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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)

Defining and differentiating parametric equations

AP Calc: CHA (BI), CHA‑3 (EU), CHA‑3.G (LO), CHA‑3.G.1 (EK)
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Parametric equations intro
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Parametric equations differentiation
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Parametric equations differentiationGet 3 of 4 questions to level up!

Second derivatives of parametric equations

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!

Finding arc lengths of curves given by parametric equations

AP Calc: CHA (BI), CHA‑6 (EU), CHA‑6.B (LO), CHA‑6.B.1 (EK)
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Parametric curve arc length
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Worked example: Parametric arc length
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Parametric curve arc lengthGet 3 of 4 questions to level up!

Defining and differentiating vector-valued functions

AP Calc: CHA (BI), CHA‑3 (EU), CHA‑3.H (LO), CHA‑3.H.1 (EK)
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Vector-valued functions intro
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Vector-valued functions differentiation
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Second 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!

Solving motion problems using parametric and vector-valued functions

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 vector
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Motion along a curve: finding rate of change
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Motion along a curve: finding velocity magnitude
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Planar 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!

Defining polar coordinates and differentiating in polar form

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 derivatives
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Worked 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!

Finding the area of a polar region or the area bounded by a single polar curve

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 curves
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Worked 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!

Finding the area of the region bounded by two polar curves

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!

Calculator-active practice

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.
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