Question 10145: Accumulation Value and Derivative at the Anchor

Original question and givens

On all R, continuous f has ascending coefficients [7, 20, 0, 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 [7, 20, 0, 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=[7, 20, 0, 4];ascending powers on all R

normalized coefficients=[7/1, 20/1, 0/1, 4/1];declared continuous real polynomial

zero primitive coefficient1 := div(7, 1)

zero primitive coefficient1 = 7

integrand power=0;coefficient=7/1

primitive power=1;coefficient=7/1

zero primitive coefficient2 := div(20, 2)

zero primitive coefficient2 = 10

integrand power=1;coefficient=20/1

primitive power=2;coefficient=10/1

zero primitive coefficient3 := div(0, 3)

zero primitive coefficient3 = 0

integrand power=2;coefficient=0/1

primitive power=3;coefficient=0/1

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=[7/1, 20/1, 0/1, 4/1]

zero-constant primitive coefficients=[0/1, 7/1, 10/1, 0/1, 1/1]

constructed P0 derivative coefficient0 := mul(1, 7)

constructed P0 derivative coefficient0 = 7

constructed P0:source power=1;coefficient=7/1

derivative power=0;coefficient=7/1

constructed P0 derivative coefficient1 := mul(2, 10)

constructed P0 derivative coefficient1 = 20

constructed P0:source power=2;coefficient=10/1

derivative power=1;coefficient=20/1

constructed P0 derivative coefficient2 := mul(3, 0)

constructed P0 derivative coefficient2 = 0

constructed P0:source power=3;coefficient=0/1

derivative power=2;coefficient=0/1

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=[7/1, 20/1, 0/1, 4/1];f=[7/1, 20/1, 0/1, 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, 0)

P0 at anchor Horner add3 = 2

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

P0 at anchor Horner multiply2 = 4

P0 at anchor Horner add2 := sum(4, 10)

P0 at anchor Horner add2 = 14

P0 at anchor Horner multiply1 := mul(14, 2)

P0 at anchor Horner multiply1 = 28

P0 at anchor Horner add1 := sum(28, 7)

P0 at anchor Horner add1 = 35

P0 at anchor Horner multiply0 := mul(35, 2)

P0 at anchor Horner multiply0 = 70

P0 at anchor Horner add0 := sum(70, 0)

P0 at anchor Horner add0 = 70

anchor a=2;P0(a)=70/1

G(t)=P0(t)-(70/1);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, 0)

f at derivative point Horner add2 = 8

f at derivative point Horner multiply1 := mul(8, 2)

f at derivative point Horner multiply1 = 16

f at derivative point Horner add1 := sum(16, 20)

f at derivative point Horner add1 = 36

f at derivative point Horner multiply0 := mul(36, 2)

f at derivative point Horner multiply0 = 72

f at derivative point Horner add0 := sum(72, 7)

f at derivative point Horner add0 = 79

anchor integral same-endpoint difference := sub(70, 70)

anchor integral same-endpoint difference = 0

G(a)=0/1;f(a)=79/1

ordered value-derivative pair=(0/1, 79/1)

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, 79/1)

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.