Download Study Guide for Exam 1.

Study Guide for Exam 1.
Math. 417
1. Theorems from Calculus:
In this course, we extensively use the Mean Value Theorem, the Mean Value
Theorem for Integrals, Rolle’s Theorem, the Intermediate Value Theorem,
and Taylor’s Theorem with error term. Review all of these theorems and
know when they can be applied.
2. Iterative Methods for One Dimensional Problems:
Understand how and why the bisection method works. Know the fixed point
iteration and the Fixed Point Theorem (Theorem 2.4) and its corollary.
Know the algorithm and derivation of Newton’s method.
3. Polynomial Interpolation:
Know how to solve low dimensional interpolation problems, e.g., on Pj ,
j = 0, 1, 2, 3. Know the error formula for polynomial interpolation and how
to derive simple error bounds using it, especially in the case of piecewise
polynomial interpolation. Know how define Lagrange interpolation polynomials and how to use them to solve the polynomial interpolation problem.
Know how to set up Hermite interpolation problems for cubics and why they
are useful.
4. Finite Differences:
Be able to compute the finite difference approximation based on a group of
nodes by differentiating the interpolating polynomial (using the Lagrange
polynomials to develop the interpolating polynomial). Know the error estimate associated with the procedure ((4.2) in the book). Be also able to
compute the finite difference approximation and the error terms by applying
undetermined coefficients.
5. Numerical Integration:
Be able to derive the weights (given the quadrature nodes) by integrating
Lagrange polynomials. Also, be able to derive the weights by undetermined
coefficients. Be able to derive simple error bounds for quadrature error
from the error estimate for polynomial interpolation. Be able to translate a
quadrature from a given interval to an interval of different length for use in
composition quadrature approximation. Be able to derive the weights and
node placement for Gaussian quadrature using undetermined coefficients.
6. MATLAB programming:
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Be able to write a simple m-file function to execute basic iterations, e.g., a
fixed point or Newton’s method iteration. You should also be able to write
a simple function for doing a composite quadrature approximation.