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M147A6

# M147A6 - Illustrate your answer with a diagram c How large...

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MATH 147 Assignment 6 1) a) Show that | sin( u ) - u |≤ | u 3 | 6 for every u R . b) Let h ( x ) = ± sin( x ) x for x 6 = 0 1 for x = 0 . Show that h ( x ) - 1 =O( x 2 ) as x 1 and then use this to show that h ( x ) is diﬀerentiable at x = 0 with h 0 ( x ) = 0 c) Show that f ( x ) = x 2 sinx 3 = x 5 +O( x 11 ) as x 0. d) Find f (5) (0) and f (8) (0). e) Find P 0 , 8 ( x ) for the function f ( x ). 2 Assume that f 00 ( x ) > 0 for every x [ a,b ]. Let c ( a,b ). Show that f ( x ) L c ( x ) for every x [ a,b ]. 3) Use Taylor’s Theorem to show that 1 - x 2 2 cos( x ) for all x [0 ]. 4 a) Let f ( x ) = e x . Use linear approximation centered at x = 0 to estimate e . 01 and show that | e . 01 - L 0 ( . 01) | < 10 - 4 b) Is your estimate in part a) larger than or smaller than the true value.
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Unformatted text preview: Illustrate your answer with a diagram. c) How large must n be so that P ,n ( . 01) provides an estimate of e . 01 with an error of at most 10-8 . 5 Find P , 7 for the function f ( x ) = x 2 cos ( x )( e x 2-1). 6 Use the Big O notation to evaluate each of the following limits a lim x → e x 3-1-x 3 x 6 b lim x → ( e x 3-1-x 3 ) sin( x 2 ) cos( x 4 )-1 1...
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