\magnification = \magstep1
\def\grad{\nabla}
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\centerline{Mathematics 241 -- Multivariable Calculus}
\centerline{Information on Final Exam}
\centerline{December 3, 2007}
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\noindent
{\it General Information}
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The final exam for this course will be given at 8:30 a.m. on Tuesday, 
December 11.  The exam will be {\it comprehensive\/}, meaning that it will
cover all the material we have studied since the beginning of the 
semester (see list of topics to be covered below).  
I will write the exam to be roughly twice the length of the 
in-class midterms.  However, you will have the full three-hour
period 8:30 - 11:30 a.m. to work on the exam if you need that much time.
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\noindent
{\it Review Session}
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I will be happy to run a daytime or evening review session to help
you prepare for the final exam.  We can set up a time in class
on Monday, December 3.
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\noindent
{\it Topics to be Covered}
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\item{a)} Vectors, the dot product, lengths and angles, orthogonal projections
\item{b)} The cross product, equations of lines and planes
\item{c)} Parametric curves and motion, tangent vectors and tangent lines 
to parametric curves, curvature
\item{d)} Vector fields, flow lines, gradient fields, curl, divergence
\item{e)} Limits and continuity for $f(x,y)$
\item{f)} Directional and partial derivatives (know the definition
of $D_u f(x_0,y_0)$ and how to compute it using the definition).
Differentiability for $f(x,y)$ (know the definition and what it means)
\item{g)} Derivative matrices and the general Chain Rule.
\item{h)} The gradient vector field $\grad f(x,y)$; the
geometric meaning of the vector $\grad f(x_0,y_0)$; the relation between 
contours of $f(x,y)$ and flow lines of $\grad f(x,y)$; the relation
between $\grad f(x_0,y_0)$ and directional derivatives. 
\item{i)} Critical points of $f(x,y)$ and their relation with
critical points of $\grad f(x,y)$ -- the ``First Derivative Test''
\item{j)} The tangent plane to a graph $z = f(x,y)$.
\item{k)} The Second Derivative Test (Hessian Criterion) for critical points of $f(x,y)$.
(Know the statement and how to apply it.)
\item{l)} The Lagrange Multiplier method for constrained max-min
problems 
\item{m)} Riemann sums and double integrals over rectangular
regions in ${\bf R}^2$.
\item{n)} Double Integrals over non-rectangular regions via
iterated integrals with variable limits of integration
\item{o)} Triple Integrals,
\item{p)} The change of variables formula for multiple integrals (know the 
statement and how to apply it). 
Polar coordinates in ${\bf R}^2$ and polar double integrals; cylindrical 
and spherical coordinates in ${\bf R}^3$ and cylindrical and spherical triple integrals.
\item{q)} Applications of double and triple integrals to volume, mass
centers of mass, etc.
\item{r)}  Line integrals and Green's Theorem (know the statement of Green's Theorem
and how to apply it).  

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\noindent
{\it Notes} 
\item{1)} The only new topic not covered on the midterm exams is the last one
(line integrals and Green's theorem).  
\item{2)} Some of the exam questions may ask you to ``put together'' several
of these topics.
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\noindent
{\it Suggestion on How to Study}
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Begin by reviewing the class notes and the discussion write-ups from 
your group's work, but don't stop there. 
Look at the in-class midterm problems and try to 
work out solutions for those again, {\it especially the ones you 
had difficulty with the first time around\/}!
As you do this, refer to your previous
work and/or my comments on your exam sheet as needed, but be sure you 
can solve problems like those on the midterm exams on your own;
the large majority of the final exam
problems will be similar.  Try  a selection of 
the practice/review problems from the review sheets for Exams
1,2,3 again also. 

Then (and only then) take a look at the practice exam questiosn and
try to work out solutions:
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\noindent
{\it Practice Final Exam Questions}
\input PracticeFinal.tex
\bye

