Function of Several Variables

[p4nda]

Sketch the level curves in the xy-plane of f(x,y)=3-y-x^2 for k=0,2,4.



Anyone mind helping/teaching me how to do these type of problems? :yuck:

 

[mjsd]

level curves mean contour map. like those isobar chart you see during the weather report. ie. you join all the points that are of the same value , here it means join all the points (x,y) giving you the same f(x,y).
by the way what is k? is it f(x,y)?

 

[HallsofIvy]

Sketch the level curves in the xy-plane of f(x,y)=3-y-x^2 for k=0,2,4.



Anyone mind helping/teaching me how to do these type of problems? :yuck:

Sketch the graphs of 3- y- x^2= 0, 3- y- x^2= 2, and 3- y- x^2= 4. Those are all parabolas.

 

[bethanyd]

just to elaborate a little... This question is asking you to draw out level curves because the level curves can give a fairly clear picture of what the surface is while dealing with less variables.

you set f(x,y) = k where k is a number (usually, but not always, an integer) you get rid of one of the variables thereby making it much easier to graph a piece of the surface.

So you set f(x,y) to each of the numbers given as k, and graph the parabolas (exactly as HallsofIvy described). From there, you should be able to have some ideas about what the surface looks like.

 

[Proof.Beh]

It is sufficent to equate k=0,2,4,... with f(x,y) to have a surface at each level.

Thanks.
Mr Beh