|Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I:||(114)||
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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 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 derivative and the following integrals:
Problem 4. In solving this problem you are not allowed to use any properties of the exponential function .
Problem 5. In solving this problem you are not allowed to use any properties of the trigonometric functions.
Problem 6. In solving this problem you are not allowed to use the irrationality of , but you are allowed, indeed advised, to borrow a few lines from the proof of the irrationality of .
Is there a non-zero polynomial defined on the interval and with values in the interval so that it and all of its derivatives are integers at both the point 0 and the point ? In either case, prove your answer in detail.