how to draw level curves

Apr 2014
73
0
australia
im given

f(x,y) = xy

and i need to draw level curves for this

have not any idea how

can anyone explain to me in details the steps i need to take when im given these types of questions??
 
Apr 2015
263
57
Somerset, England
To start with I assume you realise you need one more dimension than those named in the function.

That is if you have F(x,y) the level curves or surfaces occur in the z dimension.

Since F(x, y) is not a function of z it does not vary in the z direction.

Another way of saying this is that z = a constant.

So level curves have the form F(x,y) = a constant.

You draw them by substituting the appropriate function and working out where (in this instance)

F(x,y) = xy = c

Each different c will yield a separate level curve (also called contours)
 
Sep 2012
838
90
Canada
If we have f(x,y)=xy, that reads as z=xy (which is a surface in 3 space). This is in fact the equation of a hyperbolic paraboloid.

To draw level curves we let z=k where k is any constant. Rearranging the equation we get: y=k/x. If we let k=-1,0,1, 2, 3 etc... what are the graphs that you get? Put them all together and you should get something like this:

https://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427esqqpg2sq4d
 

Plato

MHF Helper
Aug 2006
22,491
8,653
im given
f(x,y) = xy
and i need to draw level curves for this
have not any idea how
can anyone explain to me in details the steps i need to take when im given these types of questions??
The definition corrected is;
A level curve is a function $f$ of two or three variables having the property that \(\displaystyle f \equiv K\), a constant.

The given function is a level curve if $xy=k$. Look at this plot.
 
Dec 2013
2,002
757
Colombia
The way to solve these is usually to write
$$f(x,y)=c \implies xy = c \implies y = {c \over x}$$
and sketch this curve for different values of $c$ as you would for any other function $y = g(x)$.

Sometimes you can't get $y$ as a function of $x$, in which case for each value of $c$ I would pick values of $x$ and then solve for the corresponding values of $y$. You can mark these points on the graph, and then repeat for other values of $x$. When you have sufficient points marked, you can join them.