BMCC MTH 253 - Conic Sections and Polar Coordinates

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MTH 253 Calculus (Other Topics)Graphs of Polar EquationsGraphing Polar EquationsGraphing Polar Equations Method 1: Plotting and Connecting PointsSlide 5Slide 6Slide 7Slope of Polar CurvesSlide 9Graphing Polar Equations Recognizing Common FormsSlide 11Slide 12Slide 13Slide 14Polar Graphs w/ TechnologyIntersections of Polar CurvesSlide 17MTH 253Calculus (Other Topics)Chapter 10 – Conic Sections and Polar CoordinatesSection 10.6 – Graphing in Polar CoordinatesCopyright © 2009 by Ron Wallace, all rights reserved.Graphs of Polar Equationspolar axis(r, )rpoleThe angle  may be expressed in degrees or radians.If r = f(), then the graph of this equation consists of ALL of the points whose coordinates make this equation true.Graphing Polar EquationsReminder: How do you graph rectangular equations? Method 1:Create a table of values. Plot ordered pairs. Connect the dots in order as x increases.Method 2:Recognize and graph various common forms.Examples: linear equations, quadratic equations, conics, …The same basic approach can be applied to polar equations.Graphing Polar EquationsMethod 1: Plotting and Connecting Points1. Create a table of values.2. Plot ordered pairs.3. Connect the dots in order as  increases.NOTE: Since most of these equations involve periodic functions (esp. sine and cosine), at some point the graph will start repeating itself (but not always).Graphing Polar Equationswrt x-axis•Replacing  with –  doesn’t change the function•Replacing r with –r and  with  –  doesn’t change the functionSymmetry Tests(r,)(r,-)=(-r,–)Graphing Polar Equationswrt y-axis•Replacing r with –r and  with – doesn’t change the function•Replacing  with  –  doesn’t change the functionSymmetry Tests(r,)(-r, -)=(r, -)Graphing Polar Equationswrt the origin•Replacing r with –r doesn’t change the function.•Replacing  with    doesn’t change the function.Symmetry Tests(r,)(-r,)(r,   )Slope of Polar CurvesTo find the slope of … … remember …… therefore …dymdx=( )r f q=cossinx ry rqq==( ) cos( )sinffq qq q==dy dy dmdx dx dqq= ='( )sin ( ) cos'( )cos ( )sinf ff fq q q qq q q q+=-Slope of Polar CurvesExample: Find the equation of the tangent line to the following curve when  = /4'( )sin ( ) cos'( )cos ( )sinf fmf fq q q qq q q q+=-1 sinr q= +4pGraphing Polar EquationsRecognizing Common FormsCirclesCentered at the origin: r = aradius: a period = 360Tangent to the x-axis at the origin: r = a sin center: (a/2, 90) radius: a/2 period = 180a > 0  above a < 0  belowTangent to the y-axis at the origin: r = a cos center: (a/2, 90) radius: a/2 period = 180a > 0  right a < 0  leftr = 4r = 4 sinr = 4 cosNote the SymmetriesGraphing Polar EquationsRecognizing Common FormsFlowers (centered at the origin)r = a cos n or r = a sin nradius: |a| n is even  2n petalspetal every 180/nperiod = 360n is odd  n petalspetal every 360/nperiod = 180cos  1st petal @ 0sin  1st petal @ 90/nr = 4 sin 2r = 4 cos 3Note the SymmetriesGraphing Polar EquationsRecognizing Common FormsSpiralsSpiral of Archimedes: r = k|k| large  loose |k| small  tightr =  r = ¼ Graphing Polar EquationsRecognizing Common FormsHeart (actually: cardioid if a = b … otherwise: limaçon)r = a ± b cos  or r = a ± b sin r = 3 + 3 cos r = 2 - 5 cos  r = 3 + 2 sin  r = 3 - 3 sin Note the SymmetriesGraphing Polar EquationsRecognizing Common FormsLeminscate2cos 2r a q=2sin 2r a q=a = 16Note the SymmetriesPolar Graphs w/ TechnologyTI-84WinPlotIntersections of Polar CurvesAs with Cartesian equations, solve by the substitution method.Warning: 2 polar curves may intersect, but at different values of .i.e. Setting the two equations equal to each other may not reveal ALL of the points of intersection.Solution: Always graph the equations.Intersections of Polar CurvesExample: Find the points of intersection of …cos & 1 cosr rq q= = -Note that 2 of the points are found by substitution, the third by the


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BMCC MTH 253 - Conic Sections and Polar Coordinates

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