mathematic

Your method is distorting things. You are using 10 points for 10 rectangles. You need to adjust the end points (using 1/2 the value). The result will be closer.

What is T?

A'B + AB' is not everything. However your point is correct.

There is a problem if it is. To avoid confusion, give the A's numbers and assume they are all the same. If A2εA1, then A3εA2, A4εA3, etc. How does the process ever end? Also you can up the ladder as well.

The expression doesn't look right. The first two terms add up to A'B', but the rest looks wrong, since C has no role.

You need to supply more details. A(n), f(x), integral limits (a,b)?

This is purely physics. It is changing one nucleus into another nucleus, usually by bombarding with other particles such as protons. For creating transuranium elements, heavy nuclides are smashed together.

Old fashioned plane and solid geometry are based on Euclid. Euclidean spaces generalize these notions to arbitrary finite dimensions. Next step up is Hilbert space.

sin x = x - (x^3)/6 + (x^5)/120 - .... You see that for the values of a you were using the error term for sina = a gets very small.

What do you mean by "parameter" in 3). The expression is for area. pi is a constant number. 22/7 is a simple approximate value, but it is not exact.

I took a quick look. You will be able to get a pretty good introduction to the subject from chapters 1 and 3. You would need calculus for chapter 2, but chapter 3 doesn't need chapter 2.

I can't recommend a particular book. However without calculus the only probability theory you will be able to understand is discrete probability, so it looks like the first book you mentioned should do.

Strictly speaking he is correct. The two protons don't fuse to He2, which would then decay. It is one reaction. http://en.wikipedia.org/wiki/Proton%E2%80%93proton_chain_reaction

How small? With a density ~ 5 x 10^17 kg/m^3, anything visible to the naked eye would be extremely heavy. It could easily push through the surface of the earth to a significant depth.

From Wikipedia: Quicksand is a non-Newtonian fluid: when undisturbed, it often appears to be solid ("gel" form), but a minor (less than 1%) change in the stress on the quicksand will cause a sudden decrease in its viscosity ("sol" form). After an initial disturbance — such as a person attempting to walk on it — the water and sand in the quicksand separate and dense regions of sand sediment form; it is because of the formation of these high volume fraction regions that the viscosity of the quicksand seems to increase suddenly. Someone stepping on it will start to sink. To move within the quicksand, a person or object must apply sufficient pressure on the compacted sand to re-introduce enough water to liquefy it. The forces required to do this are quite large: to remove a foot from quicksand at a speed of .01 m/s would require the same amount of force as "that needed to lift a medium-sized car."[1] Because of the higher density of the quicksand, it would be impossible for a human or animal to completely sink in the quicksand, though natural hazards present around the quicksand would lead people to believe that quicksand is dangerous. In actuality the quicksand is harmless on its own, but because it greatly impedes human locomotion, the quicksand would allow harsher elements like solar radiation, dehydration, hypothermia or tides to harm a trapped person.[2] Pulling a person from quicksand with too much force could be fatal.[citation needed] The way to escape is to wiggle the legs as slowly as possible in order to reduce viscosity, to try spreading your arms and legs far apart and lying prone to increase your surface area, which should allow you to float.[3]

|z + z0| = |z - z0 + 2z0| ≤ |z - z0| + 2|z0| < δ + 2|z0|

How did this degenerate into a question about vocabulary?

http://en.wikipedia.org/wiki/Gamma_ray_spectrometer The above may help.

L'Hopital's rule shouldn't give you 0/0 indefinitely. After each step see what cancels out.

Sort of by definition. The speed of light is the speed with which light travels.

One further suggestion - it may be tedious. Get power series expansions for the numerator and denominator separately, each stopping at the first non-zero term. If they are the some order for both the numerator and the denominator, the ratio is your answer. If they are at different orders, then the limit is 0 or infinite depending on which is the smaller order.

Have you tired L'Hopital's rule?

You could use a least square approach to get the best fitting parabola.

In general ,the two body problem has as a solution any conic section. Planet paths are ellipses, comets may be any.

The question is too vague. I suggest you give some more details.