Rates of Change in Polar Functions
On a polar curve, r is the moving point's distance from the pole, so r increasing means the point is receding from the origin and r decreasing means it is closing in, while the angle sweeps it counterclockwise either way. Average rates of change work exactly as in Unit 1 with θ as the input, in units of distance per radian, and when r dips negative the true distance is |r|, so a still-decreasing r can mean the point is actually receding on the far side of the pole.
The mistakes are translation mistakes: reading decreasing r as "the graph goes down" or "turns clockwise," treating an average rate over an interval as the rate at an endpoint, and forgetting the |r| fold once the curve crosses the pole. All three are drilled in the lesson.
The work
3 ways in · any order
Lesson
Rates of Change in Polar Functions
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Away or toward, never up or down. The lesson translates r's increase and decrease into motion relative to the pole, computes average rates of change per radian, and handles the |r| fold when curves cross the origin. Ten scenarios close it out: cardioids, spirals, tables, and interval comparisons.
Diagnostic
10-item topic check
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Ten items on the Topic 3.15 misreads: r's change taken as vertical or rotational motion, endpoint rates confused with interval averages, and the negative-r distance fold. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.