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\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Things to Know -- Exam 2}
\centerline{October 23, 2000}
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The second full-period exam of the semester will be given 
in class on Friday, October 27.   The format will be similar
to that of the first exam.
We will review in class on Thursday, October 26.  If there
is interest, Jodi Marani (our TA) has volunteered to run an
informal review/question session on Thursday afternoon or
evening as well.
You should know the following material from Chapters 2 and 4
of the text:
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\item{1)}  The Meanings of the First and Second Derivatives of 
$f(x)$
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\itemitem{a)} Know how the sign of $f'(x)$ is related to whether the function is 
increasing or decreasing at $x$, and 
\itemitem{b)}how the sign of $f''(x)$ 
is related to concavity.  
\itemitem{c)} Know how to sketch the graphs 
$y = f'(x)$ and $y = f''(x)$ qualitatively, given the graph $y = f(x)$ (or 
$y = f'(x)$).  
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\item{2)}  Computing the derivative $f'(x)$ from the definition
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\itemitem{a)} Know how to estimate a numerical value of
$f'(a)$ from a graph of $y = f(x)$, given scales on $x$ and $y$ axes 
\itemitem{b)} Know how to compute $f'(a)$ from the limit definition
$$f'(a) = \lim_{h\to 0} {f(a+h) - f(a)\over h}$$
for ``simple'' functions $f(x)$ like linear or quadratic polynomials.
\itemitem{c)}  Know how to estimate $f'(a)$ numerically by finding
${f(a+h) - f(a)\over h}$ for $h$ tending to 0.
\itemitem{d)}  Know how to compute $f'(x)$ from the limit definition
$$f'(x) = \lim_{h\to 0} {f(x+h) - f(x)\over h}$$
for ``simple'' functions $f(x)$ like linear or quadratic polynomials.
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\item{3)} Derivative Formulas (Sections 1-3 from Chapter 4)
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\itemitem{a)}$f'(x) = n x^{n-1}$ for the power function
$f(x) = x^n$
\itemitem{b)}$f'(x) = \ln(a) a^x$ for the exponential function
$f(x) = a^x$
\itemitem{c)} Sum and constant multiple rules 
\itemitem{d)} Product rule -- ${d\over dx}(f(x)g(x)) = f(x)g'(x) + g(x)f'(x)$
\itemitem{e)} Quotient rule:
$${d\over dx}\left({f(x)\over g(x)}\right) = {g(x)f'(x) - f(x)g '(x)\over (g(x))^2}$$
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\noindent
Good Review Problems:
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Review problems at the end of Chapter 2; Pages 237-238 of the text
have a large collection of practice differentiation problems.  Not
all of the functions there can be differentiated using only the rules
covered on this exam (for instance, any function involving the
trigonometric, inverse trigonometric, or logarithm functions), but many 
can.  Try a good selection of these.

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