How do I inspect the following function

[Rijad Hadzic]

Homework Statement

Reading my dif eq. book I came across the statement

"inspection of the functions f(x,y) = xy^(1/2) and partial derivative f with respect to y = x/2y^(1/2) shows that they are continuous in the upper half plane defined by y>0"

Homework Equations

The Attempt at a Solution

I'm not really sure how I can plot this on an xy graph at all.

since f(x,y) = xy^(1/2) my hunch is that if x and y are both greater than zero, f(x,y) is going to be positive, and there is nothing to indicate discontinuity..

But I'm still not sure what f(x,y) is and how I can even graph it.. I believe its just a relation and not a function? Idk.

and for partial derivative f with respect to y = x/2y^(1/2) I'm not even sure what to think. I'm in Calc 3 now which covers partial derivatives, and this dif eq class I'm taking only has a max requirement of calc 2 where we haven't seen partial derivatives yet. My teacher told me that I shouldn't worry about it but I have not knowing things, so if anyone could explain how I'm suppose to analyze the two functions I would appreciate it..

 

[andrewkirk]

The function f is from $\mathbb R^2$ to $\mathbb R$, so it can only be graphed as a 3D graph, ie a curved surface in 3D space. The same applies to its partial derivative wrt y.

 

[Rijad Hadzic]

The function f is from $\mathbb R^2$ to $\mathbb R$, so it can only be graphed as a 3D graph, ie a curved surface in 3D space. The same applies to its partial derivative wrt y.

I see. Well I'm going to have to self teach myself calc 3 I guess...thanks for the reply.

 

[PAllen]

I don’t think inspection in this context is meant to mean visual inspection of a graph. Instead, it means inspection of the functional form, combined with basic facts you already know, allows the conclusion, without need to write down a formal proof.

 

[Mark44]

and for partial derivative f with respect to y = x/2y^(1/2) I'm not even sure what to think.

What you wrote here could be interpreted as $f_y = \frac x 2 y^{1/2}$, probably not what you meant.

In this problem, I believe the meaning of "inspection" is to look at the function, $f(x, y) = xy^{1/2}$ and one of the partials, $f_y(x, y) = \frac 2 {2y^{1/2}}$, and notice that both of these functions are continuous on the half-plane $\{(x, y) | y > 0 \}$.

As already noted, you need three dimensions to graph z = f(x, y), but the problem as stated doesn't ask for a graph.