Vector-Valued Functions
Unit 4 is not assessed on the AP exam; schools include it based on local requirements, and everything here still builds directly toward calculus and physics. A vector-valued function is a position that moves with the parameter: p(t) pairs two coordinate functions into one location. Displacement over an interval is the componentwise subtraction of positions, a vector with direction and magnitude, and distance traveled can only match or exceed that magnitude.
The mistakes are lane violations: reading a negative y-change as leftward motion, comparing speeds by staring at single components, collapsing a position or displacement into a bare number, and subtracting positions in the wrong order. The lesson keeps components in their lanes and drills the displacement-versus-distance distinction.
The work
3 ways in · any order
Lesson
Vector-Valued Functions
›
Position as a vector, motion as arithmetic. The lesson evaluates moving positions, subtracts them into displacement vectors, separates displacement from distance traveled, and reads direction and average speed from change vectors. Ten scenarios close it out.
Diagnostic
10-item topic check
›
Ten items spanning the two Topic 4.9 misconceptions: components confused with each other or with speed, and position arithmetic done in the wrong order or wrong kind. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
›
Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.