Multivariable Calc Image-Graph Problem

[Loppyfoot]

Multivariable Calc Image-Graph Problem!

Homework Statement

Given a function f: R-R2 , by f(t) = (t, t^2 - cos(t)), which represents a curve in the xy plane parametrically, give a function whose GRAPH represents this same curve.

2) Also, give a function h whose level set for height k=0 represents this same curve.

The Attempt at a Solution

From what I know, the graph of this function would be in R3. How would I make a function of a graph from that function? I'm probably missing an easy point but haven't picked up on it yet.

For number 2, I'm pretty confused on where to begin this problem.

Any thoughts?

Thanks a ton!

 

[HallsofIvy]

Yes, the graph of F:t-> (x(t), y(t)) is (t, x(t), y(t)). Just put in the x(t) and y(t) you are given.

For 2 you want a function z= F(x, y) such that F(t, t^2- cos(t))= 0 for all t.

 

[Loppyfoot]

Ok Thank you HallsofIvy.

So for another example for number one would be:

If I have f(u,v) from R2-R3, = (u^2+v,u+v,u+v^2),

The graph of this function would be: (u,v, U^2+v,u+v, u+v^2).

Great, I understand now! Thanks a lot sir.

 

[Loppyfoot]

So the equation would be f(t)= (t,t,t^2-cos(t)), correct? Just making sure...

 

[Loppyfoot]

I'm having trouble with number 2.

So I understand that for level sets, I look at the outputs and set them equal to some k.

In my case, k=0.

So, (t,t^2-cos(t))= 0.

Should the first step be to solve for t?

If so, how would I create an implicit equation from this problem? Thanks!