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Unformatted text preview: (3%) (b) Using part (a), ﬁnd a power series expansion for the function h ( x ) = x ln ( 1 + x 2 ) . 1 (2%) (c) Find the radius of convergence for the power series obtained in part (b). (4%) (d) Find h ( 999 ) ( ) . (8%) 7. Suppose f is an odd function and is differentiable everywhere. Prove that for every positive number b , there exists a number c ∈ (-b , b ) such that f ( c ) = f ( b ) / b . (8%) 8. Show that f deﬁned by f ( x ) = ± e-1 | x | , x 6 = , , x = is differentiable at 0. 9. (5%) (a) Find the Taylor polynomial P 8 ( x ) for the function f ( x ) = cos x . (5%) (b) It follows that P 8 ( 1 ) = 4357 8064 estimates cos1 to within some error. Give the best estimate of the difference between cos1 and P 8 ( 1 ) that can be obtained using the formula for R 8 ( x ) given in class. 2...
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This note was uploaded on 10/14/2010 for the course MAT MAT taught by Professor Unknown during the Spring '10 term at Touro CA.
- Spring '10