Question 8540: Finite Difference Quotient at a Specified Nonzero Step

Original question and givens

On all R, f(t)=(14)t²+(2)t+(28), 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)=(14)t²+(2)t+(28), 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(14, 2)

at original x a*x = 28

at original x a*x+b := sum(28, 2)

at original x a*x+b = 30

at original x (a*x+b)*x := mul(30, 2)

at original x (a*x+b)*x = 60

at original x f(x) := sum(60, 28)

at original x f(x) = 88

f(t)=(14)*t^2+(2)*t+(28);x=2;f(x)=88;h!=0

f(x+h)-f(x)=h*(2*(14)*(2)+(2)+(14)*h)

twice quadratic coefficient := mul(2, 14)

twice quadratic coefficient = 28

twice a times x := mul(28, 2)

twice a times x = 56

linear part of quotient := sum(56, 2)

linear part of quotient = 58

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

shifted input x+h = 5

at shifted x a*x := mul(14, 5)

at shifted x a*x = 70

at shifted x a*x+b := sum(70, 2)

at shifted x a*x+b = 72

at shifted x (a*x+b)*x := mul(72, 5)

at shifted x (a*x+b)*x = 360

at shifted x f(x) := sum(360, 28)

at shifted x f(x) = 388

finite increment f(x+h)-f(x) := sub(388, 88)

finite increment f(x+h)-f(x) = 300

specified finite quotient := div(300, 3)

specified finite quotient = 100

finite-step quadratic correction := mul(14, 3)

finite-step quadratic correction = 42

expanded finite quotient := sum(58, 42)

expanded finite quotient = 100

finite quotient minus derivative := sub(100, 58)

finite quotient minus derivative = 42

h=3;finite quotient=100/1;derivative value=58

difference=42;a2*h=42

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

100/1

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.