# Chapter5 - DANANG UNIVERSITY OF TECHNOLOGY Chapter 5...

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DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II PARAMETRIC EQUATIONS AND  POLAR COORDINATES Chapter 5

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DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II I. CURVES DEFINED BY PARAMETRIC EQUATIONS Suppose C is a curve in (x,y)-plane Parametric curve
DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II It is impossible to describe C by an equation of the form y = f ( x ) because C fails the Vertical Line Test. But we can write x, y coordinates in terms of t , x=f(t) and y=g(t). Suppose that x and y are both given as functions of a third variable t (called a parameter ) by the equations

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DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II x=f(t) y=g(t) (called parametric equations ). Each value of t determines a point (x,y), which we can plot in a coordinate plane. As t varies, the point (x,y)=(f(t),g(t)) varies and traces out a curve C , which we call a parametric curve . The parameter t does not necessarily represent time and, in fact, we could use a letter other than t for the parameter.
DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Example 1 Sketch and identify the curve defined by the parametric equations 2 2 1 x t t y t = - = + Solution Each value of t gives a point on the curve,

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DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II
DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II It appears from the figure that the curve traced out by a parabola. This can be confirmed by eliminating the parameter t as follows. We obtain t = y-1 from the second equation and substitute into the first equation. This gives 2 2 2 2 ( 1) 2( 1) 4 3 x t t y y y y = - = - - - = - +

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DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II No restriction was placed on the parameter t in Example 1, so we assumed that t could be any real number. But sometimes we restrict t to lie in a finite interval. For instance, the parametric curve Note 2 2 1 0 4 x t t y t t = - = +
DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II In general, the curve with parametric equations ( ) ( ) x f t y g t a t b = =

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DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II has initial point ( f ( a ), g ( a )) and terminal point ( f ( b ), g ( b )). Example 2 Solution What curve is represented by the parametric equations cos sin 0 2 x t y t t π = = Observe that 2 2 2 2 cos sin 1 x y t t + = + =
DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Thus, the point (x,y) moves on the unit circle 2 2 1 x y + = Notice that in this example the parameter t can be interpreted as the angle (in radians)

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DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Example 3 Solution What curve is represented by the parametric equations sin 2 cos 2 0 2 ? x t y t t π = = Again 2 2 2 2 sin 2 cos 2 1 x y t t + = + =
DANANG UNIVERSITY OF TECHNOLOGY CALCULUS WITH ANALYTIC GEOMETRY II so the parametric equations again represent the unit circle 2 2 1 x y + = But as t increases from 0 to 2 h , the point (x,y) =

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## This note was uploaded on 04/29/2009 for the course EE 307 taught by Professor Dinh during the Spring '09 term at Washington State University .

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Chapter5 - DANANG UNIVERSITY OF TECHNOLOGY Chapter 5...

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