Question 8537: Finite Difference Quotient at a Specified Nonzero Step

Original question and givens

On all R, f(t)=(-11)t²+(-1)t+(22), allowing a2=0. At x=-1 and nonzero finite h=-3, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f(t)=(-11)t²+(-1)t+(22), allowing a2=0. At x=-1 and nonzero finite h=-3, 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(-11, -1)

at original x a*x = 11

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

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, 22)

at original x f(x) = 12

f(t)=(-11)*t^2+(-1)*t+(22);x=-1;f(x)=12;h!=0

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

twice quadratic coefficient := mul(2, -11)

twice quadratic coefficient = -22

twice a times x := mul(-22, -1)

twice a times x = 22

linear part of quotient := sum(22, -1)

linear part of quotient = 21

shifted input x+h := sum(-1, -3)

shifted input x+h = -4

at shifted x a*x := mul(-11, -4)

at shifted x a*x = 44

at shifted x a*x+b := sum(44, -1)

at shifted x a*x+b = 43

at shifted x (a*x+b)*x := mul(43, -4)

at shifted x (a*x+b)*x = -172

at shifted x f(x) := sum(-172, 22)

at shifted x f(x) = -150

finite increment f(x+h)-f(x) := sub(-150, 12)

finite increment f(x+h)-f(x) = -162

specified finite quotient := div(-162, -3)

specified finite quotient = 54

finite-step quadratic correction := mul(-11, -3)

finite-step quadratic correction = 33

expanded finite quotient := sum(21, 33)

expanded finite quotient = 54

finite quotient minus derivative := sub(54, 21)

finite quotient minus derivative = 33

h=-3;finite quotient=54/1;derivative value=21

difference=33;a2*h=33

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

54/1

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.