MATH1151 week7c3 - Time allowed: 20 minutes. 1. (4 marks)...

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Unformatted text preview: Time allowed: 20 minutes. 1. (4 marks) The function f is defined by f ( x ) = x 3 − x for − 1 ≤ x < 0 and f ( x + 1) = f ( x ) for all x ∈ R . (a) Find an expression for f ( x ) for 0 ≤ x < 1. (b) Prove that f is continuous at 0. That is, prove lim x → f ( x ) = f (0). (c) Does f (0) exist? Prove your answer. If it does exist, what is its value? 2. (3 marks) (a) State the Mean Value Theorem. (b) Use the Mean Value Theorem to show that, for all real x and y , | tan − 1 x − tan − 1 y | ≤ | x − y | . 3. (2 marks) It is given that 0 < < 1 2 , and the solution to | f ( x ) − 2 | < is x ∈ −∞ , − 1 − 2 ∪ 1 − 2 , ∞ . Does lim x →∞ f ( x ) exist? If so, what is its value? Give reasons for your answer. 4. (1 mark) Explain why cosh( x + h ) = cosh x + h sinh( x ) + o ( h ) as h → 0. Week 7 Friday 11-12 Class 1. f ( x ) = x 3 − x = x ( x − 1)( x + 1) for − 1 ≤ x < 0, and f ( x + 1) = f ( x ) for all x ∈ R ....
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This document was uploaded on 03/19/2012.

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MATH1151 week7c3 - Time allowed: 20 minutes. 1. (4 marks)...

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