Just assume, for the sake of argument that staying on top of your ordinary responsibilities consumes most of your entire day, e.g. you rarely have breaks or time to yourself. I think many of us have been there before.
You notice your grades are slipping and you want to improve them.
You...
Homework Statement
Let (M, *) be a monoid. Define p_m(a, b) = amb. The first part of this question was to prove that (M, p_m) defines a semigroup. The second part is to suggest what conditions on m are needed for (M, p_m) to be a monoid.
Homework Equations
The Attempt at a Solution...
Homework Statement
I've already answered the first part. The second part is what's giving me trouble.
Homework Equations
Here's how I solved #10:
Since the car is at rest, we're only worried about the force of friction and the parallel component of gravity:
F_f = F_g *...
I want to preface this by saying that these questions are not to find an exact answer, just to build intuition. If you find them ill-posed or incorrect, it would be most helpful if you could show me a "better way" of looking at it.
So, I'm trying to gather a geometric viewpoint of...
http://en.wikipedia.org/wiki/Trigonometric_substitution#Integrals_containing_a2_.2B_x2" is a special kind of u-substitution. You don't have to use an intermediate x^{-2/3} = u unless you feel more comfortable that way. But it does add more steps and may lead to making another mistake.
If trig...
Pick x^{-2/3} = \tan(\theta)^2, and then \sqrt{1 + x^{-2/3}} = \sqrt{1 + \tan(\theta)^2} = \sqrt{\sec(\theta)^2}. You'll have to be sure to find dx = \frac{d}{d\theta} \tan(\theta)^{-3} and further work for the substitution there, and later some integration by parts for what shows up afterwards...
I feel as though I've read something saying that Aristotle did not actually "believe" that A = \pi r^2, only that it was an approximation. I don't want to hijack this thread, so just take it as my mistake.
Update: Okay, wait, I found it...
I'm interested to know where you read this, Hurkyl. Since it would seem to say that they did not believe in arbitrary (infinite) number of intersections.. i.e. they believed in http://http://en.wikipedia.org/wiki/Intuitionistic_logic"
This may be true in the general case, but I vividly remember not "getting" calculus until I dropped the course and started doing epsilon delta proofs on my own. A lot of students will put up with hand waving. But not all.
I've been attracted to chaos theory well before I enrolled in college. Whether my doubts about the validity of mathematical models was put into question before then, I'm not sure.
But I'm rather happy that, even though I'm taking Diff Eq again after transferring to a 4 year school, it's more...
This thread is mostly dead. I can't be bothered to re-read what I said. But let me at least respond to the general sentiment:
When I said solve "real world problems", I may have used some bad examples, but what I was trying to point out was that the profuse use of "neglecting quantities" that...
Suppose you had some arbitrary function f : R^n \to R^p and x \in R^n. You want to know if it's continuous, so you do some epsilon-delta to find out for sure. However, only the most simple functions permit this without some extra restrictions.
Consider f(x) = x^2. To show that |x - a| < \delta...
As a sophomore / junior, I'm thinking about research more and more. I'd like to know what books in my field are the most frequently cited so I can know what average professional does.
Sure, you can look up the number of citations that any particular book has, or browse the advanced section at...