test1-03 - f ( x ) is continuous at x = a . (7%) (b)...

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Department of Mathematics, University of Toronto Term Test 1 – November 5, 2003 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes Examiners: V. Blomer, K. Consani, M. Harada, G. Leuschke, D. Miller, M. Pinsonnault, P. Rosenthal, S. Uppal, R. Wendt 1. Evaluate the following limits. (No proofs are required.) (8%) (i) lim t 1 t 3 - 1 t - 1 . (8%) (ii) lim x 3 x 2 - 5 - 1 + x x 2 - 9 . (12%) 2. Solve the inequality | x - 3 | + x 9 . Express your answer in interval notation. 3. (5%) (a) Give the formal ε , δ deﬁnition of the statement lim x a f ( x ) = L . (8%) (b) Suppose for all x , | f ( x ) | ≤ B , where B is some positive number. Prove directly from the ε , δ deﬁnition that lim x 0 x f ( x ) = 0. (12%) (c) Prove directly from the ε , δ deﬁnition that lim x 3 x 1 + x 2 = 3 10 . (12%) 4. Recall the Fibonacci sequence : F 1 = 1, F 2 = 1, and F n + 2 = F n + 1 + F n for all n 1. Prove that F 2 1 + F 2 2 + ··· + F 2 n = F n F n + 1 for all positive integers n . 5. (5%) (a) Give the deﬁnition of the statement:

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Unformatted text preview: f ( x ) is continuous at x = a . (7%) (b) Suppose f ( x ) = sin ax bx , x > , 3 , x = , a 2 x + a + b , x < . If f is continuous for all x ∈ R , what are the values of a and b ? 6. (8%) (i) Let f ( x ) = x 3 + x-3. Find an integer n such that f ( x ) contains a root on the interval [ n , n + 1 ] . Justify your answer. 1 (8%) (ii) Find the least upper bound of the set S = { x 6∈ Q | x 2 ≤ 16 } . You may use any results from your lectures or problem sets to prove your answer is correct. (7%) 7. Suppose f ( x ) is even and g ( x ) = lim t → t f ( x )-f ( x-2003 t ) exists for all x . Determine whether g ( x ) is even, odd, or neither. 2...
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This note was uploaded on 04/09/2010 for the course MAT 137 taught by Professor Uppal during the Spring '08 term at University of Toronto.

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test1-03 - f ( x ) is continuous at x = a . (7%) (b)...

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