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Calculus BC · AP · Parametric Equations, Polar Coordinates, and Vector-Valued Functions

CED focus

Second Derivatives of Parametric Equations

This topic is assessed on the AP Exam.

Multiple-choice tips

  • Identify whether the stem tests a definition, rule, or application.
  • Identify parameter derivatives before eliminating MC choices.
  • On polar area, check for the ½ factor and correct θ bounds.
  • Eliminate vector speed answers that use only one component.
  • Watch dx/dt = 0 when finding vertical tangents.

Free-response tips

  • Label f′(a) meaning in context; show the rule that justifies each step.
  • Show dy/dx = (dy/dt)/(dx/dt) with parameter labeled.
  • Set up polar area/arc length with correct bounds and factors.
  • Separate speed (scalar) from velocity (vector) in motion FRQs.
  • Sketch the path to verify orientation and starting point.

Common traps

  • Mixing average rate over an interval with instantaneous rate at a point
  • Dropping the chain rule when differentiating parametrically
  • Missing ½ in polar area formula
  • Using velocity components interchangeably with speed

Quick checklist

  • [ ] Can compute dy/dx for parametric equations
  • [ ] Can set up a polar area integral
  • [ ] Can distinguish speed from velocity

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