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Unformatted text preview: MATH 214: PARAMETRIC EQUATIONS AND POLAR COORDINATES 1. Parametric Equations Definition. A curve given by parametric equations is the set of points given by x ( t ) ,y ( t ) , t I for some functions x ( t ) ,y ( t ) : R R and some interval I R Two of the simplest examples of parametric curves are the following: Example 1. Given points ( a,b ) , ( c,d ) in the xy-plane, the straight line connecting them is given by x ( t ) = (1- t ) a + tc , y ( t ) = (1- t ) b + td , t [0 , 1] . Example 2. If a graph is given by the curve y = f ( x ) , then we can express the graph easily in parametric form as x ( t ) = t , y ( t ) = f ( t ) , t dom( f ) (i.e. we restrict t to the domain of f ). 1.1. Graphing Parametric Equations. For our purposes, we will graph parametric equations in one of two ways. Either we will plot points by plugging in different values of t into our equations x ( t ) and y ( t ), or we will eliminate the variable t to attempt to get functions or relations involving x and y that we recognize. Example 3. If x ( t ) = t and y ( t ) = 1- t , then squaring both sides of the first equation and plugging it into the second gives that y = 1- x 2 . Notice that the function x ( t ) = t implicitly assumes that both t and x are non-negative, so the parametric equations only describe the right half of the parabola. Example 4. If x ( t ) = sinh( t ) and y ( t ) = cosh( t ) , then the relation y 2- x 2 = 1 holds for all t R , so the parametric curve describes part or all of the unit hyperbola. Notice that y ( t ) = cosh( t ) for any value of t , so the parametric equations describe only the top half of the hyperbola. 1.2. Calculus with Parametric Equations. 1.2.1. Arc Length. We operate under the assumption that for the curves we will be dealing with, if we make changes in t small enough (i.e. infinitesimal), then the resulting sections of the parametric curve behave like straight lines. This leads to the following informal definition: Definition. The arc length differential is given by the formula ds...
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This note was uploaded on 07/31/2011 for the course MATH 214 taught by Professor Alexondrus during the Spring '11 term at University of Alberta.

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