Question 10127: Accumulation Value and Derivative at the Anchor

Original question and givens

On all R, continuous f has ascending coefficients [3], constant term first. For G(x)=∫_0^x f(t)dt, give (G(0),G′(0)) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

• G001 — On all R, continuous f has ascending coefficients [3], constant term first. For G(x)=∫_0^x f(t)dt, give (G(0),G′(0)) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

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

Step-by-step derivation

raw integrand f=[3];ascending powers on all R

normalized coefficients=[3/1];declared continuous real polynomial

zero primitive coefficient1 := div(3, 1)

zero primitive coefficient1 = 3

integrand power=0;coefficient=3/1

primitive power=1;coefficient=3/1

original integrand coefficients=[3/1]

zero-constant primitive coefficients=[0/1, 3/1]

constructed P0 derivative coefficient0 := mul(1, 3)

constructed P0 derivative coefficient0 = 3

constructed P0:source power=1;coefficient=3/1

derivative power=0;coefficient=3/1

P0 derivative coefficients=[3/1];f=[3/1]

P0'=f on all R in this declared polynomial model

P0 at anchor Horner multiply1 := mul(0, 0)

P0 at anchor Horner multiply1 = 0

P0 at anchor Horner add1 := sum(0, 3)

P0 at anchor Horner add1 = 3

P0 at anchor Horner multiply0 := mul(3, 0)

P0 at anchor Horner multiply0 = 0

P0 at anchor Horner add0 := sum(0, 0)

P0 at anchor Horner add0 = 0

anchor a=0;P0(a)=0/1

G(t)=P0(t)-(0/1);G'(t)=f(t)

f at derivative point Horner multiply0 := mul(0, 0)

f at derivative point Horner multiply0 = 0

f at derivative point Horner add0 := sum(0, 3)

f at derivative point Horner add0 = 3

anchor integral same-endpoint difference := sub(0, 0)

anchor integral same-endpoint difference = 0

G(a)=0/1;f(a)=3/1

ordered value-derivative pair=(0/1, 3/1)

Reverse checks

• 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 anchor integral same-endpoint difference with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute f at derivative point Horner add0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute f at derivative point Horner multiply0 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 P0 at anchor Horner add0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute P0 at anchor Horner multiply0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute P0 at anchor Horner add1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute P0 at anchor Horner multiply1 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute constructed P0 derivative coefficient0 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 — 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.

Result

(0/1, 3/1)

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.