# Term Exam 4

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.

Good Luck!

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.

1. 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.
2. Compute the volume of .
3. 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.
4. 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.

1. Compute the Taylor polynomial of degree for around 0.
2. Write the corresponding remainder term using one of the formulas discussed in class.
3. 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 .

1. Show that for every positive integer there is a positive integer
so that .
2. Show that .
3. (5 points bonus, no partial credit) Is it always true that also ?

Problem 5.

1. Compute the first 5 partial sums of the series .
2. Prove that .

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Dror Bar-Natan 2004-03-23