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March 25, 2015
(6.5) Graphs of Polar Equations
Objective: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.
Why: Graphs that have circular symmetry often have simple polar equations, that can be very helpful in calculus.
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.y = cos x
y = sin x
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.
r = cos θ
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.r = sin θ
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.
r = 3cos θ
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.Algebraic Relationship
r = cos θ
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.r = sin θ
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.
Symmetry:
1. line =
2. polar axis
3. the pole
(r,θ)
(r,θ)
(r,θ)
θπ
2
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.
Use polar symmetry tests to determine if the graph is symmetric about the x-axis, the y-axis, or the origin.
r = 4 sin 3θ
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.
Find the maximum r-value of the following graphs.
1. r = 3 + 2cos θ 2. r = 4 - sin θ
Maximum r-value: value occurring at a point the maximum distance from the pole.
*To find: find values of |r| as opposed to the directed distance r.
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.
Analyze the graph of the polar curve.
r = -3 cos 4θ
Domain:___________________Range:____________________Continuity:_________________Boundedness: ______________Symmetry: ________________Maximum r-value: ___________Asymptotes:_______________
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.
Rose Curves: r = a cos nθr = a sin nθ
Limacon Curves: r = a + b cos θr = a + b sin θ
--
Lemniscates: r2 = a2sin 2θ
r2 = a2cos 2θ
Spiral of Archimedes: r = θ
where n is an integer > 1
n is even: 2n petalsn is odd: n petals
r-value max = a + b
where a > 0, b > 0
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves, limacon curves, and lemniscates.
HW:
(HR): (6.5) Pg.500: 1, 2, 9-13, 17, 25, 27, 29