Question 8527: Finite Difference Quotient at a Specified Nonzero Step

Original question and givens

On all R, f(t)=(-1)t²+(-11)t+(2), allowing a2=0. At x=-1 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f(t)=(-1)t²+(-11)t+(2), allowing a2=0. At x=-1 and nonzero finite h=-2, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, 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-derivative

Step-by-step derivation

declared real formula on all R;derivative means the two-sided h->0 quotient limit

finite-step evaluation and derivative existence are separate

at original x a*x := mul(-1, -1)

at original x a*x = 1

at original x a*x+b := sum(1, -11)

at original x a*x+b = -10

at original x (a*x+b)*x := mul(-10, -1)

at original x (a*x+b)*x = 10

at original x f(x) := sum(10, 2)

at original x f(x) = 12

f(t)=(-1)*t^2+(-11)*t+(2);x=-1;f(x)=12;h!=0

f(x+h)-f(x)=h*(2*(-1)*(-1)+(-11)+(-1)*h)

twice quadratic coefficient := mul(2, -1)

twice quadratic coefficient = -2

twice a times x := mul(-2, -1)

twice a times x = 2

linear part of quotient := sum(2, -11)

linear part of quotient = -9

shifted input x+h := sum(-1, -2)

shifted input x+h = -3

at shifted x a*x := mul(-1, -3)

at shifted x a*x = 3

at shifted x a*x+b := sum(3, -11)

at shifted x a*x+b = -8

at shifted x (a*x+b)*x := mul(-8, -3)

at shifted x (a*x+b)*x = 24

at shifted x f(x) := sum(24, 2)

at shifted x f(x) = 26

finite increment f(x+h)-f(x) := sub(26, 12)

finite increment f(x+h)-f(x) = 14

specified finite quotient := div(14, -2)

specified finite quotient = -7

finite-step quadratic correction := mul(-1, -2)

finite-step quadratic correction = 2

expanded finite quotient := sum(-9, 2)

expanded finite quotient = -7

finite quotient minus derivative := sub(-7, -9)

finite quotient minus derivative = 2

h=-2;finite quotient=-7/1;derivative value=-9

difference=2;a2*h=2

Reverse checks

• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute finite quotient minus derivative with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute expanded finite quotient with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute finite-step quadratic correction with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute specified finite quotient with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute finite increment f(x+h)-f(x) with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute at shifted x f(x) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute at shifted x (a*x+b)*x with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute at shifted x a*x+b with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute at shifted x a*x with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute shifted input x+h with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute linear part of quotient with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute twice a times x with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute twice quadratic coefficient 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 — Recompute at original x f(x) with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute at original x (a*x+b)*x with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute at original x a*x+b with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute at original x a*x 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

-7/1

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.