MAT137Midterm1 - Give the formula and domain of . Let, with...

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1 Department of Mathematics University of Toronto MAT137Y : Calculus! Term Test 1 June 1, 2009 Time Allotted: 110 minutes Examiner: G. Richards & I. Zwiers Family Name: Given Name: Student Number: Instructions: This test consists of 12 questions , over 7 pages , including this cover. Marks for individual question are given in parenthesis, out of a total of 100 marks . Write your solutions in the space provided. Use backs of pages where necessary; always indicate clearly where solutions continue . Marks will be awarded based on the explanation of your solutions, unless otherwise indicated. No calculators or any other aid is permitted. ( For TA Use Only ) Page Marks 2 / 17 3 / 12 4 / 20 5 / 14 6 / 24 7 / 13 TOTAL / 100
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2 1. Calculate exactly. [10] [ Hint: and .] 2. [7]
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Unformatted text preview: Give the formula and domain of . Let, with , with , with 3 3. Prove that for all , [12] 4 4. (A) State the Least Upper Bound Axiom [5] (B) Give the GLB (if it exists) and the LUB (if it exists) of the set, [10] Explain each with one or two sentences. 5. (A) Let function be defined on , except perhaps at . [5] Give the precise meaning of . 5 (B) Prove by that . [14] [ Write rough work here. ] Let be given. Choose [ Write proof here. ] 6 6. Compute the following limits. Do not skip steps. You may only use theorems discussed in lecture. Names of theorems are not required. (A) [8] (B) , where is a constant. [8] (C) [8] [ Hint: Do not substitute . ] 7 7. Let . (A) Prove . [8] (B) State where is discontinuous. [5] Proof is not required....
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MAT137Midterm1 - Give the formula and domain of . Let, with...

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