Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (74) |
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University of Toronto, February 9, 2004

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 50 minutes.

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

**Problem 1. ** In a very condensed form, the definition of
integration is as follows: For bounded on and
a partition of set
,
,
and
. Then set
and
. Finally, if we say that `` is integrable on
'' and set
.

From this definition alone, without using *anything* proven in class
about integration, prove that the function given below is integrable on
and compute its integral
:

**Problem 2. ** Prove that the function

**Problem 3. ** In class we have proven that a
twice-differentiable function satisfying the equation is
determined by and . Use this fact and the known formulas for
the derivatives of and to derive a formula for
in terms of
, ,
and .

**Problem 4. ** The function is defined by .

- Compute for all .
- Explain why has a differentiable inverse for .
- Let be the inverse function of (with the domain of considered to be ). Compute and simplify your result as much as you can. Your end result may still contain in it, but not , or .

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Dror Bar-Natan 2004-02-09