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Taylor's formula

  1. Sep 7, 2014 #1
    What is taylor formula and how it is used in calculators?
     
  2. jcsd
  3. Sep 8, 2014 #2

    HallsofIvy

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    Taylor's formulas says that if f(x) is n+ 1 times differentiable in some neighborhood of [itex]x= x_0[/itex], then f(x) can be approximated by
    [tex]T(x)= f(x_0)+ f'(x_0)(x- x_0)+ \frac{f''(x_0)}{2}(x- x_0)^2+ \frac{f'''(x_0)}{6}(x- x_0)^3+ \cdot\cdot\cdot+ \frac{f^{(n)}(x_0)}{n!}(x- x_0)^n[/tex]
    where "[itex]f^{(n)}(x_0)[/itex]" indicates the nth derivative evaluated at [itex]x= x_0[/itex].


    Further, the error, |f(x)- T(x)|, will be less than
    [tex]\frac{f^{(n+1)}(x_0)}{(n+1)!}|x- x_0|^{n+1}[/tex].

    I'm surprised you did not just look it up with Google or on Wikipedia. You will get a lot more information.

    As for "how is it used in calculators"- it isn't. Calculators and Computers use a much more advanced numerical procedure called "CORDIC" to do calculations of trig functions, exponentials, etc.
    http://en.wikipedia.org/wiki/CORDIC
     
  4. Sep 10, 2014 #3
    But i am unable to understand how they calculate maximum number of values which it can accomodate before giving an error more than some specific value.
     
  5. Sep 10, 2014 #4
    "maximum number of values"? You mean, in what environment of a specific argument the Taylor polynomial supplies a "sufficiently good" approximation? There is a formula for the error term, it was already posted above. It answers your question (if I got it right) more or less directly.
    Please state your questions clearer and show a little bit more initiative.
     
  6. Sep 12, 2014 #5

    FactChecker

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    I would be surprised if the Taylor series was often used in calculators. The Taylor series has a lot of good theoretical properties and it is the first method of approximation you should learn. But it is usually not the most efficient way to approximate a given function. If it is used, @HallsOfIvy has posted the information.
     
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