Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (97) |
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University of Toronto, March 22, 2004

This document in PDF: Exam.pdf

**Solve the following 5 problems. ** Each is worth 20 points
though in question 4 you may earn a 5 points bonus that brings the
maximal possible total to 105/100. 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. ** Compute the following definite and indefinite
integrals in elementary terms:

**Problem 2. ** The ``unit ball'' in
is the result of
revolving the domain
(for
) around
the axis.

- State the general ``Cosmopolitan Integral'' formula for the volume of a body obtained by revolving a domain bounded under the graph of a function around the axis.
- Compute the volume of .
- State the general ``Cosmopolitan Integral'' formula for the surface area of a body obtained by revolving a domain bounded under the graph of a function around the axis.
- Compute the surface area of .

**Problem 3. ** Let be a real number which is not a
positive integer or 0, let
and let be a positive
integer.

- Compute the Taylor polynomial of degree for around 0.
- Write the corresponding remainder term using one of the formulas discussed in class.
- Determine (with proof) if there is an interval around 0 on which .

**Problem 4. ** Let be a ``sequence of sequences''
(an assignment of a real number to every pair of positive
integers) and assume that is a sequence so that for every we have
. Further assume that
.

- Show that for every positive integer there is a
positive integer

so that . - Show that .
- (5 points bonus, no partial credit) Is it always true that also ?

**Problem 5. **

- Compute the first 5 partial sums of the series .
- Prove that .

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Dror Bar-Natan 2004-03-23