# Search results

1. ### Here's an extension of a list posted earlier. If anybody can think of

oh yay, I'm glad people like this :).
2. ### Rational numbers and Lowest terms proof:

You're on the right track. Let r = p_1 * p_2 * ... * p_a. Let s = q_1 * q_2 * ... * q_b. Let t = p'_1 * p'_2 * ... * p'_c. Let u = q'_1 * q'_2 * ... * q'_d. Cross multiply to get (p_1 * p_2 * ... * p_k)(q'_1 * q'_2 * ... * q'_k) = (p'_1 * p'_2 * ... * p'_k)(q_1 * q_2 * ... *...
3. ### Difference hard to notice ?

The word 'homogeneous' entails that all constant terms are zero, which is not required for an algebraic equation.
4. ### Testing a series for convergence/divergence

0 \leq \frac{2+(-1)^n}{n^2+7} \leq \frac{3}{n^2+7}. You can squeeze it as such. If the series on the right converges, then so does yours.
5. ### Here's an extension of a list posted earlier. If anybody can think of

Here's an extension of a list posted earlier. If anybody can think of any additions to the list, please post :D! Perspectives of the world: ------------------------------- Optimist – The glass is half-full. Pessimist – The glass is half-empty. Existentialist – The glass is. Fatalist – The...
6. ### Science Jokes

Here's an extension of a list posted earlier. If anybody can think of any additions to the list, please post :D! Perspectives of the world: ------------------------------- Optimist – The glass is half-full. Pessimist – The glass is half-empty. Existentialist – The glass is. Fatalist – The...
7. ### Cylindrical coordinates to cartesian coordinates

The equation for the plane is just y = \sqrt{3}x.
8. ### Abstract Math, Tautology

Note: ~(P v Q) is equivalent to (~P and ~Q). So your sample statement S does not work.
9. ### Please explain the solution (Real Analysis)

Do you understand why \textbf{N}^k is countable?
10. ### Volume of a rectangle by cross-sections

In the pyramid integration, x varies from 0 to 3. However, each cross section of a rectangular prism has the same base. So you would have instead: \int_0^3 (3)^2 dx
11. ### Define the sigma-algebra generated by a partition

Yes, that should work.
12. ### Volume of a rectangle by cross-sections

First of all, there is no such thing as the "volume of a rectangle." If you mean a rectangular prism, then the volume will be different. A rectangular prism with the same square base and height would have a volume of 27 cubic units. V_{rectangular \ prism} = bh. V_{pyramid} = \frac{bh}{3}...
13. ### How to increase max speed of wave

I believe we would want (e) and not (f). A thicker string would result in a greater affinity for the string to return back to its equilibrium position. Think of it as: It takes more effort to pull a thicker string, so it'll have a 'stronger desire' to return back to its original position.
14. ### Exhibit a bijection between N and the set of all odd integers greater than 13

Almost...your injection proof is fine. But, when proving a mapping is surjective, we need to show that the domain maps all of the range, i.e., show that for each y, there's an x such that f(x) = y. You wrote "for y E N," but that's not true. y E B = set of all odd integers greater than 13, not...
15. ### Laplace transform of phase-shifted sinusoid

Use trig summation identity to simplify: f(t) = \left[ sin(4t)cos(\frac{\pi}{3}) + cos(4t)sin(\frac{\pi}{3}) \right] u(t). Can you finish from here?
16. ### REAL ANALYSIS, Mathematical Induction

phillyolly, you have (k^3 + 5k) + 3(k(k+1) + 2). We want to show this is divisible by 6, right? Well we already know (k^3 + 5k) is divisible by 6 because we assumed so in our induction proof. So now we just have to prove 3(k(k+1) + 2) is divisible by 6=3*2, i.e., prove it's divisible by 3 AND 2...
17. ### REAL ANALYSIS, Mathematical Induction

Oops, I totally overlooked that.
18. ### REAL ANALYSIS, Math Induction

Stop at = \frac{4k^3 - k + 3(2k + 1)^2}{3} and expand the numerator completely. You know you want 4(k+1)^3. So do as hunt_mat suggested and expand 4(k+1)^3 as an aside (not in the proof) so you know what it is expanded. Subtract this expanded form from your expanded numerator. It should work.
19. ### REAL ANALYSIS, Mathematical Induction

k^3 + 3k^2 + 8k + 6 = (k^3 + 5k) + 3k^2 + 3k + 6. = (k^3 + 5k) + 3(k^2 + k + 2). From our induction hypothesis, we know the first term is divisible by 6. So it remains to show that k^2 + k + 2 is divisible by 2 for all k. It's a mini-induction proof within your main induction proof. Can...
20. ### Cal ii volumes

V = \pi \int_0^4 \left[ (100 - 100e^{-x} + 25e^{-2x}) - 25 \right] dx = \pi \int_0^4 25e^{-x}(e^{-x} - 4)dx \ + \ 75\pi x|_0^4 Let u = e^{-x} - 4 \ \Rightarrow \ du = -e^{-x}dx. So V = 300\pi - 25 \pi \int u du = 300\pi - 25 \pi \left( \frac{(e^{-x} - 4)^2}{2} \right) \right|_0^4...
21. ### Epsilon Delta Proofs, finding bounds

You're finding the limit as x approaches a value. For your problem, x --> 3. Hence, you set c=3 in the generalized inequality 0<x-c< \delta. To put in more formal mathematical words, we want to prove that (as jgens said above): " \forall \epsilon > 0, \ \exists \delta > 0 \ \ni (\forall x)(0 <...
22. ### Very difficult algebra problem (real analysis)

Oh, you just need to know what you want to end up. Well you want to show in the end that y^n > x leads to contradiction. So we would want an h that will give us (y - h)^n > x, thus contradicting the fact that y is the least upper bound. We've already seen that the identity b^n - a^n < (b -...
23. ### (Real Analysis) Show the function is Bijection

I would rather you do as Mark44 suggested and solve for x in terms of y.
24. ### (Real Analysis) Find sets E\F and f(E)\f(F)

How are you going about teaching this to yourself? Which textbook are you using? Any other resources?
25. ### (Real Analysis) Find sets E\F and f(E)\f(F)

Not quite...almost there. E\F = { -1 =< x < 0 }, not f(E\F). Find f(E\F). The final step's still the same and obvious.
26. ### Cal ii volumes

Yep, that's it!
27. ### (Real Analysis) Show the function is Bijection

The process is still of course the same: Solve for x in terms of y and show that's it's a unique solution.
28. ### (Real Analysis) Show the function is Bijection

nope, the unique x part makes it a bijection. It's subtle but it's there! A surjection only guarantees there exists an x in A. A bijection tells us it's a unique x.
29. ### (Real Analysis) Show the function is Bijection

Yep, I have it backwards. Sorry, I'm a bit sleepy. It should be: \forall y \in B , \exists !x \in A \ni f(x) = y.
30. ### Subsets in Real Analysis

Mark44, that was a typo on his part. Look at his work in the latest attached thumbnail and his posts since then. He's got it pretty much now I think.