|Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:||(284)||
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Solve the following 6 problems. Each is worth 20 points although they may have unequal difficulty, so the maximal possible total grade is 120 points. 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 presiding officers.
Duration. You have 3 hours to write this exam.
Allowed Material: Any calculating device that is not capable of displaying text or graphs.
Problem 1. We say that a set of real numbers is dense if for any open interval , the intersection is non-empty.
Problem 2. Sketch the graph of the function . Make sure that your graph clearly indicates the following:
Problem 3. Compute the following integrals:
Problem 4. Agents of the CSIS have secretly developed two functions, and , that have the following properties:
Problem 6. Let be a sequence of complex numbers and let be another complex number.