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R0019-10161 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0036
On all R, continuous f has ascending coefficients [-2, 11, 0, 3], constant term first. Let lower l(x)=(2)x+(-2) and upper u(x)=(4)x+(-2). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=0. Use reduced positive-denominator fractions, retaining /1 for integers.
G001 — On all R, continuous f has ascending coefficients [-2, 11, 0, 3], constant term first. Let lower l(x)=(2)x+(-2) and upper u(x)=(4)x+(-2). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=0. 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=[-2, 11, 0, 3];ascending powers on all R
After
normalized coefficients=[-2/1, 11/1, 0/1, 3/1];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(-2, 1)
After
zero primitive coefficient1 = -2
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=-2/1
After
primitive power=1;coefficient=-2/1
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
zero primitive coefficient2 := div(11, 2)
After
zero primitive coefficient2 = 11/2
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, S003
Before
integrand power=1;coefficient=11/1
After
primitive power=2;coefficient=11/2
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, S004
Before
zero primitive coefficient3 := div(0, 3)
After
zero primitive coefficient3 = 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, S005
Before
integrand power=2;coefficient=0/1
After
primitive power=3;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, S006
Before
zero primitive coefficient4 := div(3, 4)
After
zero primitive coefficient4 = 3/4
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
integrand power=3;coefficient=3/1
After
primitive power=4;coefficient=3/4
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, S008
Before
original integrand coefficients=[-2/1, 11/1, 0/1, 3/1]
After
zero-constant primitive coefficients=[0/1, -2/1, 11/2, 0/1, 3/4]
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, S009
Before
constructed P0 derivative coefficient0 := mul(1, -2)
After
constructed P0 derivative coefficient0 = -2
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, S010
Before
constructed P0:source power=1;coefficient=-2/1
After
derivative power=0;coefficient=-2/1
Conditions: none
Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.
Rule: polynomial_derivative_coefficient_rule · Uses: G001, S011
Before
constructed P0 derivative coefficient1 := mul(2, 11/2)
After
constructed P0 derivative coefficient1 = 11
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
constructed P0:source power=2;coefficient=11/2
After
derivative power=1;coefficient=11/1
Conditions: none
Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.
Rule: polynomial_derivative_coefficient_rule · Uses: G001, S013
Before
constructed P0 derivative coefficient2 := mul(3, 0)
After
constructed P0 derivative coefficient2 = 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
constructed P0: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.
Rule: polynomial_derivative_coefficient_rule · Uses: G001, S015
Before
constructed P0 derivative coefficient3 := mul(4, 3/4)
After
constructed P0 derivative coefficient3 = 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, S016
Before
constructed P0:source power=4;coefficient=3/4
After
derivative power=3;coefficient=3/1
Conditions: none
Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.
Rule: polynomial_derivative_coefficient_rule · Uses: G001, S017
Before
P0 derivative coefficients=[-2/1, 11/1, 0/1, 3/1];f=[-2/1, 11/1, 0/1, 3/1]
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, S018
Before
lower bound m*x := mul(2, 0)
After
lower bound m*x = 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, S019
Before
lower bound m*x+b := sum(0, -2)
After
lower bound m*x+b = -2
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, S020
Before
upper bound m*x := mul(4, 0)
After
upper bound m*x = 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, S021
Before
upper bound m*x+b := sum(0, -2)
After
upper bound m*x+b = -2
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, S022
Before
H(x)=P0(u(x))-P0(l(x));P0'=f;lower slope=2;upper slope=4
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, S023
Before
f at upper Horner multiply3 := mul(0, -2)
After
f at upper Horner multiply3 = 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, S024
Before
f at upper Horner add3 := sum(0, 3)
After
f at upper Horner add3 = 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, S025
Before
f at upper Horner multiply2 := mul(3, -2)
After
f at upper Horner multiply2 = -6
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, S026
Before
f at upper Horner add2 := sum(-6, 0)
After
f at upper Horner add2 = -6
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, S027
Before
f at upper Horner multiply1 := mul(-6, -2)
After
f at upper Horner multiply1 = 12
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, S028
Before
f at upper Horner add1 := sum(12, 11)
After
f at upper Horner add1 = 23
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, S029
Before
f at upper Horner multiply0 := mul(23, -2)
After
f at upper Horner multiply0 = -46
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, S030
Before
f at upper Horner add0 := sum(-46, -2)
After
f at upper Horner add0 = -48
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, S031
Before
f at lower Horner multiply3 := mul(0, -2)
After
f at lower Horner multiply3 = 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, S032
Before
f at lower Horner add3 := sum(0, 3)
After
f at lower Horner add3 = 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, S033
Before
f at lower Horner multiply2 := mul(3, -2)
After
f at lower Horner multiply2 = -6
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, S034
Before
f at lower Horner add2 := sum(-6, 0)
After
f at lower Horner add2 = -6
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, S035
Before
f at lower Horner multiply1 := mul(-6, -2)
After
f at lower Horner multiply1 = 12
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, S036
Before
f at lower Horner add1 := sum(12, 11)
After
f at lower Horner add1 = 23
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, S037
Before
f at lower Horner multiply0 := mul(23, -2)
After
f at lower Horner multiply0 = -46
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, S038
Before
f at lower Horner add0 := sum(-46, -2)
After
f at lower Horner add0 = -48
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, S039
Before
upper chain contribution := mul(-48, 4)
After
upper chain contribution = -192
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, S040
Before
lower chain contribution := mul(-48, 2)
After
lower chain contribution = -96
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, S041
Before
moving bound derivative upper-minus-lower := sub(-192, -96)
After
moving bound derivative upper-minus-lower = -96
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, S042
Before
point=0;lower=-2;upper=-2;f(upper)=-48;f(lower)=-48
After
upper contribution=-192;lower contribution=-96;derivative=-96/1
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, S043
S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S043 · premise_and_result_audit — Recompute moving bound derivative upper-minus-lower with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute lower chain contribution with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute upper chain contribution with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute f at lower Horner add0 with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute f at lower Horner multiply0 with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute f at lower Horner add1 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute f at lower Horner multiply1 with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute f at lower Horner add2 with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute f at lower Horner multiply2 with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute f at lower Horner add3 with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute f at lower Horner multiply3 with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute f at upper Horner add0 with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute f at upper Horner multiply0 with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute f at upper Horner add1 with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute f at upper Horner multiply1 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute f at upper Horner add2 with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute f at upper Horner multiply2 with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute f at upper Horner add3 with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute f at upper Horner multiply3 with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Recompute upper bound m*x+b with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute upper bound m*x with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute lower bound m*x+b with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute lower bound m*x with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Recompute constructed P0 derivative coefficient3 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Recompute constructed P0 derivative coefficient2 with exact rational arithmetic, not rounded decimals.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 constructed P0 derivative coefficient1 with exact rational arithmetic, not rounded decimals.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 — Recompute constructed P0 derivative coefficient0 with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute zero primitive coefficient4 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Recompute zero primitive coefficient3 with exact rational arithmetic, not rounded decimals.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 — Recompute zero primitive coefficient2 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 — 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.-96/1
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.
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