Explanation The function is given by f ( x ) = e − x . Find the n th Taylor polynomial of the function f ( x ) = e − x at x = 0 , n = 3 as follows. P 3 ( x ) = f ( 0 ) + f ′ ( 0 ) ( x − 0 ) + f ″ ( 0 ) 2 ! ( x − 0 ) 2 + f ‴ ( 0 ) 3 ! ( x − 0 ) 3 (1) The derivatives are f ′ ( x ) = − e − x , f ″ ( x ) = e − x , and f ‴ ( x ) = − e − x . When x = 0 , f ( 0 ) = 1 , f ′ ( 0 ) = − 1 , f ″ ( 0 ) = 1 and f ‴ ( 0 ) = − 1 . By using the above calculation, the equation (1) becomes, P 3 ( x ) = f ( 0 ) + f ′ ( 0 ) ( x − 0 ) + f ″ ( 0 ) 2 ! ( x − 0 ) 2 + f ‴ ( 0 ) 3 ! ( x − 0 ) 3 = 1 − x + 1 2 x 2 − 1 6 x 3 Substitute x = 1 in P 3 ( x )

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ISBN: 9781305041264

Author: Tan

Publisher: CENGAGE C

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Chapter 11.1, Problem 46E

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Whether the statement “Let PN denote the Nth Taylor polynomial of the function f(x)=e−x at a. Then RN(x)=f(x)−PN(x)≠0 for all x≠0” is true or false.

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Chapter 11.1, Problem 45E

Chapter 11.2, Problem 1CQ

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Chapter 11 Solutions

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Whether the statement “Let P N denote the N th Taylor polynomial of the function f ( x ) = e − x at a . Then R N ( x ) = f ( x ) − P N ( x ) ≠ 0 for all x ≠ 0 ” is true or false.

Solution Summary: The author analyzes whether the statement "Let P_N denote the N th Taylor polynomial of the function " f(x)=e-

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Whether the statement “Let PN denote the Nth Taylor polynomial of the function f(x)=e−x at a. Then RN(x)=f(x)−PN(x)≠0 for all x≠0” is true or false.

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Chapter 11 Solutions