Calc I question...please help

[NubianPrincess]

Can someone explain to me how to do those optimization problems that are like "Find two numbers whose sum is 100 and whose product is a minimun"????

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[brandonite]

Say the first number is x. Then the second number is (1-x). The product:

y = x * (1-x), or y = x - x^2

take the derivative of that to find the critical points, and then one of those should be the maximum or minimum.

Hope that helps!

 

[brandonite]

Whoops! The other number is going to be (100 - x), right? Sorry.

So, y = (100 - x)*x.

 

[mpp]

Unfortunately that function does not have a minimum. There is a maximum at x=50, y=50, but the minimum is -infinity.

dy/dx = 100 - 2x

Set the derivative to zero.

100 - 2x = 0

x = 50

That is a maximum product (2500). If you are looking for integers, the minimuum product comes with 1 and 99. If you don't use integers you will continue to get smaller and smaller products (0.1 * 99.9 = 9.99, 0.01 * 99.99 = 0.999, etc.).

 

[brandonite]

Yup. That's right. I didn't see that.

In a first year Calculus class, you'd never come across a question that asked you for the minimum of a function like that. I presume the question must ask for the maximum...

Thanks mpp.

 

[NubianPrincess]

Oh, thank you!

 

[TechMan]

Anyone know how to parametrize an ellipsoid?

the eqn. is (4x^2) + (9y^2) + (z^2) = 36

the answer turns out to be:

r(u,v)= 3cos(v)cos(u)i + 2cos(v)sin(u)j + 6sin(v)k

I am unsure how to get this answer. any help would be appreciated.

 

[mpp]

The general formula for an ellipsoid is

((x - x0)/a)^2 + ((y-y0)/b)^2 + ((z-z0)/c)^2 = r^2

or parametrically

x = x0 + a*r*cos(theta)*cos(phi)
y = y0 + b*r*cos(theta)*sin(phi)
z = z0 + c*r*sin(theta)

The parameterization just comes from converting the cartesian coordinates into spherical coordinates

x = r*cos(theta)*cos(phi)
y = r*cos(theta)*sin(phi)
z = r*sin(theta)

And therfore for your ellipsoid

r(phi,theta) = sqrt(36/3)cos(theta)cos(phi)i + sqrt(36/9)cos(theta)sin(phi)j + sqrt(36/1)sin(theta)k

If that makes any sense...

 

[mpp]

I might have that backwards...

x = r*sin(phi)*cos(theta)
y = r*sin(phi)*sin(theta)
z = r*cos(phi)

I'm confused now. It depends on which coordinate you specify as theta and which as phi. Anyway, look for the standard conversion between cartesian and spherical coordinate systems and plug those into the equation of the ellipsoid. You can then convert coordinate systems...that is all paramterization is anyway.

 

[brandonite]

The second one is right, if my memory serves me right, though I haven't done much of this lately. You can define either one as theta or phi - the mathematicians do it one way, and the physicists do it the other, and being in an honours math/physics program, I get caught in the middle.

That question can't be from a Calculus I course, is it? We didn't talk about spherical co'ords until sophomore calculus.

 

[moo]

Hey brandonite, you're a fellow math/phys honors student eh? So am I... although I have steadily migrated toward math now... I consider myself pretty much a mathematical physicist, hence a mathematician.

 

[TechMan]

thks. Im not sure why i didn't see that. I tend to get a little weary when ive been up for days on end.

 

[brandonite]

I started out in just a physics program, and just sorta migrated towards the math as well.

If you're doing an honours physics, you end up spending all your time doing math anyway, so I just added the math. I actually like the math a lot better - some of the more abstract senior math classes I find really interesting. I'm staying away from the 'mathematical physisist' name because all my research is in astronomy (stellar spectroscopy), and I'm going into medicine, so it doesn't really make a lot of sense to just call myself one thing.

Are you doing your undergrad in Canada?

 

[kutastha]

 

[HardlyDavidson]

AMCAS = MONEY * TIME, and
MONEY = TIME, so
AMCAS = MONEY^2, wherein
MONEY = ROOT OF EVIL, therefore
AMCAS = EVIL

QED

 

[moo]

hey brandonite,

I actually started out in physics too... but I'm a total math person now doing real analysis, algebra (final tomorrow), PDE's, Calculus of variations, and green's functions... I still need to take a couple physics courses (stat mech and optics) but those should be ok... as long as I'm away from the labs I'm happy! I think Real Analysis is the coolest thing since sliced bread, to be able to finally prove all the theorems from calculus in all the full epsilon-delta-N glory and to do all that topology stuff and metric spaces is way cooL!

 

[brandonite]

Nice proof.

 

[brandonite]

I've been working on a Real Analysis proof all night, actually. I'm finding that course really cool - just the thought methods that go into it, and how it fits in so nicely with the other math I've taken over the years.

Now if I could just get this one question here...

 

[moo]

What's the question? I just had the final yesterday... (though i won't be able to answer it till at least saturday morning as I'm leaving for DC this afternoon)

 

[brandonite]

If the sequence {Sn} is bounded, prove that for any epsilon > 0 there is a closed interval J (contained in the Reals) of length epsilon such that Sn is an element of J for infinitely many values of n.

I still have no idea how to prove that...

 

[USeF]

Could you use topology to answer this?

If {Sn} is bounded, then it may either be a closed set [it contains all its limit points] or compact [for every open cover of Sn, there is a finite subcover]. Also, every neighborhood around the limit point (the bound), contains some points in {Sn}. SO, set the neighborhoods to be of radius epsilon>0 and it will have to contain infintely many points since all neighborhoods have infinite points!
Not sure if that is good enough (or understandable)
.. tell us geeks if you figure it out

 

[mpp]

Can't be of any help on this one. My degree is in physics and we didn't prove anything unless there was an experiment/observation behind it. I never liked that epsilon-delta crap...

 

[brandonite]

I haven't taken topology yet (next term). So, I picked up bits and pieces of what you said, and I tried to incorporate it into an answer with some stuff I came up with, but to be honest, it's complete BS... Hopefully the prof won't notice. If he does, maybe I'll get marks for creativity...

Ya, I'm in a physics/math program, but I've been avoiding the numerical analysis type courses that usual physics/math people like. I just don't find them interesting. I don't really like the epsilon/delta type proofs, but I do find theoretical math much more interesting than numerical analysis in general...