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Question 10197: Candidate Antiderivative Premise for the Endpoint Formula

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0019-10197 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0097

Original question and givens

On all R, continuous f has ascending coefficients [9, 22, -1, 2], constant term first. Candidate F has ascending rational coefficients [9/1, 9/1, 11/1, -1/3, 1/2]. Does F′=f satisfy the antiderivative premise for the endpoint formula?

  • G001 — On all R, continuous f has ascending coefficients [9, 22, -1, 2], constant term first. Candidate F has ascending rational coefficients [9/1, 9/1, 11/1, -1/3, 1/2]. Does F′=f satisfy the antiderivative premise for the endpoint formula?

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.

Step-by-step derivation

S001 · implies

Before

raw integrand f=[9, 22, -1, 2];ascending powers on all R

After

normalized coefficients=[9/1, 22/1, -1/1, 2/1];declared continuous real polynomial

Conditions: none

The declared integrand is a continuous real polynomial on all R,and endpoints are oriented.Use FTC only under these supplied sufficient hypotheses;finite checks are not a new general formal FTC proof.

Step identifiers and dependencies

Rule: declared_FTC_continuity_hypothesis · Uses: G001

S002 · implies

Before

raw candidate F=[[9, 1], [9, 1], [11, 1], [-1, 3], [1, 2]]

After

normalized candidate coefficients=[9/1, 9/1, 11/1, -1/3, 1/2]

Conditions: none

Every original denominator must be positive.Normalize to exact rationals while retaining the supplied candidate,not replacing it by a constructed primitive.

Step identifiers and dependencies

Rule: candidate_primitive_original_denominators · Uses: G001, S001

S003 · calculation

Before

candidate F derivative coefficient0 := mul(1, 9)

After

candidate F derivative coefficient0 = 9

Conditions: none

Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

candidate F:source power=1;coefficient=9/1

After

derivative power=0;coefficient=9/1

Conditions: none

Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.

Step identifiers and dependencies

Rule: polynomial_derivative_coefficient_rule · Uses: G001, S003

S005 · calculation

Before

candidate F derivative coefficient1 := mul(2, 11)

After

candidate F derivative coefficient1 = 22

Conditions: none

Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

candidate F:source power=2;coefficient=11/1

After

derivative power=1;coefficient=22/1

Conditions: none

Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.

Step identifiers and dependencies

Rule: polynomial_derivative_coefficient_rule · Uses: G001, S005

S007 · calculation

Before

candidate F derivative coefficient2 := mul(3, -1/3)

After

candidate F derivative coefficient2 = -1

Conditions: none

Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

candidate F:source power=3;coefficient=-1/3

After

derivative power=2;coefficient=-1/1

Conditions: none

Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.

Step identifiers and dependencies

Rule: polynomial_derivative_coefficient_rule · Uses: G001, S007

S009 · calculation

Before

candidate F derivative coefficient3 := mul(4, 1/2)

After

candidate F derivative coefficient3 = 2

Conditions: none

Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

candidate F:source power=4;coefficient=1/2

After

derivative power=3;coefficient=2/1

Conditions: none

Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.

Step identifiers and dependencies

Rule: polynomial_derivative_coefficient_rule · Uses: G001, S009

S011 · implies

Before

power=0;candidate derivative coefficient=9/1;target f coefficient=9/1

After

coefficient equality=Yes

Conditions: none

Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise.

Step identifiers and dependencies

Rule: FTC_candidate_full_derivative_identity · Uses: G001, S010

S012 · implies

Before

power=1;candidate derivative coefficient=22/1;target f coefficient=22/1

After

coefficient equality=Yes

Conditions: none

Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise.

Step identifiers and dependencies

Rule: FTC_candidate_full_derivative_identity · Uses: G001, S011

S013 · implies

Before

power=2;candidate derivative coefficient=-1/1;target f coefficient=-1/1

After

coefficient equality=Yes

Conditions: none

Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise.

Step identifiers and dependencies

Rule: FTC_candidate_full_derivative_identity · Uses: G001, S012

S014 · implies

Before

power=3;candidate derivative coefficient=2/1;target f coefficient=2/1

After

coefficient equality=Yes

Conditions: none

Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise.

Step identifiers and dependencies

Rule: FTC_candidate_full_derivative_identity · Uses: G001, S013

S015 · implies

Before

candidate derivative=[9/1, 22/1, -1/1, 2/1];integrand=[9/1, 22/1, -1/1, 2/1]

After

this candidate satisfies F'=f=Yes

Conditions: none

Failure rejects this F only,not other primitives.The candidate constant is free;coefficient identity is the requested endpoint-formula qualification.

Step identifiers and dependencies

Rule: failed_primitive_candidate_not_all_primitives · Uses: G001, S014

Reverse checks

  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Recompute candidate F derivative coefficient3 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Recompute candidate F derivative coefficient2 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Recompute candidate F derivative coefficient1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Recompute candidate F derivative coefficient0 with exact rational arithmetic, not rounded decimals.
  • 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

Yes

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: e3723881fbe57b54aa503bbf22dbbf1334dfa5f0c4433363ccec3a1105e77a25
  • candidate record SHA-256: 0c5f21afe895ffab58c97f7340b0f3fa71be0ef48daba155e2510a6ab1c63b6b
  • semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
  • question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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