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University of Toronto, February 10, 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. ** Suppose that is nondecreasing on .
Notice that is automatically bounded on , because
for any in .

- If is a partition of , write formulas for and in as simple terms as possible.
- Suppose that for each . Show that .
- Prove that is integrable.

**Problem 2. ** In each of the following, is a
continuous function on .

- Show that
- Characterize the functions that have the property that
for all .

**Problem 3. **

- Prove that if two functions and both satisfy the differential equation and if they have the same value and the same first derivative at 0, then they are equal.
- Use the above to show that for all . (Do not use the formula for the of a sum!)

**Problem 4. **

- Compute
- Use your result to estimate the difference between and . Warning: a 10 digit answer obtained with your calculator may contribute negatively to your grade. You shouldn't use any calculating device and your derivation of the answer should be simple enough that it be clear that you didn't need any machine help.

**Problem 5. ** Evaluate the following integrals in terms of
elementary functions:

- (cancelled)

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Dror Bar-Natan 2003-02-10