dr/dθ Is Not a Slope

You'll learnto convert between polar and rectangular coordinates, to recognize that one point has many polar names, to see why the rate at which a radius changes is not the steepness of a curve, to turn a polar curve into a parametric one and recover the slope from 9.1's quotient, and to read the polar area formula off the shape of a single swept sector.

The rate dr/dθ is a real quantity: how fast the radius grows as the angle turns. It is not the slope, because a slope is dy/dx and neither of those coordinates is r. The repair is not a new formula to memorise. Substitute the two conversions, treat θ as a parameter, and polar differentiation becomes Topic 9.1 with a different letter. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · A RADIUS RATE IS NOT A STEEPNESS · v2

convert first · then every rule from Unit 2 still applies

Before you start

What you're looking at

One plane at equal scale, with the origin at its center. A polar curve is drawn by turning through an angle and walking out along that ray.

The question

Two rates live on a polar curve. One says how fast the distance from the origin changes, the other how steep the curve is. Are they the same number?

Step 1 / 12