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)#?

Question

Ezra Adams · Accepted Answer

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 you found this useful.

Christopher Agresta · Answer

undefined To estimate the error in approximating a function $ y = f(x) $ on an interval $ (c - r, c + r) $ using the Taylor Remainder term, you would typically follow these steps:

1. Find the Taylor series expansion of $ f(x) $ centered at $ x = c $. This involves finding the function's derivatives at $ x = c $.

2. Write down the $ n $-th degree Taylor polynomial $ P_n(x) $ using the Taylor series expansion up to the $ n $-th term.

3. The Taylor Remainder term, denoted by $ R_n(x) $, represents the error between the actual function $ f(x) $ and the Taylor polynomial $ P_n(x) $. It is given by the formula:

$$ R_n(x) = f(x) - P_n(x) $$

4. To estimate the error on the interval $ (c - r, c + r) $, evaluate the Taylor Remainder term $ R_n(x) $ at the endpoints of the interval, i.e., $ c - r $ and $ c + r $. This gives you the maximum possible error within the interval.