test1-05 - (10(ii Prove that there is some number x such...

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Department of Mathematics, University of Toronto Term Test 1 – November 17, 2005 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes Examiners: I. Alexandrova, M. Harada, V. Ivrii, G. Lynch, M. Saprykina, A. Savage 1. (10%) (a) Solve for all x which satisfies | 2 x | + | x + 1 | = 3 2 . (10%) (b) Solve the inequality x x + 1 x - 2 x + 5 and express your answer as a union of intervals. 2. (5%) (a) Give the precise ε , δ definition of the following statement: lim x a f ( x ) = L . (13%) (b) Prove that lim x 3 5 x - 9 x 2 - 1 = 3 4 directly using the precise definition of limit. 3. (10%) (i) Evaluate lim x 5 x - 5 x 2 + 11 - 7 x + 1 . (10%) (ii) Evaluate lim x 0 f ( x ) , where | f ( x ) - 3cos x | ≤ p | x | for all x . 4. (10%) (i) Suppose f is defined as follows. f ( x ) = x - x 2 x - 1 , x < 1 , ax 2 + x + b , 1 x 2 , ( x 3 + 1 ) sin ( x - 2 ) x 2 - 3 x + 2 , x > 2 . Suppose f is continuous for all x . Find the values of a and b , or show that a and b do not exist.
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Unformatted text preview: (10%) (ii) Prove that there is some number x such that x 5 + 40 x 2 + cos 2 x + 1 = 10. (12%) 5. Recall the Fibonacci Sequence { F n } is defined recursively by F 1 = 1, F 2 = 1, and F n + 2 = F n + 1 + F n for all n ≥ 1. We define the Lucas Sequence { L n } recursively as follows: L 1 = 1 , L 2 = 3 , L n + 2 = L n + 1 + L n for all n ≥ 1 . Prove for all positive integers n ≥ 2 that L n = F n-1 + F n + 1 . 1 6. (5%) (a) State the Least Upper Bound Axiom. (5%) (b) Suppose S is a set of real numbers that obeys the following properties: (P1) S 6 = R . (P2) S 6 = ∅ . (P3) If x ∈ S and y < x , then y ∈ S . Prove that lub S exists. 2...
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test1-05 - (10(ii Prove that there is some number x such...

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