Lagrange Form of the Remainder Term in a Taylor Series

Key Questions

• How do you find the Taylor remainder term #R_n(x;3)# for #f(x)=e^(4x)#?

  By taking the derivatives,

  #f(x)=e^{4x}#
  #f'(x)=4e^{4x}#
  #f''(x)=4^2e^{4x}#
  .
  .
  .
  #f^{(n+1)}(x)=4^{n+1}e^{4x}#

  So,
  #R_n(x;3)={f^{(n+1)}(z)}/{(n+1)!}(x-3)^{n+1}={4^{n+1}e^{4z}}/{(n+1)!}(x-3)^{n+1}#,
  where #z# is between #x# and #3#.

  Wataru · · Sep 19 2014
• How do you find the remainder term #R_3(x;1)# for #f(x)=sin(2x)#?

  Remainder Term of Taylor Series

  #R_n(x;c)={f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}#,

  where #z# is a number between #x# and #c#.

  ---

  Let us find #R_3(x;1)# for #f(x)=sin2x#.

  By taking derivatives,

  #f'(x)=2cos2x#
  #f''(x)=-4sin2x#
  #f'''(x)=-8cos2x#
  #f^{(4)}(x)=16sin2x#

  So, we have

  #R_3(x;1)={16sin2z}/{4!}(x-1)^4#,

  where #z# is a number between #x# and #1#.

  ---

  I hope that this was helpful.

  Wataru · 1 · Nov 2 2014
• What is the Remainder Term in a Taylor Series?

  Taylor remainder term

  #R_n(x;c)={f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}#,

  where #z# is between #x# and #c#.

  Wataru · · Sep 20 2014

• What is the Lagrange Form of the Remainder Term in a Taylor Series?
• What is the Remainder Term in a Taylor Series?
• How do you find the Remainder term in Taylor Series?
• How do you find the remainder term #R_3(x;1)# for #f(x)=sin(2x)#?
• How do you find the Taylor remainder term #R_n(x;3)# for #f(x)=e^(4x)#?
• How do you find the Taylor remainder term #R_3(x;0)# for #f(x)=1/(2+x)#?
• 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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• How do you find the largest interval #(c-r,c+r)# on which the Taylor Polynomial #p_n(x,c)# approximates a function #y=f(x)# to within a given error?
• How do you find the smallest value of #n# for which the Taylor series approximates the function #f(x)=e^(2x)# at #c=2# on the interval #0<=x<=1# with an error less than #10^(-6)#?
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