Does anyone know any tips or ideas on visualizing the functions?

answered 2014-07-08 17:28:13 -0600

Kyouko
231 ●4 ●12

updated 2014-07-08 17:36:21 -0600

This was a pretty interesting exercise that we were asked to do in class, I must say, but I think I may be able to help you a little bit

Think back to the first problem we had which was to sketch the x-z and y-z axis interpretations of the function

$$ f(x,y) = x^2 - y^3 $$

The x-z axis will be easier to start with because we know what a parabola looks like. We take $y$ to be a constant and we will set this constant to be $y = 0$. Now we'll see that

$$ f(x,y) = z = x^2 - y^3 $$

$$ f(x,y) = z = x^2 $$

This function will produce a graph in the x-z plane in the shape of a parabola. As $y$, the constant, changes, the position of the parabola is shifted up and down the z axis.

(insert a graph here later)

Now the second piece we have to look at is what happens when we hold $x$ constant and change only $y$. By setting $x = 0$ we now have the equation:

$$ f(x,y) = z = -y^3 $$

The graph of this is a reversed hyperbolic curve in the y-z plane because we have $-y^3$ instead of just $y^3$. This curve starts in the top left corner and makes its way down to the bottom right in a hyperbolic manner

(insert another graph here later).

Now we have an idea of what the cross sections of this function look like as we hold either $x$ or $y$ constant, but this only gives us half of the picture. Two halves to be exact.

To put these two pieces together to get a rough idea of what the 3D figure should look like.

Recapping:

• in the x-z plane, we have parabolas

• in the y-z plane, we have hyperbolas starting in the negative y moving towards the positive y

The figure we get after putting this together (at least in small scale) will look something akin to a water slide.

[note if someone knows how to work matlab and can help me get a picture of these figures either in matlab or in any other language, please let me now or post an answer below]

Looking specifically at number 7 on the worksheet, I think it's good to look at the functions that are a little bit easier to figure out. Looking at (f) we are given the function:

$$ f(xy) = y^2 - x $$

We can easily compare this to the function $f(x,y) = x^2 - y^2$ which we know to be a saddle shape, but instead of having two quadratic functions we only have one. Looking at the back of the worksheet, the bottom right image matches this description pretty well, having a two hyperbolic curves on opposite ends and a rather flat section (linear) for the other two opposing ends.

Similarly we can look at the entire left hand column of graphs on the handout and due to the oscillatory nature of all three images, they can all be easily identified as the sine functions (b), (c), and (e). Subproblem (e) is simple to work out because there are no quadratics to worry about and the sine function is not multiplied by anything. The function is a sine of a simple summation producing the likewise simple image on the middle left hand side.

Trying to visualize the functions in three dimensions with or without a pick and match isn't easy, but by breaking it down into the two 2D figures and trying to recombine it either in your head or sketching it on paper, you should be able to start understanding and recognizing certain figures