Average rate of change between distinct inputs

Mathematical statement

For two permitted inputs x0≠x1 of the same specified function f, with f(x0)=y0 and f(x1)=y1, the average rate of change is (y1−y0)/(x1−x0). Equal inputs fail this entry's eligibility condition; do not replace their quotient with zero or 0/0. For f(x)=mx+b on R it equals m. For q(x)=a(x−h)²+k with a≠0 it equals a(x0+x1−2h). This finite endpoint quotient is not a derivative.

Representative formula (exact TeX)

\frac{b-a}{d-c}

Assumptions

• Both endpoints must be permitted inputs of the same specified function. Direct two-point exercises use only the explicitly provided output values.
• Except in denominator-eligibility questions, the input difference must be nonzero. Integer endpoints and coefficients yield reduced fractions with positive denominators.
• The quadratic slice requires a≠0. In the representative fraction, a,b are outputs y0,y1 and c,d are corresponding inputs x0,x1, not vertex-form coefficients.

Symbol scope

• a — first output y0
• b — second output y1
• c — first input x0
• d — second input x1

Intuition, definition, and misconceptions

Subtract outputs and inputs in matching endpoint order. Reversing both differences leaves the quotient unchanged; reversing only one changes its sign.

This quotient describes a change across two distinct inputs, not a limiting process or an instantaneous derivative. Equal inputs fail the nonzero-denominator teaching condition.

A constant affine function has zero average rate. For a quadratic, the constant k cancels and the result is a(x0+x1−2h). Distinct inputs symmetric about h give zero average rate without making the quadratic function constant.

Common errors include reversing only one difference, dividing by a zero input difference, or treating an endpoint average as a derivative.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Ratio and proportion — Fix the meaning of a quotient with a nonzero denominator.
• Graphs of real-valued functions — The endpoints belong to the same specified function.
• Affine functions and degree-one functions — Use the average-rate slice for an affine expression.
• Quadratic functions in vertex form — First fix the vertex form for the quadratic slice.

Verification boundary

Finite endpoint quotients can be checked exactly; no derivative or limit proof is claimed.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind the pinned slope API with an explicit nonzero input difference; Mathlib total division at zero does not discharge this teaching condition.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.