|© | Dror Bar-Natan: Classes: 2004-05: Math 157 - Analysis I:||(113)||
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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. This booklet has 12 pages.
Duration. You have 3 hours to write this exam.
Allowed Material: Any calculating device that is not capable of displaying text.
Problem 2. Sketch the graph of the function . Make sure that your graph clearly indicates the following:
Problem 3. Compute the following derivative and the following integrals:
Problem 4. We'll say that a function is ``bigger'' than a function (and write ) if for every large enough , . Arrange the following functions by size:
Problem 5. Let be a function which is differentiable times at some point .
Problem 6. The ``Cauchy Condensation Theorem'' says that if a sequence of positive numbers is decreasing then converges iff converges.