2007summertt - University of Toronto at Scarborough...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer and Mathematical Sciences MIDTERM TEST MATAB’T—Calculus II for Mathematical Sciences Examiner: R. Grinnell Date: Suns 23, 2007 Duration: 2 hours FAMILY NAME GIVEN NAMES: STUDENT NUMBER: SIGNATURE: CIRCLE your Teaching Assistant: Paula EHLERS Xiangqun ZOU Read these INSTRUCTIONS: 1. This examination paper has 8 numbered pages and 8 full questions. It is your responsibility to ensure that, at the beginning of the test, all of these pages and questions are included. 2. Answer all questions in the work space provided. If you need extra space, use the back of a page or the very East page and indicate cleafiy the location of your continuing work. 3. With the exception of the True / False questions, full points are awarded only for solutions that are correct, complete and that sufficiently display concepts and methods of MATA37. 4. You may use one calculator. NOTATION: R is the set of real numbers N is the set of natural numbers Q is the set of rational numbers I is the set of irrational numbers Do not write in the boxes below. They are for marking purposes only. The test questions begin here. 1. Give accurate, concise statements (inefficing appropriate mathematical notation) for each of the following: (a) The definition of Supremum and the Supremum Axiom. [4, 3 points] §~ - 2 g - V He WWW a Swift? Wk eggs/91¢ @613 ,4,” ,1; be 49¢ M4 mam. (Mm/{j Jazz/K Wm; wrists . (b) The Extreme Value theorem. ' _ ff@)£3we[€,mwcmm 4m J11)“ Egi’éfa, [03 33(5) 5. I“) 5 fff) V2“? [$219,] (6) The Fundamental Theorem of Arithmetic (PTA) and the significant®bout N that was proved carefuliy in the iectures using the FTA. [4, 3 points] QZTpflr) Vne N) 147/2) pi<pzz~~zgfi€:;i;/2 M {Q3} {2” we) fem AMA , k. may}; P 2‘ VMe/JV M be UP ‘ rte/V owe/K flag/’4 2 EM I6 1 [We [4 points] M 7 ‘ 2. In ail of this question let f 2 R —-> R be a function and let a E R. (a) Give a complete and accurate 6 — 6 definition of what it means for f to exist. j’lr’bgfi .flhg) ‘jhlkbkdg 5‘?— {4pcints} xaa . (b) Let h : B, —> R and assume f = Z and Ma?) = m. Give a complete 6 —— e proof that Egflf — 3h(2:)) exists. [9 pOintS] am 2% (Hf-Mm) 2: gm “gm ’7”... ‘ 'x ~45 0L M 270R ' (C ga’CDVZi’j/Tmig EILJIWJ .3: Wm «LI< 7: @< firm/40F! an git/Lam) '}£>03i [Afmvm/< 3;ij -~3hm "— (M "5m)( yxgofimwoe) + 3 (mflhmfl) [Jim-2U '6 3 [mvmml <a§)+3[;}$%£ 42g. 3. Let (1,12: 6 R. For n 6 N, let PM) represent the assertion: If lx—a] < 1 then Im+nal < (n+ 1)[a[ + 1 Use the PMI to prove that P(n) is true for all n E N. [7 peints] - , a 1 M [X “05/ 41' [3m Mum n . (Aumv czl>lx+a [:lxwaeélék/ S [X“‘6‘t/‘IL/gfi’/ < glee/+1 */ 5. Prove that the number 15 m {g m 3 4 is afigebraic. [8 points] ' : ' Z. ‘ L 3 “’7? 32 2: 55+ 3(5)7——92+*) + 3(5) [9914?) 4 (22+) P“? “‘"Wé'rWfiM/Saiz’ 4; 93,50 pm: M $349+ (9:0 'X“«——-[3’(9 xyle- ‘?% {pm Why w: Zgwcxw Waflfij M féHz-r/o, /Z'*""Z ’ 5 6. Let :1: E I and (1,2) E Q, L1,!)- ;é 0. Define y m as: + 6. Determine with justification whether y 6 Q or g; E I. {6 points] A «9 fg-ax—eb 7b 0W3, 19w? (>1? x:&zbw% egia)b€'®==5 a:%:>b$%i I» "will EwflJQQ @490 Carreohwx made ‘1 Quaker; need $54 We 7. Six True / False Questiens. Carefuliy ed each statement. If the aseertion must be true, then circle T (for true). Otherwise, circ} F (for false). Justification is neither required nor rewarded, but a. email workspace is given r your rough work. Each correct answer earns 3 points and each incorrect / blank. answer ea n5 0 points. ‘i‘po ' (a) if f : R we R andthen there is some 6 > 8 such that if — E] < 1 Whenever $E(a——6,a+6). mi;me xzzégz/ T @ jg X€(Ct;gjar+cg\) (b) The Weil—Ordering Property impiies the Principle of Mathematicai Induction. All @ F (0) Assume g : D -+ R is a continuous function on its domain D. If g(a)g(b) < 0 for some a and b in D then 9(0) 3 0 for some number 0 lying between a and b. (DWWM ewwaemef; T M @STflvu-Qfs m “D:(—62930)U(01°‘3) , 5 x70 SWL‘ M f* (d) Every transcendental number is irrational. g t 3% MG Jam payee) mm @‘3 F dz'm “Mi/163 WPCKJ5flAmm m (fmemm-gb? (e) Let b > 0 and B m {m2 : a: E (“5, We then have that inflB) z 0 and sup(B) m 52. @ F (f) If h is a. function with domain R and Mm + y) = Mm) + for all 23,9 6 R, then M0) = 0 and h(——:c) m ——h(:c) for all a: E R. jl[0);k[0+0] :; hIe)-+ his") #3; Mfg; :0 @ F 6 mm): mm WWW 7e 14(4) erhw) [pm at A4) 5 we! 8. Answer exactly TWO of questions A, B and C below. Points Wiii not be awarded for work shown for more than two questions. [11, 11 points] Question A Let a < b be real numbers, let D be a dense set and define A z: D D (a, 5). Use the Supremum Axiom to prove that sup(A) exists and then find (and justify with proof) the exact value of sup(A). Question 8 Let k E [2, 7]. Give a complete and accurate proof that there exists a point on the graph of the function y m e3 , Where 0 _<_ a: 3 2, Whose distance to the point (1,0) is exactly 1: units. 3+1 2$—3 18 Question C Give a complete and accurate 6 — e proof that the function f = continuous at a = 2. Sake/9W )4 Are U 0(a)!” “ Dow) Bdevflcplg) 31W 0/45 (120670: vow-6}?— 05>!fififia. “ 7 " deD)af7c/(fi&,b) M oié'Uo—‘QA (1624 M by}; ad<é>. J; fig 5/449.) A"? [Mg/b (Cantimzed answer spac for Question 8) EMT/OMB} g " (m ((9)04. ...
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This note was uploaded on 10/28/2010 for the course MATHEMATIC MATA37 taught by Professor Vadim during the Winter '08 term at University of Toronto- Toronto.

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2007summertt - University of Toronto at Scarborough...

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