The Constant Is a Vector

You'll learnto antidifferentiate a vector-valued function one component at a time, to carry a constant of integration with one entry per slot rather than a single shared scalar, to read an initial condition as two equations instead of one, to check an answer by substituting the starting parameter back into both slots, and to tell a displacement from a position and from a distance travelled.

Antidifferentiating a vector is componentwise, exactly like differentiating one, so almost nothing here is new. The single idea that costs points is what comes after. The constant of integration is a vector, with one entry per component and one equation to determine each, and a single scalar C answers half the problem. The picture is unhelpful on purpose: the constant slides the curve without changing its shape, so a sketch of a wrong answer looks exactly as plausible as a sketch of the right one.

8 STEPS · 6 QUICK CHECKS · ONE CONSTANT PER SLOT, AND THE CHECK THAT CATCHES IT · v1

antidifferentiate each slot · then put the starting parameter back in

Before you start

What you're looking at

One plane, drawn at equal scale on both axes. A particle has velocity v(t) = ⟨2t, 3t²⟩ for t from 0 to 2, and the four candidate answers between them cover y from −4 to 8 — twelve units, which is what sets the scale of this window.

The question

Antidifferentiating gives ⟨t², t³⟩ plus a constant. The whole topic is what kind of object that constant is, and the picture is built so that the shape of a curve can never answer it.

Watch for

Every curve here is drawn by integrating the given velocity, not by plotting a formula. The cards print the formulas, so if the two ever disagreed you would see it. The velocity arrow in step 7 is drawn shortened and the caption says the scale; every other arrow is at true length.

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