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Taylor–Lagrange Remainder Upper Bound

Estimate a Taylor-polynomial remainder bound from degree, center, evaluation point and a valid maximum next-derivative magnitude.

  • Formula & worked example
  • Private in your browser
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

For degree 2, |x−a| = 1 and derivative bound M = 6, the error bound is 1.

|Rₙ(x)| ≤ |M| × |x−a|ⁿ⁺¹ / (n+1)!

Worked example

Enter these known values and leave the other values blank.

Polynomial degree (n)
2
Evaluation point (x)
1
Polynomial's center (a)
0
Bound M on the next derivative magnitude
6
Remainder upper bound
1

Assumptions and limitations

  • Degree is an integer from zero through 999.
  • M must bound the absolute (n+1)th derivative over the entire interval between the center and evaluation point. A derivative value at just one point is not generally sufficient.
  • The input’s magnitude is used, preserving the captured absolute-value convention. The result is an upper bound, not a prediction of the actual signed error.
  • Arithmetic uses 64 significant decimal digits, with separate rounding for displayed values.

Common questions

Where do I find M?

Bound the absolute next derivative throughout the interval connecting a and x. This tool applies the bound you supply; it does not differentiate an arbitrary function.

References

Bookify requires an integer polynomial degree while preserving the captured Lagrange remainder formula and degree bounds.

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