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\title{Engineering and Science Mathematics I}
\author{Review for Standard Track Final Exam}
\date{Saturday, December 21, 18:30--20:30, Sports Hall I}
\maketitle


\begin{enumerate}
\item Limits of functions, limits of sequences, one-sided limits: You
should be able to solve most problems from the Chapter 2 Review on
page p.\ 96, in particular the following exercises: 15, 21, 31, 39.
For the last two problems you may (but don't need to) use Taylor
series for the functions in the numerator and denominator (as
discussed in class) or l'H\^opital's rule (not discussed in class, but
it is essentially the same thing).

Have a look at some of the Section 11.2 problems as well.  Some facts
and tricks you should know:
\begin{enumerate}
\item By setting $x=1/n$, you can always convert a limit where
$n\to\infty$ to a limit where $x\to0^+$. 
\item $\displaystyle \lim_{x \to 0} x^k \, \ln x = 0$ for every
$k>0$.  
\end{enumerate}
Things you should try: p.~632 problems 9, 13, 25, 33.  The last one
is tricky, but very instructive: First consider the logarithm of this
expression so that you can use the Taylor series of $\ln (1+x)$.

\item Continuity: When is a function continuous, can a discontinuous
function be extended to be continuous (p.~97 examples 57--60)?  More
advanced question: Can you show that differentiability implies
continuity?

\item Derivative, implicit differentiation, Taylor series.  Typical
exam problems will be similar to p.~179 no.~27, p.~208 no.~52,
``Compute the derivative of $y=\arctan x$'', or ``Compute first three
non-zero terms of the Taylor series of $f(x)=\ln (1+x)$ about the
point $x=0$.''

\item Curve sketching: Given a function, find
\begin{enumerate}
\item Domain
\item Vertical Asymptotes
\item Horizontal Asymptotes
\item Extrema
\item Points of Inflection
\end{enumerate}
Use all this information to sketch the graph.  Expect a final exam
question like Problem~1 on Midterm~II.

\emph{Important:} When you find critical points, you \emph{must} check
if and how the first derivative changes sign in order to determine
whether the critical point corresponds to a a minimum, maximum, or
neither.  The corresponding check for sign changes of the second
derivative to find points of inflection is also necessary.

\item Applied Minimax Problems: There will be (at least) one such
problem on the exam.  Review the Section 3.6 homework as well as the
corresponding questions on Midterm~I.

\item Use integration to compute the area under a curve or the area
between two curves.

\item Techniques of Integration (Chapter 9): On the exam you will not
be told which method to use, so recommended practice material are the
Chapter 9 Miscellaneous Problems on page 563.

As for Midterm II, all integration problems on the exam will be taken
literally from pages 563--566.

\item Volume, Arc Length, Surface Area: Expect at least one question
on the exam which is similar to homework/quizzes from these topic areas.

\item Initial Value Problems:  Know how to solve a separable initial
value problem.   Again, see homework/quizzes from this topic area.

\item Series, Convergence: Be able to spot a geometric series and
compute its limit.  Integral test for convergence; spot
non-convergence in cases where the sequence of terms that make up the
series does not converge to zero.

\item Basic Vector Operations: Dot product, cross product, parametric
equations for lines and planes.  See homework for Sections 12.2 and
12.3; p.~741 no.~35.

\end{enumerate}

General comments:
\begin{itemize}
\item \emph{There is a quiz on the last day of classes.}  This quiz is
supposed to be a strong hint as to which skills are the most essential
for the Final Exam.

\item Office hours during reading and final week:
 
      Friday, 13.12., 14:00, Research I Room 107 (M. Oliver) \\
      Monday, 16.12., 14:00, Research I Room 114 (S. Mittwollen) \\
      Wednesday, 18.12., 14:00, Research I Room 114 (S. Mittwollen) \\
      Friday, 20.12., 14:00, Research I Room 107 (M. Oliver)

\item Math Support Center during reading and final week:

      Sunday, 15.12.: \emph{No Math Support Center}
      Monday, 16.12.--Friday, 20.12.: 19:00--21:00 West Hall 5--8. 

\item No notes and calculators on the exam.
\item Paper will be supplied.
\end{itemize}


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