Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (51) |
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University of Toronto, December 1, 2003

This document in PDF: Exam.pdf

**Solve the following 4 problems. ** Each is worth 25 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 45 minutes.

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

**Problem 1. **
Let and be continuous functions defined for all , and
assume that . Define a new function by

**Problem 2. **
We say that a function is *locally bounded* on some interval
if for every there is an
so that is bounded on
. Prove that if a function (continuous or
not) is locally bounded on a closed interval then it is bounded
(in the ordinary sense) on that interval.

*Hint. * Consider the set
is
bounded on and think about P13.

**Problem 3. **

- Prove that if a function satisfies on then for some constant .
- A certain function was differentiated twice, and to everybody's surprise, the result was back the function again, except with the sign reversed: . It was also found that . Set and compute , and (making sure that you explain every step of your computation).

**Problem 4. **
Draw a detailed graph of the function

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Dror Bar-Natan 2003-12-01