# Tips when trying to visualize shape of graphs for functions of two variables?

I've been struggling to determine the shape of most graphs when sketching level curves for functions of two variables. E.g.

$$f(x,y) = \log\sqrt{x^2+y^2}$$

I can find the equation of the curves for $z=-1, 0, ...$ just fine but when it comes to actually sketching the graph, I dont know how to figure out its shape without plugging in heaps of numbers for $x$ and $y$. Which is obviously time inefficient in an exam context. Any tips or tricks will be much appreciated!

## 3 Answers

In a exam most of the time you explore the function properties, the derivatives. A level curve is just as a regular 1 variable function on each level, so you put dots on min max points. Then u have to ask yourself if the line between them is convex or concave. This you know from the second derivative

Most of the time it is simply experience. In your example above, one would recognize that the part $x^2+y^2$ makes the function rotational symmetric. Further, $$\sqrt{x^2+y^2}$$ is the distance from the origin (0,0), as in 1D a positive $x$ is also just the distance from the origin $0$. So the final function is just a rotation surface of the form of the logarithm.

One trick might be to set $y=0$ and look what the remaining function will look like. In your case $$\log\sqrt {x^2+0^2}=\log|x|.$$

You will find the same if you set $x=0$. In general, fixing some of the variables (as you might have thought if already) will help you find intersections of the graph (surface) with a shifted coordinate plane and this might reveal the over all shape to you.

If $f(x,y)$ is a function of $x^2+y^2$ alone then your surface will be symmetric with respect to rotations about the z axis. In that case you only need to determine the shape of the curve along 1 axis. Once you know that shape rotate it around $z$ to generate the complete surface.

It is very important to think about stationary points. If you find local maxima/minima you can draw circles around them in countour plots. If you find saddle points then you can draw lines through them that cross.

You should also consider points where the function is not defined (the origin in your example. )

When preparing and practising, I would advise plotting the surface with software if you are able to check your results.