Question 10158: Derivative of an Oriented Integral with Moving Bounds

Original question and givens

On all R, continuous f has ascending coefficients [-5, 8, 0, 4], constant term first. Let lower l(x)=(-2)x+(-3) and upper u(x)=(0)x+(-5). 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 [-5, 8, 0, 4], constant term first. Let lower l(x)=(-2)x+(-3) and upper u(x)=(0)x+(-5). 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=[-5, 8, 0, 4];ascending powers on all R

normalized coefficients=[-5/1, 8/1, 0/1, 4/1];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(8, 2)

zero primitive coefficient2 = 4

integrand power=1;coefficient=8/1

primitive power=2;coefficient=4/1

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=[-5/1, 8/1, 0/1, 4/1]

zero-constant primitive coefficients=[0/1, -5/1, 4/1, 0/1, 1/1]

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, 4)

constructed P0 derivative coefficient1 = 8

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

derivative power=1;coefficient=8/1

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=[-5/1, 8/1, 0/1, 4/1];f=[-5/1, 8/1, 0/1, 4/1]

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

lower bound m*x := mul(-2, 1)

lower bound m*x = -2

lower bound m*x+b := sum(-2, -3)

lower bound m*x+b = -5

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

upper bound m*x = 0

upper bound m*x+b := sum(0, -5)

upper bound m*x+b = -5

H(x)=P0(u(x))-P0(l(x));P0'=f;lower slope=-2;upper slope=0

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

f at upper Horner multiply3 := mul(0, -5)

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, -5)

f at upper Horner multiply2 = -20

f at upper Horner add2 := sum(-20, 0)

f at upper Horner add2 = -20

f at upper Horner multiply1 := mul(-20, -5)

f at upper Horner multiply1 = 100

f at upper Horner add1 := sum(100, 8)

f at upper Horner add1 = 108

f at upper Horner multiply0 := mul(108, -5)

f at upper Horner multiply0 = -540

f at upper Horner add0 := sum(-540, -5)

f at upper Horner add0 = -545

f at lower Horner multiply3 := mul(0, -5)

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, -5)

f at lower Horner multiply2 = -20

f at lower Horner add2 := sum(-20, 0)

f at lower Horner add2 = -20

f at lower Horner multiply1 := mul(-20, -5)

f at lower Horner multiply1 = 100

f at lower Horner add1 := sum(100, 8)

f at lower Horner add1 = 108

f at lower Horner multiply0 := mul(108, -5)

f at lower Horner multiply0 = -540

f at lower Horner add0 := sum(-540, -5)

f at lower Horner add0 = -545

upper chain contribution := mul(-545, 0)

upper chain contribution = 0

lower chain contribution := mul(-545, -2)

lower chain contribution = 1090

moving bound derivative upper-minus-lower := sub(0, 1090)

moving bound derivative upper-minus-lower = -1090

point=1;lower=-5;upper=-5;f(upper)=-545;f(lower)=-545

upper contribution=0;lower contribution=1090;derivative=-1090/1

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

-1090/1

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.