Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: | (135) |
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University of Toronto, December 2, 2002.

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. ** Prove that there is a real number so that

**Problem 2. **

- Define in precise terms `` is differentiable at ''.
- Let

**Problem 3. ** Calculate in each of the following
cases. Your
answer may be in terms of , of , or of both, but reduce
it algebraically to a reasonably simple form. You do not need
to specify the domain of definition.

(a) | (c) | |||

(b) | (d) |

**Problem 4. **

- Prove that if on some interval then is increasing on that interval.
- Sketch the graph of the function .

**Problem 5. ** Write a formula for
in terms of
, and . Under what conditions does your formula hold?

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Dror Bar-Natan 2002-12-02