10.3: Polar functions* Learning Goals: Graph using polar form

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Presentation transcript:

10.3: Polar functions* Learning Goals: Graph using polar form Convert polar coordinates to rectangular and vice versa Calculus of polar forms *BC Calculus Only ©2008 Roy L Gover (www.mrgover.com)

Definition A Polar Coordinate System is a method of locating a point (r, ) in a plane where r is a distance from a point called the pole directed at an angle θ measured counterclockwise from a polar axis.

Example r=10 Pole Polar Axis Polar Grid (r,θ )=

Example Find the polar coordinates: (r,θ )=

Example Find the polar coordinates: (r,θ )=

Example Graph: r=6

Example Graph: θ=

Some Classic Polar Graphs Example Some Classic Polar Graphs 8 petal rose

Some Classic Polar Graphs Example Some Classic Polar Graphs limacon

Some Classic Polar Graphs Example Some Classic Polar Graphs Spiral of Archimedes

Try This Use your graphing calculator to graph CARDIOID MODE POL Y=

Important Idea 8 4 petals Polar Roses

Important Idea 10 8 petals Polar Roses

Try This Name the angle in rads? Polar Roses

Try This Name the angle in rads? Polar Roses

Important Idea What do you observe about the number of petals?

Try This Describe what you would do to find the angle.

Try This Find the angle.

Important Idea Polar coordinates are related to rectangular coordinates as follows: r y θ x

Try This Change the polar coordinates to rectangular coordinates

Try This Change the rectangular coordinates to polar coordinates

Example Change the polar relation to a rectangular (cartesian) relation.

Important Idea The slope of the tangent line (derivative-rate of change) to the graph of at the point is: Remember:

Example Find the slope of at and use it to find the equation of the tangent line. Demana text ex 5 p.552

Try This Use the nDerive feature of your calculator to find the slope of the graph of the polar curve at Hint: find -.667

Analysis S r in radians Area Formula

Analysis Find the area from 0 to Area1=

Analysis Find the area from 0 to Area2=

Analysis Find the area from 0 to Area3=

Analysis Find the area from 0 to Riemann Sum for total area

Analysis Find the area from 0 to

Example Find the area enclosed by Demana Study Guide ex 6. p173

Try This Find the area of the region bounded by 703/10 18.850 sq u

Set up the integral to find the shaded area. Try This Set up the integral to find the shaded area. Hint: solve for when r=0 [PI/2,5PI/6]

Find the exact area of one petal of Try This Hint: 703/4 sq. units

Lesson Close This lesson completes the curriculum for BC Calculus. The topics I have taught have included everything in the College Board syllabus for BC Calculus.