Family Overview

In Plain Language

Topic 3.15 describes how polar radius changes with angle.

• Distance trends: rules for when the point moves farther from or closer to the origin (3.15.A.1-A.2).
• Extrema: turning points in monotonicity give closest/farthest distances (3.15.A.3).
• Average rate: (Delta r)/(Delta theta) estimates how fast radius changes per radian (3.15.A.4-A.5).

Why It Matters

This connects polar graphs to motion and shape analysis—where a path moves outward, inward, or reaches extreme distance.

How Families Can Help

• Ask: "Is the point getting farther from or closer to the origin?"
• Compute a simple (Delta r)/(Delta theta) from two table entries.
• Mark closest/farthest points on a polar sketch together.

What Not to Do

• Do not confuse increasing theta with increasing distance.
• Do not ignore negative r when judging distance.
• Do not skip justification on extremum claims.