Question 10116: Point Derivative of a Continuous Accumulating Integral

Original question and givens

On all R, the polynomial f is continuous and f has ascending coefficients [3, 16, -1, 4], constant term first. Let G(x)=∫_-1^x f(t)dt with oriented endpoints. Find G′(-2). Use reduced positive-denominator fractions, retaining /1 for integers.

• G001 — On all R, the polynomial f is continuous and f has ascending coefficients [3, 16, -1, 4], constant term first. Let G(x)=∫_-1^x f(t)dt with oriented endpoints. Find G′(-2). 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, 16, -1, 4];ascending powers on all R

normalized coefficients=[3/1, 16/1, -1/1, 4/1];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

zero primitive coefficient2 := div(16, 2)

zero primitive coefficient2 = 8

integrand power=1;coefficient=16/1

primitive power=2;coefficient=8/1

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=[3/1, 16/1, -1/1, 4/1]

zero-constant primitive coefficients=[0/1, 3/1, 8/1, -1/3, 1/1]

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

constructed P0 derivative coefficient1 := mul(2, 8)

constructed P0 derivative coefficient1 = 16

constructed P0:source power=2;coefficient=8/1

derivative power=1;coefficient=16/1

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=[3/1, 16/1, -1/1, 4/1];f=[3/1, 16/1, -1/1, 4/1]

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)

P0 at anchor Horner add4 = 1

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

P0 at anchor Horner multiply3 = -1

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

P0 at anchor Horner add3 = -4/3

P0 at anchor Horner multiply2 := mul(-4/3, -1)

P0 at anchor Horner multiply2 = 4/3

P0 at anchor Horner add2 := sum(4/3, 8)

P0 at anchor Horner add2 = 28/3

P0 at anchor Horner multiply1 := mul(28/3, -1)

P0 at anchor Horner multiply1 = -28/3

P0 at anchor Horner add1 := sum(-28/3, 3)

P0 at anchor Horner add1 = -19/3

P0 at anchor Horner multiply0 := mul(-19/3, -1)

P0 at anchor Horner multiply0 = 19/3

P0 at anchor Horner add0 := sum(19/3, 0)

P0 at anchor Horner add0 = 19/3

anchor a=-1;P0(a)=19/3

G(t)=P0(t)-(19/3);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, 16)

f at derivative point Horner add1 = 34

f at derivative point Horner multiply0 := mul(34, -2)

f at derivative point Horner multiply0 = -68

f at derivative point Horner add0 := sum(-68, 3)

f at derivative point Horner add0 = -65

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

-65/1

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.