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\centerline{Mathematics 241 -- Multivariable Calculus}
\centerline{Make-up Exam 1}
\centerline{October 3, 2007}
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{\it Directions}
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Do all work in the blue exam booklet.  Do not place anything you want
considered for credit on this sheet.  There are 100 possible points.
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I.  All parts of this question refer to the three points 
$P = (1,0,-1)$, $Q = (3,2,0)$, and $R = (2,-1,6)$ in ${\bf R}^3$.
\item{A)}  (10) Give a parametrization of the {\it line segment} from $P$ to $Q$
including the proper range of $t$-values.  
\item{B)}  (5)  Is $R$ on the line from part A?  (Take the whole line here, not
just the line segment.)
\item{B)}  (15) Find the equation of the plane containing $R$ and perpendicular
to the line from part A.
\item{C)}  (10) What is the angle between $v = R - Q$ and $w = P - Q$?
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II. (15)  Let ${\bf x} = (x,y,z)$ be the position vector of a point in ${\bf R}^3$, 
let ${\bf i} = (1,0,0)$, ${\bf j} = (0,1,0)$, and consider the equation 
$${{\bf x} \cdot {\bf i}\over {\bf x} \cdot {\bf j}} = 1.$$
Rewrite the equation using spherical coordinates, and
describe the set of all points $(x,y,z)$ satisfying the equation.
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III.  Consider the function 
$$f(x,y) = \cases{x^3 + 3x^2 y\over x^2 + 4 y^2 & if $(x,y) \ne (0,0)$\cr
                        0 & if $(x,y) = (0,0)$\cr}$$
\item{A)} (5)  What would have to be true to say $f$ is {\it continuous} at $(0,0)$?
\item{B)} (10) If we let $(x,y)$ tend to $(0,0)$ along any curve $y = m x$, where
$m > 0$, what is the limiting value?  Does your answer here answer the question
of whether $f$ is continuous at $(0,0)$?  Explain.
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IV.  All parts of this question refer to the function $f : {\bf R}^2 \to {\bf R}$
defined by $f(x,y) = \sin\left(x^2 + y^2\right)$.
\item{A)} (10) Find the partial derivative ${\partial f\over \partial x}$ at
$(x,y) = \left(0,\sqrt{\pi}\right)$.
\item{B)}  (10)  If we used cylindrical coordinates in ${\bf R}^3$ instead of 
rectangular coordinates, what would be the equation of the graph of $f$? 
\item{C)} (10) Describe or sketch the level curve of $f$ for the value $c = -1$.  
(Careful -- there is more than one connected piece!)  
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{\it Extra Credit}  (10) Let $S$ be the surface in ${\bf R}^3$
defined by $2 x^2 - 8 y^2 - 4z^2  = 8$.  Describe
the slices of $S$ in planes of each of the forms
$x = a$, $y = b$ and $z = c$.  
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