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\textbf{2250 - Reveiw for test 1}

\textbf{Topics:}

\begin{enumerate}

\item  Test is Monday, June 24.
\item  Test will cover Chapters 1-4. (Everything except 1.6, 3.7)
\item  Test will be closed book, closed notes.  
  You can use your calculator, but I expect to be able to follow your work as if you had no calculator.
\item  First order differential equations:
  \begin{enumerate}
  \item  Solving: Separable equations
  \item  Solving: first order linear equations
  \item  Graphing: direction fields
  \item  Existance and uniqueness theorems
  \item  Numerical methods (Euler's method, nothing about improved Euler or Runge-Kutta on test)
  \end{enumerate}

\item  Mathematical models
  \begin{enumerate}
  \item  Populations (natural growth, logistic equation, harvesting, etc)
  \item  Acceleration and velocity (including air resistance)
  \end{enumerate}

\item  Linear systems (and matrices)
  \begin{enumerate}
  \item  Solving linear systems (using inverses or Gaussian elimination)
  \item  Determining if there are no solutions, many solutions or one solution.
  \item  Finding inverses
  \item  Finding determinants (and using the determinant)
  \end{enumerate}

\item  Vector Spaces
  \begin{enumerate}
  \item  Definition of a vector space
  \item  Subspaces
  \item  Examples of vector spaces and subspaces (the usual examples as well as function spaces!)
  \item  Linear independant and dependant vectors
  \item  Basis for a vector space or subspace.
  \end{enumerate}

\end{enumerate}

\newpage
\textbf{Some things you should make sure you can do (``sample test,'' not an exhaustive list)}

\begin{enumerate}
\item  Solve a first order differential equation (Sections 1.1, 1.2, 1.4, 1.5)
\item  Determine if a first order equation has a solution (without solving it) (Section 1.3)
\item  Set up and solve a logistic equation for a population model (Section 2.1)
\item  Determine equilibrium solutions, determine stability of solutions, draw phase diagrams (Section 2.2)
\item  Solve a linear system using elimination (Section  3.2)
\item  Determine how many solutions a given linear system has (Chapter 3)
\item  Understand what reduced echelon form is (and how to use row operations) (Section 3.3)
\item  Compute the inverse of a matrix (Section 3.5)
\item  Compute the determinant of a matrix (Section 3.6)
\item  Understand what the determinant tells you.(Section 3.6)
\item  Be able to determine if a set of vectors in linearly independant or linearly dependant.
\item  Be able to compute the dimension of a vector space. (Section 4.4)
\item  What is a basis?  If you are given a set of vectors, is it a basis for a vector space? (Section 4.4)
\item  Given a set of vectors, is it a vector space (or a vector subspace of a known vector space)? (Section 4.2)
\end{enumerate}






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