Question 8547: Finite Difference Quotient at a Specified Nonzero Step

Original question and givens

On all R, f(t)=(-21)t²+(9)t+(42), allowing a2=0. At x=-1 and nonzero finite h=-1, compute (f(x+h)−f(x))/h. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f(t)=(-21)t²+(9)t+(42), allowing a2=0. At x=-1 and nonzero finite h=-1, 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(-21, -1)

at original x a*x = 21

at original x a*x+b := sum(21, 9)

at original x a*x+b = 30

at original x (a*x+b)*x := mul(30, -1)

at original x (a*x+b)*x = -30

at original x f(x) := sum(-30, 42)

at original x f(x) = 12

f(t)=(-21)*t^2+(9)*t+(42);x=-1;f(x)=12;h!=0

f(x+h)-f(x)=h*(2*(-21)*(-1)+(9)+(-21)*h)

twice quadratic coefficient := mul(2, -21)

twice quadratic coefficient = -42

twice a times x := mul(-42, -1)

twice a times x = 42

linear part of quotient := sum(42, 9)

linear part of quotient = 51

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

shifted input x+h = -2

at shifted x a*x := mul(-21, -2)

at shifted x a*x = 42

at shifted x a*x+b := sum(42, 9)

at shifted x a*x+b = 51

at shifted x (a*x+b)*x := mul(51, -2)

at shifted x (a*x+b)*x = -102

at shifted x f(x) := sum(-102, 42)

at shifted x f(x) = -60

finite increment f(x+h)-f(x) := sub(-60, 12)

finite increment f(x+h)-f(x) = -72

specified finite quotient := div(-72, -1)

specified finite quotient = 72

finite-step quadratic correction := mul(-21, -1)

finite-step quadratic correction = 21

expanded finite quotient := sum(51, 21)

expanded finite quotient = 72

finite quotient minus derivative := sub(72, 51)

finite quotient minus derivative = 21

h=-1;finite quotient=72/1;derivative value=51

difference=21;a2*h=21

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

72/1

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.