\magnification=\magstep1
\centerline{Mathematics 172 -- Numerical Analysis 2}
\centerline{Review Sheet -- Midterm Exam}
\centerline{February 25, 1999}
\bigskip
\noindent
{\bf General Information}
\bigskip
As announced in the course syllabus, the midterm exam for this
course will be given in class on Thursday, March 4.  You will have
the full class period to work on the exam.  The format of the exam
will be similar to that of the inclass exams last semester -- four
or five problems, each with several parts.  The material to 
be covered includes the following topics, through the material on
Gaussian Elimination from class on February 23 (and Lab 3).  Specifically:
\bigskip
\noindent
{\bf Topics To Be Covered}
\bigskip
The exam will cover the following material:
\item{1)}  Adaptive Quadrature (know the error estimate
that is the basis of the algorithm and how to 
apply the algorithm in relatively simple cases like integrals in problems
1,2 from section 4.6).
\item{2)}  Gaussian Quadrature (know the theory behind the Gaussian
Quadrature method -- see the Theorems to Know A below -- 
and how to apply it to approximate a definite integral if you 
are given a table of the roots $x_{n,i}$ and the weights $c_{n,i}$.
I will not ask you to compute roots or weights as we did in Lab 1, though.)
\item{3)}  Methods for ODE initial value problems: Euler's Method,
the Modified Euler and Midpoint Methods, the Classical 4th order 
Runge-Kutta method.  Be prepared to carry out several steps of 
each of these methods by hand on a reasonably simple equation.
Also know how to estimate their one-step or local truncation errors.
\item{4)}  Gaussian Elimination for linear systems $Ax = b$ (be able
to carry out the algorithm on a $3\times 3$ or $4 \times 4$ system, 
with or without maximal column pivoting.
\bigskip
\noindent
{\bf Theorems to Know}
\bigskip
\item{A)}  Show that if $\{p_0(x), \ldots ,p_n(x)\}$ are polynomials
orthogonal with respect to the inner product
$$\langle f,g\rangle = \int_a^b f(x)g(x)\ dx,$$
with $\deg(p_i) = i$ all $i$, and such that $p_n(x) = 0$ has $n$ real roots
$x_1, \ldots, x_n$,  then with weights
$$c_i = \int_a^b {\prod_{k\ne i} (x - x_k) \over \prod_{k\ne i} (x_i - x_k)}\ dx,$$
the approximate integration rule
$$\int_a^b f(x)\ dx \doteq c_1f(x_1) + \cdots + c_n f(x_n)$$
has degree of precision (at least) $2n - 1$.
\item{B)}  The Midpoint Method for the ODE IVP 
$$\cases{y' = f(t,y) & \cr
         y(t_0) = \alpha &\cr}$$
has one-step error $O(h^3)$ (or local truncation error $O(h^2)$.
\bigskip
\noindent
{\bf Some Review Problems}
\bigskip
\item{} Section 4.6/1,2
\item{} Section 4.7/1,2,3,4 (given the information in 
the table on page 227), 6,8
\item{} (possible Extra Credit problem) section 5.1/7 (Picard iteration)
\item{} Section 5.2/3,4, 5 a,b
\item{} Section 5.4/1,3,10, 13
\item{} Section 6.2/Problems like 5, 7 ("partial pivoting" = "maximal
column pivoting")
\bigskip
\noindent
{\bf Additional Problem}
\bigskip
A projectile of mass $m = 0.11$kg is shot vertically upward with
initial velocity $v(0) = 8$m/sec, from an initial height of 6m.  The projectile
encounters the usual gravitational force $F_g = -mg$ where $g = 9.8$m/sec$^2$,
and air resistance $F_r = 0.002 v^2$.  Thus the differential
equation for the velocity function of the projectile is:
$$0.11\ v^\prime(t) = -0.002(v(t))^2 - 1.078$$
\item{A)}  Use Euler's Method to find the velocity of the projectile
at $t = 0.1, 0.2, \ldots, 1.0$ (step size $h = .1$).
\item{B)}  Repeat A), but using the Runge-Kutta method of order 4.
Compare your results.  Which should be closer to the true values, and why?
\item{C)}  From the Runge-Kutta values, interpolate to find an approximation
to the time when the projectile reaches its {\it maximum} height.  Explain
how and why you arrived at your answer.
\bigskip
\noindent
{\bf Review Session}
\bigskip
I will be happy to hold an afternoon or evening review session for the 
exam either Monday or Tuesday, March 1, or 2.
\bye
