I'm self-learning multivariable calculus, I have two questions I don't really understand about sketching the parametric curve.
1) Are there only two ways to sketch the parametric curve? One is using the table, the other is eliminating the parameter right? Do we have more methods to do so?
2) For doing the table, I have confronted some situations where I have no idea what values I should use for parameter t, are there any tricks or methods to determine the value of t?
Thanks
Typically those are the two ways you need to know. As for hints, just try to make $t$ simplify the problem as much as possible. Sometimes that implies that $x(t) = t$ and then derive $y(t)$.
There are some examples where this may not be the best approach. For instance, take the function $x^2 + y^2 = 1$. For a parameterization centered around $x=t$, this would be super messy. However, making $x = \cos t, y = \sin t$ simplifies the problem. (Note that this is similar to just using polar coordinates). If you know the problem/curve gets simplified in polar, cylindrical, or spherical coordinates, just use those.
Most times that you use weird parameterizations is when you use a jacobian change of variables, but like previously, its used to simplify the problem.