How Can Taylor Polynomials Approximate Third Derivatives with Reduced Error?

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SUMMARY

The discussion focuses on deriving a method to approximate the third derivative, f'''(x0), using a fourth Taylor polynomial expansion around the point x0. The error term for this approximation is established to be of order h². Participants suggest starting the solution by applying Taylor's formula to the expressions f(x0 + h) and f(x0 + 2h) to derive the necessary coefficients for the polynomial.

PREREQUISITES
  • Understanding of Taylor series and polynomial expansions
  • Familiarity with derivatives and their notation
  • Knowledge of error analysis in numerical methods
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the derivation of Taylor polynomials, specifically fourth-order expansions
  • Learn about error terms in numerical approximations, focusing on order of error
  • Explore practical applications of Taylor series in numerical analysis
  • Investigate methods for improving derivative approximations using higher-order Taylor polynomials
USEFUL FOR

Students in calculus or numerical analysis courses, mathematicians interested in approximation methods, and anyone looking to enhance their understanding of Taylor series and derivative approximations.

Ryuuken
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Homework Statement


Derive a method for approximating f'''(x0) whose error term is of order h^{2} by expanding the function f in a fourth taylor polynomial about x0 and evaluating at x_{0} \pm h and x_{0} \pm 2h.


Homework Equations





The Attempt at a Solution



I'm not sure where to start.
 
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Ryuuken said:
I'm not sure where to start.
Sounds like you should start by using Taylor's formula on the two expressions f(x_0+h) and f(x_0+2h).
 

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