# Question 10192: Candidate Antiderivative Premise for the Endpoint Formula

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0019-10192 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0092



## Original question and givens

On all R, continuous f has ascending coefficients [4, 17, 0, 1], constant term first. Candidate F has ascending rational coefficients [4/1, 4/1, 17/2, 0/1, 5/4]. Does F′=f satisfy the antiderivative premise for the endpoint formula?


- G001 — On all R, continuous f has ascending coefficients [4, 17, 0, 1], constant term first. Candidate F has ascending rational coefficients [4/1, 4/1, 17/2, 0/1, 5/4]. 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.


- [mko-fundamental-theorem-calculus](https://ocme.evemisslab.com/en/mko/mko-fundamental-theorem-calculus/)





## Step-by-step derivation

S001 · impliesBefore

raw integrand f=[4, 17, 0, 1];ascending powers on all R

After

normalized coefficients=[4/1, 17/1, 0/1, 1/1];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 dependenciesRule: declared_FTC_continuity_hypothesis · Uses: G001





S002 · impliesBefore

raw candidate F=[[4, 1], [4, 1], [17, 2], [0, 1], [5, 4]]

After

normalized candidate coefficients=[4/1, 4/1, 17/2, 0/1, 5/4]

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 dependenciesRule: candidate_primitive_original_denominators · Uses: G001, S001





S003 · calculationBefore

candidate F derivative coefficient0 := mul(1, 4)

After

candidate F derivative coefficient0 = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

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

After

derivative power=0;coefficient=4/1

Conditions: none

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

Step identifiers and dependenciesRule: polynomial_derivative_coefficient_rule · Uses: G001, S003





S005 · calculationBefore

candidate F derivative coefficient1 := mul(2, 17/2)

After

candidate F derivative coefficient1 = 17

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

candidate F:source power=2;coefficient=17/2

After

derivative power=1;coefficient=17/1

Conditions: none

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

Step identifiers and dependenciesRule: polynomial_derivative_coefficient_rule · Uses: G001, S005





S007 · calculationBefore

candidate F derivative coefficient2 := mul(3, 0)

After

candidate F derivative coefficient2 = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

candidate F:source power=3;coefficient=0/1

After

derivative power=2;coefficient=0/1

Conditions: none

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

Step identifiers and dependenciesRule: polynomial_derivative_coefficient_rule · Uses: G001, S007





S009 · calculationBefore

candidate F derivative coefficient3 := mul(4, 5/4)

After

candidate F derivative coefficient3 = 5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

candidate F:source power=4;coefficient=5/4

After

derivative power=3;coefficient=5/1

Conditions: none

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

Step identifiers and dependenciesRule: polynomial_derivative_coefficient_rule · Uses: G001, S009





S011 · impliesBefore

power=0;candidate derivative coefficient=4/1;target f coefficient=4/1

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 dependenciesRule: FTC_candidate_full_derivative_identity · Uses: G001, S010





S012 · impliesBefore

power=1;candidate derivative coefficient=17/1;target f coefficient=17/1

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 dependenciesRule: FTC_candidate_full_derivative_identity · Uses: G001, S011





S013 · impliesBefore

power=2;candidate derivative coefficient=0/1;target f coefficient=0/1

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 dependenciesRule: FTC_candidate_full_derivative_identity · Uses: G001, S012





S014 · impliesBefore

power=3;candidate derivative coefficient=5/1;target f coefficient=1/1

After

coefficient equality=No

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 dependenciesRule: FTC_candidate_full_derivative_identity · Uses: G001, S013





S015 · impliesBefore

candidate derivative=[4/1, 17/1, 0/1, 5/1];integrand=[4/1, 17/1, 0/1, 1/1]

After

this candidate satisfies F'=f=No

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 dependenciesRule: 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

No

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: ede51bffe9716fbf376dc2a84477f2b5e9ce632eeaef7382795d3d299754dd8a
- candidate record SHA-256: 210caa8cb40c166e409c2b9bb4c3ea134e242a15b7cbf1ba7599fe9a7224a7f4
- semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
- question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
- Lean: NOT_RUN · human review: NOT_PERFORMED

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