The proof to both of these relies on INDUCTION.
The basic idea behind induction is the following:
1) Prove that a statement P(n) is true for n=1.
2) Prove that if P(n) is true, then P(n+1) is true, so long as n is at least 1.
If we can prove the above two statements, then we have...
Let's compile a list of theorems we think every mathematician ought to know!
I'll start:
Stoke's Theorem: If M is a smooth n-dimensional manifold, and \omega is a compactly supported (n-1) form on M, then \int_{M} d\omega = \int_{\partial M} \omega
I'm currently registered for:
Complex Variables II
Differential Geometry II
Basic Probability
Functional Analysis
Philosophy of Mind
Advanced Logic
I'm actually really excited for all my courses this semester!
It's going to be so much work, though. I'll have to be sure to devote...
You know, I agree with you that take-home finals better test your ability than in-class finals.
I have a much better understanding of real analysis than I do of differential geometry. Differential Geometry had a take-home final, and I got an A. Real Variables had an in-class final, and I got a...
OK, I'll give you a few questions, and let's see if you can answer it. If you can't answer it without looking at a book, I'll just tell you the answer, and ask you a similar question.
Question 1: Is it possible for |x| to be less than 1/n for any natural number n, but for x to not equal zero...
The probability is actually (3/5)^10, which is about 0.006, which is 0.6%
3/5 is the probability that any given ticket will NOT be given on a Monday or a Friday, and there are 10 independent tickets being given.
The probability of something just as surprising happening is actually 10 times...
If P(A) is small, then the probability of [receiving 10 tickets overall but 0 tickets on Monday or Friday] is small.
If the probability of not receiving any tickets on Monday or Friday is very low under the assumption that tickets are given out uniformly across all days, then this gives us...
How do you define "equidistant"?
Here's the definition I assume you are using:
If C is a curve, and E is an ellipse, then C is "equidistant from E" if there is some positive real number r such that for all x in C, inf{|x-y| : y in E} = r.
This is actually an interesting question.
There is a...
FAIL.
From wikipedia: http://en.wikipedia.org/wiki/Outer_measure#Formal_definitions
Defining properties of an outer measure:
* The empty set has measure 0.
* Monotonicity: If A is a subset of B, then the measure of A is at most the measure of B.
* Countable Subadditivity: The measure of a...
Because in a CRT, there is no diffraction taking place.
The narrowness of the slit through which the electrons pass through determines the wideness of the diffraction pattern. If the slit were very narrow, the electrons would leave a very wide diffraction pattern.
The slit itself is what...
Ah, I read the paper, and the Glan-Thompson prism is actually just there to separate the entangled photons exiting the BBO. The BBO creates entagled photons with different polarizations (but not in different directions), and then the Glan-Thompson prism directs the two entangled photons in...
No, that text you are quoting is referring to the prism PS, not the Glan-Thompson prism. The Glan-Thompson prism somehow affects polarization, and I'm trying to figure out why it's necessary (it's not mentioned in the wikipedia description).
Yes, but the light is coherent to begin with! The...
Wait a second, how can this be true?
After "PS", the "red" photons and "blue" photons are traveling on entirely different paths. Detectors D3 and D4 could only possibly detect photons that traveled through the "blue" and "red" slits, respectively.
I don't see where the phase information...
After reading about the
http://en.wikipedia.org/wiki/Delayed_choice_quantum_eraser" [Broken] experiment,
I thought of two interesting questions.
Here I have taken http://en.wikipedia.org/wiki/File:Kim_EtAl_Quantum_Eraser.svg" [Broken] and simplified it:
In the normal delayed choice quantum...
http://www.thefreedictionary.com/unicorn
Now you're just being silly.
There is no physical test one can do on the brain to determine with any accuracy whether one will be considered a genius by society.
Einstein worked tremendously hard. Edison is quoted as saying, "Genius is one per cent...
Mugaliens, there is no such thing as a genius.
There is nothing special which could separate an ordinary person from a "genius". Their brains are not more powerful. The only thing that the people whom we call "geniuses" have in common is working hard.
The idea of a "genius" only does our...
Universities have tenure because they are based on community.
The fundamental idea of a university is not to produce results, but to provide an environment in which people can study and advance their own knowledge, as well as that of humanity. A university is not a machine.
Having a...
Don't be like that. You made a claim about the nature of life, which I didn't think accurately represented it. I offered an alternative definition.
Perhaps "complicated" could be replaced by something more accurate and meaningful. I want a condition that captures the fact that life seems to...
And it's questionable whether that would be considered life, or just part of the non-life processes which eventually led to the more complicated processes we call life. I'd go with the latter.
If we were to try to pinpoint the absolute first process which one could call life under my...
A refrigerator does not locally decrease entropy in a roundabout way.
It does so in a rather direct way. Similarly, when a crystal forms, it does so in a direct way, involving simple processes.
If you look at a living cell, on the other hand, it involves complicated enzymes and catalysts...
We already have computer programs which can self-reproduce: viruses.
Some even "evolve", randomly changing bits to avoid detection.
The only ways one might separate computer viruses from "life" proper are the following:
1) Computer viruses depend on an existing system (namely, computers of...
Your constants are wrong. Also, that .5x should be a -.5x.
You want to find a constant such that 2*x + b1 = x^2 + 2 for x = 2.
You also want to find a constant such that -0.5*x + b2 = x^2 + 2 for x = -0.25
Well, f'(1) is the slope of the tangent line to f(x) at x = 1, and f'(-0.25) is the slope of the tangent line to f(x) at x = -0.25.
So, you are looking for two lines:
y = a1x + b1
and
y = a2x + b2
That are tangent to f(x) at x=x1 and x = x2, respectively, but which are...
Well, normally you define a function to have three parts:
1) The domain.
2) The codomain.
3) A rule assigning to each element of the domain an element of the codomain.
This is why you often write, f: A --> B when f maps the set A into the set B.
f(n) = n^2 between the naturals and the...
Well, n --> n/2 is also a bijection between the two sets, just in the other direction.
In fact we can make this a theorem:
THEOREM: If f is a bijection between A and B, then f-1 exists and is a bijection between B and A.
(Remember, bijection <==> one-to-one and onto).
Now, a mapping like...
Yes. Let f(n) = n/2.
For any even natural number, f(n) is a natural number.
2 -> 1
4 -> 2
6 -> 3
8 -> 4
etc...
Notice that it is defined for each and every even natural number. f is clearly one-to-one. Moreover, for each number in N, we can find an even natural number m such that f(m) = n...
TO THE OP: I think you may be misunderstanding the question you were asked. The question is: find two tangent lines to f(x) such that the two are perpendicular to each other. In other words, they will both be parallel to the function f(x) at their respective points, but they will be...