How do you use the Taylor Remainder term to estimate the error in approximating a function #y=f(x)# on a given interval #(c-r,c+r)#?

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Answer 1 · 2014-10-16T21:39:42

Assume that there exists a finite #M>0# such that

#|f^{(n+1)}(x)| le M#

for all #x# in #(c-r,c+r)#.

The error of approximating #f(x)# by the Taylor polynomial #p_n(x;c)# can be estimated by

#|f(x)-p_n(x;c)|#

#=|R_n(x;c)|#

#=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|#, where #z# is between #x# and #c#

#le M/{(n+1)!}r^{n+1}#

I hope that this was helpful.