Explanation Definition used: The nth Taylor Polynomial: Let f be a function and its first n derivatives are defined at a. Then the n th Taylor polynomial of f at a is the polynomial, P n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x − a ) n . Calculation: The function is given by f ( x ) = ln x . Find the fourth Taylor polynomial of the function f ( x ) = ln x at x = 1 as follows by using the above mentioned definition. P n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x − a ) n (1) Let f ( x ) = ln x . The derivative of f ′ ( x ) is calculated as follows. f ′ ( x ) = 1 x The derivative of f ″ ( x ) is calculated as follows. f ″ ( x ) = − 1 x 2 The derivative of f ‴ ( x ) is calculated as follows. f ‴ ( x ) = − ( − ) 2 x 3 = 2 x 3 When x = 1 , find f ( 1 ) as follows. f ( 1 ) = ln 1 = 0 When x = 1 , find f ′ ( 1 ) as follows. f ′ ( 1 ) = 1 1 = 1 When x = 1 , find f ″ ( 1 ) as follows

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ISBN: 9781337825597

Author: Tan

Publisher: INTER CENG

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Chapter 11.1, Problem 24E

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To find: The third Taylor polynomial of f(x)=lnx at x=1 and use it to estimate ln1.1.

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Chapter 11.1, Problem 23E

Chapter 11.1, Problem 25E

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Chapter 11 Solutions

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Ch. 11.1 - Prob. 1CQ Ch. 11.1 - Prob. 2CQ Ch. 11.1 - Prob. 1E Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E

Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 10E Ch. 11.1 - Prob. 11E Ch. 11.1 - Prob. 12E Ch. 11.1 - Prob. 13E Ch. 11.1 - Prob. 14E Ch. 11.1 - In Exercises 1120, find the nth Taylor polynomial...Ch. 11.1 - Prob. 16E Ch. 11.1 - Prob. 17E Ch. 11.1 - Prob. 18E Ch. 11.1 - Prob. 19E Ch. 11.1 - Prob. 20E Ch. 11.1 - Prob. 21E Ch. 11.1 - Prob. 22E Ch. 11.1 - Prob. 23E Ch. 11.1 - Prob. 24E Ch. 11.1 - Prob. 25E Ch. 11.1 - Prob. 26E Ch. 11.1 - Prob. 27E Ch. 11.1 - Prob. 28E Ch. 11.1 - Prob. 29E Ch. 11.1 - Prob. 30E Ch. 11.1 - Prob. 31E Ch. 11.1 - Prob. 32E Ch. 11.1 - Prob. 33E Ch. 11.1 - Prob. 34E Ch. 11.1 - Let f(x) = ex. a. 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The third Taylor polynomial of f ( x ) = ln x at x = 1 and use it to estimate ln 1.1 .

Solution Summary: The author explains how to find the third Taylor polynomial of f(x)=mathrmlnx and its value.

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To find: The third Taylor polynomial of f(x)=lnx at x=1 and use it to estimate ln1.1.

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Chapter 11 Solutions