- #1
Aftermarth
- 74
- 0
Ok I am doing 1st year maths at uni and I am finding the differential calculus course really hard, i was hoping people here could just help me with the ideas.
firstly:
Level curves - I am 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?
firstly:
Level curves - I am 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?