must also increase. Therefore, we must be moving up the curve from bottom to top as tincreases as that is the only direction that will always give an increasing yas tincreases. Note that the xderivative isn’t as useful for this analysis as it will be both positive and negative and hence xwill be both increasing and decreasing depending on the value of t. That doesn’t help with direction much as following the curve in either direction will exhibit both increasing and decreasing xIn some cases, only one of the equations, such as this example, will give the direction while in other cases either one could be used. It is also possible that, in some cases, both derivatives would be needed to determine direction. It will always be dependent on the individual set of parametric equations. The second problem with eliminating the parameter is best illustrated in an example as we’ll be running into this problem in the remaining examples.

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Calculus II © 2007 Paul Dawkins 10 Example 4 Sketch the parametric curve for the following set of parametric equations. Clearly indicate direction of motion. 5cos2sin02xtyttπ==≤≤SolutionBefore we proceed with eliminating the parameter for this problem let’s first address again why just picking t’s and plotting points is not really a good idea. Given the range of t’s in the problem statement let’s use the following set of t’s. t x 0 5 2π0 π-5 32π0 -2 2π5 The question that we need to ask now is do we have enough points to accurately sketch the graph of this set of parametric equations? Below are some sketches of some possible graphs of the parametric equation based only on these five points. y 0 2 0 0

Calculus II © 2007 Paul Dawkins 11 Given the nature of sine/cosine you might be able to eliminate the diamond and the square but there is no denying that they are graphs that go through the given points. The last graph is also a little silly but it does show a graph going through the given points. Again, given the nature of sine/cosine you can probably guess that the correct graph is the ellipse. However, that is all that would be at this point. A guess. Nothing actually says unequivocally that the parametric curve is an ellipse just from those five points. That is the danger of sketching parametric curves based on a handful of points. Unless we know what the graph will be ahead of time we are really just making a guess. So, in general, we should avoid plotting points to sketch parametric curves. The best method, provided it can be done, is to eliminate the parameter. As noted just prior to starting this example there is still a potential problem with eliminating the parameter that we’ll need to deal with. We will eventually discuss this issue. For now, let’s just proceed with eliminating the parameter. We’ll start by eliminating the parameter as we did in the previous section. We’ll solve one of the of the equations for tand plug this into the other equation. For example, we could do the following, 11