $A$ is a $n \times n$ matrix with $2$'s on the diagonal and 1's elsewhere.  What is the characteristic polynomial of $A$?



\begin{multicols}{3}
\begin{enumerate}
\item $(x-2)^n$.
\item $(x-1)^{n/2}(x-2)^{n/2}$.
\item $x\cdot (x-n+1)^{n-1}$.
\item $x^n - nx + 2$.
\item $(x-1)^{n-1}(x-n-1)$.
\end{enumerate}
\end{multicols}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}





\item Which of the statements below is \textbf{not} equivalent to the logical formula
\begin{align*}
((A \lor B) \land (A \to C) \land (B \to C)) \to C~~ ?
\end{align*}

\begin{enumerate}
\item $(A \to B) \Leftrightarrow (\lnot B \to \lnot A)$
\item $((\lnot A \to B) \land (\lnot A \to \lnot B)) \to A$
\item $\lnot(A \land B) \Leftrightarrow (\lnot A \lor \lnot B)$
\item $((A\to B) \lor (B \to A) \lor (A \to C) \lor (C \to A)) \to C$
\item $((A \to B) \land (B \to C)) \to (A \to C)$
\end{enumerate}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}



\item What is the integral of the vector field $F(x,y,z) = (x,y,z)$ over the outward-oriented surface $S$ formed by intersecting the ellipsoid $\displaystyle \frac{x^2}{4} + y^2 + z^2 = 1$ with the positive half-space $\{(x,y,z) ~|~ z \geq 0\}$?



\begin{multicols}{5}
\begin{enumerate}
\item $\displaystyle \frac{8\pi}{3}$.
\item $\displaystyle -\frac{8\pi}{3}$.
\item $16$.
\item $0$.
\item $8\pi$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Suppose that $A$ is a $n \times n$ matrix with $n$ distinct eigenvalues, and let $\vec{x}$ denote the vector $(1,2,3,\ldots n)$.   Suppose that for any permutation $\sigma \in S_n$, we have $\lim_{n \to \infty} A^n\cdot \sigma(\vec{x}) = \vec{0}.$  What must always be true about $A$?



\begin{enumerate}
\item $A$ is nilpotent.
\item More than half of the entries of $A$ are zero.
\item $A$ has no nonzero eigenvalues.
\item All of $A$'s eigenvectors are in $(-1,1)$.
\item $A$ is invertible.
\end{enumerate}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}



\item What is the length of the curve $\gamma(t) = (t, \ln(\sec(t))$ on the interval $[0, \pi/4]$?



\begin{multicols}{3}
\begin{enumerate}
\item $\ln(\sqrt{2}/2 + 1)$.
\item $\ln(\sqrt{2} + 1)$.
\item $\ln(\sqrt{2}/2)$.
\item $\pi \cdot \ln(\sqrt{2})$.
\item $1-\ln(\sqrt{2})$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\item Which of the following five objects has a bijection with $\mathbb{R}$?

\begin{enumerate}
\item The Gaussian integers.
\item The collection of all finite subsets of $\mathbb{N}$.
\item The collection of all co-finite subsets of $\mathbb{Z}$.
\item The collection of all ordinal numbers.
\item The set of all uncountable subsets of the real line.
\end{enumerate}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\item Let $f(x) = y$ be the solution to the the differential equation 
\begin{align*}
 \frac{dy}{dx} + 3 y= x^2+1,
\end{align*}
given the boundary condition that at $x = 0$ we want $y = 0$.

What is $f(0)$?
\begin{multicols}{5}
\begin{enumerate}
\item $0$.
\item $1/3$.
\item $5/9$.
\item $6/7$.
\item $-2/3$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}



\item How many roots does the equation $2x^3 + 3x^2 + 6x + c = 0$ have?


\begin{multicols}{3}
\begin{enumerate}
\item 0.
\item 1.
\item 2.
\item 3.
\item It depends on $c$.
\end{enumerate}
\end{multicols}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}





\item Find the volume of the tetrahedron spanned by the three vectors $(1,2,3), (2,3,1), (3,1,2)$.


\begin{multicols}{5}
\begin{enumerate}
\item 6.
\item 9.
\item 12.
\item 18.
\item 36.
\end{enumerate}
\end{multicols}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\item Which of the following series converge?
\begin{multicols}{3}
\begin{enumerate}
\item[I.] $\displaystyle \sum^\infty_{n=1} \frac{n^3}{3^n}$.
\item[II.]  $\displaystyle \sum^\infty_{n=1} \frac{n!}{n^n}$.
\item[III.]  $\displaystyle \sum^\infty_{n=1} \frac{1}{n\ln(n)}.$
\end{enumerate}
\end{multicols}
\begin{multicols}{3}
\begin{enumerate}
\item I and II only.
\item I and III only.
\item II and III only.
\item I, II and III.
\item None of the above.
\end{enumerate}
\end{multicols}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\item As $n$ goes to infinity, which of the five expressions listed below is the \textbf{largest}?


\begin{enumerate}
\item The number of degree-$n$ polynomials with coefficients in $\{1,\ldots n\}$.
\item The number of even-order permutations in $S_n$.
\item The number of subsets of the set $\{1,2\ldots n\}$.
\item The reciprocal of the coefficient of $x^n$ in $e^x$'s Taylor series.
\item $\binom{2n}{n}$.
\end{enumerate}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Suppose that $G$ is a set endowed with a binary operation $\cdot: G \times G \to G$, such that the following properties hold:
\begin{itemize}
\item For all $a,b,c \in G$, we have $(a \cdot (b \cdot c)) = (a \cdot b) \cdot (a \cdot c)$.
\item There is an element $e \in G$ such that for all $a \in G$, $e \cdot a = a \cdot e = a$.
\end{itemize}

Which of the following properties must be true?

\begin{enumerate}
\item[I.] $G$ is a group.
\item[II.] $G$ is infinite.
\item[III.] For any $g \in G$, there is some $n \in \mathbb{N}$ such that $g^n = e$.
\end{enumerate}
\begin{multicols}{3}
\begin{enumerate}
\item II only.
\item III only.
\item I and III only.
\item I, II and III.
\item None of the above.
\end{enumerate}
\end{multicols}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\item What is the average value of the function $f(x,y) = xy$ over the set $S = \{(x,y) ~|~ x^2 + y^2 = 1, x, y \geq 0\}$ of points on the unit circle with positive co\"ordinates?



\begin{multicols}{5}
\begin{enumerate}
\item $1/2\pi$.
\item 1.
\item $1/\pi$.
\item 0.
\item $2\pi/3$.
\end{enumerate}
\end{multicols}

\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}



\item Consider the following algorithm, that takes in as input two positive natural numbers $n, m$ and outputs another natural number $p$:
\begin{Verbatim}
input (n) 
input (m)
p:= 0

while n > 0
	begin
		if (n mod 2 == 1)
			p := p + m
			n := (n-1)/2
			m:= 2*m
		else
			n := n/2
			m := 2*m
	end
output(p)
\end{Verbatim}

On input $n = 100, m = 2$, what is the output of this process?
\begin{multicols}{5}
\begin{enumerate}
\item 0.
\item 2.
\item 50.
\item 100.
\item 200.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Let $A$ be a $n \times n$ matrix.  Which of the following statements is false?

\begin{enumerate}
\item The dimension of the null space of $A$ plus the dimension of the kernel of $A$ is $n$.
\item The dimension of the row space of $A$ is the dimension of the image of $A$.
\item The dimension of the column space of $A$ is never less than the dimension of the row space of $A$.
\item The rank of $A$ plus the dimension of the null space of $A$ is at least the dimension of the row space.
\item The sum of the dimensions of $A$'s row, column and null spaces is always less than $2n$.
\end{enumerate}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Determine the radius of convergence of the following power series.
\begin{align*}
\sum_{n=1}^\infty \frac{\cos(\pi n)x^n}{n}
\end{align*}


\begin{multicols}{3}
\begin{enumerate}
\item $\infty$.
\item $0$.
\item $1$.
\item $\pi$.
\item $n$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Suppose that $A, B, C$ are three pairwise independent events.  Which of the following \textbf{is not necessarily} true?

\begin{enumerate}
\item $P(A) = P(A|B)$.
\item $P(A)\cdot P(B)\cdot P(C) = P(A \cap B \cap C) $.
\item $\displaystyle P(A) = \frac{P(A \cap C)}{P(C)}$.
\item $(P(A)\cdot P(B)\cdot P(C))^2 = P(A \cap B)\cdot P(A \cap C)\cdot P(B \cap C)$.
\item $P(B\cap C) = P(B)\cdot P(C)$.
\end{enumerate}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item If $\begin{bmatrix} 2 & 1 \\ 3 & 0 \end{bmatrix} \cdot \begin{bmatrix}x \\ y \end{bmatrix}$ is the vector $(2,0)$, what is $x$?

\begin{multicols}{5}
\begin{enumerate}
\item 0.
\item $1/3$.
\item $1$.
\item $2/3$.
\item $3$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Find the limit as $x$ goes to 0 of the following fraction:
\begin{align*}
\frac{ 6\sin(x) - 6x +x^3}{x^5}.
\end{align*}

\begin{multicols}{5}
\begin{enumerate}
\item 0.
\item 1.
\item $1/20$.
\item $6/20$.
\item $-1/120$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}



\item Given a set $S$, let $\mathcal{F}(S)$ denote the collection of all finite subsets of $S$, and $\mathcal{I}(S)$ denote the collection of all infinite subsets of $S$.  Which of the following are true? 
\begin{enumerate}
\item[I.] $\mathcal{F}(\mathbb{N})$ is a topology on the natural numbers.
\item[II.] $\mathcal{I}(\mathbb{N})$ is a topology on the natural numbers.
\item[III.] The union $\mathcal{F}(\mathbb{N}) \cup \mathcal{I}(\mathbb{N})$ is a topology on the natural numbers.
\end{enumerate}
\begin{multicols}{3}
\begin{enumerate}
\item I only.
\item II only.
\item III only.
\item II and III only.
\item I, II and III only.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}



\item We call a group $G$ ``pleasant'' if it is abelian and every non-identity element $g$ in $G$ has prime order.  Up to isomorphism, how many pleasant groups are there of order at most $25$?

\begin{multicols}{5}
\begin{enumerate}
\item 0.
\item 9.
\item 16.
\item 25.
\item 31.
\item Infinitely many.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item What is the antiderivative of 
\begin{align*}
\frac{x-1}{x^2+x} ~?
\end{align*}

\begin{multicols}{3}
\begin{enumerate}
\item $\displaystyle \frac{\tan(x)}{x}$.
\item $\ln(x^2+x)$.
\item $2\ln(x+1)-\ln(x)$.
\item $\displaystyle \frac{1}{x^2-x}$.
\item $\displaystyle \frac{\cos(x)}{x\sin(x)}$.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}







\item A sequence of numbers $d_1, \ldots d_n$ is called \textbf{graphic} if there is a simple undirected graph $G$ on $n$ vertices $\{v_1,\ldots v_n\}$ where the degree of each vertex $v_i$ is equal to $d_i$.  

Which of the following sequences are graphic?
\begin{enumerate}
\item[I.] $d_1 = 1, d_2 = 2, d_3 = 3, d_5 = 4, d_5 = 5$.
\item[II.] $d_1 = 1, d_2 = 1, d_3 = 2, d_5 = 2, d_5 = 3$.
\item[III.] $d_1 = 3, d_2 = 3, d_3 = 3, d_5 = 3, d_5 = 3$.
\end{enumerate}

\begin{multicols}{3}
\begin{enumerate}
\item I only.
\item III only.
\item II and III only.
\item I and II only.
\item None of the above.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}

\newpage
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\item Find $2015^{2015}$ mod 7.

\begin{multicols}{5}
\begin{enumerate}
\item 0.
\item 1.
\item 2.
\item 4.
\item 6.
\end{enumerate}
\end{multicols}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}



\item Which of these trigonometric expressions is equal to $\cot(x) + \tan(x)$?

\begin{enumerate}
\item $\cos(x)\sin(x) - \sin(x)\cos(x)$.
\item $\sin^2(x) - \cos^4(x)$.
\item $\tan(x)\sin(x)$
\item $\tan(x) - \csc(x)$.
\item $\sec(x)\csc(x).$
\end{enumerate}
\noindent\makebox[\linewidth]{\rule{\textwidth}{2pt}}


\end{enumerate}







\end{document}











-sequence limitxxx
-seriesxxxx
-more seriesxx
-function limitxxx
-continuityxxxx
-derivatives as linear approx
-tricky derivatives to evaluatexxx
-higher order derivatives
-curve sketching and derivatives
-mean val thm
-max/minxxx
-related rates
-integration by subxxxxx
-int by partsxxxxxxx
-trig subsxxxx
-partial fractionsxxxxxxx
-fundamental thm of calc
-integral as averagexxxxx
-area between curves
-polarxxxx
-volume of solids of rev
-arc lengthxxxxxxxxx
-lhopitalxxxxxx
-power series, convergencexxxxxxx
-taylor seriesxxxx
-error terms on taylor
-dot/cross productsxxxx
-lines, planes in 3d
-other shapes in 3dxxx
-cyl/sph coordsxxx
-partial derivatives
-tangent planesxxxx
-chain rule for higher-orderxxxx
-dir. derivatives, gradientxxxx
-max/min problemsxxxxxx
-Lagrangexxxxx
-line ints of functions, vfields
-multiple intsxxx
-gauss green stokesxxx
-DEs of one kindxxxxxx

