Question 10171: Derivative of an Oriented Integral with Moving Bounds

Original question and givens

On all R, continuous f has ascending coefficients [8, 21, 1, 1], constant term first. Let lower l(x)=(3)x+(0) and upper u(x)=(6)x+(8). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=-2. Use reduced positive-denominator fractions, retaining /1 for integers.

• G001 — On all R, continuous f has ascending coefficients [8, 21, 1, 1], constant term first. Let lower l(x)=(3)x+(0) and upper u(x)=(6)x+(8). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=-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=[8, 21, 1, 1];ascending powers on all R

normalized coefficients=[8/1, 21/1, 1/1, 1/1];declared continuous real polynomial

zero primitive coefficient1 := div(8, 1)

zero primitive coefficient1 = 8

integrand power=0;coefficient=8/1

primitive power=1;coefficient=8/1

zero primitive coefficient2 := div(21, 2)

zero primitive coefficient2 = 21/2

integrand power=1;coefficient=21/1

primitive power=2;coefficient=21/2

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(1, 4)

zero primitive coefficient4 = 1/4

integrand power=3;coefficient=1/1

primitive power=4;coefficient=1/4

original integrand coefficients=[8/1, 21/1, 1/1, 1/1]

zero-constant primitive coefficients=[0/1, 8/1, 21/2, 1/3, 1/4]

constructed P0 derivative coefficient0 := mul(1, 8)

constructed P0 derivative coefficient0 = 8

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

derivative power=0;coefficient=8/1

constructed P0 derivative coefficient1 := mul(2, 21/2)

constructed P0 derivative coefficient1 = 21

constructed P0:source power=2;coefficient=21/2

derivative power=1;coefficient=21/1

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, 1/4)

constructed P0 derivative coefficient3 = 1

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

derivative power=3;coefficient=1/1

P0 derivative coefficients=[8/1, 21/1, 1/1, 1/1];f=[8/1, 21/1, 1/1, 1/1]

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

lower bound m*x := mul(3, -2)

lower bound m*x = -6

lower bound m*x+b := sum(-6, 0)

lower bound m*x+b = -6

upper bound m*x := mul(6, -2)

upper bound m*x = -12

upper bound m*x+b := sum(-12, 8)

upper bound m*x+b = -4

H(x)=P0(u(x))-P0(l(x));P0'=f;lower slope=3;upper slope=6

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

f at upper Horner multiply3 := mul(0, -4)

f at upper Horner multiply3 = 0

f at upper Horner add3 := sum(0, 1)

f at upper Horner add3 = 1

f at upper Horner multiply2 := mul(1, -4)

f at upper Horner multiply2 = -4

f at upper Horner add2 := sum(-4, 1)

f at upper Horner add2 = -3

f at upper Horner multiply1 := mul(-3, -4)

f at upper Horner multiply1 = 12

f at upper Horner add1 := sum(12, 21)

f at upper Horner add1 = 33

f at upper Horner multiply0 := mul(33, -4)

f at upper Horner multiply0 = -132

f at upper Horner add0 := sum(-132, 8)

f at upper Horner add0 = -124

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

f at lower Horner multiply3 = 0

f at lower Horner add3 := sum(0, 1)

f at lower Horner add3 = 1

f at lower Horner multiply2 := mul(1, -6)

f at lower Horner multiply2 = -6

f at lower Horner add2 := sum(-6, 1)

f at lower Horner add2 = -5

f at lower Horner multiply1 := mul(-5, -6)

f at lower Horner multiply1 = 30

f at lower Horner add1 := sum(30, 21)

f at lower Horner add1 = 51

f at lower Horner multiply0 := mul(51, -6)

f at lower Horner multiply0 = -306

f at lower Horner add0 := sum(-306, 8)

f at lower Horner add0 = -298

upper chain contribution := mul(-124, 6)

upper chain contribution = -744

lower chain contribution := mul(-298, 3)

lower chain contribution = -894

moving bound derivative upper-minus-lower := sub(-744, -894)

moving bound derivative upper-minus-lower = 150

point=-2;lower=-6;upper=-4;f(upper)=-124;f(lower)=-298

upper contribution=-744;lower contribution=-894;derivative=150/1

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

150/1

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.