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Calculus II: Dr. Staples Section 9.1 Approximating Functions with Polynomials
Definition: Taylor Polynomials centered at ๐ = ๐ Let f be a function with derivatives, ๐!, ๐!! , ๐!!!, โฆ๐! defined at a. Then the nth-order Taylor polynomial centered at ๐ฅ = ๐ is given by ๐! ๐ฅ = ๐ ๐ + ๐!(๐)(๐ฅ โ ๐)+ !
!!(!)!!
(๐ฅ โ ๐)! + !!!!(!)!!
(๐ฅ โ ๐)! +โฏ !! !!!
(๐ฅ โ ๐)! Note that you only need to compute the coefficients to determine the Taylor Polynomial. Observe that coefficient ๐! of (๐ฅ โ ๐)! follows the rule ๐! =
!! !!! and ๐! ๐ฅ = ๐!!
!!! (๐ฅ โ ๐)! The nth-order Taylor polynomial centered at ๐ = ๐ matches f in value, slope, and all derivatives at a. Thus it serves as a very useful approximation function. In fact, the error term or so-called remainder that arises from approximating a function with a Taylor Polynomial is well understood. Definition: Let ๐! ๐ฅ be the Taylor polynomial of order n for f. The remainder (i.e. error) in using ๐! ๐ฅ to approximate f at the point x is
๐ ! ๐ฅ = ๐ ๐ฅ โ ๐! ๐ฅ .
Estimating the Remainder Let n be a fixed positive integer. Suppose that there exists a number M such that the (n+1)st derivative satisfies | ๐!!! ๐ | โค ๐ for all c in [a,x]. Then the remainder in the nth-order Taylor polynomial centered at ๐ฅ = ๐, satisfies
|๐ ! ๐ฅ | = |๐ ๐ฅ โ ๐! ๐ฅ | โค ๐|๐ฅ โ ๐|!!!
๐ + 1 !
If you work with numerical algorithms or computer programs, it is important to be able to estimate the accuracy of your approximation.
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