Explanation Given: The function f ( x ) is, f ( x ) = x − 2 The value of a is equal to 2 . Concept used: The n th Taylor polynomial T n forfunction f that is centered at x = a is, T n ( x ) = f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x − a ) n Here, f ( n ) is the n th derivative of the function f ( x ) . Calculation: The given function f ( x ) is, f ( x ) = x − 2 It is given that the value of a is 2 . The n th Taylor polynomial T n centered at x = 2 is, T n ( x ) = f ( 2 ) + f ′ ( 2 ) 1 ! ( x − 2 ) + f ″ ( 2 ) 2 ! ( x − 2 ) 2 + f ‴ ( 2 ) 3 ! ( x − 2 ) 3 + ⋯ + f ( n ) ( 1 ) n ! ( x − 2 ) n = f ( 2 ) + f ′ ( 2 ) 1 ( x − 2 ) + f ″ ( 2 ) 2 ⋅ 1 ( x − 2 ) 2 + f ‴ ( 2 ) 3 ⋅ 2 ( x − 2 ) 3 + ⋯ + f ( n ) ( 2 ) n ! ( x − 2 ) n Simplify the above n th Taylor polynomial as, T n ( x ) = f ( 2 ) + ( x − 2 ) f ′ ( 2 ) + ( x − 2 ) 2 f ″ ( 2 ) 2 + ( x − 2 ) 3 f ‴ ( 2 ) 6 + ⋯ + f ( n ) ( 2 ) n ! ( x − 2 ) n ....... (1) Substitute 2 for x in the function f ( x ) to obtain the value of f ( 2 ) as, f ( 2 ) = 2 − 2 = 1 2 2 = 1 4 Differentiate the function f ( x ) with respect to x as, f ′ ( x ) = d d x ( x − 2 ) = − 2 ⋅ x − 2 − 1 = − 2 x − 3 Substitute 2 for x in above equation to obtain the value of f ′ ( 2 ) as, f ′ ( 2 ) = − 2 ( 2 ) − 3 = − 2 ⋅ 1 2 3 = − 1 4 Differentiate the function f ′ ( x ) with respect to x as, f ″ ( x ) = d d x ( − 2 x − 3 ) = − 2 ⋅ d d x ( x − 3 ) = − 2 ( − 3 x − 3 − 1 ) = 6 x − 4 Substitute 2 for x in above equation to obtain the value of f ″ ( 2 ) as, f ″ ( 2 ) = 6 ⋅ 2 − 4 = 6 ⋅ 1 2 4 = 6 <

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 9.4, Problem 22E

To determine

To find: The nth Taylor polynomial Tn for the function f(x)=x−2 centered at x=a for all n .

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Chapter 9.4, Problem 21E

Chapter 9.4, Problem 23E

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Chapter 9 Solutions

Calculus - Standalone book

Ch. 9.1 - Prob. 1PQ Ch. 9.1 - Prob. 2PQ Ch. 9.1 - Prob. 3PQ Ch. 9.1 - Prob. 4PQ Ch. 9.1 - Prob. 5PQ Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E

Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E Ch. 9.1 - Prob. 11E Ch. 9.1 - Prob. 12E Ch. 9.1 - Prob. 13E Ch. 9.1 - Prob. 14E Ch. 9.1 - Prob. 15E Ch. 9.1 - Prob. 16E Ch. 9.1 - Prob. 17E Ch. 9.1 - Prob. 18E Ch. 9.1 - Prob. 19E Ch. 9.1 - Prob. 20E Ch. 9.1 - Prob. 21E Ch. 9.1 - Prob. 22E Ch. 9.1 - Prob. 23E Ch. 9.1 - Prob. 24E Ch. 9.1 - Prob. 25E Ch. 9.1 - Prob. 26E Ch. 9.1 - Prob. 27E Ch. 9.1 - Prob. 28E Ch. 9.1 - Prob. 29E Ch. 9.1 - Prob. 30E Ch. 9.1 - Prob. 31E Ch. 9.1 - Prob. 32E Ch. 9.1 - Prob. 33E Ch. 9.1 - Prob. 34E Ch. 9.1 - Prob. 35E Ch. 9.1 - Prob. 36E Ch. 9.1 - Prob. 37E Ch. 9.1 - Prob. 38E Ch. 9.1 - Prob. 39E Ch. 9.1 - Prob. 40E Ch. 9.1 - Prob. 41E Ch. 9.1 - Prob. 42E Ch. 9.1 - Prob. 43E Ch. 9.1 - Prob. 44E Ch. 9.1 - Prob. 45E Ch. 9.1 - Prob. 46E Ch. 9.1 - Prob. 47E Ch. 9.1 - Prob. 48E Ch. 9.1 - Prob. 49E Ch. 9.1 - Prob. 50E Ch. 9.1 - Prob. 51E Ch. 9.1 - Prob. 52E Ch. 9.1 - Prob. 53E Ch. 9.1 - Prob. 54E Ch. 9.1 - Prob. 55E Ch. 9.1 - Prob. 56E Ch. 9.1 - Prob. 57E Ch. 9.2 - Prob. 1PQ Ch. 9.2 - Prob. 2PQ Ch. 9.2 - Prob. 3PQ Ch. 9.2 - Prob. 4PQ Ch. 9.2 - Prob. 5PQ Ch. 9.2 - Prob. 1E Ch. 9.2 - Prob. 2E Ch. 9.2 - Prob. 3E Ch. 9.2 - Prob. 4E Ch. 9.2 - Prob. 5E Ch. 9.2 - Prob. 6E Ch. 9.2 - Prob. 7E Ch. 9.2 - Prob. 8E Ch. 9.2 - Prob. 9E Ch. 9.2 - Prob. 10E Ch. 9.2 - Prob. 11E Ch. 9.2 - Prob. 12E Ch. 9.2 - Prob. 13E Ch. 9.2 - Prob. 14E Ch. 9.2 - Prob. 15E Ch. 9.2 - Prob. 16E Ch. 9.2 - Prob. 17E Ch. 9.2 - Prob. 18E Ch. 9.2 - Prob. 19E Ch. 9.2 - Prob. 20E Ch. 9.2 - Prob. 21E Ch. 9.2 - Prob. 22E Ch. 9.2 - Prob. 23E Ch. 9.2 - Prob. 24E Ch. 9.2 - Prob. 25E Ch. 9.2 - Prob. 26E Ch. 9.2 - Prob. 27E Ch. 9.2 - Prob. 28E Ch. 9.3 - Prob. 1PQ Ch. 9.3 - Prob. 2PQ Ch. 9.3 - Prob. 3PQ Ch. 9.3 - Prob. 4PQ Ch. 9.3 - Prob. 5PQ Ch. 9.3 - Prob. 6PQ Ch. 9.3 - Prob. 1E Ch. 9.3 - Prob. 2E Ch. 9.3 - Prob. 3E Ch. 9.3 - Prob. 4E Ch. 9.3 - Prob. 5E Ch. 9.3 - Prob. 6E Ch. 9.3 - Prob. 7E Ch. 9.3 - Prob. 8E Ch. 9.3 - Prob. 9E Ch. 9.3 - Prob. 10E Ch. 9.3 - Prob. 11E Ch. 9.3 - Prob. 12E Ch. 9.3 - Prob. 13E Ch. 9.3 - Prob. 14E Ch. 9.3 - Prob. 15E Ch. 9.3 - Prob. 16E Ch. 9.3 - Prob. 17E Ch. 9.3 - Prob. 18E Ch. 9.3 - Prob. 19E Ch. 9.3 - Prob. 20E Ch. 9.3 - Prob. 21E Ch. 9.3 - Prob. 22E Ch. 9.3 - Prob. 23E Ch. 9.3 - Prob. 24E Ch. 9.3 - Prob. 25E Ch. 9.3 - Prob. 26E Ch. 9.3 - Prob. 27E Ch. 9.3 - Prob. 28E Ch. 9.3 - Prob. 29E Ch. 9.3 - Prob. 30E Ch. 9.3 - Prob. 31E Ch. 9.3 - Prob. 32E Ch. 9.3 - Prob. 33E Ch. 9.3 - Prob. 34E Ch. 9.3 - Prob. 35E Ch. 9.3 - Prob. 36E Ch. 9.3 - Prob. 37E Ch. 9.3 - Prob. 38E Ch. 9.3 - Prob. 39E Ch. 9.3 - Prob. 40E Ch. 9.3 - Prob. 41E Ch. 9.3 - Prob. 42E Ch. 9.3 - Prob. 43E Ch. 9.3 - Prob. 44E Ch. 9.3 - Prob. 45E Ch. 9.3 - Prob. 46E Ch. 9.3 - Prob. 47E Ch. 9.4 - Prob. 1PQ Ch. 9.4 - Prob. 2PQ Ch. 9.4 - Prob. 3PQ Ch. 9.4 - Prob. 4PQ Ch. 9.4 - Prob. 1E Ch. 9.4 - Prob. 2E Ch. 9.4 - Prob. 3E Ch. 9.4 - Prob. 4E Ch. 9.4 - Prob. 5E Ch. 9.4 - Prob. 6E Ch. 9.4 - Prob. 7E Ch. 9.4 - Prob. 8E Ch. 9.4 - Prob. 9E Ch. 9.4 - Prob. 10E Ch. 9.4 - Prob. 11E Ch. 9.4 - Prob. 12E Ch. 9.4 - Prob. 13E Ch. 9.4 - Prob. 14E Ch. 9.4 - Prob. 15E Ch. 9.4 - Prob. 16E Ch. 9.4 - Prob. 17E Ch. 9.4 - Prob. 18E Ch. 9.4 - Prob. 19E Ch. 9.4 - Prob. 20E Ch. 9.4 - Prob. 21E Ch. 9.4 - Prob. 22E Ch. 9.4 - Prob. 23E Ch. 9.4 - Prob. 24E Ch. 9.4 - Prob. 25E Ch. 9.4 - Prob. 26E Ch. 9.4 - Prob. 27E Ch. 9.4 - Prob. 28E Ch. 9.4 - Prob. 29E Ch. 9.4 - Prob. 30E Ch. 9.4 - Prob. 31E Ch. 9.4 - Prob. 32E Ch. 9.4 - Prob. 33E Ch. 9.4 - Prob. 34E Ch. 9.4 - Prob. 35E Ch. 9.4 - Prob. 36E Ch. 9.4 - Prob. 37E Ch. 9.4 - Prob. 38E Ch. 9.4 - Prob. 39E Ch. 9.4 - Prob. 40E Ch. 9.4 - Prob. 41E Ch. 9.4 - Prob. 42E Ch. 9.4 - Prob. 43E Ch. 9.4 - Prob. 44E Ch. 9.4 - Prob. 45E Ch. 9.4 - Prob. 46E Ch. 9.4 - Prob. 47E Ch. 9.4 - Prob. 48E Ch. 9.4 - Prob. 49E Ch. 9.4 - Prob. 50E Ch. 9.4 - Prob. 51E Ch. 9.4 - Prob. 52E Ch. 9.4 - Prob. 53E Ch. 9.4 - Prob. 54E Ch. 9.4 - Prob. 55E Ch. 9.4 - Prob. 56E Ch. 9.4 - Prob. 57E Ch. 9.4 - Prob. 58E Ch. 9.4 - Prob. 59E Ch. 9.4 - Prob. 60E Ch. 9.4 - Prob. 61E Ch. 9.4 - Prob. 62E Ch. 9.4 - Prob. 63E Ch. 9.4 - Prob. 64E Ch. 9.4 - Prob. 65E Ch. 9.4 - Prob. 66E Ch. 9.4 - Prob. 67E Ch. 9.4 - Prob. 68E Ch. 9.4 - Prob. 69E Ch. 9.4 - Prob. 70E Ch. 9.4 - Prob. 71E Ch. 9 - Prob. 1CRE Ch. 9 - Prob. 2CRE Ch. 9 - Prob. 3CRE Ch. 9 - Prob. 4CRE Ch. 9 - Prob. 5CRE Ch. 9 - Prob. 6CRE Ch. 9 - Prob. 7CRE Ch. 9 - Prob. 8CRE Ch. 9 - Prob. 9CRE Ch. 9 - Prob. 10CRE Ch. 9 - Prob. 11CRE Ch. 9 - Prob. 12CRE Ch. 9 - Prob. 13CRE Ch. 9 - Prob. 14CRE Ch. 9 - Prob. 15CRE Ch. 9 - Prob. 16CRE Ch. 9 - Prob. 17CRE Ch. 9 - Prob. 18CRE Ch. 9 - Prob. 19CRE Ch. 9 - Prob. 20CRE Ch. 9 - Prob. 21CRE Ch. 9 - Prob. 22CRE Ch. 9 - Prob. 23CRE Ch. 9 - Prob. 24CRE Ch. 9 - Prob. 25CRE Ch. 9 - Prob. 26CRE Ch. 9 - Prob. 27CRE Ch. 9 - Prob. 28CRE Ch. 9 - Prob. 29CRE Ch. 9 - Prob. 30CRE Ch. 9 - Prob. 31CRE Ch. 9 - Prob. 32CRE Ch. 9 - Prob. 33CRE Ch. 9 - Prob. 34CRE Ch. 9 - Prob. 35CRE Ch. 9 - Prob. 36CRE Ch. 9 - Prob. 37CRE

Knowledge Booster

The n th Taylor polynomial T n for the function f ( x ) = x − 2 centered at x = a for all n .

Solution Summary: The author explains the nth Taylor polynomial, T_n, for the function f(x) that is centered at x=

To find: The nth Taylor polynomial Tn for the function f(x)=x−2 centered at x=a for all n .

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Chapter 9 Solutions