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%%%%%%%%%%%%%%%%%%%%% Start /document/ma227f95.tex %%%%%%%%%%%%%%%%%%%%

%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
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\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Mon Apr 06 18:20:37 1998}
%TCIDATA{CSTFile=Exam.cst}
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\input{tcilatex}
\begin{document}


\section{\protect\vspace{1pt}Ma 227\qquad Final Exam\qquad 10 May 1995}

\vspace{1pt}

\textbf{Directions: \ }This examination is in two parts. In Part I you must
answer all six (6) problems. In Part II choose any two questions. \ Indicate
on your blue book(s) which questions you have chosen.

\subsection{Problem 1}

\paragraph{a) \ (8 points)}

Find the first four nonzero terms of the Fourier \emph{sine} series of\qquad
\qquad \qquad \qquad \qquad

\vspace{1pt}

$f(x)=\left\{ 
\begin{array}{c}
0\qquad 0<x<\pi \\ 
-2\text{ \ \ \ \ \ \ }\pi <x<2\pi
\end{array}
\right. $

\vspace{1pt}

\vspace{1pt}

\paragraph{b) \ (8 points)}

Sketch the graph of the function to which the Fourier series in (a) converges

on $-2\pi <x<4\pi $.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad

\vspace{1pt}

\vspace{1pt}

\paragraph{\protect\vspace{1pt}c) \ (9 points)}

Find the eigenvalues and eigenfunctions for the problem

$\vspace{1pt}$

$y^{\prime \prime }+\lambda y=0$, \ \ $y^{\prime }(0)=y^{\prime }(1)=0$

\vspace{1pt}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\ 

Be sure to check the cases $\lambda <0$, $\lambda =0$, \ and $\lambda >0$ .

\subsection{\protect\vspace{1pt}}

\subsection{Problem 2}

\paragraph{a) \ (10 points)}

Use separation of variables, $u(x,t)=X(x)T(t)$ , to find ordinary \qquad
\qquad

differential equations which $X(x)$ and $T(t)$ must satisfy if $u(x,t)$

is to be a solution of\vspace{1pt}

\vspace{1pt}

\qquad \qquad $11t^{2}x^{9}u_{xx}-(t-3)(x+2)u_{ttt}=0\qquad \qquad \qquad
\qquad $

\vspace{1pt}

Do \emph{not} solve these equations.

\vspace{1pt}\vspace{1pt}

\paragraph{b) \ (15 points)}

Solve:

\qquad P.D.E.: \ $u_{xx}-4u_{tt}=0$

\vspace{1pt}

\qquad B.C.'s: \ \ $u_{x}(0,t)=0\qquad u_{x}(\pi ,t)=0$

\qquad

\qquad I.C.'s: \ \ $u(x,0)=0\qquad u_{t}(x,0)=-8\cos (4x)+17\cos (8x)\qquad
\qquad \qquad $

\vspace{1pt}\newpage

\subsection{\protect\vspace{1pt}}

\subsection{Problem 3}

\paragraph{a) \ (15 points)}

Solve the system $AX=B$, where

\vspace{1pt}

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad

\qquad $A=\left[ 
\begin{array}{llll}
1 & -3 & 0 & 1 \\ 
1 & 0 & -1 & 0 \\ 
3 & -6 & -1 & 2 \\ 
0 & -1 & 1 & 2
\end{array}
\right] $ \ \ and\qquad $B=\left[ 
\begin{array}{l}
2 \\ 
1 \\ 
5 \\ 
0
\end{array}
\right] \qquad \qquad \qquad $

\vspace{1pt}\vspace{1pt}

\vspace{1pt}

\vspace{1pt}

\paragraph{b) \ (10 points)}

Find the inverse of the matrix:

\vspace{1pt}

\qquad \qquad $A=\left[ 
\begin{array}{lll}
1 & 1 & 1 \\ 
1 & 2 & 2 \\ 
1 & 2 & 3
\end{array}
\right] \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad $

\vspace{1pt}

\vspace{1pt}

\subsection{Problem 4}

\paragraph{a.}

Let D be the solid cut from the sphere of radius $2$ centered at the origin
by

the plane $z=1$. (For D, $z\geq 1$ . See figure)

\vspace{1pt}

\vspace{1pt}$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \FRAME{itbpF}{%
136.25pt}{146.0625pt}{0in}{}{}{Figure }{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
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\vspace{1pt}Express the volume of D as an iterated triple integral in

\subparagraph{i) \ (4 points)\ }

Rectanglar coordinates.\qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad\ \ \ \qquad \qquad \qquad

\subparagraph{\protect\vspace{1pt}ii) \ (4 points)}

Cylindrical coordinates.\qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad\ \ \ \qquad \qquad \qquad

\subparagraph{\protect\vspace{1pt}iii) \ (4 points)}

Spherical coordinates.\qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad

\subparagraph{\protect\vspace{1pt}iv) \ (3 points)}

Then find the volume by evaluating one of the three triple

integrals you obtained above.\qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad\ \ \ 

\paragraph{\protect\vspace{1pt}}

\paragraph{b)}

Let $\overrightarrow{F}=(y\cos z-yze^{x})\overrightarrow{i}+(x\cos z-ze^{x})%
\overrightarrow{j}-(xy\sin z+ye^{x}+2)\overrightarrow{k}$ .

\subparagraph{\protect\vspace{1pt}}

\subparagraph{i) \ (3 points)}

Calculate the curl of $\overrightarrow{F}$.\qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad

\subparagraph{\protect\vspace{1pt}ii) (7 points)}

Does there exist a function $f(x,y,z)$ such that $\overrightarrow{F}=%
\overrightarrow{\nabla }f$ ? If yes,

\vspace{1pt}find $f(x,y,z)$ .\qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\ \ \ 

\vspace{1pt}

\subsection{Problem 5}

\paragraph{a) \ (10 points)}

The iterated integral

\vspace{1pt}

$I=\int_{0}^{2}\int_{\frac{y}{2}}^{1}ye^{x^{3}}$ $dxdy$

\vspace{1pt}

cannot be evaluated by first integrating with

respect to $x$. Sketch the region $R$ over which the integration is to be

performed. Write another iterated integral for $I$\ and

carry out the integration.\qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad

\vspace{1pt}

\paragraph{b)}

\subparagraph{\protect\vspace{1pt}i) \ (10 points)}

Use Green's theorem to evaluate the line integral

\vspace{1pt}

$\oint_{C}(x^{3}+\sin x\sin y)dx+(\tan y-\cos x\cos y)dy\qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad $

\vspace{1pt}

\vspace{1pt}where $C$ is the circle $x^{2}+y^{2}=1$ .

\subparagraph{\protect\vspace{1pt}}

\subparagraph{ii) \ (5 points)}

Does the answer in (i) change if $C$ is any closed curve in the $xy$ -plane?

\vspace{1pt}

Explain.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad

\subsection{\protect\vspace{1pt}}

\subsection{Problem 6}

\paragraph{a) \ (15 points)}

Verify Stokes' Theorem for the vector $\overrightarrow{v}=y\overrightarrow{i}%
-x\overrightarrow{j},$ where $S$ is the

\vspace{1pt}hemisphere $x^{2}+y^{2}+z^{2}=9$, $z\geq 0$ .\qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad

\vspace{1pt}\vspace{1pt}

\paragraph{b) \ (10 points)}

Find the value of the line integral $\oint_{C}xdx+(x+y)dy+(x+y+z)dz$,

\vspace{1pt}where $C$ is the line segment from $(1,0,-1)$ to $(2,3,4)$%
.\qquad \qquad \qquad \qquad \qquad \qquad \qquad

\vspace{1pt}\newpage

\textbf{Part II}: Solve \emph{two entire} problems.

\subsection{\protect\vspace{1pt}Problem 7}

\paragraph{a) \ (10 points)}

Find the surface area of the caps cut from the sphere $x^{2}+y^{2}+z^{2}=4$

by the cylinder $x^{2}+y^{2}=1$ . Sketch the surface.\qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad

\paragraph{\protect\vspace{1pt}}

\paragraph{b. \ (15 points)}

Let $S$ be the surface of the solid cylinder $T$ bounded by $z=0$ and

\vspace{1pt}

$z=3$ and $x^{2}+y^{2}=4$. Evaluate$\int \int_{S}\overrightarrow{F}\cdot 
\overrightarrow{n}dS$, where

\vspace{1pt}

$\overrightarrow{F}=(x^{2}+y^{2}+z^{2})(x\overrightarrow{i}+y\overrightarrow{%
j}+z\overrightarrow{k})$ and $\overrightarrow{n}$ is the outward unit normal.

\vspace{1pt}

Sketch the surface.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad

\subsection{\protect\vspace{1pt}}

\subsection{Problem 8}

\paragraph{a) \ (13 points)}

Let $S$\ be the surface of the region $V$ bounded by $z=0$, $y=0$, $y=2,$

and the paraboloid $z=1-x^{2}$ . Apply the divergence theorem to

\vspace{1pt}

compute $\int \int_{S}\overrightarrow{F}\cdot \overrightarrow{n}dS$, where $%
\overrightarrow{n}$ is the outer unit normal to $S$ and

\vspace{1pt}

$\overrightarrow{F}=(x+\cos y)\overrightarrow{i}+(y+\sin z)\overrightarrow{j}%
+(z+e^{x})\overrightarrow{k}$ .\qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad\ 

\paragraph{\protect\vspace{1pt}}

\paragraph{b) \ (12 points)}

Find the volume of the region $T$ that is bounded by the parabolic

\vspace{1pt}cylinder $x=y^{2}$ and the planes $z=0$ and $x+z=1$.

\subsection{\protect\vspace{1pt}}

\subsection{Problem \protect\vspace{1pt}9}

\paragraph{a) \ (15 points)}

Find the eigenvalues of the matrix

\vspace{1pt}

\qquad \qquad $\left[ 
\begin{array}{lll}
2 & 0 & 0 \\ 
1 & 0 & 2 \\ 
0 & 0 & 3
\end{array}
\right] $

\vspace{1pt}

Find the eigenvectors corresponding to the \emph{largest} and \emph{smallest}

\vspace{1pt}eigenvalues.\qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad

\paragraph{\protect\vspace{1pt}}

\paragraph{\protect\vspace{1pt}b) \ (10 points)}

Let $\overrightarrow{F}$\ be such that $\oint_{C}\overrightarrow{F}\cdot 
\overrightarrow{dr}=0$ for any closed path $C$. Prove that

$\int \overrightarrow{F}\cdot \overrightarrow{dr}$ is path
independent.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad\ \ \ 

\end{document}

%%%%%%%%%%%%%%%%%%%%%% End /document/ma227f95.tex %%%%%%%%%%%%%%%%%%%%%

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