If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Main content

Unit: Parametric equations, polar coordinates, and vector-valued functions

0

Possible mastery points

Parametric equations intro

Learn

Practice

Parametric equations differentiationGet 3 of 4 questions to level up!

Second derivatives of parametric equations

Learn

Second derivatives (parametric functions)
(Opens a modal)

Practice

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

Arc length: parametric curves

Learn

Practice

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

Quiz 1

Level up on the above skills and collect up to 240 Mastery points

Vector-valued functions

Learn

Practice

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

Quiz 2

Level up on the above skills and collect up to 160 Mastery points

Planar motion

Learn

Practice

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!

Quiz 3

Level up on the above skills and collect up to 240 Mastery points

Polar functions

Learn

Practice

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

Area: polar regions (single curve)

Learn

Practice

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!

Area: polar regions (two curves)

Learn

Worked example: Area between two polar graphs
(Opens a modal)

Practice

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

Arc length: polar curves

Learn

Practice

Arc length of polar curvesGet 3 of 4 questions to level up!

Calculator-active practice

Learn

Evaluating definite integral with calculator
(Opens a modal)

Practice

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

Quiz 4

Level up on the above skills and collect up to 560 Mastery points

Up next for you:

Unit test

Level up on all the skills in this unit and collect up to 1500 Mastery points!

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.