Slopes on Polar Curves

Calculus 2 · Axiom Academy

A polar curve is given as r( ), not y of x — yet it still has a tangent line at every point. Watch one swing around, then steer it yourself. Every point on the curve still has a tangent line A polar curve like r = 2 + cos(2 ) is drawn by a distance that depends on the angle — there is no y as a function of x to differentiate. But the curve is still a smooth path in the plane, so at every point it has a slope. The trick is that the slope depends on two things changing at once: how fast r grows, and how the whole point is being swung around the origin. Watch the point ride once around the curve. At each angle the tangent line is drawn straight from the slope formula, and the readout shows the running value of dy/dx. Notice where the tangent goes flat (a horizontal tangent) and where it stands straight up (a vertical tangent) — those are the special points the formula will pin down exactly. The tangent keeps turning as the point goes around — flat at six angles, vertical at two. That changing direction is what the slope formula computes. Drive the point and read the slope Drag the angle (or grab the point on the curve). The radial line shows where the point sits; the tangent line shows which way the curve is heading there. The slope comes straight from the formula below, built from r and r = dr/d at that angle. The tangent line really is tangent — both it and the point are computed from the same r( ), so the line kisses the curve exactly at the dot.

This is the written version of the interactive lesson above. See the full Calculus 2 course.