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Similar to C4 parametric curves_lesson
C4 parametric curves_lesson
- 1. A2 Mathematics: C4Core Maths Curves and TangentsParametric Curves
- 2. ObjectivesWe will beable to Plot Graphs defined by parametric equationsby hand and by calculatorUse algebra to eliminate the parameter and find the Cartesian equation of the curve.Find the gradient of the curve for any value of the parameter.Find the equation of the tangent or normal to the curve at any value of the parameter.
- 3. What is aParametric Graph ? To plot a graphwe could follow a point as it crawls along the curveespeciallyIf the point obeys a ruleIf it gives x and yIn terms of time Or other parameter
- 4. Tracing out aParametric Graph http://www.flashandmath.com/mathlets/calc/param2d/param_advanced.html
- 5. Tracing out aParametric Graph This also shown in your WEC text-bookOn page 323
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- 9. Parametric Equationsfor a Curvex=2t, y=15t– 5t²
- 10. Plotting x andy via parameters
- 11. Curves definedby parametric equations
- 12. Parametric Equationsfor a Curvex = 3cosθ, y = 3sinθ
- 13. Plotting x andy via parameters
- 14. Curves definedby parametric equations
- 15. Plotting Parametric CurvesUseSharp EL9900 calculatorParametric settings on next slideUse AutographEquation entry via x=2t, y=t^2Separated by a comma
- 16. Parametric Settingsfor EL9900
- 17. Parametric Entryfor EL9900
- 18. Parametric Displayson the EL9900
- 19. Cartesian Equation fora CurveWe have x as a function of t or θ Andyas a function of t or θWe need to eliminate t or θLeaving only x and y.MethodsEliminate t by substitution and algebraEliminate θ via trigonometric Identities and algebra
- 20. Cartesian Equation –Eliminate tx = t2, y = t – t2
- 21. Cartesian Equation –identities in θx = 3cosθ, y = sinθ
- 22. Activity step 1Usetable of values to plot each curve (and/or use your calculator).Match each parameter formula and its curve with correct curve card.Match each curves with its correct Cartesian equation.
- 23. Parametric Equationsfor a Curvex = 3cosθ, y = 3sinθ
- 24. Cartesian Equationfor a Curvex2 + y2 = 9
- 25. Curves definedby parametric equations
- 26. Parametric Equationsfor a Curvex=2t, y=15t– 5t²
- 27. Cartesian Equationfor a Curve4y = 15x– 4.9x2
- 28. Curves definedby parametric equations
- 29. Parametric Equationsfor a Curve x=t²–4, y=t³–4t
- 30. Cartesian Equationfor a Curvey = x√(x+4) y = x(x+4)0.5
- 31. Curves definedby parametric equations
- 32. Parametric Equationsfor a Curvex=sinθ, y=sin2θ
- 33. Cartesian Equationfor a Curvey = 2x√(1-x2)y = 2x(1-x2)0.5
- 34. Curves definedby parametric equations
- 35. Parametric Equationsfor a Curvex=t2, y=t3
- 36. Cartesian Equationfor a Curvey=x√x
- 37. Curves definedby parametric equations
- 38. Parametric Equationsfor a Curvex=t, y=1/t
- 39. Cartesian Equationfor a Curve y = 1/x
- 40. Curves definedby parametric equations
- 41. Parametric Equationsfor a Curvex = 1+ t, y = 2 - t
- 42. Cartesian Equationfor a Curvex + y = 3
- 43. Curves definedby parametric equations
- 44. Parametric Equationsfor a Curvex=(2+3t)/(1+t), y=(3–2t)/(1+t)
- 45. Parametric Equationsfor a Curve y=13–5x
- 46. Curves definedby parametric equationsStops here !
- 47. Extension: Try theseParametersx= t + 1/t, y= t - 1/tx = 3cosθ, y= sinθInvestigate/Create your own
- 48. Parametric Equationsfor a Curvex=………...., y=…….……..
- 49. Curves definedby parametric equations
- 50. Tangents to thecurve?How do we find dy/dx ?How do we find the equation of the tangent at one particular point on the curve – for example when t=1
- 51. Parametric Equationsfor a Curvex = 3cosθ, y = 3sinθ
- 52. Gradient ofTangents to the CurveWe know (why?) that
- 53. Gradient ofTangents to the Curvex = 3cosθ, y = 3sinθ...so.....
- 54. Gradient ofTangents to the CurvePutting it together.......
- 55. How do wefind a particular tangent?Given a particular t value find x and y, and dy/dxNow we have the gradient of the tangent and
- 56. the co-ordinates whereit touches the curve.......so.....
- 57. Equation of one Tangent to Circle
- 58. Equation of one Tangent to Circle x = 3cosπ/4, y = 3sin π/4 ...so.....
- 59. Image of oneTangent to the Curve
- 60. Activity step 2Usex and y parameter functions, to match dy/dx equation one tangent equation with previous cards
- 61. Parametric Equationsfor a Curvex=2t, y=15t– 5t²
- 62. Gradient ofTangents to the Curve
- 63. Image ofone Tangent to the Curve
- 64. Equation ofone Tangent to the Curve
- 65. Parametric Equationsfor a Curve x=t²–4, y=t³–4t
- 66. Gradient of Tangentsto the Curve
- 67. Image of one Tangent to the Curve
- 68. Equation of OneTangent to the Curve
- 69. Parametric Equationsfor a Curvex=sinθ, y=sin2θ
- 70. Gradient of Tangentsto the Curve
- 71. Image of Tangentto Curve
- 72. Equation of OneTangent to the Curve
- 73. Parametric Equationsfor a Curvex=t2, y=t3
- 74. Gradient ofTangents to the Curve
- 75. Image ofTangent to Curve
- 76. Equation ofone Tangent to the Curve
- 77. Parametric Equationsfor a Curvex=t, y=1/t
- 78. Gradient ofTangents to the Curve
- 79. Image ofTangent to Curve
- 80. Equation ofone Tangent to the Curve
- 81. Parametric Equationsfor a Curvex = 1+ t, y = 2 - t
- 82. Gradient of Tangentsto the Curve
- 83. Image ofTangent to Curve
- 84. Equation ofone Tangent to the Curve
- 85. Parametric Equationsfor a Curvex=(2+3t)/(1+t), y=(3–2t)/(1+t)
- 86. Gradient of Tangentsto the Curve
- 87. Image of Tangentto Curve
- 88. Equation ofone Tangent to the Curve
- 89. Parametric Equationsfor a Curvex=………...., y=…….……..
- 90. Curves definedby parametric equations
- 91. Gradient of Tangentsto the Curve
- 92. Equation ofone Tangent to the Curve
- 93. Image of Tangentto Curve