# aB - | c-d |< 1 2 Β 10-4 2 SHE 11.5 48 3 Evaluate the...

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Department of Mathematics, University of Toronto Problem Set Supplement #2 MAT 137Y, 2008-09 Winter Session Not to be handed in. Assignment Posted/Revised: January 5, 2009, 14:55 While the following problems do not have to be handed in, you will be responsible for this material for the second term test. 1. Consider the function f ( x ) = x 3 - x + 1. (a) Show that f ( x ) has exactly one root c , and that c ( - 2 , - 1 ) . (b) Using a calculator, apply Newton’s Method several times until the last ﬁve decimal places remain the same after iteration – show your work. Round off to get a 4 decimal place approximation d for c . (c) By evaluating f ( d ± 1 2 · 10 - 4 ) , show that
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Unformatted text preview: | c-d | < 1 2 Β· 10-4 . 2. SHE 11.5: 48. 3. Evaluate the following limits. (i) lim x β sin x ( 1-cos x ) x-sin x . (ii) lim x β β sin x ( 1-cos x ) x-sin x . (iii) lim x β x-sin x x 3 . (iv) lim x β cos x-1 + x 2 2 x 4 . (v) lim x β cos ( Ο 2 cos x ) sin 2 x . (vi) lim x β x-tan x x 3 . (vii) lim x β sec x-1-x 2 2 x 4 . (viii) lim x β β ( x 5 + x 4 ) 1 / 5-( x 3 + x ) 1 / 3 ( x 2 + x ) 1 / 2-x . 4. Evaluate lim x β ( x-sin x )( cos x-1 + x 2 2 ) ( sec x-1-x 2 2 )( x-tan x ) without using LβHΛopitalβs Rule....
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