Question 10174: Derivative of an Oriented Integral with Moving Bounds

Original question and givens

On all R, continuous f has ascending coefficients [11, 24, 1, 4], constant term first. Let lower l(x)=(-3)x+(-3) and upper u(x)=(0)x+(11). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=1. 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. Let lower l(x)=(-3)x+(-3) and upper u(x)=(0)x+(11). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=1. 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

lower bound m*x := mul(-3, 1)

lower bound m*x = -3

lower bound m*x+b := sum(-3, -3)

lower bound m*x+b = -6

upper bound m*x := mul(0, 1)

upper bound m*x = 0

upper bound m*x+b := sum(0, 11)

upper bound m*x+b = 11

H(x)=P0(u(x))-P0(l(x));P0'=f;lower slope=-3;upper slope=0

H'(x)=f(u(x))*upper_slope-f(l(x))*lower_slope

f at upper Horner multiply3 := mul(0, 11)

f at upper Horner multiply3 = 0

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

f at upper Horner add3 = 4

f at upper Horner multiply2 := mul(4, 11)

f at upper Horner multiply2 = 44

f at upper Horner add2 := sum(44, 1)

f at upper Horner add2 = 45

f at upper Horner multiply1 := mul(45, 11)

f at upper Horner multiply1 = 495

f at upper Horner add1 := sum(495, 24)

f at upper Horner add1 = 519

f at upper Horner multiply0 := mul(519, 11)

f at upper Horner multiply0 = 5709

f at upper Horner add0 := sum(5709, 11)

f at upper Horner add0 = 5720

f at lower Horner multiply3 := mul(0, -6)

f at lower Horner multiply3 = 0

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

f at lower Horner add3 = 4

f at lower Horner multiply2 := mul(4, -6)

f at lower Horner multiply2 = -24

f at lower Horner add2 := sum(-24, 1)

f at lower Horner add2 = -23

f at lower Horner multiply1 := mul(-23, -6)

f at lower Horner multiply1 = 138

f at lower Horner add1 := sum(138, 24)

f at lower Horner add1 = 162

f at lower Horner multiply0 := mul(162, -6)

f at lower Horner multiply0 = -972

f at lower Horner add0 := sum(-972, 11)

f at lower Horner add0 = -961

upper chain contribution := mul(5720, 0)

upper chain contribution = 0

lower chain contribution := mul(-961, -3)

lower chain contribution = 2883

moving bound derivative upper-minus-lower := sub(0, 2883)

moving bound derivative upper-minus-lower = -2883

point=1;lower=-6;upper=11;f(upper)=5720;f(lower)=-961

upper contribution=0;lower contribution=2883;derivative=-2883/1

Reverse checks

• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Recompute moving bound derivative upper-minus-lower with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute lower chain contribution with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute upper chain contribution with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute f at lower Horner add0 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute f at lower Horner multiply0 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute f at lower Horner add1 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute f at lower Horner multiply1 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute f at lower Horner add2 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute f at lower Horner multiply2 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute f at lower Horner add3 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute f at lower Horner multiply3 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute f at upper Horner add0 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute f at upper Horner multiply0 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute f at upper Horner add1 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute f at upper Horner multiply1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute f at upper Horner add2 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute f at upper Horner multiply2 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute f at upper Horner add3 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute f at upper Horner multiply3 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Recompute upper bound m*x+b with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute upper bound m*x with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute lower bound m*x+b with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute lower bound m*x 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

-2883/1

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.