Question 10159: Derivative of an Oriented Integral with Moving Bounds

Original question and givens

On all R, continuous f has ascending coefficients [-4, 9, 1, 1], constant term first. Let lower l(x)=(3)x+(0) and upper u(x)=(6)x+(-4). 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 [-4, 9, 1, 1], constant term first. Let lower l(x)=(3)x+(0) and upper u(x)=(6)x+(-4). 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=[-4, 9, 1, 1];ascending powers on all R

normalized coefficients=[-4/1, 9/1, 1/1, 1/1];declared continuous real polynomial

zero primitive coefficient1 := div(-4, 1)

zero primitive coefficient1 = -4

integrand power=0;coefficient=-4/1

primitive power=1;coefficient=-4/1

zero primitive coefficient2 := div(9, 2)

zero primitive coefficient2 = 9/2

integrand power=1;coefficient=9/1

primitive power=2;coefficient=9/2

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=[-4/1, 9/1, 1/1, 1/1]

zero-constant primitive coefficients=[0/1, -4/1, 9/2, 1/3, 1/4]

constructed P0 derivative coefficient0 := mul(1, -4)

constructed P0 derivative coefficient0 = -4

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

derivative power=0;coefficient=-4/1

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

constructed P0 derivative coefficient1 = 9

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

derivative power=1;coefficient=9/1

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=[-4/1, 9/1, 1/1, 1/1];f=[-4/1, 9/1, 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, -4)

upper bound m*x+b = -16

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, -16)

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, -16)

f at upper Horner multiply2 = -16

f at upper Horner add2 := sum(-16, 1)

f at upper Horner add2 = -15

f at upper Horner multiply1 := mul(-15, -16)

f at upper Horner multiply1 = 240

f at upper Horner add1 := sum(240, 9)

f at upper Horner add1 = 249

f at upper Horner multiply0 := mul(249, -16)

f at upper Horner multiply0 = -3984

f at upper Horner add0 := sum(-3984, -4)

f at upper Horner add0 = -3988

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, 9)

f at lower Horner add1 = 39

f at lower Horner multiply0 := mul(39, -6)

f at lower Horner multiply0 = -234

f at lower Horner add0 := sum(-234, -4)

f at lower Horner add0 = -238

upper chain contribution := mul(-3988, 6)

upper chain contribution = -23928

lower chain contribution := mul(-238, 3)

lower chain contribution = -714

moving bound derivative upper-minus-lower := sub(-23928, -714)

moving bound derivative upper-minus-lower = -23214

point=-2;lower=-6;upper=-16;f(upper)=-3988;f(lower)=-238

upper contribution=-23928;lower contribution=-714;derivative=-23214/1

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

-23214/1

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.