AP Calculus AB Unit 4: Understanding Rates of Change Through Derivatives

Derivative

A measure of how an output quantity changes as an input quantity changes; gives the instantaneous rate of change (slope of the tangent line) at a point.

Instantaneous rate of change

The “right now” rate at a specific input value; equal to the derivative at that input (slope of the tangent line).

Average rate of change

Change in function value over an interval divided by change in input: (f(b)−f(a))/(b−a); slope of the secant line.

Secant line

A line that intersects a graph at two points; its slope represents the average rate of change over an interval.

Tangent line

A line that just touches a curve at a point with matching local direction; its slope equals the derivative at that point.

Limit definition of the derivative

f'(a)=lim(h→0)[f(a+h)−f(a)]/h; the instantaneous rate of change as the interval shrinks to a point.

Difference quotient

The expression [f(a+h)−f(a)]/h (or [f(b)−f(a)]/(b−a)) used to compute an average rate of change; approaches the derivative as h→0.

Units of a derivative

Derivative units are “output units per input units” (e.g., meters per second, dollars per item), which help interpret meaning correctly.

Notation: dy/dx

A common notation meaning “the derivative of y with respect to x,” i.e., the rate at which y changes as x changes.

Function value vs. derivative value

f(a) is the amount/value of the quantity at a; f'(a) is the rate of change at a (not the amount).

Sign of the derivative

If f'(x)>0 the function is increasing; if f'(x)<0 the function is decreasing (locally).

Magnitude of the derivative

|f'(x)| indicates how fast the function is changing; larger magnitude means a steeper graph and faster change.

Horizontal tangent

A point where f'(a)=0; indicates a slope of zero but does not automatically guarantee a local maximum or minimum.

Marginal cost

C'(q), interpreted as the approximate additional cost per additional item when producing around q items (units: dollars per item).

Marginal revenue

R'(q), interpreted as the approximate additional revenue per additional item sold/produced around q items (units: dollars per item).

Sensitivity (in modeling)

Using a derivative to describe how sensitive an output is to small changes in an input (e.g., temperature change per time, pressure change per altitude).

Rate in/out interpretation

When a quantity like volume V(t) changes over time, V'(t) describes how fast it’s increasing/decreasing; negative means decreasing.

Position function (straight-line motion)

s(t), the location of an object along a line at time t (units: distance, such as meters).

Velocity

v(t)=s'(t); the signed rate of change of position with respect to time (units: distance per time).

Acceleration

a(t)=v'(t)=s''(t); the rate of change of velocity with respect to time (units: distance per time²).

Speed

The magnitude of velocity: |v(t)|; always nonnegative and ignores direction.

At rest

An object is at rest at time t when v(t)=0 (velocity is zero), regardless of its position value.

Direction change

Occurs when velocity changes sign (e.g., from positive to negative); v(t)=0 is necessary but not sufficient unless the sign changes.

Speeding up vs. slowing down

Speeding up when v(t) and a(t) have the same sign (|v| increasing); slowing down when v(t) and a(t) have opposite signs (|v| decreasing).

Symmetric difference quotient

An estimate of f'(a) from data: f'(a)≈[f(a+h)−f(a−h)]/(2h); often more accurate than one-sided estimates.