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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 1112 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.
 Three '11

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