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\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Things to Know -- Final Exam}
\centerline{December 4, 2000}
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\noindent
{\it General Information}
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\item{$\bullet$} The final exam will be given in our regular
class room 8:30am - 11:30am on Friday, December 8.  
\item{$\bullet$} This will be a {\it comprehensive exam} -- it will
cover all of the material from Chapters 1 - 5 of the text
that we have studied this semester.  See the
detailed list of topics below.
\item{$\bullet$} The final will be similar in format to the 
three full period midterm exams, but roughly twice as long.
\item{$\bullet$} If you are well prepared and work steadily, it
should be possible to complete the exam in about 2 hours.  But you
will have the full-three hour period to work on the exam if 
you need that much time.
\item{$\bullet$}  If there is interest, I can schedule a daytime 
or evening review session for the final during the reading period.
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\noindent
{\it Topics to Be Covered}
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\item{A)}  The ``library of functions'' and their properties
\itemitem{1)} Linear functions, the slope and its meaning
(linear means constant rate of change), 
slope-intercept and point-slope forms.
\itemitem{2)}  Exponential functions $f(x) = ba^x$, the shapes
of the graphs, how to determine a formula from a graph or
a table of values, exponential growth vs. exponential decay
\itemitem{3)}  Power functions $f(x) = cx^p$.  ``Which
function dominates?''
\itemitem{4)}  Inverse functions (when inverse functions
exist, how to determine the graph, etc.)
\itemitem{5)}  Logarithm functions (Most important:  $\log_a(x)$ is 
the inverse function of $a^x$, so $y = \log_a(x)$ means the 
same thing as $x = a^y$.)  Natural logarithm is logarithm
with base $a = e\doteq 2.71828\cdots$.
\itemitem{6)}  Trigonometric functions:  $\sin(x),\cos(x),\tan(x)$.
Shifted, scaled sinusoidal functions $y = A\sin(Bx) + C$ and 
$y = A\cos(Bx) + C$.
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\item{B)} The definition and meaning of the derivative of a function.
\itemitem{1)}  The derivative of $f(x)$ at $x$ is 
$$f'(x) = \lim_{h\to 0} {f(x+h) - f(x)\over h}$$
I may ask you to compute a derivative (of a relatively
simple function, of course) using this definition
\itemitem{2)} Know how to estimate the derivative of 
a function given by a table of values numerically
\itemitem{3)} Be able to sketch the 1st and 2nd derivatives
of a function given by a graph.  
\itemitem{4)} $f'(a)$ gives the {\it instantaneous rate of change
of} $f(x)$ at $x=a$.  Geometrically, this is the same as
the {\it slope of the tangent line to the graph} $y = f(x)$ at
$x = a$.  
\itemitem{5)}  $f'(x) > 0$ on an interval implies $f$ is increasing
on that interval; $f'(x) < 0$ implies $f$ is decreasing on that
interval. 
\itemitem{6)}  $f''(x) > 0$ on an interval implies that $y = f(x)$
is concave up on that interval; $f''(x) < 0$ on an interval 
implies that $y = f(x)$ is concave down on that interval.
\itemitem{7)}  If $f'(c) = 0$ or $f'(c)$ does not exist, we say 
that $c$ is a critical point.  Know how to find critical points
of functions given either by formulas or by graphs,
and be able to apply the First and Second Derivative
Tests for determining the type of the critical point (local maximum,
local minimum, or neither).
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\item{C)}  Derivative Rules
\itemitem{1)}  The power rule: ${d\over dx}\left(x^n\right) = nx^{n-1}$
\itemitem{2)}  The exponential rule: ${d\over dx}\left(a^x\right) = 
\ln(a)a^x$ (the derivative of the natural exponential $e^x$ is a 
special case of this).
\itemitem{3)}  The rules for trigonometric functions:
${d\over dx}(\sin(x)) = \cos(x)$ and ${d\over dx}(\cos(x)) = -\sin(x)$.
\itemitem{4)}  The rules for inverse functions:
${d\over dx}(\ln(x)) = {1\over x}$,
$${d\over dx}(\arcsin(x)) = {1\over\sqrt{1 - x^2}}, \qquad 
  {d\over dx}(\arctan(x)) = {1\over 1 + x^2}$$
\itemitem{5)}  Sum and constant multiple rules
\itemitem{6)}  Product rule:  ${d\over dx}(f(x)g(x)) = f(x)g'(x)+f'(x)g(x)$
\itemitem{7)}  Quotient rule: 
$${d\over dx}\left({f(x)\over g(x)}\right)
= {g(x)f'(x) - f(x)g'(x)\over (g(x))^2}$$
\itemitem{8)} Chain rule:  ${d\over dx}(f(g(x))) = f'(g(x))g'(x)$.
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\item{D)}  Applications of Derivatives
\itemitem{1)} Problems about rates of change, etc.
\itemitem{2)} Maximum/Minimum problems (again, see handout from
class on Monday, November 20.  I will ask a problem similar
to one of those!)
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\item{E)} Integrals
\itemitem{1)}  Left- and Right-hand sums
\itemitem{2)} The definite integral
\itemitem{3)}  Areas, average value
\itemitem{4)}  The Fundamental Theorem of Calculus:  If $f(x)$
is continuous on $[a,b]$ and $F'(x) = f(x)$, then
$\int_a^b f(x)\ dx = F(b) - F(a)$.
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\noindent
{\it Suggestions on How to Study}
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\item{A)}  Reread your class notes and work through the 
examples we did in class -- everything on the final 
will be very close to something we did along the way somewhere!
\item{B)}  Also look at your graded problem sets and the
write-ups from the group discussion days. 
\item{C)}  Look over your quizzes and exams, especially any 
questions you had difficulty with.  Read the comments and 
corrections -- they're there to help you figure out what you 
did wrong and how to solve the problem correctly.
{\it Everything on the final 
will be very close to something we did along the way somewhere!}
\item{D)}  Look over the practice exams for the three full-period
midterms.
\item{E)}  Don't panic!  There is a lot of material here, but 
not all that many key ideas. 
\bye    
