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\section{Differentiation} 
\begin{itemize}
  \item $\frac{d}{dt} \;  e^{kt}   \; = k e^{kt}$
  \item $\frac{d}{dt} \; \cos(t)   \; = - \sin(t)$
  \item $\frac{d}{dt} \;  \sin(t)    \; =  \cos(t)$
  \item $\frac{d}{dt} \;  t^n   \; = n t^{n-1}  $
  \item $\frac{d}{dt} \;  \ln(t)   \; =  1/t  $
  \item $\frac{d}{dt} \;  
	\arctan(t)   \; =  \frac{1}{1+t^2} $
  \item $\frac{d}{dt} \;  uv   \; = 
		u \frac{dv}{dt} +v \frac{du}{dt} $
  \item $\frac{d}{dt} \;  f(w)   \; = 
		f'(w) \frac{dw}{dt}  $
  \item $\frac{d}{dt} \;  (u/v)   \; = 
		\frac{v u' - u v'}{v^2}  $
  \item $\frac{d}{dt} \; c u(t)    \; =  
		c \frac{du}{dt}  $
  \item $\frac{d}{dt} \; \left[ 
	\begin{array}{c} u(t) + v(t)\end{array} \right] \; 
                     = \frac{du}{dt} + \frac{dv}{dt}  $
\end{itemize}
Note: $c$, $k$, and $n$ are {\em constants}; 
$u$, $v$, and $w$ are {\em functions}.

\section{Integration}
\begin{itemize}
  \item $\int e^{w}  \, dw = e^w + C$
  \item $\int \sin(w) \, dw = -\cos(w) + C $
  \item $\int \cos(w) \, dw = \sin(w) + C $
  \item $\int w^{n}  \, dw = 
	\frac{1}{n+1} w^{n+1}+ C \quad 
	\mbox{for} \quad n\neq -1$
  \item $\int \frac{1}{w}  \, dw = \ln(|w|) + C $
  \item $\int \frac{1}{1+w^2} \, dw = \arctan (w) + C $
  \item $\int ( f(w) + g(w) ) \, dw = 
		\int f(w) \, dw + \int g(w)\, dw $
  \item $\int c f(w) \, dw = c  \int f(w) \, dw$
  \item $\int  u \, dv = u v -  \int v \, du$
\end{itemize}
Note:
$n$ and $c$ are {\em constants}; 
$w$ is a {\em variable};
$u$, $v$, $f$, and $g$ are {\em functions}.

\subsection{Substitution:}
If you substitute $w = f(t)$
you must also substitute
$dw = f'(t)\, dt$.

\section{Linear Algebra}
You should be able to do the following:
\begin{itemize} 
\item Convert
linear systems of equations to matrix equtions.
\item Solve linear systems by hand using 
row reduction.
\item Solve linear systems using a calculator and 
{\it Mathematica}.
\item Write down the definition 
$$
A.\vec{v} =\lambda \vec{v} 
$$ 
of an eigenvalue-eigenvector pair $\{ \lambda,\vec{v} \}$
for a square matrix A.
\item Compute the eigenvalues and eigenvectors of small
matrices by hand.
\item Check a possible eigenvalue-eigenvector 
pair $\{ \lambda,\vec{v} \}$ for a matrix A by hand.
\item Compute the eigenvalues and eigenvectors of 
matrices using a calculator and {\it Mathematica}.
\end{itemize} 

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