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# Parametric equations, polar coordinates, and vector-valued functions

AP Calc: CHA (BI), CHA‑3 (EU), CHA‑5 (EU), CHA‑6 (EU), FUN (BI), FUN‑3 (EU), FUN‑8 (EU)

AP Calc: CHA (BI), CHA‑3 (EU), CHA‑3.G (LO), CHA‑3.G.1 (EK)

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Parametric equations introParametric equations differentiation

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Parametric equations differentiationGet 3 of 4 questions to level up!

AP Calc: CHA (BI), CHA‑3 (EU), CHA‑3.G (LO), CHA‑3.G.3 (EK)

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Second derivatives (parametric functions)

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Second derivatives (parametric functions)Get 3 of 4 questions to level up!

AP Calc: CHA (BI), CHA‑6 (EU), CHA‑6.B (LO), CHA‑6.B.1 (EK)

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Parametric curve arc lengthWorked example: Parametric arc length

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Parametric curve arc lengthGet 3 of 4 questions to level up!

AP Calc: CHA (BI), CHA‑3 (EU), CHA‑3.H (LO), CHA‑3.H.1 (EK)

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Vector-valued functions introVector-valued functions differentiationSecond derivatives (vector-valued functions)

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Vector-valued functions differentiationGet 3 of 4 questions to level up!

Second derivatives (vector-valued functions)Get 3 of 4 questions to level up!

AP Calc: FUN (BI), FUN‑8 (EU), FUN‑8.B (LO), FUN‑8.B.1 (EK), FUN‑8.B.2 (EK)

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Planar motion example: acceleration vectorMotion along a curve: finding rate of changeMotion along a curve: finding velocity magnitudePlanar motion (with integrals)

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Planar motion (differential calc)Get 3 of 4 questions to level up!

Motion along a curve (differential calc)Get 3 of 4 questions to level up!

Planar motion (with integrals)Get 3 of 4 questions to level up!

AP Calc: FUN (BI), FUN‑3 (EU), FUN‑3.G (LO), FUN‑3.G.1 (EK), FUN‑3.G.2 (EK)

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Polar functions derivativesWorked example: differentiating polar functions

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Differentiate polar functionsGet 3 of 4 questions to level up!

Tangents to polar curvesGet 3 of 4 questions to level up!

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.D (LO), CHA‑5.D.1 (EK), CHA‑5.D.2 (EK)

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Area bounded by polar curvesWorked example: Area enclosed by cardioid

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Area bounded by polar curves introGet 3 of 4 questions to level up!

Area bounded by polar curvesGet 3 of 4 questions to level up!

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.D (LO), CHA‑5.D.1 (EK), CHA‑5.D.2 (EK)

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Worked example: Area between two polar graphs

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Area between two polar curvesGet 3 of 4 questions to level up!

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.D (LO), CHA‑5.D.1 (EK), CHA‑5.D.2 (EK)

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Evaluating definite integral with calculator

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Area with polar functions (calculator-active)Get 3 of 4 questions to level up!

### About this unit

We are used to working with functions whose output is a single variable, and whose graph is defined with Cartesian, i.e., (x,y) coordinates. But there can be other functions! For example, vector-valued functions can have two variables or more as outputs! Polar functions are graphed using polar coordinates, i.e., they take an angle as an input and output a radius! Learn about these functions and how we apply the concepts of the derivative and the integral on them.AP® is a registered trademark of the College Board, which has not reviewed this resource.