10.1_LA_solutions

# 10.1_LA_solutions - MthSc 108-Learning Activity Solutions...

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Unformatted text preview: MthSc 108-Learning Activity Solutions 10.1 Approximating Functions with Polynomials 1. For the function 2 ( ) x f x e = a. Find the Taylor polynomials of order 0, 1, 2, 3, and 4 generated by f at a = . Construct a Table: n ( ) ( ) n f x ( ) (0) n f ( ) ( ) ! n f n 0 2 ( ) x f x e = 2(0) (0) 1 f e = = 1 1 0! = 1 2 '( ) 2 x f x e = 2(0) '(0) 2 2 f e = = 2 2 1! = 2 2 ''( ) 4 x f x e = 2(0) ''(0) 4 4 f e = = 4 2 2! = 3 2 '''( ) 8 x f x e = 2(0) '''(0) 8 8 f e = = 8 4 3! 3 = 4 (4) 2 ( ) 16 x f x e = (4) 2(0) (0) 16 16 f e = = 16 2 4! 3 = Therefore, p ( x ) = 1 p 1 ( x ) = 1 + 2 x p 2 ( x ) = 1 + 2 x + 2 x 2 p 3 ( x ) = 1 + 2 x + 2 x 2 + 4 3 x 3 p 4 ( x ) = 1 + 2 x + 2 x 2 + 4 3 x 3 + 2 3 x 4 b. Use each of the Taylor polynomials in part (a) to approximate 1 e . Give your answers as exact values. What is the approximation (to 5 decimal places) that your calculator gives for e ? Note: e 1 = e 2 x ⇒ 1 = 2 x ⇒ x = 1 2 . e 1 ≈ p 1 2 = 1 e 1 ≈ p 1 1 2 = 1 + 2 1 2 = 2 e 1 ≈ p 2 1 2 = 1 + 2 1 2 + 2 1 2 2 = 2.5 e 1 ≈ p 3 1 2...
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10.1_LA_solutions - MthSc 108-Learning Activity Solutions...

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