AP Calculus BC

Advanced Placement Calculus BC including series, parametric equations, and polar functions.

Limits and ContinuityDifferentiation and ApplicationsIntegration and Its Uses
Infinite Series and ConvergenceParametric Equations and MotionPolar Coordinates and Graphs
Analyzing Population GrowthDesigning Roller Coasters with Parametric EquationsSignal Processing and Polar Graphs
Mastering Multiple-Choice QuestionsEffective Free-Response Techniques
Limits and Continuit...Differentiation and ...Integration and Its ...Infinite Series and ...Parametric Equations...Polar Coordinates an...Analyzing Population...Designing Roller Coa...Signal Processing an...Mastering Multiple-C...Effective Free-Respo...
Advanced Topics

Parametric Equations and Motion

The Power of Parameters

Parametric equations use a third variable (often \( t \)) to describe how \( x \) and \( y \) change over time or another parameter.

  • Instead of \( y = f(x) \), write \( x = f(t) \), \( y = g(t) \).
  • Useful for describing motion, curves, or anything where both \( x \) and \( y \) depend on something else.

Calculus with Parametric Curves

  • Find derivatives: \( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \).
  • Calculate arc length and area under parametric curves.

Real-World Motion

Planets orbiting the sun, roller coasters, and animation paths all use parametric equations.

Examples

  • Projectile paths: \( x = v_0 t \cos \theta \), \( y = v_0 t \sin \theta - \frac{1}{2}gt^2 \).

  • Drawing a circle: \( x = r \cos t, y = r \sin t \).

In a Nutshell

Parametric equations let us track motion and complex curves using a parameter like time.

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Polar Coordinates and Graphs
Parametric Equations and Motion - AP Calculus BC Content | Practice Hub