Homework Help: How to draw graphs and level curves?

Use a computer ;)

But even then you need to work out exactly what you want to plot before you can start. You haven't provided that information: neither of the examples can be plotted with just the information provided.

By "level curve" do you mean "contour"?
In which case you need to decide which direction is "up" and what value to apply to f(x,y,z) - or the function has 4 axes.

 

OK - you are given ##f:\mathbb{R}^3\rightarrow \mathbb{R}, (x,y,z)\rightarrow f(x,y,z)##

the problem is to "sketch or describe the surfaces in ##\small \mathbb{R}^3## which correspond to the mapping. i.e. f(x,y,z) represent 3D surfaces ... sets of them. You should have, in an earlier part of the same text, examples of various types of 3D surfaces and their equations. Compare. i.e. what is the equation for a 3D ellipsoid?

 

x^2+y^2+z^2=1 would be "a unit radius sphere". That's it's description.

for: f(x,y,z)=x^2+y^2+z^2 ... the surfaces in R^3 would be described as "spheres".

 

The problem with "drawing graphs" for these is that you need three orthogonal axes for the independent variables, x, y, and z, and an axes perpendicular to all of those for the function value, f. That is, you will need a four dimensional graph.

Level curves will help you reduce a dimension by treating the function value as a constant. That is, the level curves (more correctly "level surfaces") for for f(x,y,z)= 4x^2+ y^2+ 9z^2 will be the three dimensionl graphs 4x^2+ y^2+ 9z^2= C for different values of C. Those will be a number of ellipsoids, of different sizes, one inside the other.

An added problem here is that you will probably want to draw them on paper which is only two-dimensional!