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R0001-01386 · q-20261007-advance-20261008-r06-antiderivative-0086
On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(-2), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.
G001 — On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(-2), retaining the actual additive constant. Give a reduced fraction with positive denominator, 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
domain=R; coefficient lists are in ascending powers
After
constant term first; rational pairs=[numerator,denominator]
Conditions: none
Fix coefficient order;do not read ascending coefficients as descending or swap numerator and denominator.
Rule: polynomial_representation · Uses: G001
Before
f coefficients ascending=[-2, 11, 0, 3]
After
seek F₀ with constant coefficient0 and F₀′=f
Conditions: none
Antiderivatives are not generally unique; this task additionally fixes the constant coefficient at0.
Rule: canonical_primitive_domain · Uses: G001, S001
Before
primitive coefficient at degree1 := div(-2, 1)
After
primitive coefficient at degree1 = -2
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
differentiate back coefficient0 := mul(-2, 1)
After
differentiate back coefficient0 = -2
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
primitive coefficient at degree2 := div(11, 2)
After
primitive coefficient at degree2 = 11/2
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
differentiate back coefficient1 := mul(11/2, 2)
After
differentiate back coefficient1 = 11
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
primitive coefficient at degree3 := div(0, 3)
After
primitive coefficient at degree3 = 0
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
differentiate back coefficient2 := mul(0, 3)
After
differentiate back coefficient2 = 0
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
primitive coefficient at degree4 := div(3, 4)
After
primitive coefficient at degree4 = 3/4
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
differentiate back coefficient3 := mul(3/4, 4)
After
differentiate back coefficient3 = 3
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
[0, -2, 11/2, 0, 3/4]
After
all derivative coefficients agree with f
Conditions: none
Coefficient agreement proves the polynomial identity on all reals,not merely equality at a few points.
Rule: global_polynomial_identity · Uses: G001, S010
Before
constant coefficient+C := sum(0, -2)
After
constant coefficient+C = -2
Conditions: none
Change only the constant coefficient;the derivative remains the same f.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
F(x) multiply at degree4 := mul(0, 2)
After
F(x) multiply at degree4 = 0
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
F(x) add coefficient4 := sum(0, 3/4)
After
F(x) add coefficient4 = 3/4
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
F(x) multiply at degree3 := mul(3/4, 2)
After
F(x) multiply at degree3 = 3/2
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
F(x) add coefficient3 := sum(3/2, 0)
After
F(x) add coefficient3 = 3/2
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
F(x) multiply at degree2 := mul(3/2, 2)
After
F(x) multiply at degree2 = 3
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
F(x) add coefficient2 := sum(3, 11/2)
After
F(x) add coefficient2 = 17/2
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
F(x) multiply at degree1 := mul(17/2, 2)
After
F(x) multiply at degree1 = 17
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
F(x) add coefficient1 := sum(17, -2)
After
F(x) add coefficient1 = 15
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
F(x) multiply at degree0 := mul(15, 2)
After
F(x) multiply at degree0 = 30
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
F(x) add coefficient0 := sum(30, -2)
After
F(x) add coefficient0 = 28
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
x=2; additive constant=-2
After
F(x)=28/1
Conditions: none
Use a reduced positive-denominator fraction,retaining /1 even for integers.
Rule: primitive_value_format · Uses: G001, S022
S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Recompute F(x) add coefficient0 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute F(x) multiply at degree0 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute F(x) add coefficient1 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute F(x) multiply at degree1 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute F(x) add coefficient2 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute F(x) multiply at degree2 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute F(x) add coefficient3 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute F(x) multiply at degree3 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute F(x) add coefficient4 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute F(x) multiply at degree4 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute constant coefficient+C 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 differentiate back coefficient3 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute primitive coefficient at degree4 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute differentiate back coefficient2 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute primitive coefficient at degree3 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute differentiate back coefficient1 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute primitive coefficient at degree2 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute differentiate back coefficient0 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute primitive coefficient at degree1 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.28/1
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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