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# A8_2011 - review our lecture proof of IET 4 Let f be a...

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University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A37H Summer 2011 Assignment # 8 You are expected to work on this assignment prior to your tutorial during the week of July 11th. You may ask questions about this assignment in that tutorial. At the beginning of your TUTORIAL during the week of July 18th you need to submit the following homework problems. STUDY: Chapter 10, Chapter 11 (only up to pg 192, inclusive), reference Chapter 12 material as needed. HOMEWORK PROBLEMS: 1. Spivak # 10.31 2. Spivak # 10.18(c) 3. Let a , b R ∪ {±∞} with a < b . Let f be defined on ( a, b ). If a point c ( a, b ) is a local maximum point of f , and f is differentiable at c , then prove that f 0 ( c ) = 0. Do NOT use or cite any theorems in your proof. You must work entirely from defi- nitions and ’allowable manipulations/contstructions’. Hint: You may find it helpful to
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Unformatted text preview: review our lecture proof of IET. 4. Let f be a function deﬁned by f ( x ) = x 5 + x 4 + x 3 + x 2 + x + 1. Why does f-1 ( x ) exist for all x ∈ (0 , ∞ )? Find the value of ( f-1 ) (6). EXERCISES: You do not need to submit these questions but you should make sure that you are able to answer them; 1. Spivak # 10.2, 10.16(a)(b), 11.1, 11.5(i)-(iii), 11.9, 12.8, 12.11, 12.16(a), 12.17, 12.19 2. Let g be diﬀerentiable. Find a formula for h ( x ) where h ( x ) = g ( x + g ( x )) + e 2 . 3. Suppose that f is diﬀerentiable at 2 and 4 with f (2) = 2, f (4) = 3, f (2) = π , and f (4) = e , If g ( x ) = f 2 ( √ x ), ﬁnd the value of g (4)....
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