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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