dr/dθ Is Not a Slope

You'll learnto convert between polar and rectangular coordinates, to recognise that one point has infinitely 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. What is genuinely new arrives at the end, where the area of a thin slice turns out to be a sector rather than a rectangle.

8 STEPS · 6 QUICK CHECKS · A RADIUS RATE IS NOT A STEEPNESS · v1

convert first · then every rule from Unit 2 still applies

Before you start

What you're looking at

One plane, drawn at equal scale on both axes, with the origin at its centre. The running curve is r = 2cos θ, which is a circle of radius 1 centred at (1, 0) — not obvious from the equation and immediate once a few points are plotted.

The question

Two rates live on a polar curve. One says how fast the distance from the origin is changing; the other says how steep the curve is. They are different numbers, and the whole topic is not mistaking the first for the second.

Watch for

Every tangent line here is built from the two component rates and never from a slope, so the one drawn at a vertical tangent is drawn the same way as all the others. The amber arrow in step 5 is the radial rate, drawn at true length, and it points inward because that rate is negative.

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