Help with Differential Calculus Course Struggles

In summary, the level curves for the function f(x,y) = 2x+y-5 are confusing and the limits for the function are not defined.
  • #1
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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?
 
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  • #2
oh, and what are cosh and sinh?

all we got is that:

coshT = (e^T + e^-T) / 2
sinhT = (e^T - e^-T) / 2

but what are they supposed to symbolize? i don't see any logic in it, and when i don't see logic i don't understand.
 
  • #3
For first,
I don't think you need to get any concepts; just see lots of 3-d graphs ... and some practice - you would good at it.
You think plane cannot cross the the axis?

2.
Squeeze law: see your previous calculus notes. You should have learned squeeze theorem for two-D functions.
Try looking at the proof why sin(x)/x = 1 ..as x --> 0 (I think this uses squeeze theorm) and it would be easier to understand in 2-d
It just means .. two non-intersection functions touch at one point .. and then another function is always between them .. so the limit of that ...

3. Have you missed one calc course?
You should have learned those too in the previous calculus course..
They are just there are as such .. After spending some time, I started feeling comfortable with them.. They are very similar to sin and cos ..
and even more interesting sin (jx) = j.sinh(x) .. (there's relationship between them and trig) =P
 

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