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\centerline{Mathematics 171--Numerical Analysis I}
\centerline{Review Sheet--Midterm 2}
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{\bf General Information}
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 The second midterm will be given in class {\bf Thursday,
November 19}.  This will be a 75 minute in-class exam. 
You may (and certainly should) use a hand calculator.
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{\bf Topics to be Covered}
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The exam will cover the material from the first exam through
the material on extrapolation from the discussion we did
in class on Thursday, November 12.  This includes the material from
 sections 2.5, 3.1, 3.2, 3.4, 4.1, 4.2 in Burden and Faires:
\item{1.}  Aitken acceleration and Steffensen's method for 
root finding.
\item{2.}  Polynomial interpolation -- the Lagrange,
divided difference forms, and Newton forms for the interpolating 
polynomial of degree $\le n$ for $S = \{(x_0,f(x_0)),\ldots,(x_n,f(x_n))\}$.
\item{3.} The error formula for interpolation.
\item{4.} Cubic spline interpolation (know the definition of a
cubic spline and how to use them to approximate values of a function, 
its derivative, etc.  I will {\it not\/} ask you to compute the coefficients
in a cubic spline, though.  As you probably realize, that could
take the whole 75 minutes by hand!)
\item{5.} Numerical differentiation methods, their error terms, 
Richardson Extrapolation.
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{\bf Proofs to Know}
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The following are somewhat loose statements of the major theoretical
points in the material we have studied.  You should review and 
know exact statements and proofs of these results.  
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\item{1.} Given any set $S = \{(x_0,f(x_0)),\ldots,(x_n,f(x_n))\}$
with $x_i$ distinct, there exists a unique polynomial of degree $\le n$
interpolating $S$.
\item{2.} The error formula for interpolation:
$$f(x) = p_n(x) + {f^{(n+1)}(\xi)\over (n+1)!} (x-x_0)(x-x_1)\cdots (x-x_n)$$
for some $\xi$.
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{\bf Review Problems}
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From Burden and Faires:
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\item{} Section 2.5: 7, 8, 11, 14, 15
\item{} Section 3.1: Problems like 3 and 7, 15 where the interpolation
points $(x_i,f(x_i))$ come from functions where you can compute and
bound the 
derivatives needed for the error formula (e.g. by graphing), 17
\item{} Section 3.2: 1,2,6,7,10,13
\item{} Section 3.4: 7,8,9,10, problems like 11,12 if you are
given the spline function.
\item{} Section 4.1: 1,2,16,21,22
\item{} Section 4.2: 5, 8, 9
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{\bf Review Session}
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Either Tuesday or Wednesday evenings next week are possible.  We can
set this up in class November 12.
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