Parametric Equations and Curves

Parametric Equations and Curves - Parametric Equations and...

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Parametric Equations and Curves To this point (in both Calculus I and Calculus II) we’ve looked almost exclusively at functions in the form or and almost all of the formulas that we’ve developed require that functions be in one of these two forms. The problem is that not all curves or equations that we’d like to look at fall easily into this form. Take, for example, a circle. It is easy enough to write down the equation of a circle centered at the origin with radius r . However, we will never be able to write the equation of a circle down as a single equation in either of the forms above. Sure we can solve for x or y as the following two formulas show but there are in fact two functions in each of these. Each formula gives a portion of the circle. Unfortunately we usually are working on the whole circle, or simply can’t say that we’re going to be working only on one portion of it. Even if we can narrow things down to only one of these portions the function is still often fairly unpleasant to work with.
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There are also a great many curves out there that we can’t even write down as a single equation in terms of only x and y . So, to deal with some of these problems we introduce parametric equations . Instead of defining y in terms of x ( ) or x in terms of y ( ) we define both x and y in terms of a third variable called a parameter as follows, This third variable is usually denoted by t (as we did here) but doesn’t have to be of course. Sometimes we will restrict the values of t that we’ll use and at other times we won’t. This will often be dependent on the problem and just what we are attempting to do. Each value of t defines a point that we can plot. The collection of points that we get by letting t be all possible values is the graph of the parametric equations and is called the parametric curve . Sketching a parametric curve is not always an easy thing to do. Let’s take a look at an example to see one way of sketching a parametric curve. This example will also illustrate why this method is usually not the best. Example 1 Sketch the parametric curve for the following set of parametric equations. Solution At this point our only option for sketching a parametric curve is to pick values of t , plug them into the parametric equations and then plot the points. So, let’s plug in some t ’s. t x y -2 2 -5 -1 0 -3 -2 0 0 -1 1 2 1 The first question that should be asked as this point is, how did we know to use the values
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of t that we did, especially the third choice? Unfortunately there is no real answer to this question. We simply pick t ’s until we are fairly confident that we’ve got a good idea of what the curve looks like. It is this problem with picking “good” values of t that make this method of sketching parametric curves one of the poorer choices. Sometimes we have no choice, but if we do have a choice we should avoid it. We’ll discuss an alternate graphing method in later examples. We have one more idea to discuss before we actually sketch the curve. Parametric curves have
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This note was uploaded on 11/10/2011 for the course MATH 136 taught by Professor Prellis during the Fall '08 term at Rutgers.

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Parametric Equations and Curves - Parametric Equations and...

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