# Term Exam 2

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.

Good Luck!

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

Is continuous for all ? Prove or give a counterexample.

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.

1. Prove that if a function satisfies on then for some constant .
2. 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

Your drawing must clearly indicate the domain of definition of , all intersections of the graph of with the axes, the behaviour of far out near and near the boundaries of its domain of definition, the regions on which it is increasing or decreasing, the regions on which it is convex or concave and all local and global minima and maxima of .

The generation of this document was assisted by LATEX2HTML.

Dror Bar-Natan 2003-12-01