Explore related knowledge and questions →
← Question derivation index · Original question batch
R0019-10176 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0076
On all R, continuous f has ascending coefficients [0], constant term first. Candidate F has ascending rational coefficients [5/1]. Does F′=f satisfy the antiderivative premise for the endpoint formula?
G001 — On all R, continuous f has ascending coefficients [0], constant term first. Candidate F has ascending rational coefficients [5/1]. Does F′=f satisfy the antiderivative premise for the endpoint formula?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.
Before
raw integrand f=[0];ascending powers on all R
After
normalized coefficients=[0/1];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.
Rule: declared_FTC_continuity_hypothesis · Uses: G001
Before
raw candidate F=[[5, 1]]
After
normalized candidate coefficients=[5/1]
Conditions: none
Every original denominator must be positive.Normalize to exact rationals while retaining the supplied candidate,not replacing it by a constructed primitive.
Rule: candidate_primitive_original_denominators · Uses: G001, S001
Before
candidate F:constant coefficient=5/1
After
derivative coefficients=[0/1]
Conditions: none
A constant differentiates to the zero polynomial,represented as[0/1],not a fabricated negative power.
Rule: constant_polynomial_derivative_zero · Uses: G001, S002
Before
power=0;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.
Rule: FTC_candidate_full_derivative_identity · Uses: G001, S003
Before
candidate derivative=[0/1];integrand=[0/1]
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.
Rule: failed_primitive_candidate_not_all_primitives · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — 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.Yes
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 0bd483b3f9d2f5837fd37bccac028f86f251834f15db8f8245cc18a289600583candidate record SHA-256: c3259daecbc91067b6eff9486c8b841dd9e211e1cf33ad0936816a7f0bf15a0esemantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED