Multivariable Calculus: Functions and Limits in R2 and R3

[psycho81]

Define f : R2 -> R by

f (x, y) = x²y
x4+y2 (x, y) ≠ (0, 0)

0 (x, y) = (0, 0).


(i)What value does f (x, y) take on the coordinate axes?

(ii) Define g : R -> R2 by

g(t) = ( t )
( kt )

k is an arbitrary nonzero constant. Describe the image of g. Calculate g(t) = f  g(t) . Is
g(t) continuous?

(iii) Define g1 : R -> R2 by

g1(t) = ( t )
( t^2 )


Calculate h(t) = f  g1(t). Also calculate lim t->0 h(t) for t ≠ 0. Explain clearly what
you have found out about the function h(t). Also explain what your calculations tell you
about the function f (x, y).

 

[psycho81]

thats supposed to be (x^2)*y / x^4 + y^2 at the top.

 

[micromass]

As in my other reply, you will need to show some attempt. This will allow us to see what is troubling you about it...

 

[psycho81]

for i) would you set z=0 to get the coordinate axis?

 

[micromass]

No, in fact the domain of f is only two-dimensional. So there is no z-coordinate to set 0 in the domain.

Note that a point lies on the coordinate axes of it has the form (x,0) or (0,y). So to find what values the function takes on the coordinate axes, you'll need to calculate f(x,0) and f(0,y). And don't forget to include the special case (0,0)!