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Parametric Equations

Parametric Equations. Cartesian Equations. Equations defined in terms of x and y . These may or may not be functions. Some examples include: x 2 + y 2 = 4 y = x 2 + 3x + 2. Parametric Equations. Equations where x and y are functions of a third variable, such as t . That is,

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Parametric Equations

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  1. Parametric Equations

  2. Cartesian Equations Equations defined in terms of x and y. These may or may not be functions. Some examples include: x2 + y2 = 4 y = x2 + 3x + 2

  3. Parametric Equations Equations where x and y are functions of a third variable, such as t. That is, x = f(t) and y = g(t). The graph of parametric equations are called parametric curves and are defined by (x, y) = (f(t), g(t)).

  4. Example The path of a particle in two-dimensional space can be modeled by the parametric equations: x = 2 + cos t and y = 3 + sin t. Sketch a graph of the path of the particle for 0  t  2.

  5. Plot of x = 2 + cos t and y = 3 + sin t How is t represented on this graph?

  6. Plot of x = 2 + cos t and y = 3 + sin t   t =   t = 0 

  7. Parametric Equations and Technology Graphing calculators and other mathematical software can plot parametric equations much more efficiently then we can. Put your graphing calculator and plot the following equations. In what direction is t increasing? (a) x = t2, y = t3 (b) (c) x = sec θ, y = tan θ; -/2 < θ < /2

  8. Converting Equations Parametric equations can easily be converted to Cartesian equations by solving one of the equations for t and substituting the result into the other equation.

  9. You Try It You try it for x = sec θ, y = tan θ where -/2 < θ < /2 Hint: sec2θ – tan2θ = 1

  10. Cycloids A cycloid is the graph of the path of a fixed point P on a circle of radius r that rolls along a straight line. x = r( – sin ) y = r(1 – cos )

  11. only to be crushed by a rolling wheel. An ant is walking along... Question: What is the path traced out by its bloody splat? Why would we ask such a question? Mathematicians are sick !!!

  12. Problem Posed Again(in a less gruesome manner) A wheel with a radius of r feet is marked at its base with a piece of tape. Then we allow the wheel to roll across a flat surface. a) What is the path traced out by the tape as the wheel rolls? b) Can the location of the tape be determined at any particular time?

  13. Questions: • What is your prediction for the shape of the curve? • Is the curve bounded? • Does the curve repeat a pattern?

  14. Picture of the Problem

  15. Finding an Equation • f(x) = y may not be good enough to express the curve. • Instead, try to express the location of a point, (x,y), in terms of a third parameter to get a pair of parametric equations. • Use the properties of the wheel to our advantage. The wheel is a circle, and points on a circle can be measured using angles. WARNING: Trigonometry ahead!

  16. Diagram of the Problem 2r We would like to find the lengths of OX and PX, since these are the horizontal and vertical distances of P from the origin. r C q P Q rq O X T

  17. The Parametric Equations r C |OX| = |OT| - |XT| r = |OT| - |PQ| q r cosq x(q) = rq - r sinq P Q r sinq |PX| = |CT| - |CQ| y(q) = r - r cosq rq T X O rq

  18. Graph of the Function If the radius r=1, then the parametric equations become: x(q)=q-sinq, y(q)=1-cosq

  19. Real-World Example: Gears

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