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%% TITLE

\noindent
\textbf{MATH 225: Calculus III}\hfill
Name:\makebox[2.5in][l]{\rule{2in}{.5pt}}

\medskip\noindent
\textbf{Exam I}\quad{\sl September 30, 2003}\hfill
Instructor:\makebox[2.5in][l]{\rule{2in}{.5pt}}

\vspace{.5in}
\noindent
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this page. There are 8 multiple
choice questions worth 5 points each and 4 partial credits problems worth
10 points each.  You start with 20 points.  On the partial credit problems
try to simplify your answer and indicate your final answer clearly.
\textit{You must show your work and all important steps to receive credit}.

\medskip\noindent
You may use a calculator if you wish.

\vspace{.5in}

\def\answerspace{0.25in}


%% PROBLEM 1

Let $\a = \langle1,-1,1\rangle$ and $\b = \langle-1,1,1\rangle$.
Compute $\proj_{\a}\b$, the vector projection of $\b$ onto $\a$.

%% CORRECT

$\langle-1/3,1/3,-1/3\rangle$

%% WRONG

$\langle-1/3,1/3,1/3\rangle$

%% WRONG

$\langle-1,1,-1\rangle$

%% WRONG

$\langle1/\sqrt{3},1/\sqrt{3},1/\sqrt{3}\rangle$

%% WRONG

$\langle2,1,0\rangle$


%% PROBLEM 2

Determine which of the following is true for all unit vectors $\a$ and $\b$
with the property that $\a$ and $\b$ are perpendicular.

%% CORRECT

$(\a + \b)\cdot (\a + \b)=2$.

%% WRONG

$(\a + \b)\cdot (\a - \b)=2$.

%% WRONG

$(\a + \b)\cdot (\a + \b)=4$.

%% WRONG

$(\a - \b)\cdot (\a - \b)=-1$.

%% WRONG

$(\a + \b)\cdot (\a - \b)=1$.


%% PROBLEM 3

Find the distance from the point $(1,0,3)$ to the plane $2x-y-2z=-1$.

%% CORRECT

$1$

%% WRONG

$1/\sqrt{6}$

%% WRONG

$4/3$

%% WRONG

$5/3$

%% WRONG

$1/\sqrt{3}$


%% PROBLEM 4

Find the direction in which $f(x,y,z) = z^2 e^{y(1-x)}$ has the maximum rate
of change at the point $(1,1,1)$.

%% CORRECT

$\langle-1,0,2\rangle$

%% WRONG

$\langle1,1,1\rangle$

%% WRONG

$\langle1,1,2\rangle$

%% WRONG

$\langle-1,0,1\rangle$

%% WRONG

$\langle-1,1,2\rangle$


%% PROBLEM 5

Find all the values of $t$, if any, where the curve parameterized by
$$\r(t) =\langle te^{-t^2/2}, t^3 - 3t, \cos(\pi t)\rangle$$ is {\sl not}
smooth.

%% CORRECT

$t = -1, 1$

%% WRONG

$t = 1$

%% WRONG

$t = $ any integer

%% WRONG

$t = 0, 1/2, -1/2, \sqrt{3}, -\sqrt{3}$

%% WRONG

No values of $t$.


%% PROBLEM 6

Find the equation of the osculating plane (i.e., the plane determined by
$\T$ and $\N$) for the curve parameterized by $\r(t) = \langle
t^2-t,t^3,t^2+t\rangle$ at the point $(2,8,6)$.

%% CORRECT

$9x-y-3z=-8$

%% WRONG

$3x+12y+5z=132$

%% WRONG

$x+3y+3z=44$

%% WRONG

$3x-y=-2$

%% WRONG

$31x-54y+111z = 296$


%% PROBLEM 7

Determine the limit
$\lim_{(x,y)\to (0,0)} \frac{\sqrt{4+x^2+y^2}-2}{x^2+y^2}$.

%% CORRECT

$\frac{1}{4}$

%% WRONG

$\frac{1}{2}$

%% WRONG

$1$

%% WRONG

$0$

%% WRONG

\textit{does not exist}


%% PROBLEM 8

Let $f(x,y)=x\sin (x+y)$.
Calculate $f_{xx}+f_{xy}+f_{yx}$.

%% CORRECT

$4\cos(x+y)-3x\sin(x+y)$;

%% WRONG

$2\cos(x+y)+3x\sin(x+y)$;

%% WRONG

$4\cos(x+y)-2x\sin(x+y)$;

%% WRONG

$2\cos(x+y)-2x\sin(x+y)$;

%% WRONG

$2\sin(x+y)-3x\cos(x+y)$.

%% END

\newpage
%\bigskip

\noindent\llap{9.\ }Find a unit vector in the first octant and in the plane
$z = x$ that makes a $45^\circ$ angle with the $y$-axis.

\newpage
%\bigskip

\noindent\llap{10.\ }Find the parametric equations of the line of
intersection of the planes $x -y +2z =2$ and $x+2y+z=-1$.

\newpage
%\bigskip

\noindent\llap{11.\ }Find the domain and range of the function $f(x,y) =
\frac{4}{1+x^2+y^2}$. Then sketch several level curves of $f$ and use them
to sketch the graph of $f$. Be sure to show all your work and compare the
level curves to the graph.

\newpage
%\bigskip

\noindent\llap{12.\ }Suppose $f(x,y,z) = x^2 y + y^3 z + z^4 x$ where $x =
x(t)$, $y = y(t)$, $z = z(t)$ are the components of a vector function
$\r(t) = \langle x(t),y(t),z(t)\rangle$ satisfying $\r(1) =
\langle2,0,1\rangle$ and $\r'(1) = \langle3,-4,6\rangle$. Find
$\frac{df}{dt}$ at $t = 1$.

\end{document}
