\magnification = \magstep1
\def\grad{\nabla}
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\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Information on Final Exam}
\centerline{November 26, 2013}
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\noindent
{\it General Information}
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The final exam for this course will be given starting at 11:30 a.m. on Saturday, 
December 14.  The exam will be {\it comprehensive\/}, meaning that it will
cover all the material we have studied since the beginning of the 
semester, {\it including the material from Problem Set 10 that was not included
on Exam III} (see list of topics to be covered below).  
I will write the exam to be doable in about twice the time for 
one of the midterm exams (i.e. about 100 minutes).  
However, you will have the full two and one half-hour
(i.e. 150 minute) period 11:30 a.m. - 2:00 p.m. to work on the 
exam if you need that much time.
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\noindent
{\it Some Advice}
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The scheduling of this exam {\it will not be working in your favor}.  In my
experience, students tend to get ``burned out'' for exams that 
fall at the end of the period and since the final is an important component
of your course grade, that grade can suffer as a result.  To counteract 
that, it will be vital to start reviewing early and work steadily on this 
material so you don't need to put in maximum effort the last few days before
the exam.  Said a different way, don't put off studying for this course while 
you work on material for other classes because you might not have the focus 
and energy you will need when the time comes to prepare for this.  
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\noindent
{\it Review Session}
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I will be happy to run an (optional) review session to help
you prepare for the final exam, but due to the timing of our exam
and several other commitments I have, I think it will have to be
in the evening on one of the days during exam week.  We can discuss
this in class after Thanksgiving.  
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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
\item{d)} Vector fields, flow lines, critical points (including the 
``sink-source-saddle-center'' classification), stability of 
critical points 
\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)} 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{h)} Critical points of $f(x,y)$ and their relation with
critical points of $\grad f(x,y)$ -- the ``First Derivative Test''
\item{i)} The tangent plane to a graph $z = f(x,y)$.
\item{j)} The Second Derivative Test for critical points of $f(x,y)$.
\item{k)} The Lagrange Multiplier method for constrained max-min
problems 
\item{l)} Riemann sums and double integrals over rectangular
regions in ${\bf R}^2$ 
\item{m)} Double Integrals over non-rectangular regions via
iterated integrals with variable limits of integration
\item{n)} Polar coordinates in ${\bf R}^2$, and polar double
integrals 
\item{o)} Triple Integrals,
\item{p)} Applications of double and triple integrals to mass, volume, 
centers of mass, etc.
\item{q)} Cylindrical and spherical coordinates in ${\bf R}^3$.
\item{r)} Line integrals and Green's theorem.
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\noindent
Note -- 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, the problem sets and computer labs, 
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, my comments on your exam sheet, and the posted solutions 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 following questions from
 previous MATH 241 finals and try to work out solutions:
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\noindent
{\it Practice Final Questions}
\input PracticeFinal.tex
\bye

