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Dror Bar-Natan

May 4, 2004

This document in PDF: Final.pdf

**Solve the following 6 problems. ** Each is worth 20 points
although they may have unequal difficulty, so the maximal possible
total grade is 120 points. Write your answers in the space below the
problems and on the front sides of the extra pages; use the back of the
pages for scratch paper. Only work appearing on the front side of
pages will be graded. Write your name and student number on each page.
If you need more paper please ask the presiding officers. This booklet has
12 pages.

**Duration. ** You have 3 hours to write this exam.

**Allowed Material: ** Any calculating device that is not
capable of displaying text.

**Problem 1. **

- Define ``the function is continuous at a point ''.
- Define `` ''.
- Prove from these definitions that if is continuous at and if , then .
- Find an example for funtions and defined over all of and for which and yet (of course, will not be continuous at ).

**Problem 2. ** Sketch the graph of the function
. Make sure that your graph clearly indicates the following:

- The domain of definition of .
- The behaviour of near the points where it is not defined (if any) and as .
- The intervals over which is increasing and the intervals over which is decreasing.
- The exact coordinates of the - and -intercepts and all minimas and maximas of .

**Problem 3. ** Compute the following derivative and the
following integrals:

- The final answer here may still have an integral (which
you don't need to evaluate)

**Problem 4. ** We'll say that a function is ``bigger''
than a function (and write ) if for every large enough
, . Arrange the following functions by size:

**Problem 5. **
Let be a function which is differentiable times at some point
.

- Define ``The th Taylor polynomial of at ''.
- Prove that if is the th Taylor polynomial of at , then .
- For a certain function it is known that , and that . Prove that the point is a local max of .

**Problem 6. ** The ``Cauchy Condensation Theorem'' says
that if a sequence of positive numbers is decreasing then
converges iff
converges.

- Prove the Cauchy Condensation Theorem.
- Use the Cauchy Condensation Theorem to show that diverges.
- Use the Cauchy Condensation Theorem to show that diverges.

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Dror Bar-Natan 2005-05-12