A Slope Built From Two Rates

You'll learnto build the slope of a parametric curve as a quotient with the right rate on top, to read a flat tangent off the numerator and an upright one off the denominator, and to see why the parameter is the thing you have to keep.

A particle moves in the plane and one number drives both of its coordinates. There is still a tangent line at nearly every point, but the single derivative that would normally give its slope is not available. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE SLOPE IS A QUOTIENT OF TWO RATES · v2

the slope is the climb divided by the run, both measured against t

Before you start

What you're looking at

One plane at equal scale. A particle traces a looped curve as t runs from −2 to 2; a ring marks the start and a filled dot is where it is now.

The question

Neither coordinate is a function of the other here, so where does the slope of this curve come from?

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