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R0019-10152 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0027
On all R, continuous f has ascending coefficients [0], constant term first. Let lower l(x)=(1)x+(0) and upper u(x)=(3)x+(0). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=1. Use reduced positive-denominator fractions, retaining /1 for integers.
G001 — On all R, continuous f has ascending coefficients [0], constant term first. Let lower l(x)=(1)x+(0) and upper u(x)=(3)x+(0). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=1. Use reduced positive-denominator fractions, retaining /1 for integers.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
zero primitive coefficient1 := div(0, 1)
After
zero primitive coefficient1 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
integrand power=0;coefficient=0/1
After
primitive power=1;coefficient=0/1
Conditions: none
For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive.
Rule: polynomial_primitive_coefficient_rule · Uses: G001, S002
Before
original integrand coefficients=[0/1]
After
zero-constant primitive coefficients=[0/1]
Conditions: none
This is one convenient representative,not a claim that the primitive is unique without its constant family.
Rule: zero_constant_polynomial_primitive · Uses: G001, S003
Before
constructed P0:constant coefficient=0/1
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, S004
Before
P0 derivative coefficients=[0/1];f=[0/1]
After
P0'=f on all R in this declared polynomial model
Conditions: none
The full coefficient identity establishes the primitive relation in the declared model.With the supplied continuity premise use the FTC endpoint difference,not a function guessed from sample points.
Rule: constructed_primitive_coefficient_identity · Uses: G001, S005
Before
lower bound m*x := mul(1, 1)
After
lower bound m*x = 1
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
lower bound m*x+b := sum(1, 0)
After
lower bound m*x+b = 1
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
upper bound m*x := mul(3, 1)
After
upper bound m*x = 3
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
upper bound m*x+b := sum(3, 0)
After
upper bound m*x+b = 3
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
H(x)=P0(u(x))-P0(l(x));P0'=f;lower slope=1;upper slope=3
After
H'(x)=f(u(x))*upper_slope-f(l(x))*lower_slope
Conditions: none
Use the oriented endpoint difference,then the chain rule for both compositions.Subtract the whole lower-bound term;constant bounds have slope0.Equal bound values at one point do not mean equal velocities or force a zero derivative.
Rule: FTC_moving_bounds_two_chain_factors · Uses: G001, S010
Before
f at upper Horner multiply0 := mul(0, 3)
After
f at upper Horner multiply0 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
f at upper Horner add0 := sum(0, 0)
After
f at upper Horner add0 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
f at lower Horner multiply0 := mul(0, 1)
After
f at lower Horner multiply0 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
f at lower Horner add0 := sum(0, 0)
After
f at lower Horner add0 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
upper chain contribution := mul(0, 3)
After
upper chain contribution = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
lower chain contribution := mul(0, 1)
After
lower chain contribution = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
moving bound derivative upper-minus-lower := sub(0, 0)
After
moving bound derivative upper-minus-lower = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
point=1;lower=1;upper=3;f(upper)=0;f(lower)=0
After
upper contribution=0;lower contribution=0;derivative=0/1
Conditions: none
Retain both actual endpoints,integrand values and slopes,then subtract with sign.Do not take an absolute area or discard zero/negative velocities.
Rule: FTC_moving_bounds_actual_values · Uses: G001, S018
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Recompute moving bound derivative upper-minus-lower with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute lower chain contribution with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute upper chain contribution with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute f at lower Horner add0 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute f at lower Horner multiply0 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute f at upper Horner add0 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute f at upper Horner multiply0 with exact rational arithmetic, not rounded decimals.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 — Recompute upper bound m*x+b with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute upper bound m*x with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute lower bound m*x+b with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute lower bound m*x 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 — 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 — Recompute zero primitive coefficient1 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.0/1
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: 5ed0fdb526cf7cfa8b9d04f068d219c9ae6327664b45f1a471611105be17c9eccandidate record SHA-256: 7daed810e5fc832ac9fe32ca3e1e6db5e1b90db5091756243a9e0d2fb361193asemantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED