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MTH 253 – CalculusConics and Quadratic EquationsSlide 3Slide 4Slide 5Slide 6Slide 7Polar CoordinatesGraphs of Polar EquationsPolar: AreasPolar: Length of CurveChapter 10ReviewMTH 253 – CalculusConics and Quadratic EquationsConicsParabolaEllipseCircleHyperbola2 20Ax Bxy Cy Dx Ey F+ + + + + =24 0B AC- =No B term!xy24 0B AC- <24 0B AC- >Conics and Quadratic EquationsConicsRotate (eliminate Bxy)Translate (eliminate Dx and Ey) … complete the squaresParabolaEllipseCircleHyperbola2 20Ax Bxy Cy Dx Ey F+ + + + + =112tanA CBq--� �=� �� �24x py=24y px=2 22 21x ya b+ =2 2 2 (i.e. ellipse with )x y r a b+ = =2 22 21x ya b- =2 22 21y xa b- =2 2 2c a b= -2 2 2c a b= +2 22 21x yb a+ = & x h y k- -Conics and Quadratic EquationsParabola( ) ( )24x h p y k- = -F: (h,k+p)P: (h+2p,k+p)V: (h,k)dir: y = k–pEccentricity = 1 PF e PD= gConics and Quadratic EquationsEllipse( ) ( )2 22 21x h y ka b- -+ =(h,k+b)V: (h+a,k)C: (h,k)dir: x = h+a2/c = h+a/e2 2 2 Eccentricity = cc a ba= -F: (h+c,k) PF e PD= gHyperbolaConics and Quadratic Equations( ) ( )2 22 21x h y ka b- -- =(h,k+b)V: (h+a,k)C: (h,k)dir: x = h+a2/c = h+a/e2 2 2 Eccentricity = cc a ba= +F: (h+c,k)asy: y = (b/a)(x-h)+k PF e PD= gConics and Quadratic EquationsPolar Forms11 cos coseek kre q q= =+ +EccentricityParabola: e = 1Ellipse: 0 < e < 1Hyperbola: e > 1F: (0,0)V: (ek/(1+e),0)dir: x = kF: (0,0)Other Orientations:1 sinekre q=-Directrix below.1 sinekre q=+Directrix above.1 cosekre q=-Directrix left.Polar Coordinatespolar axis(r, )rpole( ), 2r kq p�( ), (2 1)r kq p- � +2 2 2r x y= +C  PP  Ctanyxq =cosx r q=siny r q=Conversions between Polar and CartesianGraphs of Polar EquationsGraph common polar curvescircles, limaçons, flowers/roses, lemniscateinequalitiesslopes (formula will be given)Intersectionssolve as system of equationscheck graph'( )sin ( )cos'( ) cos ( )sinf fmf fq q q qq q q q+=-Polar: Areas( )r f q=[ ]212( )A f dbaq q=�1( )r f q=2( )r g q=[ ] [ ] [ ] [ ]( )2 2 2 21 1 12 2 2( ) ( ) ( ) ( )gA f d g d f g da a ab b bq q q q q q q= - = -� � �Polar: Length of Curve( )r f q=22drdL r dbqaq= +�[ ]22( )drdf dbqaq q=


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BMCC MTH 253 - Review

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