AP Calculus BC Unit 2 Notes: Understanding the Derivative from First Principles

Rate of change

A measure of how one quantity changes in response to another, often expressed as a ratio of output change to input change (Δf/Δx).

Change in input (Δx)

The difference in input values: Δx = x₂ − x₁.

Change in output (Δf)

The difference in function values: Δf = f(x₂) − f(x₁).

Difference quotient

The ratio (f(b) − f(a)) / (b − a) (or similar forms) that computes average rate of change; foundational for defining derivatives.

Average rate of change

The function’s overall change over an interval [a,b]: (f(b) − f(a)) / (b − a).

Secant line

A line through two points on a curve, (a,f(a)) and (b,f(b)); its slope equals the average rate of change on [a,b].

Instantaneous rate of change

How fast f(x) changes at a specific input x=a; the limit of secant slopes as the interval shrinks to that point.

Tangent line

A line that touches a curve at a point and matches its local direction there; its slope equals the derivative at that point (when it exists).

Limit (as used in derivatives)

A process of evaluating what a quantity approaches as a variable (like h) approaches a value (like 0), avoiding direct division by zero.

Derivative at a point

The instantaneous rate of change/slope of the tangent line at x=a, defined by f'(a) = lim_{h→0} (f(a+h) − f(a)) / h (if the limit exists).

Differentiable at a point

A function is differentiable at x=a if the derivative limit exists as a finite real number at that point.

Two-point derivative definition

An equivalent derivative definition: f'(a) = lim_{x→a} (f(x) − f(a)) / (x − a).

Derivative function

The function formed by taking the derivative at every x where it exists: f'(x) = lim_{h→0} (f(x+h) − f(x)) / h.

Notation f'(x)

Common algebraic notation meaning “the derivative of f with respect to x,” evaluated at x.

Notation dy/dx

Derivative notation emphasizing variables/units; treated as a single symbol meaning “the derivative,” not an ordinary fraction in this unit.

Operator notation d/dx (f(x))

An operator form meaning “take the derivative of f(x) with respect to x.”

One-sided difference quotient (estimate)

An estimate of f'(a) using values from one side: right-hand (f(a+h)−f(a))/h or left-hand (f(a)−f(a−h))/h.

Symmetric difference quotient

A typically better table-based estimate of f'(a) using points on both sides: (f(a+h) − f(a−h)) / (2h).

Continuity at a point

f is continuous at x=a if (1) f(a) is defined, (2) lim{x→a} f(x) exists, and (3) lim{x→a} f(x) = f(a).

Differentiability implies continuity

Key fact: if f is differentiable at x=a, then f must be continuous at x=a (but not vice versa).

Corner

A point where a function is continuous but the left-hand and right-hand slopes are finite and unequal, so the derivative does not exist there (e.g., |x| at 0).

Cusp

A pointed tip where slopes become unbounded in opposite directions; the function may be continuous but the derivative fails to exist as a finite value.

Vertical tangent

A point where the slope becomes infinite/undefined as a finite real number; the function can be continuous but not differentiable there (in AP context).

Discontinuity

A break in the graph (hole, jump, or infinite behavior) that prevents continuity and therefore prevents differentiability at that point.

Conjugate method

An algebra technique (often for roots) that multiplies by the conjugate to simplify a difference quotient and resolve an indeterminate form like 0/0 (e.g., for f(x)=√x at x=4).