Question 24081: A Taylor Jet Includes Every Derivative Order from Zero through n

Original question and givens

Use the derivative values=[6, 12, 18, 24] at the fixed center=-7, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

• G001 — Use the derivative values=[6, 12, 18, 24] at the fixed center=-7, through declared order=3≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, derivatives, jet_length, complete_jet, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-taylor-polynomial

Step-by-step derivation

source roles = [taylor-jet-qualification]

derived = [[center, order, derivatives, jet_length, complete_jet, certificate]]

source roles = [-7, 3, [6, 12, 18, 24]]

derived = [[0, 1, 2, 3], 4, Yes]

source roles = [0]

derived = [1]

Taylor coefficient0 derivative over factorial := div(6, 1)

Taylor coefficient0 derivative over factorial = 6

Taylor k1 factorial step1 := mul(1, 1)

Taylor k1 factorial step1 = 1

Taylor coefficient1 derivative over factorial := div(12, 1)

Taylor coefficient1 derivative over factorial = 12

Taylor k2 factorial step1 := mul(1, 1)

Taylor k2 factorial step1 = 1

Taylor k2 factorial step2 := mul(1, 2)

Taylor k2 factorial step2 = 2

Taylor coefficient2 derivative over factorial := div(18, 2)

Taylor coefficient2 derivative over factorial = 9

Taylor k3 factorial step1 := mul(1, 1)

Taylor k3 factorial step1 = 1

Taylor k3 factorial step2 := mul(1, 2)

Taylor k3 factorial step2 = 2

Taylor k3 factorial step3 := mul(2, 3)

Taylor k3 factorial step3 = 6

Taylor coefficient3 derivative over factorial := div(24, 6)

Taylor coefficient3 derivative over factorial = 4

source roles = [-7, 3, [6, 12, 9, 4]]

derived = [T_n(x)=sum c_k*(x-center)^k, T_n^(j)(center)=declared derivative[j] for EVERY0<=j<=n, no remainder or infinite-series certificate]

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute Taylor coefficient3 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Taylor k3 factorial step3 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Taylor k3 factorial step2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Taylor k3 factorial step1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Taylor coefficient2 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Taylor k2 factorial step2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Taylor k2 factorial step1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Taylor coefficient1 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Taylor k1 factorial step1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Taylor coefficient0 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(-7, 3, [6, 12, 18, 24], 4, Yes, complete_zero_through_n_derivative_jet)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.