# Midterm I Review

To prepare for this midterm I can only remind you of the topics that we have covered and suggest ways in which you should prepare.

Methods:

1. Make sure you know how to do all the homework problems given to date.
2. Make sure you know the solution to the quiz.
4. Discuss the material with your peers. I highly recommend this as a method of studying.

Material:

1. You should totally grok the calculus review part of the course. While there will not be a question specifically asking about this part, you can be sure that many problems will have this material as required skill. Example problem: $\sum_{i=1}^N \tfrac1i \qquad$ can be bounded using an an appropriate integral of a smooth function. Draw the relevant picture and find the bounds (plural because I want 2 bounds, one from above and one from below).
Another example:
Explain a derivation of the sum $\sum_{i=0}^N \lambda^i$.
2. Taylor Series, with/without remainder term, go over all our derivations of error for various expressions. Example for question: Calculate the error in the approximation $f'(x) \approx \frac{f(x+h)-f(x-h)}{2h} + a h^p$. (That is, find a and p.)
3. Fixed point methods. You should understand the conditions for convergence and the methods of proof. You should remember the definition of order of convergence.
4. Root finding. Understand the connection between fixed point methods and root finding methods. Example problem:
In the derivation of Newton's Method we implicitly assume that the root is a regular root. If the root is not regular, so that $f(x)\approx A (x-\alpha)^p$ for some p > 1 and for x near α, the standard newtons method doesn't have second order convergence.
5. Find the order of convergence of Newton's method in this case.
6. Show that the second order convergence can be recovered by using $x_{n+1}=x_n-\frac{p f(x_n)}{f'(x_n)}$
• Polynomial interpolation. Make sure you know the different forms of writing the interpolating polynomial. Remember the formula for the remainder term. Example problem:
The function cos(x) is approximated by an n-th order polynomial on n+1 nodes (equally distributed) between 0 and π / 2. Estimate the error of this interpolation.