Midterm I Review
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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. | 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. | ||
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Material: | Material: | ||
− | #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 | + | #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: |
− | + | #:<math>\sum_{i=1}^N \tfrac1i \qquad</math> 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: | #:Another example: | ||
− | + | #:Explain a derivation of the sum <math> \sum_{i=0}^N \lambda^i</math>. | |
− | #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 <math> f'(x) \approx \frac{f(x+h)-f(x-h)}{2h} + a h^p </math>. | + | #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 <math> f'(x) \approx \frac{f(x+h)-f(x-h)}{2h} + a h^p </math>. (That is, find a and p.) |
#Fixed point methods. You should understand the conditions for convergence and the methods of proof. You should remember the definition of order of convergence. | #Fixed point methods. You should understand the conditions for convergence and the methods of proof. You should remember the definition of order of convergence. | ||
+ | #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 | ||
+ | #:<math>f(x)\approx A (x-\alpha)^p</math> for some <math>p>1</math> and for x near <math>\alpha</math>, the standard newtons method doesn't have second order convergence. | ||
+ | #:<ol style="list-style-type:lower-alpha"> | ||
+ | #: <li>Find the order of convergence of Newton's method in this case.</li> | ||
+ | #: <li>Show that the second order convergence can be recovered by using <math>x_{n+1}=x_n-\frac{p f(x_n)}{f'(x_n)}</math> </li> | ||
+ | #:</ol> | ||
+ | #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 <math>\cos(x)</math> is approximated by an n-th order polynomial on n+1 nodes (equally distributed) between 0 and <math>\pi/2</math>. Estimate the error of this interpolation. |
Latest revision as of 22:35, 4 March 2009
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:
- Make sure you know how to do all the homework problems given to date.
- Make sure you know the solution to the quiz.
- Read the relevant chapters in the reader.
- Discuss the material with your peers. I highly recommend this as a method of studying.
Material:
- 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:
- 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 .
- 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 . (That is, find a and p.)
- Fixed point methods. You should understand the conditions for convergence and the methods of proof. You should remember the definition of order of convergence.
- 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
- for some p > 1 and for x near α, the standard newtons method doesn't have second order convergence.
-
- Find the order of convergence of Newton's method in this case.
-
- Show that the second order convergence can be recovered by using
- 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.