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University of Toronto, March 24, 2003

this document in PDF: Exam.pdf

**Solve the following 5 problems. ** Each is worth 20 points
although they may have unequal difficulty. 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 tutors. You have an
hour and 50 minutes.

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

**Problem 1. ** Is there a non-zero polynomial
defined on the interval and with values in the interval
so that it and all of its derivatives are integers at
both the point 0 and the point ? In either case, prove your
answer in detail. (Hint: How did we prove the irrationality of ?)

**Problem 2. ** Compute the volume of the ``Black Pawn''
on the right -- the volume of the solid obtained by revolving the
solutions of the inequalities
and
about the axis (its vertical axis of symmetry). (Check that
and hence the height of the pawn is ).

**Problem 3. **

- Compute the degree Taylor polynomial of the function around the point 0.
- Write a formula for the remainder in terms of the derivative evaluated at some point .
- Show that at least for very small values of , .

**Problem 4. **

- Prove that if and the function is continuous at , then
- Let be a number, and define a sequence via the relations and for . Assuming that this sequence is convergent to a positive limit, determine what this limit is.

**Problem 5. ** Do the following series converge? Explain
briefly why or why not:

- .
- .
- .
- .
- .

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Dror Bar-Natan 2003-03-21