Question 10149: Accumulation Value and Derivative at the Anchor

Original question and givens

On all R, continuous f has ascending coefficients [11, 24, 1, 4], constant term first. For G(x)=∫_2^x f(t)dt, give (G(2),G′(2)) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

• G001 — On all R, continuous f has ascending coefficients [11, 24, 1, 4], constant term first. For G(x)=∫_2^x f(t)dt, give (G(2),G′(2)) 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=[11, 24, 1, 4];ascending powers on all R

normalized coefficients=[11/1, 24/1, 1/1, 4/1];declared continuous real polynomial

zero primitive coefficient1 := div(11, 1)

zero primitive coefficient1 = 11

integrand power=0;coefficient=11/1

primitive power=1;coefficient=11/1

zero primitive coefficient2 := div(24, 2)

zero primitive coefficient2 = 12

integrand power=1;coefficient=24/1

primitive power=2;coefficient=12/1

zero primitive coefficient3 := div(1, 3)

zero primitive coefficient3 = 1/3

integrand power=2;coefficient=1/1

primitive power=3;coefficient=1/3

zero primitive coefficient4 := div(4, 4)

zero primitive coefficient4 = 1

integrand power=3;coefficient=4/1

primitive power=4;coefficient=1/1

original integrand coefficients=[11/1, 24/1, 1/1, 4/1]

zero-constant primitive coefficients=[0/1, 11/1, 12/1, 1/3, 1/1]

constructed P0 derivative coefficient0 := mul(1, 11)

constructed P0 derivative coefficient0 = 11

constructed P0:source power=1;coefficient=11/1

derivative power=0;coefficient=11/1

constructed P0 derivative coefficient1 := mul(2, 12)

constructed P0 derivative coefficient1 = 24

constructed P0:source power=2;coefficient=12/1

derivative power=1;coefficient=24/1

constructed P0 derivative coefficient2 := mul(3, 1/3)

constructed P0 derivative coefficient2 = 1

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

derivative power=2;coefficient=1/1

constructed P0 derivative coefficient3 := mul(4, 1)

constructed P0 derivative coefficient3 = 4

constructed P0:source power=4;coefficient=1/1

derivative power=3;coefficient=4/1

P0 derivative coefficients=[11/1, 24/1, 1/1, 4/1];f=[11/1, 24/1, 1/1, 4/1]

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

P0 at anchor Horner multiply4 := mul(0, 2)

P0 at anchor Horner multiply4 = 0

P0 at anchor Horner add4 := sum(0, 1)

P0 at anchor Horner add4 = 1

P0 at anchor Horner multiply3 := mul(1, 2)

P0 at anchor Horner multiply3 = 2

P0 at anchor Horner add3 := sum(2, 1/3)

P0 at anchor Horner add3 = 7/3

P0 at anchor Horner multiply2 := mul(7/3, 2)

P0 at anchor Horner multiply2 = 14/3

P0 at anchor Horner add2 := sum(14/3, 12)

P0 at anchor Horner add2 = 50/3

P0 at anchor Horner multiply1 := mul(50/3, 2)

P0 at anchor Horner multiply1 = 100/3

P0 at anchor Horner add1 := sum(100/3, 11)

P0 at anchor Horner add1 = 133/3

P0 at anchor Horner multiply0 := mul(133/3, 2)

P0 at anchor Horner multiply0 = 266/3

P0 at anchor Horner add0 := sum(266/3, 0)

P0 at anchor Horner add0 = 266/3

anchor a=2;P0(a)=266/3

G(t)=P0(t)-(266/3);G'(t)=f(t)

f at derivative point Horner multiply3 := mul(0, 2)

f at derivative point Horner multiply3 = 0

f at derivative point Horner add3 := sum(0, 4)

f at derivative point Horner add3 = 4

f at derivative point Horner multiply2 := mul(4, 2)

f at derivative point Horner multiply2 = 8

f at derivative point Horner add2 := sum(8, 1)

f at derivative point Horner add2 = 9

f at derivative point Horner multiply1 := mul(9, 2)

f at derivative point Horner multiply1 = 18

f at derivative point Horner add1 := sum(18, 24)

f at derivative point Horner add1 = 42

f at derivative point Horner multiply0 := mul(42, 2)

f at derivative point Horner multiply0 = 84

f at derivative point Horner add0 := sum(84, 11)

f at derivative point Horner add0 = 95

anchor integral same-endpoint difference := sub(266/3, 266/3)

anchor integral same-endpoint difference = 0

G(a)=0/1;f(a)=95/1

ordered value-derivative pair=(0/1, 95/1)

Reverse checks

• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Recompute anchor integral same-endpoint difference with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute f at derivative point Horner add0 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute f at derivative point Horner multiply0 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute f at derivative point Horner add1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute f at derivative point Horner multiply1 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute f at derivative point Horner add2 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute f at derivative point Horner multiply2 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute f at derivative point Horner add3 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute f at derivative point Horner multiply3 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Recompute P0 at anchor Horner add0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute P0 at anchor Horner multiply0 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute P0 at anchor Horner add1 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute P0 at anchor Horner multiply1 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute P0 at anchor Horner add2 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute P0 at anchor Horner multiply2 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute P0 at anchor Horner add3 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute P0 at anchor Horner multiply3 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute P0 at anchor Horner add4 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute P0 at anchor Horner multiply4 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.

Result

(0/1, 95/1)

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.