An Arrow Has Two Numbers. A Speed Has One.

You'll learnto differentiate a vector-valued function one component at a time with every ordinary rule intact, to tell velocity from speed by the type of the answer, to compute a magnitude rather than adding components or measuring the wrong arrow, to read acceleration as the derivative of velocity, and to see constant speed and nonzero acceleration coexist.

A vector-valued function is two parametric equations wearing one bracket, and its derivative is taken one component at a time with every rule from Units 2 and 3 unchanged. Almost nothing here is new machinery. The one genuinely new idea is small and costs a great many points: velocity is a vector and speed is a number, and a question asking for one will not accept the other.

8 STEPS · 6 QUICK CHECKS · VELOCITY IS AN ARROW, SPEED IS ITS LENGTH · v1

differentiate each slot · then decide what type the answer is

Before you start

What you're looking at

One plane, drawn at equal scale on both axes. The path is r(t) = ⟨3t², 2t³⟩ for t from 0 to 1 — the same curve as Topic 9.3, so the geometry is already familiar. Everything happens at t = 1, where the particle is at (3, 2).

The question

Three quantities come out of one function: where the particle is, how it is moving, and how fast. Two of those are arrows and one is a number, and telling them apart is most of this topic.

Watch for

The velocity and acceleration arrows are drawn shortened so they fit — their directions are exact and the caption says the scale each time. The one arrow drawn at full length is the acceleration on the circle in step 6, and where its tip lands is the point of that step.

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