How Do You Correctly Apply Taylor Series Expansion for f(x-dx)?

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SUMMARY

The correct application of Taylor Series Expansion for f(x-dx) requires understanding that substituting x-dx directly into the series for f(x) is incorrect. A Taylor series is typically expressed as f(a+x) = f(a) + xf'(a) + x²f''(a)/2!, with the Maclaurin series being a specific case where a=0. The discussion emphasizes the importance of specifying the point of expansion, as different points yield different coefficients in the series. Resources such as Wilmott and Wikiversity provide valuable insights into the derivation and application of Taylor series.

PREREQUISITES
  • Understanding of Taylor Series and Maclaurin Series
  • Familiarity with calculus concepts such as derivatives
  • Basic knowledge of function expansion techniques
  • Ability to interpret mathematical notation and formulas
NEXT STEPS
  • Study the derivation of Taylor Series from first principles
  • Explore the application of Taylor Series in numerical methods
  • Learn about convergence criteria for Taylor Series
  • Investigate the differences between Taylor Series and other series expansions, such as Fourier Series
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Students and professionals in mathematics, physics, and engineering who require a solid understanding of Taylor Series for function approximation and analysis.

saadsarfraz
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Hi, how would you find the taylor series for f(x-dx). i know that substituting x-dx in the series for f(x) is not correct.
 
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You are using an unusual terminology. Typically a Taylor series looks like:

f(a+x)=f(a) + xf'(a) + x2 f''(a)/2! + ...

When a=0, it is called a MacLaurin series.
 
what happens if you have dx instead of a. the second post on this website http://www.wilmott.com/messageview.cfm?catid=19&threadid=17563
has a formula but i don't know how it was derived.
 
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u are using dx in the sense of a second point of x, or a change in x
check out this page from wikiversity that breaks everything down really well.
normally i know that posts from wiki are frowned upon, but i learned from this site and i think they break it down really rigorously and guide you nicely through the expansion of a function into a summed power series and then into taylors series at different points.
http://en.wikiversity.org/wiki/Taylor's_series
 
Saying "Taylor Series" is not enough, you have to attach a point around which you are expanding the function. Notice that around a different point, the taylor series will also have different coefficients.

In your example, f(x+dx) was calculated around a point x, so of course substitution won't work for the exapnasion of f(x) around 0.
 

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