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C4 parametric curves_lesson
 

C4 parametric curves_lesson

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A2-level maths UK

A2-level maths UK

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    C4 parametric curves_lesson C4 parametric curves_lesson Presentation Transcript

    • A2 Mathematics: C4 Core Maths
      Curves and Tangents
      Parametric Curves
    • Objectives
      We will be able to
      Plot Graphs defined by parametric equations
      by hand and
      by calculator
      Use 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.
    • What is a Parametric Graph ?
      To plot a graph
      we could follow a point
      as it crawls
      along the curve
      especially
      If the point obeys a rule
      If it gives x and y
      In terms of time
      Or other parameter
    • Tracing out a Parametric Graph
      http://www.flashandmath.com/mathlets/calc/param2d/param_advanced.html
    • Tracing out a Parametric Graph
      This also shown in your WEC text-book
      On page 323
    • Parametric Curve examples
    • Parametric Curve examples
    • Parametric Curve examples
    • Parametric Equations for a Curve
      x=2t, y=15t– 5t²
    • Plotting x and y via parameters
    • Curves defined by parametric equations
    • Parametric Equations for a Curve
      x = 3cosθ, y = 3sinθ
    • Plotting x and y via parameters
    • Curves defined by parametric equations
    • Plotting Parametric Curves
      Use Sharp EL9900 calculator
      Parametric settings on next slide
      Use Autograph
      Equation entry via x=2t, y=t^2
      Separated by a comma
    • Parametric Settings for EL9900
    • Parametric Entry for EL9900
    • Parametric Displays on the EL9900
    • Cartesian Equation for a Curve
      We have
      x as a function of t or θ
      And
      yas a function of t or θ
      We need to eliminate t or θ
      Leaving only x and y.
      Methods
      Eliminate t by substitution and algebra
      Eliminate θ via trigonometric Identities and algebra
    • Cartesian Equation – Eliminate t
      x = t2, y = t – t2
    • Cartesian Equation – identities in θ
      x = 3cosθ, y = sinθ
    • Activity step 1
      Use table 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.
    • Parametric Equations for a Curve
      x = 3cosθ, y = 3sinθ
    • Cartesian Equation for a Curve
      x2 + y2 = 9
    • Curves defined by parametric equations
    • Parametric Equations for a Curve
      x=2t, y=15t– 5t²
    • Cartesian Equation for a Curve
      4y = 15x– 4.9x2
    • Curves defined by parametric equations
    • Parametric Equations for a Curve
      x=t²–4, y=t³–4t
    • Cartesian Equation for a Curve
      y = x√(x+4)
      y = x(x+4)0.5
    • Curves defined by parametric equations
    • Parametric Equations for a Curve
      x=sinθ, y=sin2θ
    • Cartesian Equation for a Curve
      y = 2x√(1-x2)
      y = 2x(1-x2)0.5
    • Curves defined by parametric equations
    • Parametric Equations for a Curve
      x=t2, y=t3
    • Cartesian Equation for a Curve
      y=x√x
    • Curves defined by parametric equations
    • Parametric Equations for a Curve
      x=t, y=1/t
    • Cartesian Equation for a Curve
      y = 1/x
    • Curves defined by parametric equations
    • Parametric Equations for a Curve
      x = 1+ t, y = 2 - t
    • Cartesian Equation for a Curve
      x + y = 3
    • Curves defined by parametric equations
    • Parametric Equations for a Curve
      x=(2+3t)/(1+t), y=(3–2t)/(1+t)
    • Parametric Equations for a Curve
      y=13–5x
    • Curves defined by parametric equations
      Stops here !
    • Extension: Try these Parameters
      x= t + 1/t, y= t - 1/t
      x = 3cosθ, y= sinθ
      Investigate/Create your own
    • Parametric Equations for a Curve
      x=………...., y=…….……..
    • Curves defined by parametric equations
    • Tangents to the curve?
      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
    • Parametric Equations for a Curve
      x = 3cosθ, y = 3sinθ
    • Gradient of Tangents to the Curve
      We know (why?) that
    • Gradient of Tangents to the Curve
      x = 3cosθ, y = 3sinθ
      ...so.....
    • Gradient of Tangents to the Curve
      Putting it together.......
    • How do we find a particular tangent?
      Given a particular t value
       find x and y, and dy/dx
      Now we have
      • the gradient of the tangent and
      • the co-ordinates where it touches the curve
      .......so.....
    • Equation of one Tangent to Circle
    • Equation of one Tangent to Circle
      x = 3cosπ/4, y = 3sin π/4
      ...so.....
    • Image of one Tangent to the Curve
    • Activity step 2
      Use x and y parameter functions, to match
      dy/dx equation
      one tangent equation
      with previous cards
    • Parametric Equations for a Curve
      x=2t, y=15t– 5t²
    • Gradient of Tangents to the Curve
    • Image of one Tangent to the Curve
    • Equation of one Tangent to the Curve
    • Parametric Equations for a Curve
      x=t²–4, y=t³–4t
    • Gradient of Tangents to the Curve
    • Image of one Tangent to the Curve
    • Equation of One Tangent to the Curve
    • Parametric Equations for a Curve
      x=sinθ, y=sin2θ
    • Gradient of Tangents to the Curve
    • Image of Tangent to Curve
    • Equation of One Tangent to the Curve
    • Parametric Equations for a Curve
      x=t2, y=t3
    • Gradient of Tangents to the Curve
    • Image of Tangent to Curve
    • Equation of one Tangent to the Curve
    • Parametric Equations for a Curve
      x=t, y=1/t
    • Gradient of Tangents to the Curve
    • Image of Tangent to Curve
    • Equation of one Tangent to the Curve
    • Parametric Equations for a Curve
      x = 1+ t, y = 2 - t
    • Gradient of Tangents to the Curve
    • Image of Tangent to Curve
    • Equation of one Tangent to the Curve
    • Parametric Equations for a Curve
      x=(2+3t)/(1+t), y=(3–2t)/(1+t)
    • Gradient of Tangents to the Curve
    • Image of Tangent to Curve
    • Equation of one Tangent to the Curve
    • Parametric Equations for a Curve
      x=………...., y=…….……..
    • Curves defined by parametric equations
    • Gradient of Tangents to the Curve
    • Equation of one Tangent to the Curve
    • Image of Tangent to Curve