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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
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(b) The Extreme Value theorem. ' _ [email protected])£3we[€,mwcmm 4m J11)“ Egi’éfa, [03 33(5) 5. I“) 5 fff) V2“? [$219,] (6) The Fundamental Theorem of Arithmetic (PTA) and the signiﬁcant®bout N that
was proved carefuliy in the iectures using the FTA. [4, 3 points] QZTpﬂr) Vne N) 147/2) pi<pzz~~zgﬁ€:;i;/2
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7 ‘ 2. In ail of this question let f 2 R —> R be a function and let a E R. (a) Give a complete and accurate 6 — 6 deﬁnition of what it means for f to exist. j’lr’bgﬁ .ﬂhg) ‘jhlkbkdg 5‘?— {4pcints} xaa . (b) Let h : B, —> R and assume f = Z and Ma?) = m. Give a complete 6 —— e proof that Egﬂf — 3h(2:)) exists. [9 pOintS]
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[Jim2U '6 3 [mvmml <a§)+3[;}$%£ 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 $349+ (9:0 'X“«——[3’(9 xyle ‘?% {pm Why w: Zgwcxw Waﬂﬁj
M féHzr/o, /Z'*""Z ’ 5 6. Let :1: E I and (1,2) E Q, L1,!) ;é 0. Deﬁne y m as: + 6. Determine with justiﬁcation whether
y 6 Q or g; E I. {6 points] A
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egia)b€'®==5 a:%:>b$%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
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dz'm “Mi/163 WPCKJ5ﬂAmm m (fmemmgb? (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
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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: vow6}?— 05>!ﬁﬁﬁa. “ 7
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J; ﬁg 5/449.) A"? [Mg/b (Cantimzed answer spac for Question 8) EMT/OMB} g " (m ((9)04. ...
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 Winter '08
 Vadim
 Calculus, Supremum, Irrational number, mathematical sciences, Computer and Mathematical Sciences

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