Unit 4

Contextual Applications

10–15% AB & BCCED Unit 4

Motion & Rates of Change

Derivatives as rates: position, velocity, acceleration, and units in context.

The Derivative as Rate of Change

The derivative

represents the instantaneous rate of change of

with respect to

. The units of

are always:

Derivative Units

(units of

) / (units of

)

Straight-Line Motion (Rectilinear Motion)

For a particle moving along a line with position function

Quantity	Definition	Meaning
Position	Given	Location at time
Velocity		Rate of change of position
Acceleration		Rate of change of velocity
Speed	Absolute value	How fast (always ≥ 0)

Speeding Up vs. Slowing Down

Speeding up:

and

have the same sign (both positive or both negative)

Slowing down:

and

have opposite signs

⚠ Watch Out

Velocity vs. Speed: Velocity can be negative (indicates direction). Speed is always positive (magnitude only). "Speeding up" means speed is increasing, NOT "velocity is increasing."

Units in Context Problems

has units of meters and

has units of seconds, then

has units of meters per second (m/s).

⚠ Watch Out

Forgetting Units: Rates have units! Always include units in your final answer. Common units: ft/sec, m/s, cm²/min, gal/hr, etc.

AP® Calculus AB & BC · Unit 4 Overview · Mr. Brantley