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# a2 - Q in R(b Use deﬁnitions and ideas from your lecture...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences MATA37 Calculus II for Mathematical Sciences Summer 2007 Assignment #2 Work on the text material, current lectures and the problems below in preparation for your 2nd tutorial which takes place during the week of May 21-25. Remember: there is a quiz on Assignment 1 in the 2nd tutorial. You will have a quiz on Assignment 2 or related concepts/materials at your TA’s discretion in your 3rd tutorial which takes place during the week of May 28-June 1. STUDY: Chapters 2-6. Omit pages 75-89. PROBLEMS: 1. Pages 30-32 # 10; 18(b) (also show that the given number is algebraic). 2. Use the Principle of Mathematical Induction (PMI) to prove that for each natural number n , if A has n elements, then A has 2 n subsets. 3. Pages 52-53 # 21(a), (c); 26; 27(c) (In # 26 and 27 assume all domains are R .) 4. (a) Complete the proof of Case 2 for the density of
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Unformatted text preview: Q in R . (b) Use deﬁnitions and ideas from your lecture notes to prove that I is dense in R . (c) If D is a dense subset of R , prove that D must be inﬁnite. Proof by contradiction is useful here. 5. Pages 106-111 # 1(iv), (vi) (Just do the calculations; no proofs); 3(i), (ii); 10(d); 13; 17(a); 18; 19. 6. Give a proof of Theorem 1 on pages 98-99 that does not involve contradiction. 7. Let f ( x ) = x + 3 2 x . Prove that lim x → a f ( x ) = f ( a ) for all a 6 = 0. 8. Prove carefully that for all n ∈ N , lim x → n [ x ] does not exist. 9. Let D ⊂ R and let f : D → R be a function. Given a real number a , if lim x → a f ( x ) = l and l > 0, prove there exists a δ > 0 such that f ( x ) > 0 for all x ∈ D ∩ ( a-δ,a + δ ) but x 6 = a . 10. Page 117 # 1(i); 3; 4....
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