hw7 - and denominator: lim x e x-e-x sin 5 x , lim x sin 4...

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Math 135A, Winter 2008 Homework #7 (due 2/22/2008) Routine problems: § 12.6: #1, #2, #5, #9, #10, #11, #13, #17; § 12.7: #3, #7, #20, #31, #34; § 12.8: #1, #3, #4, #5, #8, #19, #27, #41, #44. Also, make use of the material in the notes “More on Taylor Polynomials” to do the following: For each of the functions below, used the known expansions ofr e x , sin x , cos x , and (1 - x ) - 1 , together with thek techniques from the notes to find the 5th order Taylor polynomial (about a = 0: ( x 2 + 1) e x , cos( x 3 ) , sin( x + x 2 ) , cos( e x - 1) . In each of the following, compute the limit by using Taylor expansions of the numerator
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Unformatted text preview: and denominator: lim x e x-e-x sin 5 x , lim x sin 4 x 1-cos( x 2 ) . To hand in: (1) Problem 66 on page 63 of the text. Note: There is a typo: s q = q k =0 1 k ! (2) (a) Find the 4th order Taylor polynomial (about a = 0 of cos(sin x ). (b) Use the result of (a) (not lHospitals rule) to compute lim x cos(sin x )-cos x x 2 sin( x 2 ) . (3) Find all values of x for which the followins series (a) converges absolutely, (b) condition-ally f ( x ) = X n =0 (-1) n ( x-3) n n + 2 . 1...
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