Explanation Given: The function f ( x ) is, f ( x ) = cos x The value of a is equal to π 4 . 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 ) = cos x It is given that the value of a is π 4 . The n th Taylor polynomial T n centered at x = π 4 is, T n ( x ) = ( f ( π 4 ) + f ′ ( π 4 ) 1 ! ( x − π 4 ) + f ″ ( π 4 ) 2 ! ( x − π 4 ) 2 + f ‴ ( π 4 ) 3 ! ( x − π 4 ) 3 + ⋯ + f ( n ) ( π 4 ) n ! ( x − π 4 ) n ) = ( f ( π 4 ) + f ′ ( π 4 ) 1 ( x − π 4 ) + f ″ ( π 4 ) 2 ⋅ 1 ( x − π 4 ) 2 + f ‴ ( π 4 ) 3 ⋅ 2 ( x − π 4 ) 3 + ⋯ + f ( n ) ( π 4 ) n ! ( x − π 4 ) n ) Simplify the above n th Taylor polynomial as, T n ( x ) = f ( π 4 ) + ( x − π 4 ) f ′ ( π 4 ) + ( x − π 4 ) 2 f ″ ( π 4 ) 2 + ( x − π 4 ) 3 f ‴ ( π 4 ) 6 + ⋯ + f ( n ) ( π 4 ) n ! ( x − π 4 ) n ......

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 9.4, Problem 23E

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

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Chapter 9.4, Problem 22E

Chapter 9.4, Problem 24E

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

Solution Summary: The author explains how the nth Taylor polynomial is centered at x=a for the function f(x)=mathrmco

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

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Chapter 9 Solutions