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\title{Entry Quiz for M3150, Fall 2012 \\ Due day: M, August 27  }
\author{Sign your name:}
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\maketitle
This quiz lists  topics in the prerequisite courses M1220 and M2250 or equivalent that are  critical for understanding the material of this course M3150. Solve the problems and read  the textbooks if you meet difficulties. 
\section{Integration}
Evaluate the integrals, review the method of evaluation
\begin{equation}
\int_a^b\cos(c\, x)  \ dx, \quad \int_{-\pi}^\pi \sin(n\, x) \ dx, \quad n \mbox{ is an integer}
\label{1}
\end{equation}

\begin{equation}
 \int_a^b\cos^2(c\, x)  \ dx, \quad \int_{-\pi}^\pi \sin^2(n\, x) \ dx, \quad n \mbox{ is an integer}
\label{2}
\end{equation}

\begin{equation}
\int_a^b\cos(c\, x)\, \cos(d\, x) \ dx \quad \int_a^b\sin(c\, x)\ \cos(d\, x) \ dx
\label{3}
\end{equation}

\begin{eqnarray}
& &\int_{-\pi}^\pi \sin(n\, x)\sin( m\, x) \ dx, \quad n, m \mbox{ are integers. Consider cases } n=m, n \neq m
\label{4}
\\
& & \int_{-\pi}^\pi \sin(n\, x)\cos( m\, x) \ dx, \quad n, m \mbox{ are integers } 
\label{5}
\end{eqnarray}

\begin{equation}
\int_a^b x\, \cos(c\, x) \ dx, \quad \int_a^b x^2\cos(c\, x) \ dx, \quad \int_a^b e^x \sin(c\, x) \ dx
\label{6}
\end{equation}

\section{Linear algebra}
Define parameter $a$ so that vectors $p$ and $q$ are orthogonal,
\begin{equation}
p= (1, \ 2), \quad q=(a, -1).
\label{7}
\end{equation}

For a symmetric matrix $A$,  
\begin{equation}
A=\left(\begin{matrix} 2 &1 \cr 1 & 2 \end{matrix}\right),
\label{8}
\end{equation}
a) find the eigenvectors and show that they form an {\em orthogonal basis},
\newline
b) represent  vector $( 2, 3)$ in that basis. 

\section{Differential equations}
Solve the equations (find a general solution):

\begin{equation}
\frac{d u}{dx} + c u=0,
\label{9}
\end{equation}

\begin{equation}
\frac{d^2 u}{dx^2} + c^2 u=0, \quad \frac{d^2 u}{dx^2} - c^2 u=0,
\label{10}
\end{equation}

Find all solutions of the second-order differential equation with one initial condition. 
\begin{equation}
\frac{d^2 u}{dx^2} + c^2 u=0, \quad u(0)=0.
\label{11}
\end{equation}
Sketch the graph of that family. 


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