Question 10118: Point Derivative of a Continuous Accumulating Integral

Original question and givens

On all R, the polynomial f is continuous and f has ascending coefficients [5, 18, 1, 2], constant term first. Let G(x)=∫_1^x f(t)dt with oriented endpoints. Find G′(0). Use reduced positive-denominator fractions, retaining /1 for integers.

• G001 — On all R, the polynomial f is continuous and f has ascending coefficients [5, 18, 1, 2], constant term first. Let G(x)=∫_1^x f(t)dt with oriented endpoints. Find G′(0). 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=[5, 18, 1, 2];ascending powers on all R

normalized coefficients=[5/1, 18/1, 1/1, 2/1];declared continuous real polynomial

zero primitive coefficient1 := div(5, 1)

zero primitive coefficient1 = 5

integrand power=0;coefficient=5/1

primitive power=1;coefficient=5/1

zero primitive coefficient2 := div(18, 2)

zero primitive coefficient2 = 9

integrand power=1;coefficient=18/1

primitive power=2;coefficient=9/1

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(2, 4)

zero primitive coefficient4 = 1/2

integrand power=3;coefficient=2/1

primitive power=4;coefficient=1/2

original integrand coefficients=[5/1, 18/1, 1/1, 2/1]

zero-constant primitive coefficients=[0/1, 5/1, 9/1, 1/3, 1/2]

constructed P0 derivative coefficient0 := mul(1, 5)

constructed P0 derivative coefficient0 = 5

constructed P0:source power=1;coefficient=5/1

derivative power=0;coefficient=5/1

constructed P0 derivative coefficient1 := mul(2, 9)

constructed P0 derivative coefficient1 = 18

constructed P0:source power=2;coefficient=9/1

derivative power=1;coefficient=18/1

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, 1/2)

constructed P0 derivative coefficient3 = 2

constructed P0:source power=4;coefficient=1/2

derivative power=3;coefficient=2/1

P0 derivative coefficients=[5/1, 18/1, 1/1, 2/1];f=[5/1, 18/1, 1/1, 2/1]

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

P0 at anchor Horner multiply4 := mul(0, 1)

P0 at anchor Horner multiply4 = 0

P0 at anchor Horner add4 := sum(0, 1/2)

P0 at anchor Horner add4 = 1/2

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

P0 at anchor Horner multiply3 = 1/2

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

P0 at anchor Horner add3 = 5/6

P0 at anchor Horner multiply2 := mul(5/6, 1)

P0 at anchor Horner multiply2 = 5/6

P0 at anchor Horner add2 := sum(5/6, 9)

P0 at anchor Horner add2 = 59/6

P0 at anchor Horner multiply1 := mul(59/6, 1)

P0 at anchor Horner multiply1 = 59/6

P0 at anchor Horner add1 := sum(59/6, 5)

P0 at anchor Horner add1 = 89/6

P0 at anchor Horner multiply0 := mul(89/6, 1)

P0 at anchor Horner multiply0 = 89/6

P0 at anchor Horner add0 := sum(89/6, 0)

P0 at anchor Horner add0 = 89/6

anchor a=1;P0(a)=89/6

G(t)=P0(t)-(89/6);G'(t)=f(t)

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

f at derivative point Horner multiply3 = 0

f at derivative point Horner add3 := sum(0, 2)

f at derivative point Horner add3 = 2

f at derivative point Horner multiply2 := mul(2, 0)

f at derivative point Horner multiply2 = 0

f at derivative point Horner add2 := sum(0, 1)

f at derivative point Horner add2 = 1

f at derivative point Horner multiply1 := mul(1, 0)

f at derivative point Horner multiply1 = 0

f at derivative point Horner add1 := sum(0, 18)

f at derivative point Horner add1 = 18

f at derivative point Horner multiply0 := mul(18, 0)

f at derivative point Horner multiply0 = 0

f at derivative point Horner add0 := sum(0, 5)

f at derivative point Horner add0 = 5

Reverse checks

• 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

5/1

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.