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\centerline{Mathematics 136, section 2 -- AP Calculus}
\centerline{Information on Final Exam}
\centerline{December 7, 2009}
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\noindent
{\it Background}
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The final exam for this section of AP Calculus will be given {\it at 
8:30 a.m. on Friday, December 18, in our regular classroom -- Stein 316\/}. 
The exam will be {\it comprehensive}, covering all the material we have 
studied since the beginning of the semester.  In addition to the material 
covered on the three midterms, this includes the topics of geometric and 
Taylor series from the last week.

I will write the exam to be roughly twice the length of the 
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.
The questions will be mostly similar to those on the 3 in-class exams,
although there might be some that combine topics in somewhat different
ways (see II, IV in the Practice Final below).  
I may include some questions asking for definitions
of terms or statements of theorems (see III in the Practice
Final below).  
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\noindent
{\it Topics to be Covered}
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\item{1)}  Functions defined by graphs, tables, formulas,
the ``library of functions''
\item{2)}  Derivatives (know the definition):  know how to sketch 
the graph of the derivative of a function defined by a graph, 
how to approximate values of the derivative given a table for the 
function, how to use the derivative rules on functions defined by 
formulas.  
\item{3)}  The meaning of the signs of $f'(x), f''(x)$.  
\item{4)}  Differentiability = local linearity, tangent lines, 
etc.
\item{5)} Total change of a function, Riemann sums, the definite
integral (know the definition), antiderivatives
\item{6)} The Fundamental Theorem of Calculus, and
\item{7)} Integration by substitution, by parts, by trigonometric
substitution, by partial fractions, using the 
table.  A copy of a suitable portion of the table of integrals
from the text will be provided with the exam; any integral you 
need to compute will be do-able by some combination of substitution, 
integration by parts, trigonometric substitution, partial fractions,
and/or consultation with the table).  
\item{8)}  Applications of integration (setting up problems via
Riemann sums; in limit a definite integral is obtained) -- volumes
by slices, arclengths, physical examples like mass from
non-constant density functions, center of mass of a wire or a plate,
work.
\item{9)}  Differential equations -- slope fields and solutions
\item{10)}  Solving differential equations via separation of variables
and integration, including population growth models. 
\item{11)}  Geometric series, Taylor series, Taylor polynomials, approximations
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\noindent
{\it Philosophical Comments and Suggestions on How to Prepare}
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\item{$\bullet$}  The reason we give final exams  
in almost all mathematics classes is to encourage students
to ``put whole courses together'' in their minds. 
Also, preparing for the final should help to 
make the ideas ``stick'' so you will have the material at your 
disposal to use in later courses.  
\item{$\bullet$}  It may not be necessary to say this, but
here goes anyway:  {\it You should take this exam seriously} -- 
it is worth $30\%$ of your course average and it can 
pull your course grade up {\it or down} depending on how you do.  
\item{$\bullet$}  Get started reviewing early
and do some work on this {\it every day} between now 
and the date of the final.  Don't try to ``cram'' at the end.  
There's too much stuff that you need to know to approach 
preparing that way!
\item{$\bullet$}  Reread your class notes in addition to the
text, especially for topics where you lost points on the midterms.  
There are a lot of worked-out examples and discussions of all of 
the topics we have covered there.
\item{$\bullet$}  
Look over the midterm exams with the solutions. Go over your 
corrected problem sets.  If there were questions where you lost a lot 
of points, be sure you understand why what you did was not
correct, and how to solve those questions. 
\item{$\bullet$}  Be sure you actually do 
enough practice problems so that you have the 
facility to solve exam-type questions in a limited
amount of time.   
\item{$\bullet$} Look over some of the problems from the text from the 
previous review sheets, and the following problems for item 
11 above:  p. 572-3/13-18, p.616/5,7,9,11,13,15.
Also be sure you understand how you can 
use a Taylor polynomial to approximate a value of a function
at an $x$ close to the $a$ used to construct the polynomial.
\item{$\bullet$}
Then take a couple of hours and try the practice
exam problems below.
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\noindent
{\it Practice Exam}
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\input PracticeFinal.tex

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\noindent
{\it Review Session}
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I will be happy to run an evening review session for the 
final during exam week if there is interest. 
We can discuss this in class on Monday, December 7.
\bye  

