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AP Calculus AB Presentation: Lemniscates (Eight Curve).

AP Calculus AB Presentation: Lemniscates (Eight Curve).
Credits: Annie Tan, Johnny Chang, Alice Hu, Junwei Kwee

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  • 1. Eight Curve Annie Tan Alice Hu Junwei Kwee Johnny Chang
  • 2. Lemniscates
    • The lemniscate is official term describing the eight curve.
    • The equation of a lemniscate is
    • x 4 =a 2 (x 2 -y 2 )
  • 3. The Problem: P.161 #7
    • The graph of the eight curve, (x 2 +y 2 ) 2 =2a 2 (x 2 -y 2 ), is shown.
    • A. explain how you could use a graphing utility to graph this curve
    • B. Use the graphing utility to graph the curve for various values of the constant a. Describe how a affects the shape of the curve
    • C. Determine the points on the curve where the tangent line is horizontal.
    -a a
  • 4. Solving (a)
    • A. explain how you could use a graphing utility to graph this curve
    • x 4 =a 2 (x 2 -y 2 )
    • X 4 =a 2 x 2 -a 2 y 2
    • A 2 y 2 =a 2 x 2 -x 4
    • Y 2 =(a 2 x 2 -x 4 )/a 2
    • Y= +/- ((a 2 x 2 -x 4 )/a 2 ) 1/2 = +/- ((a 2 x 2 -x 4 ) 1/2 )/a
    • Plug + ((a 2 x 2 -x 4 )/a 2 ) 1/2 and - ((a 2 x 2 -x 4 )/a 2 ) 1/2 into the calculator. Substitute values for a because a is a constant.
  • 5. Solving (b)
    • B. Use the graphing utility to graph the curve for various values of the constant a. Describe how a affects the shape of the curve
  • 6. Solving (b) cont.
    • As we can see from the previous graphs of the curve, as a increases, the curve will get taller and thinner and the y-values of relative extrema will increase.
  • 7. Solving (c)
    • C. Determine the points on the curve where the tangent line is horizontal.
    • x 4 =a 2 (x 2 -y 2 )
    • 4x 3 =a 2 (2x-2yy’)
    • 4x 3 =2a 2 x-2a 2 yy’
    • 2x 3 =a 2 x-a 2 yy’
    • Y’=(a 2 x-2x 3 )/a 2 y
    • 0=(a 2 x-2x 3 )/a 2 y
    • a 2 x-2x 3 =0
    • a 2 =2x 2
    • X=+/- a/(2) 1/2
  • 8. Solving (c) cont.
    • X 4 =a 2 (x 2 -y 2 )
    • a 4 /4=a 2 (a 2 /2)-a 2 y 2
    • a 2 y 2 =a 4 /2 – a 4 /4
    • a 2 y 2 = a 4 /4
    • y 2 = a 2 /4
    • y= +/- a/2
    • Points: (a/(2) 1/2 , a/2), (a/(2) 1/2 , -a/2), (-a/(2) 1/2 , a/2), (-a/(2) 1/2 , -a/2)
  • 9. Real World Application
    • Racetracks
    • We can also use this curve for a car racing track. For the racers, the relative extrema are important points because that is where the motion of the cars change the most.
      • Y 2 =(a 2 x 2 -x 4 )/a 2
      • 2yy’=(2a 2 x-4x 3 )
  • 10. Real World Application
    • Ice Skating
    • http://www.youtube.com/watch?v=Y3jsq-u8pWM
    • A figure-eight in ice skating resembles the shape of an eight curve.
  • 11. Facebook Integration
    • Alice: http://www.facebook.com/#!/aliice?ref=ts
    • Annie: http://www.facebook.com/#!/annietanniebannie?ref=ts
    • Johnny: http://www.facebook.com/#!/AzNkiSEkiStaR
    • Junwei: http://www.facebook.com/#!/profile.php?id=100001146442174&ref=ts
  • 12. Importance of Facebook
    • It’s scientifically proven that people cannot live without social interaction. Websites like Facebook and Myspace help people interact with people
    • Companies view Facebook profiles during application and interview processes. Basically, what one displays on their Facebook page is what the interviewers and managers see.