Parametric Equations and their Derivatives

Question

Hi!

I am a Calculus AB student and I am asked to do a BC Project. My topic is Parametric Equations and their Derivatives. I have to come up with an activity to teach the class the topic, so I thought about doing Jeopardy. However, I am having problems thinking of questions to ask to cover these 3 topics:

Eliminate the Parameter

Parametric Chain Rule

Parametric Derivatives

If possible, can you please give me a total of 15 samples questions so that I can use it for my Jeopardy? I really don't have too much time, so I could really use a hand here. Also, if you could work out each problem, that would be great.

Note: Not one person has to answer. It can be several different people. Please help me get a total of 15 questions. Thank you very much for your help.

Answer 1

This is a big ask that seems like you want others to do your project for you. Do you want to learn about parameterization so that you can design the questions? I'll get you started:

If you have a curve traced out in the x,y plane by the functions x=x(t), y=y(t), a common way of eliminating the parameter is to solve either x or y for t, then substitute that expression into the 2nd equation for t, getting an equation that relates x and y directly.

For example, if x=t-7, and y=2t+5, then t=x+7, so y=2(x+7)+5 => 2x-y=-19,

However this method only works if one of the equations can be solved for t algebraically. Try using it for x=t*sin(t), y=t*e^t, and you'll quickly see its limitations.

In some cases a simpler or different method might work. For example, if x=cos(t), and y=sin(t), then x^2+y^2=1. But, again, this technique, that is used often when x and y are simple trig functions, based on various trig identities, cannot always be used. A good project would mention these methods, but show their limitations with examples where they will not work.

It should also be noted, that though the graph of the parameterized form will be restricted to the curve defined by the non-parameterized equation that relates x and y directly, it may not trace out the entire graph defined by the non-parameterized curve; domain and range restrictions of the parameterized form must be examined. For example, if x=(cos(t))^2, y=(sin(t))^2, x+y=1, but the original parameterized form does not trace out the entire line defined by x+y=1; only the portion of the graph in quadrant 1, and the two coordinate axes intercepts, because the functions (cos(t))^2 and (sin(t))^2 are confined to values in [0,1].