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,

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

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)

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.1, Problem 20E

Chapter 5.1, Problem 22E

Students have asked these similar questions

Please help me answer the following questions from this problem.

Find the general solution of the given differential equation. y' + 4x3y = x3 y(x) = ? Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution.

Please help me find the sample variance for this question.

Chapter 5 Solutions

Numerical Analysis

Ch. 5.1 - Use the two-point forward-difference formula to...Ch. 5.1 - Use the three-point centered-difference formula to...Ch. 5.1 - Use the two-point forward-difference formula to...Ch. 5.1 - Carry out the steps of Exercise 3, using the...Ch. 5.1 - Use the three-point centered-difference formula...Ch. 5.1 - Use the three-point centered-difference formula...Ch. 5.1 - Develop a formula for a two-point...Ch. 5.1 - Prove the second-order formula for the first...Ch. 5.1 - Develop a second-order formula for the first...Ch. 5.1 - Find the error term and order formula for the...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

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.

Concept explainers

Videos

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)

Want to see the full answer?

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.

Concept explainers

Videos

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)

Want to see the full answer?

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)