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\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Exam 3 -- Things To Know}
\centerline{November 27, 2000}
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\noindent
{\it General Information}
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As announced in the course syllabus, the third and last full-period
midterm exam of the semester will be given in class on Friday, 
December 1.  The format will be similar to that of the first
two midterm exams.  The exam will cover the material since the second
midterm (Quizzes 6,7,8), including the following material from 
Chapter 4 sections 4,5,6, Chapter 5 sections 1,3,5, and 
Chapter 3 sections 1,2,3 of the text:
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\item{1)}  The Chain Rule for derivatives:  
$${d\over dx}f(g(x)) = f'(g(x))g'(x)$$
\item{2)}  Derivatives of trigonometric and inverse functions:
$$\eqalign{
{d\over dx} \sin(x) &= \cos(x)\cr
{d\over dx} \cos(x) &= -\sin(x)\cr
{d\over dx} \ln(x) &= {1\over x}\cr
{d\over dx} \arcsin(x) &= {1\over \sqrt{1-x^2}}\cr
{d\over dx} \arctan(x) &= {1\over 1 + x^2}\cr}$$
Be able to use these in combination with the sum, product, quotient,
and chain rules.
\item{3)}  Critical points of functions, the First and Second 
Derivative Tests
\item{4)}  Applied max/min problems -- I will ask one very close
to the ones in the handout given in class on Monday, November 20.
\item{5)}  Left- and Right-hand sums for functions, the definite 
integral, area and average value of a function, the equation 
$|RHS - LHS| = |f(b) - f(a)|\Delta x$ if $f$ is either increasing
or decreasing on the whole interval $[a, b]$.
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\noindent
We will review for the exam in class on Thursday, November 30.  If
there is interest, I will also try to set up an evening session with Jodi
Marani Wednesday or Thursday for other questions.
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\noindent
{\it Sample Exam}
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\noindent
I.  Find the derivatives of each of the following functions.  Say
which rules you are applying in each case:
\item{A)}  $f(x) = (1 - x^2)^{5/2}$
\item{B)}  $g(x) = {\cos(x)\over 1 - \sin(x)}$
\item{C)}  $h(x) = \arctan(x \ln(x))$
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\noindent
II.  The following graph shows $y = f'(x)$ (the {\it derivative}
of $f$).  
\vglue 1.9truein
Use the information there to find the critical points of $f$,
and classify each as a local maximum, local minimum, or neither. 
{\it Note:  I may also ask a question of this kind where the 
given information is a formula for $f$, and you must find $f'(x)$,
determine the critical points, and the sign of $f'(x)$ between
each consecutive pair of critical points to apply the First
Derivative test and/or compute $f''(x)$ to apply the Second
Derivative Test.}
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\noindent
III.  Holy Cross is building a new running track next to Fitton Field.  
It is to be the perimeter of the region obtained by putting together 
two identical semicircles on opposite sides of a rectangle.  Due 
to the ongoing parking crunch on campus, the area surrounded by 
the track will be used as a parking lot when track practice is
not being held.  If the track is to be 
500 yards long, determine the dimensions of the track to maximize 
the area of the parking lot. 
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IV.  The following graph shows $y = f(x)$.
\vglue 1.9truein
\item{A)}  Use the information from the graph to compute the 
left- and right-hand sums for $f(x)$ on $[0,3]$ with $\Delta x = 1/2$.
\item{B)}  Use the information from the graph to compute
the integral $\int_0^3 f(x)\ dx$.
\item{C)}  Your answer from B should be negative.  What 
does this mean in terms of areas?
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\noindent
Also see Problem Sets 6,7,8, the last three quizzes, and 
the Review Problems from the ends of Chapters 3,4,5 in
the text for other practice problems.
\bye
