Description
Return the leading central-difference derivative error term.
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 the leading central-difference derivative error term.
When to use this tool
Use Calculus: Scalar Central Difference Error Term when checking the stated approximation term or derivative-bound calculation.
How to use the tool and how it works
- Enter thirdDerivative, step. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Return the leading central-difference derivative error term. Require step > 0; leadingCentralDifferenceError = thirdDerivative*step²/6.1
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 |
|---|---|---|
| thirdDerivative | Third Derivative | Required. Finite number. |
| step | Step | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Returned calculated value or nested object under the stated output convention. | Required. Object. |
| result.leadingCentralDifferenceError | Leading central-difference approximation-minus-derivative term, not a rigorous bound on total error. | Calculated result member: number. |
{
"input": {
"thirdDerivative": 6,
"step": 0.1
},
"output": {
"result": {
"leadingCentralDifferenceError": 0.01
}
}
}Limitations and assumptions
- The sign describes central-difference approximation minus the true derivative's leading term; higher-order and rounding errors are omitted.
- Outputs are nested inside result. 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.
- Scalar Second Order Taylor Remainder Bound
Return a quadratic Taylor remainder bound from a supplied third-derivative bound.