Calc1231W9T2 - Taylor polynomial of a function

Tuesday, 18 September 2018

10:07 PM

Machine generated alternative text: If the first n derivatives of a function f all exist at = a then the Taylor polynomial pn of degree n about a is f(2) (a) pn(x) _ f(O) (a) + f(l) _ a) + 2 2! Note- f (n) (a) denotes the nth derivative of f evaluated at x = a, with f (0) (a) f(a) and f@) = sin(x) Then Suppose a 2 . f(l) ( # ) . f(2) ( # ) • f(3) ( # ) Since sin@), the pattern repeats itself after n

 

Machine generated alternative text: The Taylor polynomial of degree 5 for f@) = sin(x) about — is equal to P5(c) — The Taylor polynomials pn@) for n = 1, 2, 3, 4 and 5 are pictured in the GeoGebra app below Which looks like the best approximation to sin@) T ne Taylor polynomial ot degree 5 Note: the Maple syntax for T is pi. near

 

Machine generated alternative text: to find a rational approximation to In Now use the Taylor polynomial of degree 3 for sin@) to find a rational approximation to sin Use the Taylor polynomial of degree 2 for ln@) near x —1 —o near x — P'2(x) — 10 In 9 1 1 17/162 1 485/4374 3

 

Machine generated alternative text: The Taylor polynomial of degree 4 for f@) = In(l + x) about = 0 exists, and is equal to The first few Taylor polynomials for In(l + x) near x = O are shown in the GeoGebra app below; they are reasonable approximations to In(l + x) provided we choose < P30@) n — 3D

 

Machine generated alternative text: The Taylor polynomialw of degrees 1, • • , 5 for f(x) ln@) about x 1 are shown in the GeoGebra app below, along with their polynomial equations. f@) + 24 . 3! These Taylor polynomial have an incremental nature. For example, the degree 6 Taylor polynomial for f(x) about = 1 is

 

 

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