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University of Toronto, February 7, 2005

This document in PDF: Exam.pdf

**Solve all of the following 4 problems. **
Each problem is worth 25 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 tutors. You have an hour and 50 minutes.

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

**Problem 1. **

- Compute .
- Compute .
- For , compute .

**Problem 2. **

- Perhaps using L'Hôpital's law, compute and .
- Use these results to give educated guesses for the values of and (no calculators, please).

**Problem 3. **

- State the ``one partition for every '' criterion of the integrability of a bounded function defined on an interval .
- Let be an increasing function on and let be the partition defined by , for . Write simple formulas for and for .
- Under the same conditions, write a very simple formula for .
- Prove that an increasing function on is integrable.

**Problem 4. **

- Show that the function is monotone on the interval .
- Deduce that for every the equation has a unique solution in the range .
- For , let be the unique in the range for which . Write a formula for and simplify it as much as you can. Your end result may still contain in it, but not , or .

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Dror Bar-Natan 2005-02-04