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[MagmaMcFry]

When you rotate a curve around an axis, you get a surface, therefore your parametric function needs two parameters.
If y = f(x) is your old graph, your function (x, y, z) = Rf(u, v) is defined as Rf(u, v) = (u, cos(v)f(u), sin(v)f(u)).

[Mego]

When you rotate a curve around an axis, you get a surface, therefore your parametric function needs two parameters.
If y = f(x) is your old graph, your function (x, y, z) = Rf(u, v) is defined as Rf(u, v) = (u, cos(v)f(u), sin(v)f(u)).

Err, yeah, surface. My brain isn't fully functioning today. Thanks!

[Pnx]

Ok, so I'm seriously struggling with getting a working comprehension of this stuff here.

Right now this book is asking me to prove a certain limit is true using the epsilon-delta definition of limits. I think I understand how to do this algebraically, but I really don't understand how this actually proves the limit to be true.

I feel like a moron...

[MagmaMcFry]

Okay, here's the basics: If a number y is the limit value of a function f at point x, then the function must converge to y as the point approaches x, right? This means that f([x-d, x+d]) must converge against {y} as d approaches zero. The epsilon-delta method shows that this is in fact true, by showing that the range of f([x-d, x+d]) becomes arbitrarily small (smaller than every positive 2e) around y as d approaches zero.

[Pnx]

Yes, but how exactly does it prove that the range around y becomes arbitrarily small? I understand what it's supposed to do, but I'm not really understanding how it does it.

[MagmaMcFry]

Yes, but how exactly does it prove that the range around y becomes arbitrarily small? I understand what it's supposed to do, but I'm not really understanding how it does it.

It shows that the range around y becomes arbitrarily small by, for every arbitrary epsilon interval around y, finds a delta interval around x whose image fits inside there.

[Pnx]

I don't know... this is still not making sense to me.

[MagmaMcFry]

Imagine drawing a horizontal bar on the graph of f that includes (x, y). If f converges, any such horizontal bar must contain all the graph points (except maybe (x, f(x))) in a certain horizontal distance from x, right? That's exactly what you need to show. For every horizontal bar [y-e, y+e], there must exist a range [x-d, x+d] in which the bar contains all graph points.

[Pnx]

I think I might be getting it now.

So basically we say that the limit of X as it approaches A, is L. However it's not enough to just say this, we also need to be able to prove it to the gods of mathematics. To do this we make our case by creating two variables, Epsilon and Delta, which are defined by their distance from L and A respectively. Since L acts as a reference point in terms of A, we can use it to set Epsilon and Delta in the same... err, (my command of standard mathematical language is failing me here) logic space? Which allows us to equate the two together, and form a function of Delta which should hold true for any distance from A, within certain possible limitations of distance.

And I guess if it's not a true limit, it wouldn't work out then?

[Skyrunner]

Is there any good reason I shouldn't write cos x as 'c' and sin x as 's', tan x as 't'? O_o My teachers tell me that even if I say sin x := s etc etc at the beginning of a solution they'll mark the whole thing as wrong. Just curious.

[Vector]

Because there's so many common functions named s, t, r, and c that it's extremely bad practice.

[MonkeyHead]

Because there's so many common functions named s, t, r, and c that it's extremely bad practice.

Lol, I am so sick and tired of arguing with students who demand that current should be a "c" in thier work, not an "I", or that wavelength should be "w" not "λ"... and so on.

[Skyrunner]

Because there's so many common functions named s, t, r, and c that it's extremely bad practice.

There are common one-letter functions named s, t, c? O_o

Also, I'm asking this because time is at a premium on math tests and writing cos cos cos sin sin sin over and over feels like a huge time-waster.

[Jim Groovester]

But cos, sin, tan are already abbreviations.

Moreover, they're completely unambiguous.

s, c, and t are not. E.G., c is frequently used as a constant, t is frequently a variable for time, s is fairly frequently a variable for position. You're liable to confuse people who are expecting the usual notations, which is pretty much everybody.

[Skyrunner]

That sounds weird.

So I'm not allowed to assign "cos x" to c, "sin x" to s, and "tan x" to t, but I can assign any arbitrary algebraic function to t (eg, t = sqrt ( x^2 + 2 x - 7) ). Isn't this a double standard in a way?