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1. Quick question about inequality notation

Which of the following is correct, if x = 4? n\leq3x^{2} \leq3(4)^{2} \leq48 or n\leq3x^{2} =3(4)^{2} =48 In other words, in what situations is it necessary to switch from inequality to equality, and it is always required to have only one comparison symbol per line?
2. Proof by Induction

This is generally a good way to think about it in my opinion.
3. Differentiation of abs(x)

You just go back to the definition of abs[(f(x)] abs[f(x)] = f(x) \forall f(x) \geq 0 abs[f(x)] = -f(x) \forall f(x) < 0 And differentiate separately. In other words, 'splitting the domains', as someone above said.
4. Uniting quantum physics with the theory of relativity

I think he's trying to ask why it's not possible?
5. Other Should I Become an Engineer?

I would like some advice on choosing an engineering specialisation. I have a love of mathematics so preferably I would like an engineering degree/profession with as much math as possible. I usually like a layer of abstraction - I am not one who likes physically working with technology as such...
6. Quantifying Entropy - an analogy

Yes I assume it'd be non-subjective shuffling
7. Quantifying Entropy - an analogy

Although I note Lambert looks at Entropy literally not as an analogy
8. Heat transfer rate to heating time

could you fix the latex?
9. Pressure inside a sphere

This is correct - inside hollow shell there is no gravity from the shell
10. Need help for toboggan speed

It is relatively simple if you assume no air resistance/friction etc. But it will depend on how much space you have and how quickly you want to reach your target speed. \frac{1}{2}m (\Delta v)^{2} = mg \Delta h (kinetic energy gained = gravitational potential energy lost) \Delta h =...
11. What is a particle?

Firstly mass and energy are equivalent; they are the same thing. 'Is there an accurate model that encompasses everything that we have observed about particles? - Yes, well to a sufficient accuracy anyway. Whilst there is not one 'theory of everything' that descirbes everything we know...
12. Magnetic force vs gravity

You require the force on the magnet to be equal to the force of gravity, as you said. F = I \Delta l B sin \theta = \frac{Gm_{1}m_{2}}{r^{2}}
13. Gravitation qns

1. The force of attraction is perpendicular to the velocity of the moon's orbit - ie the moon is travelling tangential to the orbit, assuming a circular orbit. Thus there is no component of the force in the direction of motion. The force remains perpendicular to the motion as it moves so it...
14. Quantifying Entropy - an analogy

My friend and I were playing cards, and he was shuffling the deck. He claimed that the longer he shuffled, the 'more random' the arrangement of the cards would become. I argued that at a certain point the cards would be sufficiently disorganised such that they were in a state with no pattern...
15. News Liberal fascism

Sounds like a crap sensationalist book that Ann Coulter would write... slipperly slope arguments, reductio ad absurdum etc...
16. Energy to move a car

By Newton's first law once the car is moving at 55mph you will not need to add any more energy as you will not be accelerating it any more. K=\frac{1}{2}mv^{2} \Delta{K} = \frac{1}{2}m(\Delta{v})^{2} as \frac{dm}{dt} is constant \approx\frac{1}{2}(\frac{1000}{2.2})(55*1.6)^{2} \approx\1.76 *...
17. Ready for some LHC rap?

hah i like it
18. Properties of the Absolute Value

let a = mcis(kpi) where m => 0, k is integer and b = ncis(hpi) where n => 0, h integer (clearly a,b are real) |a||b|= mn |ab|=|mcis(kpi)*ncis(hpi)| = |mncis[pi(k+h)]| = mn as required