Lagrange Remainder Bound
Lagrange Remainder Bound: calculate results from explicit inputs, with a formula and worked example.
Formula
- Use the units shown beside each field.
- Results update automatically whenever you change an input.
- Displayed values use up to four decimal places, or scientific notation for very small or large numbers. Calculations use standard floating-point arithmetic; exact integer and fraction tools identify their own precision rules.
- Real finite inputs. Numeric results use floating-point arithmetic; this is not a symbolic computer algebra system. M must bound the absolute (n+1)th derivative throughout the interval between a and x. This tool does not verify that bound.
Example Calculation
- Bound M on |f⁽ⁿ⁺¹⁾|: 1
- Distance |x−a|: 0.1
- Taylor degree n: 3
With M=1, |x−a|=0.1 and degree 3, the bound is 0.1⁴/24≈0.0000041667.
Frequently asked questions
How do I use this calculator?
Enter bound m on |f⁽ⁿ⁺¹⁾|, distance |x−a|, taylor degree n. The result updates automatically. Use Reset to restore the example values.
What method does it use?
|Rₙ(x)| ≤ M |x−a|^(n+1) / (n+1)!. Real finite inputs. Numeric results use floating-point arithmetic; this is not a symbolic computer algebra system. M must bound the absolute (n+1)th derivative throughout the interval between a and x. This tool does not verify that bound.
Why might a rounded result differ?
The calculation keeps full numeric precision internally, then rounds the displayed result. Rounding intermediate steps by hand can produce a slightly different answer.
Are my inputs saved online?
No. This calculator processes your inputs in your browser. A recently used list stores only calculator names on this device, not your entered values.
Check the assumptions and units before using this result. Read about calculation methods.