Question 10153: Derivative of an Oriented Integral with Moving Bounds

Original question and givens

On all R, continuous f has ascending coefficients [3, 1], constant term first. Let lower l(x)=(2)x+(1) and upper u(x)=(2)x+(1). 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 [3, 1], constant term first. Let lower l(x)=(2)x+(1) and upper u(x)=(2)x+(1). 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=[3, 1];ascending powers on all R

normalized coefficients=[3/1, 1/1];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(1, 2)

zero primitive coefficient2 = 1/2

integrand power=1;coefficient=1/1

primitive power=2;coefficient=1/2

original integrand coefficients=[3/1, 1/1]

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

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, 1/2)

constructed P0 derivative coefficient1 = 1

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

derivative power=1;coefficient=1/1

P0 derivative coefficients=[3/1, 1/1];f=[3/1, 1/1]

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, 1)

lower bound m*x+b = -1

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

upper bound m*x = -2

upper bound m*x+b := sum(-2, 1)

upper bound m*x+b = -1

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

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

f at upper Horner multiply1 := mul(0, -1)

f at upper Horner multiply1 = 0

f at upper Horner add1 := sum(0, 1)

f at upper Horner add1 = 1

f at upper Horner multiply0 := mul(1, -1)

f at upper Horner multiply0 = -1

f at upper Horner add0 := sum(-1, 3)

f at upper Horner add0 = 2

f at lower Horner multiply1 := mul(0, -1)

f at lower Horner multiply1 = 0

f at lower Horner add1 := sum(0, 1)

f at lower Horner add1 = 1

f at lower Horner multiply0 := mul(1, -1)

f at lower Horner multiply0 = -1

f at lower Horner add0 := sum(-1, 3)

f at lower Horner add0 = 2

upper chain contribution := mul(2, 2)

upper chain contribution = 4

lower chain contribution := mul(2, 2)

lower chain contribution = 4

moving bound derivative upper-minus-lower := sub(4, 4)

moving bound derivative upper-minus-lower = 0

point=-1;lower=-1;upper=-1;f(upper)=2;f(lower)=2

upper contribution=4;lower contribution=4;derivative=0/1

Reverse checks

• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Recompute moving bound derivative upper-minus-lower with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute lower chain contribution with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute upper chain contribution with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute f at lower Horner add0 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute f at lower Horner multiply0 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute f at lower Horner add1 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute f at lower Horner multiply1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute f at upper Horner add0 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute f at upper Horner multiply0 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute f at upper Horner add1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute f at upper Horner multiply1 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 upper bound m*x+b with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute upper bound m*x with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute lower bound m*x+b with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute lower bound m*x with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute constructed P0 derivative coefficient1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute constructed P0 derivative coefficient0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

0/1

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.