Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (26) |
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University of Toronto, October 20, 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. **
All that is known about the angle is that
. Can you find
and
? Explain your reasoning in full detail.

**Problem 2. **

- State the definition of the natural numbers.
- Prove that every natural number has the property that whenever is natural, so is .

**Problem 3. **
Recall that a function is called ``even'' if
for all
and ``odd'' if
for all , and let be some
arbitrary function.

- Find an even function and an odd function so that .
- Show that if where and are even and and are odd, then and .

**Problem 4. **
Sketch, to the best of your understanding, the graph of the function

**Problem 5. **

- Suppose that for all , and that the limits and both exist. Prove that .
- Suppose that for all , and that the limits and both exist. Is it always true that ? (If you think it's always true, write a proof. If you think it isn't always true, provide a counterexample).

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Dror Bar-Natan 2003-10-20