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Ways to Describe a Curve in the Plane An equation in two variables Example: x + y − 2 x − 6 y + 8 = 0 2 2This equation describes a circle.
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A Polar Equation r =θ This polar equationdescribes a double spiral. We’ll study polar curves later.
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Parametric Equations Example: x = t − 2t 2 y = t +1 The “parameter’’ is t.It does not appear in the graph of thecurve!
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Why?The x coordinates of points on the curve are given by a function. x = t − 2t 2 The y coordinates of points on the curve are given by a function. y = t +1
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Two Functions, One Curve? Yes. If x = t − 2t and y = t + 1 2then in the xy-plane the curve looks like this, for values of t from 0 to 10...
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Why use parametric equations?• Use them to describe curves in the plane when one function won’t do.• Use them to describe paths.
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Paths?A path is a curve, together with a journey traced along the curve.
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Huh? When we write x = t − 2t 2 y = t +1we might think of x as the x-coordinate of the position on the path at time t and y as the y-coordinate of the position on the path at time t.
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From that point of view... The path described by x = t − 2t 2 y = t +1is a particular route along the curve.
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As t increasesfrom 0, x firstdecreases,then increases. Path moves right! Path moves left!
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More PathsTo designate one route around the unit circle use x = cos(t ) y = sin(t )
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Where do you get that? Think of t as an angle.If it starts at zero, and increases to 2π , then the path starts at t=0, where x = cos(0) = 1, and y = sin(0) = 0.
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To start at (0,1)... Use x = sin(t ) y = cos(t )
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How Do You Find The Path• Plot points for various values of t, being careful to notice what range of values t should assume• Eliminate the parameter and find one equation relating x and y• Use the TI82/83 in parametric mode
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Plotting Points• Note the direction the path takes• Use calculus to find – maximum points – minimum points – points where the path changes direction• Example: Consider the curve given by x = t + 1, y = 2t , − 5 ≤ t ≤ 5 2
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Consider x = t 2 + 1, y = 2t , − 5 ≤ t ≤ 5• The parameter t ranges from -5 to 5 so the first point on the path is (26, -10) and the last point on the path is (26, 10)• x decreases on the t interval (-5,0) and increases on the t interval (0,5). (How can we tell that?)• y is increasing on the entire t interval (-5,5). (How can we tell that?)
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Note Further x = t + 1, y = 2t , − 5 ≤ t ≤ 5 2• x has a minimum when t=0 so the point farthest to the left on the path is (1,0).• x is maximal at the endpoints of the interval [-5,5], so the points on the path farthest to the right are the starting and ending points, (26, -10) and (26,10).• The lowest point on the path is (26,-10) and the highest point is (26,10).
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Eliminate the ParameterStill use x = t + 1, y = 2t , − 5 ≤ t ≤ 5 2Solve one of the equations for t Here we get t=y/2Substitute into the other equation Here we get x = ( y / 2) + 1 or x = ( y / 4) + 1 2 2
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Summary• Use parametric equations for a curve not given by a function.• Use parametric equations to describe paths.• Each coordinate requires one function.• The parameter may be time, angle, or something else altogether...
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