Recent content by JamesGold

Homework Statement A starship is moving toward a space station at half the speed of light. When it is 7.00 seconds away from reaching a space station (as measured on the ship's clocks) the starship fires a projectile toward the space station at 0.600 c (as measured by the ship's crew). When...

The proof goes as follows: For contradiction, assume there exists |s|< ∞ such that s = {e1, e2, ... , en} and span(s} = ℝ^∞. The above makes at least some sense to me. The proof goes on... Let m > n and u = en+1 + en+2 + ... + em u \notin span(s), u \in s Because {e1, e2, ... ...

I see - that makes sense. Are there any examples of relations where a given input can have two different outputs? Does this model anything?

In Larson's precalculus textbook, he says This seems to imply that there are other relations for which this restriction need not hold. My first question is: what are some examples of such relations? And second, if there are other relations out there, then why are functions so important? Why are...

I'm sorry; I'm still not understanding. Maybe I need to be more clear on what it is I was doing. I'm testing this axiom: For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0. In this case, there are only two vectors in V: 0 and 1. So I...

What do you mean by this?

Thanks for the response. I've got a couple more questions: Does the axiom that says u + (v + w) must equal (u + v) + w apply in this case as there are only two vectors in the set? This set seems to fail the test when you try to show that -u + u = 0. Using it on the first vector in the set gets...

Homework Statement Let V = {0,1} with addition defined modulo 2 (i.e. the remainder upon division by 2), and scalar multiplication given by ku = u^k for all k in the real numbers and u in V. Is the set V a vector space? Homework Equations The 10 axioms! The Attempt at a Solution I...

Every resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?

Cool! Three questions: Can that be represented by an integral? Will this work for any two points on any (continuous) function? Must n go to infinity, or can it be any finite number and still get the right answer? If n = 1, isn't that the same as putting the tangent lines at A and B end to end?

But doesn't the dx serve an actual purpose when finding the area under a function as opposed to just indicating that the integration is done with respect to x? When finding the area under a function, the dx represents the width of an infinitely thin rectangle of height f(x). So why doesn't the...

Thanks for the responses. Intuitively, why the dx? arctan(f'(x)) gives you an angle, so why multiply it by dx? Also, arctan(f'(x)) gives you the angle with respect to the x-axis, right? Would it be possible to do this problem by finding the change in angle between a tangent line at point x = a...

That is, adding up the differential changes in angle between two arbitrarily chosen points on a function, to find the total change in angle between the tangent lines of those two points. How can this be done?

Bump!

Could someone please explain why the meteor's gravitational potential energy is converted to heat after the collision? I understand why the meteor's kinetic energy is converted to heat; after all, it's moving very quickly and then stops. That kinetic energy must go somewhere! But why is there...