M147A5

M147A5 - c ∈ [0 , 1] such that e c-3 c = 0. b) Use...

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MATH 147 Assignment 5 Due: Monday, November 30 1) a) Let F ( x ) be the function the gives the area bounded by the graph of f ( t ) = 1 t , the t - axis, the line t = 1 and the line t = x . Recall that we showed that for any x > 1 we have that F 0 ( x ) = 1 x . Use this to show that F ( x ) = ln ( x ) for all x 1. b) Show that 1 n +1 < ln( n + 1) - ln( n ) < 1 n for each n N . 2) Use L’Hˆ o pital’s Rule to evaluate the following limits. a) lim x 0 e x - 1 - x x 2 b) lim x 0 + x sin( x ) c) lim x 0 tan( x ) - x x 3 d) lim x →∞ xe 1 x - x 3) Let f ( x ) = ± e - 1 x 2 for x 6 = 0 0 for x = 0 . a) Show that for any k N lim x 0 f ( x ) x k = 0 . 1

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b) Show by induction that there exists a polynomial p n ( x ) and an integer k n 0 such that f ( n ) ( x ) = p n ( x ) f ( x ) x k n for every x 6 = 0 and for every n N . c) Prove that f ( n ) (0) = 0 for every n N . 4) Show that Newton’s Method fails for the function f ( x ) = x 1 3 by proving that if x 1 6 = 0, then the iterates always diverge. 5) a) Show that there exists a
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Unformatted text preview: c ∈ [0 , 1] such that e c-3 c = 0. b) Use Newtons Method to approximate c with an accuracy of at least 8 decimal places. 6) a) Show that if f ( x ) is continuous at x = a and if f ( a ) > 0, then there is a δ > 0 such that f ( x ) is strictly increasing on [ a-δ,a + δ ]. b) Let f ( x ) = ± x + x 2 sin( 1 x 2 ) for x 6 = 0 for x = 0 . Show that f (0) > 0 but that there is no interval [-δ,δ ] on which f ( x ) is strictly increasing. 7) Two runners begin and ﬁnish a 100m race at exactly the same time. Show that at some point they must have been traveling with the same velocity. 2...
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This note was uploaded on 10/21/2010 for the course MATH 147 taught by Professor Wolzcuk during the Fall '09 term at Waterloo.

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M147A5 - c ∈ [0 , 1] such that e c-3 c = 0. b) Use...

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