MASSIVE NUMBER OF QUESTIONS REGARDING NUM SENSE TRICKS

[BlasianAsian]

Hey! Long time no see! I'll be using this thread as a base for all of the questions I'll have this summer regarding different tricks that I have yet to know of. Please answer them! I don't know how many of you are active during the summer, so... yeah. Anyway, without further ado, enjoy this mountain of stupidity! =D

Binomial Expansion Tricks

I know how to do binomial expansions, just not the num sense way...

UIL HS SAC 2009 Test

64) The simplified coefficient of the [unparseable or potentially dangerous latex formula] term in the expansion of [unparseable or potentially dangerous latex formula] is __________.

Turning Repeating Decimals into Fractions

I know the "99" trick and also the "90" trick for questions like: .32222222222, but what about:

Random Question: .5255555555555555....?

Cartesian Products and other Assorted Numerical Terms

Yeah, please explain this clearly. Also, are there any related terms with this?

Remainder Tricks

HS SAC 2012 Test

55) [unparseable or potentially dangerous latex formula] has a remainder of __________.

Randomness

Is there an easy way to do this following problem?

59) If A is 20% more than B, and B is 25% less than C, then A is what % less than C? ________%

Also, can someone please explain derivitives, and their (opposite?) companion, integral?

Please use this expression as an example: [unparseable or potentially dangerous latex formula]

THANK YOU! =D

[darksaber21]

[darksaber21]

[88bobcat]

[88bobcat]

[nsguy1350]

[nsguy1350]

Integration is used for finding areas under curves. There are two (basic) kinds of integrals: definite integrals, and indefinite integrals.
The symbol for both is the same, except in one aspect: definite integrals have upper and lower "limits", or points of integration. On the number sense test, you will only see definite integrals. Definite integrals are used for finding areas, and require indefinite integration (or a numerical approximation. That's how your calculator integrates, unless it has a computer algebra system that can do definite integration the "right" way). Indefinite integration deals with finding antiderivatives. An example of an indefinite integral is [unparseable or potentially dangerous latex formula], and an example of a definite integral is [unparseable or potentially dangerous latex formula].

Just as it sounds, an antiderivative is basically the opposite of a derivative. And so, [unparseable or potentially dangerous latex formula]. (The dx is used to say you're integrating with respect to x, but don't worry about it here. It can be thought of as the derivative of x for now.)

But wait; [unparseable or potentially dangerous latex formula]. If you add a constant to the function, and take the derivative, you get the same derivative! So which one should you pick when you find an antiderivative? Surely, we do not say [unparseable or potentially dangerous latex formula] (same derivative as the other two). You have one choice out of an infinite number! Because of this, we indicate the antiderivative of a function [unparseable or potentially dangerous latex formula] as [unparseable or potentially dangerous latex formula], where C is some arbitrary constant. Therefore, [unparseable or potentially dangerous latex formula].

Well that's indefinite integrals (indefinite = finding antiderivatives) for you. Here's definite integration.

Definite integration is where you find areas under curves. An example of a definite integral is [unparseable or potentially dangerous latex formula]. Here, you are "integrating 2x from 0 to 2," or finding the area under 2x from 0 to 2. 0 is your lower limit here, while 2 is your upper limit. First, geometrically, find the area under 2x (it's a triangle. Since this is in the NS forum, do it in your head without using pen, calculator, or anything besides your brain :P) from 0 to 2 (oh yea, one more thing: area means from the x-axis to the function. If a function is negative, it means "negative area" from the function to the x-axis. Yes, in Calculus, you can have "negative area", in a sense. No area is negative, but we can treat it as negative when doing things such as integrating velocity functions and such.)

Alright, well hopefully you got that. Now, the actual process of definite integration.
To integrate a function from a lower limit to an upper limit, we use the Fundamental Theorem of Calculus (FTC). There are actually two parts; the one we are going to talk about deals with finding areas using antiderivatives. The other part, basically, deals with how differentiation/antidifferentiation are opposite of each other, and how doing one to the other reverses the whole thing --- find the derivative of the antiderivative of [unparseable or potentially dangerous latex formula] to see what I mean, for example.
Alright, so basically, [unparseable or potentially dangerous latex formula]. (Ahh, I used \displaystyle and a space, looks better now :D) That's one part of the FTC for you.
So, going back to our [unparseable or potentially dangerous latex formula], we see that [unparseable or potentially dangerous latex formula]. Since we evaluate the antiderivative at two points when we integrate, we'll be using that antiderivative. Thus, [unparseable or potentially dangerous latex formula], where | indicates evaluation. This simplifies to [unparseable or potentially dangerous latex formula] As you can see, the answer 4 is what you got geometrically (I would hope). Also, do you see how the constants cancel? This means that when finding the area under a curve, do not worry about constants at all; they will always cancel. Constants of integration are only for indefinite integration. Thus, we can write that a bit more concisely:
[unparseable or potentially dangerous latex formula]

Also, taking derivatives is easy. Taking antiderivatives is a bit harder, and some functions are either REALLY hard or impossible (as far as we know) to integrate, at least using elementary functions (go look up on integration if you wish to know more.) Basically, you reverse differentiation. Get used to doing both, and then you'll become fit to do them during the NS test. I won't go too far into rules of integration, as you can just reverse the differentiation ones (remember, this is for the basics, there are many more advanced techniques!)

So yea, I won't go too far into rules and things. If there is something you need clearance on, ask, or if there is a mistake, please tell me.

For practice: antidifferentiate [unparseable or potentially dangerous latex formula] (Hint: Reverse the chain rule!). Then, find the area under each from 1 to 3.

[88bobcat]

[88bobcat]

[nsguy1350]

88bobcat is correct. I basically said the things you need to know, but I didn't go into why things work very much. If you enter a calculus class, or read a calculus textbook, the limit definition of a derivative and several other proofs and definitions should be available. A big portion of Calculus is about infinitesimals. For derivatives, as 88bobcat said, you're taking a secant line over two points of a curve, and then pushing them closer and closer together. As they get closer and closer, the change in the slope of the line approaches zero, and you approach an actual slope, which is the slope of the tangent line. By the way, the triangle symbol is the Greek letter delta (uppercase? I'm not too familiar with my Greek symbols, besides the very common ones,) which stands for "change in". So [unparseable or potentially dangerous latex formula] change in x. As the change in x goes to zero (pushing the points on the curve closer and closer together,) you get the value of the derivative at that point.

As far as integrals, remember when I talked about numerical approximations? For a given curve, you can evaluate that curve at several points. Now, let's have an interval [unparseable or potentially dangerous latex formula] that you want to integrate (find the area) the curve over. For a really rough approximation, we could find some value on this curve over the interval [unparseable or potentially dangerous latex formula], and then say the area is approximately the rectangle which has a height of that value, and a length corresponding to the length of the interval, [unparseable or potentially dangerous latex formula]. (Take a curve, for example [unparseable or potentially dangerous latex formula]. Let's say we want to integrate from 0 to [unparseable or potentially dangerous latex formula], or [unparseable or potentially dangerous latex formula]. Draw the curve over that interval, and you can very roughly approximate the area as a rectangle that runs from 0 to [unparseable or potentially dangerous latex formula], and whose height corresponds to a value of [unparseable or potentially dangerous latex formula] on that interval. The rectangle isn't perfect (unless the height is the average value of that function, which you can find through integration.)) The 1 rectangle approximation is very rough. To make it better, add another rectangle. Then, add more and more rectangles; as you can see, this will make the approximation better and better. Now, just as with derivatives, make these rectangles increasingly thin; also, their respective heights will differ: a rectangle in the middle of the interval, where sin has its maximum, will be taller than one at the ends, where sin is close to zero. As we add more and more rectangles, the individual lengths of these rectangles becomes smaller and smaller, and eventually infinitely small; as we approach an infinite number of rectangles, we can find the value that the area approaches. That is your definite integral. Also, the dx stands for an infinitesimal length, and sin gives us the height at every point (I think, I might be a bit off on that though, as I am rusty on the basic definitions.) When you do this from 0 to [unparseable or potentially dangerous latex formula], you get the actual area.

[BlasianAsian]

Thanks for the information, but what I really want to know is how to do the trick, not the theories and other assorted goodness behind it. I'm afraid I'm not at that level yet to understand what you're giving me here... D:

Regarding darksaber21's post, I was wondering whether or not there were any terms related to a Cartesian Product I would need to know. A quick search on google provided me of the information I sought after for Cartesian Product's, but I was worried that Larry White might throw some bonus stuff in there to throw me off, you know.

And as for the derivatives and integrals, what I was really wondering was which one was which. Like, you know you have to pull down the exponent and multiply it with the coefficient and stuff. Like this problem:

HS SAC 2012 Test

63) If [unparseable or potentially dangerous latex formula] then [unparseable or potentially dangerous latex formula]?

Also, can you explain to me when you're supposed to do the opposite, like integrals? Like when you pull the exponent up or something.

If you could, please rephrase everything, especially the [unparseable or potentially dangerous latex formula] problem in a language that I can understand, please. I am indeed sorry that my knowledge of mathematics is not at your level. Please excuse my incompetence. =D

[PokémonMaster]

[nsguy1350]

[88bobcat]

[darksaber21]

[BlasianAsian]

O.O Oh, fail. Thanks for telling me the difference!

I don't know. I just don't think it's NECESSARY to learn the fundamentals and theories behind the tricks. What good is that going to do you on the test, anyhow? And that's fine. It was never a requirement to help me anyway. They earned them! No need to share.

Right now, I'm just trying to raise my score so I can make it to state. After I've achieved that, then I might consider learning the other stuff. I don't need a 5-year history on the trick and other things. I just want to know the trick.

Yeah, what nsguy gave me is exactly what I asked. Which one is the pull down and multiply and vice versa? I guess it's not really a trick, but oh well.

At least give me an example on a number sense test where a problem is not used to test a person's knowledge of tricks, if that's where your'e going. Ignorance is bliss, ain't it?

[88bobcat]

[BlasianAsian]

Yeah, well, I'm not even close to that point. I'm working on my accuracy and speed first, and in order to do that, I need to know the tricks. Theories come later. Thanks for the help, though............ And yeah, I'll enjoy it.