Question 8546: Finite Difference Quotient at a Specified Nonzero Step

Original question and givens

On all R, f(t)=(20)t²+(8)t+(40), allowing a2=0. At x=-2 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)=(20)t²+(8)t+(40), allowing a2=0. At x=-2 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(20, -2)

at original x a*x = -40

at original x a*x+b := sum(-40, 8)

at original x a*x+b = -32

at original x (a*x+b)*x := mul(-32, -2)

at original x (a*x+b)*x = 64

at original x f(x) := sum(64, 40)

at original x f(x) = 104

f(t)=(20)*t^2+(8)*t+(40);x=-2;f(x)=104;h!=0

f(x+h)-f(x)=h*(2*(20)*(-2)+(8)+(20)*h)

twice quadratic coefficient := mul(2, 20)

twice quadratic coefficient = 40

twice a times x := mul(40, -2)

twice a times x = -80

linear part of quotient := sum(-80, 8)

linear part of quotient = -72

shifted input x+h := sum(-2, 3)

shifted input x+h = 1

at shifted x a*x := mul(20, 1)

at shifted x a*x = 20

at shifted x a*x+b := sum(20, 8)

at shifted x a*x+b = 28

at shifted x (a*x+b)*x := mul(28, 1)

at shifted x (a*x+b)*x = 28

at shifted x f(x) := sum(28, 40)

at shifted x f(x) = 68

finite increment f(x+h)-f(x) := sub(68, 104)

finite increment f(x+h)-f(x) = -36

specified finite quotient := div(-36, 3)

specified finite quotient = -12

finite-step quadratic correction := mul(20, 3)

finite-step quadratic correction = 60

expanded finite quotient := sum(-72, 60)

expanded finite quotient = -12

finite quotient minus derivative := sub(-12, -72)

finite quotient minus derivative = 60

h=3;finite quotient=-12/1;derivative value=-72

difference=60;a2*h=60

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

-12/1

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.