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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\begin{document}


\subsection{Mathematical Computations}

\subsubsection{Arithmetic}

\begin{enumerate}
\item  Add the fractions $\frac{3}{71}$ and $\frac{13}{101}$.

\item  Compute $100$ factorial.

\item  Factor your social security number.

\item  Is the number $1429\,553$ prime?

\item  Is the number $123123111231$ prime?
\end{enumerate}

\subsubsection{Trigonometric Functions}

\begin{enumerate}
\item  Compute the cotangent of $37\unit{\UNICODE{0xb0}}$.

\item  Express $\sin 5x$ in terms of powers of $\cos x$ and $\sin x.$

\item  Expand $\sin (x+y)+\sin (x-y).$

\item  Simplify ($\cos 3t+2\cos t)\sec t.$
\end{enumerate}

\subsubsection{Polynomials}

\begin{enumerate}
\item  Factor the expression $\allowbreak \allowbreak 6x^{2}-5xy-4y^{2}.$

\item  Find the greatest common divisor of the polymonials $%
x^{4}+x^{2}-x^{3}-x$ and $x^{3}+x+x^{2}+1.$

\item  Find the roots of the polynomial $x^{5}+x+1.$
\end{enumerate}

\subsubsection{Limits}

Evaluate the following limits:

\begin{enumerate}
\item  $\lim\limits_{x\rightarrow 0}\dfrac{1+\sin x}{1+\cos x}$

\item  $\lim\limits_{x\rightarrow 0}\dfrac{\sin x}{2x^{2}-x}$

\item  $\lim_{x\rightarrow \infty }\frac{3x^{3}+4x^{2}+5}{7x^{3}-21x^{2}+1}$
\end{enumerate}

\subsubsection{Derivatives}

Evaluate the following derivatives:

\begin{enumerate}
\item  $\dfrac{d}{dx}(3x^{2}+2x+1)$
\end{enumerate}

For $g(x)=24x^{3}+3x^{2}+7x+1$,

\begin{enumerate}
\item  Find $g^{\prime }(x)$

\item  Find $g^{\prime \prime }(x)$
\end{enumerate}

Evaluate the following derivatives

\begin{enumerate}
\item  $D_{x}\left( \dfrac{x}{x^{2}-2}\right) $

\item  $D_{x}\left( x\sqrt{1-x^{2}}\right) $

\item  $\dfrac{d^{2}}{dx}\left( \dfrac{\cos ^{2}x}{1+\tan x}\right) $

\item  $\dfrac{d}{dx}\sqrt[4]{\dfrac{x^{2}+x+1}{x^{2}-x+1}}$
\end{enumerate}

\subsubsection{Integrals}

Evaluate the following integrals

\begin{enumerate}
\item  $\int \tan ^{6}x\sec ^{4}xdx$

\item  $\int x^{5}\sqrt{x^{2}+1}dx$

\item  $\int_{0}^{3}\dfrac{dx}{1+x^{4}}$
\end{enumerate}

\subsubsection{Plotting}

\begin{enumerate}
\item  Trigonometric functions

\begin{enumerate}
\item  Plot $\sin x$

\item  Add a plot of $\cos x$ to the first plot

\item  Make the plot of $\sin x$ blue and the plot of $\cos x$ red
\end{enumerate}

\item  Polynomials

\begin{enumerate}
\item  Plot $3x^{2}+1$ and $2x^{3}-4$ on the same graph

\item  The graphs intersect at a point just greater than 2. Use the zoom
facility to find the intersection point to two decimal places.
\end{enumerate}
\end{enumerate}

\subsubsection{Miscellaneous}

Evaluate the following sums

\begin{enumerate}
\item  $\sum_{n=1}^{\infty }\frac{1}{n^{2}}$

\item  $\sum_{n=1}^{100}n^{4}$

\item  Find the cube roots of the number $1+i.$

\item  Compute the determinant of the matrix $\left( 
\begin{array}{lll}
1 & 2 & 3 \\ 
2 & 4 & 5 \\ 
3 & 5 & 6
\end{array}
\right) .$

\item  Solve the equation $\left( 
\begin{array}{ll}
2 & 3 \\ 
4 & 5
\end{array}
\right) X=\left( 
\begin{array}{ll}
3 & 7 \\ 
8 & 9
\end{array}
\right) .$

\item  Find a basis for the nullspace of $\allowbreak \left( 
\begin{array}{ccc}
3 & 1 & 0 \\ 
3 & 1 & 6 \\ 
6 & 2 & 0
\end{array}
\right) $.

\item  Solve the system of equations$
\begin{array}{l}
x+2y+3z=1 \\ 
2x+4y+5z=1 \\ 
3x+5x+6z=1
\end{array}
$.

\item  Use Newton's method to find a solution to the equation $\sin x-x^{3}=0
$ for $x>0.$
\end{enumerate}

\subsubsection{Units}

How many inches are there in a light year?

\end{document}
