- #1

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firstly:

Level curves - im having trouble drawing out the level curves for functions of two variables.

For f(x,y) = 2x + y - 5

the graph of the level curves supplied is something like so:

\ \ |\ \

\ \ | \ \

\ | \ \

\| \ \ \

_______________________________

| \ \ \ \

| \ \ \ \

(pardon the bad drawing but u get the idea)

where the left-most line is c=-6, and the rightmost is c = 3, increments of 3,

yet they state that the function is a plane, so why is it crossing the axes (which i might add are not labled!)?

2. Limits

for the function F(x,y) = (x^3 + 3(x^2)y + y^3) / (x^2 + y^2)

it is not defined at (0,0)

but the limit as f approaches (0,0) does exist (given).

the lecturer changed it to polar coordinates so now:

f(x,y) = r^3(cosT^3 + 3(cosT^2)sinT + sinT^3) / r^2(cosT^2 + sint^2)

where i have used T as a replacement for theta

so, sinT^2 + cosT^2 = 1, and the r's cancel

then the function is given in terms of r and T

f(x,y) = r(cosT^3 + 3(cosT^2)sinT + sinT^3)

so he gives the function in terms of r and T

= f(r, T)

then, lim{(x,y)>(0,0)} F(x,y) = lim {(r,T)>(0,0)} F(r,T)

= 0 by the Squeeze Law

WHAT!? squeeze law? how does that work?