Explanation Calculation: Let Δ x be a small increment in x , h be the step size. Thus, h = Δ x According to the Taylor’s theorem, if f is five times continuously differentiable, the Taylor series expansion of f ( x + h ) is given by, f ( x + h ) = f ( x ) + h f ′ ( x ) + h 2 2 f ″ ( x ) + h 3 6 f ‴ ( x ) + h 4 24 f i v ( x ) + O ( h 5 ) Similarly, f ( x − h ) = f ( x ) − h f ′ ( x ) + h 2 2 f ″ ( x ) − h 3 6 f ‴ ( x ) + h 4 24 f i v ( x ) + O ( h 5 ) Add 4 f ( x + h ) and 4 f ( x − h ) 4 f ( x + h ) + 4 f ( x − h ) = 4 f ( x ) + 4 h f ′ ( x ) + 2 h 2 f ″ ( x ) + 2 h 3 3 f ‴ ( x ) + h 4 6 f i v ( x ) + 4 O ( h 5 ) + 4 f ( x ) − 4 h f ′ ( x ) + 2 h 2 f ″ ( x ) − 2 h 3 3 f ‴ ( x ) + h 4 6 f i v ( x ) + 4 O ( h 5 ) 4 f ( x + h ) + 4 f ( x − h ) = 8 f ( x ) + 4 h 2 f ″ ( x ) + h 4 3 f i v ( x ) + 8 O ( h 5 ) 4 h 2 f ″ ( x ) = 4 f ( x + h ) + 4 f ( x − h ) − 8 f ( x ) − h 4 3 f i v ( x ) − 8 O ( h 5 ) → ( 1 ) According to the Taylor’s theorem, if f is five times continuously differentiable, the Taylor series expansion of f ( x + 2 h ) is given by, f ( x + 2 h ) = f ( x ) + 2 h f ′ ( x ) + ( 2 h ) 2 2 f ″ ( x ) + ( 2 h ) 3 6 f ‴ ( x ) + ( 2 h ) 4 24 f i v ( x ) + O ( h 5 ) = f ( x ) + 2 h f ′ ( x ) + 2 h 2 f ″ (</

3rd Edition

ISBN: 9780134696454

Author: Sauer, Tim

Publisher: Pearson,

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Chapter 5.1, Problem 21E

To determine

Prove the second-order formula for the fourth derivative

fiv(x)=f(x−2h)−4f(x−h)+6f(x)−4f(x+h)+f(x+2h)h4+O(h2)

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Chapter 5.1, Problem 20E

Chapter 5.1, Problem 22E

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Prove the second-order formula for the fourth derivative f i v ( x ) = f ( x − 2 h ) − 4 f ( x − h ) + 6 f ( x ) − 4 f ( x + h ) + f ( x + 2 h ) h 4 + O ( h 2 )

Solution Summary: The author explains the second-order formula for the fourth derivative.

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Prove the second-order formula for the fourth derivative

fiv(x)=f(x−2h)−4f(x−h)+6f(x)−4f(x+h)+f(x+2h)h4+O(h2)

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Chapter 5 Solutions

Prove the second-order formula for the fourth derivative f i v ( x ) = f ( x − 2 h ) − 4 f ( x − h ) + 6 f ( x ) − 4 f ( x + h ) + f ( x + 2 h ) h 4 + O ( h 2 )

Solution Summary: The author explains the second-order formula for the fourth derivative.

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Prove the second-order formula for the fourth derivative fiv(x)=f(x−2h)−4f(x−h)+6f(x)−4f(x+h)+f(x+2h)h4+O(h2)

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Chapter 5 Solutions

Prove the second-order formula for the fourth derivative

fiv(x)=f(x−2h)−4f(x−h)+6f(x)−4f(x+h)+f(x+2h)h4+O(h2)