Description
Return a quadratic Taylor remainder bound from a supplied third-derivative bound.
A truncation term and a rigorous error bound require different information. Taylor and quadrature bounds need derivative bounds over the entire relevant interval; a derivative sampled at one point or a leading asymptotic term does not automatically control the full approximation error.
Return a quadratic Taylor remainder bound from a supplied third-derivative bound.
When to use this tool
Use Scalar Second Order Taylor Remainder Bound when checking the stated approximation term or derivative-bound calculation.
How to use the tool and how it works
- Enter thirdDerivativeBound, displacement. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read bound. Use the output convention below.
Return a quadratic Taylor remainder bound from a supplied third-derivative bound. Require thirdDerivativeBound >= 0; bound = M*abs(displacement)³/6.1
Taylor remainder control requires the appropriate derivative bound over the relevant interval; a leading local expansion term need not bound the full approximation error.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| thirdDerivativeBound | Supplied nonnegative bound on the absolute third derivative across the complete relevant interval. | Required. Finite number. |
| displacement | Displacement | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| bound | Bound | Required. Finite number. |
{
"input": {
"thirdDerivativeBound": 3,
"displacement": 2
},
"output": {
"bound": 4
}
}Limitations and assumptions
- M must bound the absolute third derivative over the whole interval under Taylor's hypotheses; the tool does not verify that bound.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Establish the required smoothness and interval-wide bounds for a rigorous estimate; analyze the complete numerical approximation when only local derivative data are available.
References
Similar or alternative tools
- Scalar Derivative Error Propagation Factor
Return first-order output half-width from a supplied derivative.
- Calculus: Scalar Central Difference Error Term
Return the leading central-difference derivative error term.