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# Eight Curve[1][1]

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

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### Eight Curve[1][1]

1. 1. Eight Curve Annie Tan Alice Hu Junwei Kwee Johnny Chang
2. 2. Lemniscates <ul><li>The lemniscate is official term describing the eight curve. </li></ul><ul><li>The equation of a lemniscate is </li></ul><ul><li>x 4 =a 2 (x 2 -y 2 ) </li></ul>
3. 3. The Problem: P.161 #7 <ul><li>The graph of the eight curve, (x 2 +y 2 ) 2 =2a 2 (x 2 -y 2 ), is shown. </li></ul><ul><li>A. explain how you could use a graphing utility to graph this curve </li></ul><ul><li>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 </li></ul><ul><li>C. Determine the points on the curve where the tangent line is horizontal. </li></ul>-a a
4. 4. Solving (a) <ul><li>A. explain how you could use a graphing utility to graph this curve </li></ul><ul><li>x 4 =a 2 (x 2 -y 2 ) </li></ul><ul><li>X 4 =a 2 x 2 -a 2 y 2 </li></ul><ul><li>A 2 y 2 =a 2 x 2 -x 4 </li></ul><ul><li>Y 2 =(a 2 x 2 -x 4 )/a 2 </li></ul><ul><li>Y= +/- ((a 2 x 2 -x 4 )/a 2 ) 1/2 = +/- ((a 2 x 2 -x 4 ) 1/2 )/a </li></ul><ul><li>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. </li></ul>
5. 5. Solving (b) <ul><li>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 </li></ul>
6. 6. Solving (b) cont. <ul><li>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. </li></ul>
7. 7. Solving (c) <ul><li>C. Determine the points on the curve where the tangent line is horizontal. </li></ul><ul><li>x 4 =a 2 (x 2 -y 2 ) </li></ul><ul><li>4x 3 =a 2 (2x-2yy’) </li></ul><ul><li>4x 3 =2a 2 x-2a 2 yy’ </li></ul><ul><li>2x 3 =a 2 x-a 2 yy’ </li></ul><ul><li>Y’=(a 2 x-2x 3 )/a 2 y </li></ul><ul><li>0=(a 2 x-2x 3 )/a 2 y </li></ul><ul><li>a 2 x-2x 3 =0 </li></ul><ul><li>a 2 =2x 2 </li></ul><ul><li>X=+/- a/(2) 1/2 </li></ul>
8. 8. Solving (c) cont. <ul><li>X 4 =a 2 (x 2 -y 2 ) </li></ul><ul><li>a 4 /4=a 2 (a 2 /2)-a 2 y 2 </li></ul><ul><li>a 2 y 2 =a 4 /2 – a 4 /4 </li></ul><ul><li>a 2 y 2 = a 4 /4 </li></ul><ul><li>y 2 = a 2 /4 </li></ul><ul><li>y= +/- a/2 </li></ul><ul><li>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) </li></ul>
9. 9. Real World Application <ul><li>Racetracks </li></ul><ul><li>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. </li></ul><ul><ul><li>Y 2 =(a 2 x 2 -x 4 )/a 2 </li></ul></ul><ul><ul><li>2yy’=(2a 2 x-4x 3 ) </li></ul></ul>
10. 10. Real World Application <ul><li>Ice Skating </li></ul><ul><li>http://www.youtube.com/watch?v=Y3jsq-u8pWM </li></ul><ul><li>A figure-eight in ice skating resembles the shape of an eight curve. </li></ul>