Multivariable Calculus: f(x,y) and g(t) Functions and their Images

[psycho81]

Homework Statement

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).

Homework Equations

The Attempt at a Solution

do i just set z=0 for (i)

help with the other parts would be good too.

 

[lanedance]

for i) i think it means set x=0, then seprately set y=0 and consider the values the function takes in each case