text:
Topology/Topological Spaces

In this section, we will define what a topology is and give some examples and basic constructions.

Motivation
In Abstract Algebra, a field generalizes the concept of operations on the real number line. This general definition allows concepts about quite different mathematical objects to be grasped intuitively by comparison with the real numbers. Likewise, the concept of a topological space is concerned with generalizing the structure of sets in Euclidean spaces. Of course, for many topological spaces the similarities are remote, but aid in judgment and guide proofs. Interesting differences in the structure of sets in Euclidean space, which have analogies in topological spaces, are connectedness, compactness, dimensionality, and the presence of "holes".

If we begin with an arbitrary set, it may not be immediately obvious what is needed to imbue it with an interesting structure. One possibility might be to define a metric on the set, but as it turns out, requiring a metric is overly restrictive. In fact, there are many equivalent ways to define what we will call a topological space just by defining families of subsets of a given set. The properties of the topological space depend on the number of subsets and the ways in which these sets overlap. Topological spaces can be fine or coarse, connected or disconnected, have few or many dimensions.

The most popular way to define a topological space is in terms of open sets, analogous to those of Euclidean Space. (In Euclidean space, an open set is intuitively seen as a set that does not contain its "boundary").

Definition of a topological space
Suppose we are given two sets, $$X$$ and $$\mathcal{T}$$, where $$\mathcal{T}$$ is a collection of subsets of $$X$$.

If $$\mathcal{T}$$ has the following properties: then $$\mathcal{T}$$ is called a topology on $$X$$. The ordered pair $$(X, \mathcal{T})$$ is called a topological space.
 * the empty set and $$X$$ are both in $$\mathcal{T}$$,
 * any (finite or infinite) union of sets in $$\mathcal{T}$$ is itself in $$\mathcal{T}$$, and
 * any finite intersection of sets in $$\mathcal{T}$$ is itself in $$\mathcal{T}$$,

This definition of a topological space allows us to redefine open sets as well. Previously, we defined a set to be open if it contained all of its interior points, and the interior of a set was defined by open balls, which required a metric. That is, we needed some notion of distance in order to define open sets. Topological spaces have no such requirement. In fact, the three properties given above— and them alone — are enough to define an open set. Our new definition is this: Or, in other words, an open set is an element of a topology. So a topology is really a collection of open sets. The rules above are descriptions of how open sets behave: a collection of sets can be called open if the union of an arbitrary number of open sets is open, the intersection of a finite number of open sets is open, and the space itself is open. (The empty set is considered open by default). We also say that a set is closed if its complement is open. As mentioned before, a set can be both open and closed at the same time: the space itself is open, but since its complement (the empty set) is open, it is also closed.
 * the elements of $$\mathcal{T}$$ are called open sets.

As it happens, the properties we associate with open sets, which allow us to study many topological ideas (like continuity and convergence, which were defined earlier using open sets), are encoded entirely by the three properties described above, without any need for a distance metric at all. This is in fact a very abstract definition, using only the most basic ideas of set theory (subsets, unions and intersections), and it allows enormous flexibility in what can be studied as a topological space (as well as how something can be seen as a topological space; there are many different ways a topology can be chosen on a given set). This definition is so general, in fact, that topological spaces appear naturally in virtually every branch of mathematics, and topology is considered one of the great unifying topics of mathematics.

So, to recap: a topology on a set is a collection of subsets which contains the empty set and the set itself, and is closed under unions and finite intersections. The sets that are in the topology are open and their complements are closed. A topological space is a set together with a topology on it.

Some things to note:

 * A set is only open under a particular topology. It does not strictly make sense to merely say that a set is open. In practice, however, the topology under consideration is usually clear from context. For instance, suppose we have the set $$X = \{1,2,3\}$$, and the set $$\mathcal{T} = \{\emptyset, \{1\}, \{1, 2, 3\}\}$$. $$\mathcal{T}$$ is a topology on $$X$$, and one would say that the set $$\{1\}$$ is open. This is inferred to mean that $$\{1\}$$ is open under $$\mathcal{T}$$. If we were given another topology $$\mathcal{T}_2 = \{\emptyset, \{1, 2, 3\}\}$$, then the set $$\{1\}$$ would be open under $$\mathcal{T}$$ but not open under $$\mathcal{T}_2$$. In this case it would be ambiguous to merely say that $$\{1\}$$ is open. (Incidentally, it is not closed under $$\mathcal{T}_2$$ either, because its complement $$\{2, 3\}$$ is not open)
 * While the term topological space strictly refers to the ordered pair $$S = (X, \mathcal{T})$$ (where $$X$$ is a set and $$\mathcal{T}$$ is a topology on $$X$$), often the topological space $$S$$ is used interchangeably with the underlying set $$X$$, or the topology $$\mathcal{T}$$. Which one is meant is usually clear from context. For instance, one might say "suppose $$S \subseteq \mathbb{R}$$ is a topological space and the interval $$I$$ is a subset of $$S$$. Now, suppose $$I$$ is open..." and so on. In this case (assuming the author has not made a mistake), it is meant that the topological space $$S$$ is the pair $$(X, \mathcal{T})$$, where $$X$$ is a subset of $$\mathbb{R}$$, and $$I$$ is a subset of $$X$$. In the case of the real numbers, usually the topology $$\mathcal{T}$$ is the usual topology on $$\mathbb{R}$$, where the open sets are either open intervals, or the union of open intervals.
 * Infinite intersections of open sets do not need to be open. For example, consider open intervals of the form $$\left ( -\frac{1}{n}, \frac{1}{n} \right )$$. The intersection of these intervals for all positive integer values of $$n$$ is the set $$\{0\}$$, which is not open in the real numbers.

Examples of topological spaces
For any set $$X$$, there are two topologies we can always define on $$X$$:
 * The Discrete topology - the topology consisting of all subsets of a set $$X$$.
 * The Indiscrete topology (also known as the trivial topology) - the topology consisting of just $$X$$ and the empty set, $$\emptyset$$.

Metric Topology
Given a metric space $$ \ (X, d)\ $$, its metric topology is the topology induced by using the set of all open balls as the base. One can also define the topology induced by the metric, as the set of all open subsets defined by the metric. We denote the topology $$\mathcal{T}$$ induced from the metric d with $$\mathcal{T} = top(d)$$

This forms a topological space from a metric space.

If for a topological space $$(X, \mathcal{T})$$, we can find a metric $$d$$, such that $$\mathcal{T} = top(d)$$, then the topological space is called metrizable.

The usual topology on the real numbers
We can define a topology $$\mathcal{U}$$ on $$\mathbb{R}$$ by defining $$U\subseteq\mathbb{R}$$ to be in $$\mathcal{U}$$ if for every point $$x\in U$$, there is an $$\varepsilon>0$$ such that $$(x-\varepsilon,x+\varepsilon)\subseteq U$$. We call this topology the standard topology, or usual topology on $$\mathbb{R}$$.

The cofinite topology on any set
Let $$X$$ be a non-empty set. Define $$\mathcal{T}$$ to be the collection of all subsets $$G$$ of $$X$$ satisfying the following:

(In other words, the open sets are formed by removing a finite number of elements from $$X$$)
 * 1) Either $$G = \empty$$
 * 2) Or $$X \setminus G $$ is finite.

Then $$\mathcal{T}$$ is a topology on $$X$$ called the cofinite topology (or "finite complement topology") on $$X$$. Further, this topology turns out to be discrete if and only if $$X$$ is finite.

The cocountable topology on any set
Let $$X$$ be a non-empty set. Define $$\mathcal{T}$$ to be the collection of all subsets $$G$$ of $$X$$ satisfying the following:

(In other words, the open sets are formed by removing a countably infinite number of elements from $$X$$)
 * 1) Either $$G = \emptyset$$
 * 2) Or $$X \setminus G $$ is countable.

Then $$\mathcal{T}$$ is a topology on $$X$$ called the cocountable topology (or "countable complement topology") on $$X$$. Further, this topology turns out to be discrete if and only if $$X$$ is countable.

Sets in topological spaces
Let $$X$$ be a topological space. There are many types of sets we can define on $$X.$$
 * The complement of a set A in X, denoted by $$A^C$$, is $$A^C = X \setminus A$$ (that is, the entire space except for A).
 * A subset $$C$$ is called closed if the set $$C^C$$ is open. Notice that the intersection of any non-zero number of closed sets is closed and the union of finitely many closed sets is closed.
 * Note also that a set can be both closed and open. The trivial examples are the empty set $$\emptyset$$ and the entire set $$X$$, each of which is both closed and open. By definition, $$\emptyset$$ is open, so its complement, $$X$$, is closed. But $$X$$, by definition, is an open set, so $$X$$ is both open and closed.
 * A set $$N$$ is called a neighborhood of a point $$x \in X$$ if it includes an open set containing $$x$$. In other words, $$N$$ is a neighborhood of $$x$$ if there is an open set $$U$$ such that $$x\in U\subseteq N.$$

We now investigate some commonly occurring sets in the study of Topology.

Definition
In a topological space, a $G_{\delta }$ set is a countable intersection of open sets. A $$F_{\sigma }$$ set is a countable union of closed sets.

Theorem
The complement of a $$F_{\sigma }$$ set is $$G_{\delta }$$, and vice versa.

Proof: Let A be a $$F_{\sigma }$$ set and let $$n\in \mathbb{N}$$. Then A is a countable union of closed sets, $$\bigcup\limits_{n}^ – {A_{n}}$$ such that $$A_{n}$$ is closed for all n. Then $$A^{c}=\bigcap\limits_{n}^ – {\left( A_{n} \right)^{c}}$$. Since $$A_{n}$$ is closed, $$\left( A_{n} \right)^{c}$$ is open, so we have a countable intersection of open sets. Hence $$A^{c}$$ is $$G_{\delta }$$. The entirely similar proof of the other implication is left to the reader.

Theorem
In any metric space, a closed set is a $$G_{\delta }$$ set.  Proof: Let X be a metric space and let $$A\subseteq (X,d)$$. Define $$O_{n}=\bigcup\limits_{n}{\left\{ \beta_{1/n}(x)~\left| ~x\in A \right. \right\}}$$. Observe that $$O_{n}$$ is open for any n, and hence the union is open. Now our goal is to show that $$\bar{A}=\bigcap\limits_{n}{O_{n}}$$ to show that a closed set is the intersection of countably many open sets. $$\subseteq $$: Let $$x\in \bar{A}$$. Then $$\beta_{1/n}(x)$$ intersects A at some $$x_{0}$$ which implies $$x\in \beta _{1/n}(x_{0})\subseteq O_{n}.$$. This is true for any n so $$x\in \bigcap\limits_{n}{O_{n}}$$. $$\supseteq $$: Let $$x\in \bigcap\limits_{n}{O_{n}}$$ and $$\varepsilon >0$$. Then $$\exists ~n\in \mathbb{N}$$ such that $$1/n<\varepsilon $$. So $$x\in O_{n}\Rightarrow \exists ~x_{0}$$ in A such that $$x\in \beta _{1/n}(x_{0})$$, which implies $$x\in \beta _{\varepsilon }(x_{0})$$. Thus $$x\in \bar{A}$$. Therefore $$\bar{A}=\bigcap\limits_{n}{O_{n}}$$ and is a $$G_{\delta }$$ set.

Theorem
In usual $$\mathbb{R}$$, $$\mathbb{Q}$$ is a $$F_{\sigma }$$ set.

Proof: Since $$\mathbb{Q}$$ with the usual topology is a metric space, every singleton  such that $$x\in \mathbb{Q}$$ is closed. Thus, we have a countable union of closed sets, and hence $$\mathbb{Q}$$ is a $$F_{\sigma }$$ set.

Exercises

 * 1) Prove the following are topologies:
 * 2) *The discrete topology on any set.
 * 3) *The indiscrete topology on any set.
 * 4) *The cofinite topology on any set.
 * 5) *The cocountable topology on any set.
 * 6) Show that the cofinite (respectively, cocountable) topology on a set $$X$$ equals the discrete topology if and only if $$X$$ is finite (respectively, countable).
 * 7) Prove that a set is open if and only if for every element within the set, there is a neighborhood contained within the set.

 * 1) Show that the discrete topology is the topology induced by the discrete metric. (This is also a splendid way of remembering the discrete and the indiscrete topology)

Espace topologique 位相空間論

title: Topology/Topological Spaces


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Template:Featured recipe

Create a new featured recipe by entering its name in the field below without the Cookbook:, then add it to the list in alphabetical order, and then update the total at Main Page/Recipe.

|Image credit

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Template:Featured recipe


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User talk:"Lina Rincon"

Limites
Estos ejercicios fueron tomados de un taller

Evalue los siguientes limites

1. $$\lim_{x\to 1} \frac {x^2+2x-3} {x^2-5x+4}$$ =$$\lim_{x\to 1} \frac {(x+3)(x-1)} {(x-1)(x-4)}$$ =$$\lim_{x\to 1} \frac {x+3} {x-4}$$ =$$\frac {1+3} {1-4}$$ =$$\frac {4} {-3}$$ 2. $$\lim_{x\to 1} \frac {x^4-x^5} {1-x}$$ =$$\lim_{x\to 1} \frac {x^4(1-x)} {1-x}$$ =$$\lim_{x\to 1} \frac {x^4}{1}$$ =$$\ 1^4 $$    =$$\ 1 $$

Derivadas
Estos ejercicios fueren tomados del calculo de Purcell, octava edicion.

1. En el problema esta dibujada una recta tangente a una curva, evalue su pendiente.

$$\lim_{h\to 0} \frac {f(c+h) - f(c)} {h}$$
 * Respuesta:
 * $$\lim_{h\to 0}\frac {(5+h)- (5)} {h}$$
 * $$\lim_{h\to 0}\frac {25+10+h^2-25} {h}$$
 * $$\lim_{h\to 0}\frac {10h +h^2} {h}$$
 * $$\lim_{h\to 0}\frac {h(10 +h)} {h}$$
 * $$\ {10+0} $$
 * $$\ 10 $$
 * "Lina Rincon"

2. Encuentre las pendientes de las recta tangentes a la curva $$\ y = x^2-1 $$ en los puntos en donde x = -2, -1, 0, 1, 2.

 * $$\lim_{h\to 0} \frac {f(c+h) - f(c)} {h}$$
 * Respuesta:
 * $$\lim_{h\to 0} \frac {((c+h)^2-1) - (c^2-1)} {h}$$
 * $$\lim_{h\to 0}\frac {c^2+2ch+h^2-1-c^2+1} {h}$$
 * $$\lim_{h\to 0}\frac {2ch+h^2} {h}$$
 * $$\lim_{h\to 0}\frac {h(2c+h)} {h}$$
 * $$\ {2c+0} $$
 * $$\ 2c $$
 * (2.-2=-4), (2.-1=-2),  (2.0=0),  (2.1=2),  (2.2=4)
 * "Lina Rincon"

3. En los problemas siguientes utilice la definicíón $$f`(x)=lim_{h\to 0}\frac{f(c+h)-f(c)}{h}$$
 * a. f(1) si f(x)=x^2/

 * Respuesta:

 * $$\lim_{h\to 0}\frac{f(1+h)-f(1)}{h}$$
 * $$\lim_{h\to 0} \frac {(1+h)^2-(1)^2} {h}$$
 * $$\lim_{h\to 0} \frac {1+2h+h^2-1} {h}$$
 * $$\lim_{h\to 0} \frac {2h+h^2} {h}$$
 * $$\lim_{h\to 0} \frac {h(2+h)} {h}$$
 * $$\ 2+0 $$
 * $$\ 2 $$

 * b. f(2) si f(t)=(2t)^2/

 * Respuesta:

 * $$\lim_{h\to 0} \frac {f(c+h) - f(c)} {h}$$
 * $$\lim_{h\to 0} \frac {f(2+h) - f(2)} {h}$$
 * $$\lim_{h\to 0}\frac {2(2+h)^2-2(2)^2}{h}$$
 * $$\lim_{h\to 0}\frac{(2(4+4h+h^2)-2(4)}{h}$$
 * $$\lim_{h\to 0} \frac {8+8h+2h^2-8} {h}$$
 * $$\lim_{h\to 0} \frac {8h+2h^2} {h}$$
 * $$\lim_{h\to 0} \frac {h(8+2h)} {h}$$
 * $$\ {8+0}$$
 * $$\ 8$$

 * c. f(3) si f(t)=t^2-t/

 * Respuesta:

 * $$\lim_{h\to 0} \frac {f(c+h) - f(c)} {h}$$
 * $$\lim_{h\to 0} \frac {f(3+h) - f(3)} {h}$$
 * $$\lim_{h\to 0}\frac{((3+h)^2-(3+h))-(3^2-3)}{h}$$
 * $$\lim_{h\to 0} \frac {(9+6h+h^2-3-h)-(9-3)} {h}$$
 * $$\lim_{h\to 0} \frac {9+6h+h^2-3-h-6)} {h}$$
 * $$\lim_{h\to 0} \frac {5h+h^2)} {h}$$
 * $$\lim_{h\to 0} \frac {h(5+h)} {h}$$
 * $$\ {5+0}$$
 * $$\ 5$$

4.En los siguientes problemas utilice las reglas para encontrar derivadas.

a. $$\ Y = {2x^2} $$
 * Respuesta:
 * $$\ Dx Y= {2.2x^{2-1}} $$
 * $$\ Dx Y= {4x} $$

"Lina Rincon"

b. $$\ Y = {3x^3} $$
 * Respuesta:
 * $$\ Dx Y = {3.3x^{2-1}} $$
 * $$\ Dx Y = {9x^2} $$

"Lina Rincon"

c. $$\ Y = {pi.x} $$
 * $$\ pi DxY = {x} $$
 * $$\ pi DxY = {1} $$
 * $$\ DxY = {pi.1} $$
 * $$\ DxY = {pi} $$

"Lina Rincon"

d. $$\ Y = {pi.x^3} $$
 * $$\ piDxY = {3.1x^{3-1}} $$
 * $$\ piDxY = {3x^2}$$
 * $$\ DxY = {3pix^2}$$

"Lina Rincon"

e. $$\ DxY = \frac {500x^4} {x^10}$$
 * $$\ DxY = {2.(-2)x^{-2-1}} $$
 * $$\ DxY = {-4x^{-3}} $$

"Lina Rincon"

f. $$\ Y = {-3x^{-4}} $$
 * $$\ DxY = {-4.(-3)x^{-4-1}} $$
 * $$\ DxY = {-12x^{-5}} $$

"Lina Rincon"

g. $$\ Y = \frac {pi} {x}$$
 * $$\ DxY = \frac {(0.pi)-(1.pi)} {x^2}$$
 * $$\ DxY = \frac {0-pi} {x^2}$$
 * $$\ DxY = \frac {-pi} {x^2}$$

"Lina Rincon"

h. $$\ Y = \frac {100} {x^5}$$
 * $$\ DxY = \frac {(0.x^5)-(100.5x^4)} {{(x^5)}^2}$$
 * $$\ DxY = \frac {0-500x^4} {x^10}$$
 * $$\ DxY = \frac {500x^4} {x^10}$$
 * $$\ DxY = \frac {500} {x^6}$$

"Lina Rincon"

i. $$\ Y = {x^2+2x} $$
 * $$\ DxY = {Dx(x^2)+Dx(2x)} $$
 * $$\ DxY = {Dx(2x)+Dx(2)} $$
 * $$\ DxY = {2x+2} $$

"Lina Rincon"

j. $$\ Y = {3x^4+x^3} $$
 * $$\ DxY = {Dx(3x^4)+Dx(x^3)} $$
 * $$\ DxY = {Dx(12x^3)+Dx(3x^2)} $$
 * $$\ DxY = {12x^3+3x^2} $$

"Lina Rincon"

i. $$\ Y = {x^4+x^3+x^2+x+1} $$
 * $$\ DxY = {Dx(x^4)+Dx(x^3)+Dx(x^2)+Dx(x)+Dx(1)} $$
 * $$\ DxY = {Dx(4x^3)+Dx(3x^2)+Dx(2x)+Dx(1)+Dx(0)} $$
 * $$\ DxY = {4x^3+3x^2+2x+1} $$

"Lina Rincon"

title: User talk:"Lina Rincon"


text:
Cookbook:Biscuits I

| Bread

Biscuits are a traditional bread from the American South and a staple of soul food. This recipe is in the Yankee style, which is to say, it contains an egg and the end result is less dependent on your pastry technique. These biscuits are tall, light, and sturdy. They are less salty than many store-bought mixes. They taste good plain or with butter, honey, jam, or sausage/ham gravy. Because they are sturdy, this recipe is particularly well-suited for making a sandwich. Try splitting the biscuit open and putting a fried egg inside, or filling it with cold deli meats and other sandwich fixings.

For the more tender Southern-style (eggless) version, try rich biscuits. If you are trying to substitute gluten-free flour, you might have better luck with this recipe, because the egg will help hold the dough together.

Procedure

 * 1) Preheat oven to 450°F (230°C).
 * 2) Sift all dry ingredients into a bowl.
 * 3) Cut in shortening until it has the texture of coarse meal.
 * 4) Beat egg lightly and add to milk.
 * 5) Add liquid to dry ingredients and mix with fork just until dough holds together.
 * 6) Turn out onto floured board and knead lightly with floured fingers.
 * 7) Roll out ¾ inch (2 cm) thick, and cut out biscuits with a round, floured cutter.
 * 8) Place biscuits on baking sheet and bake at 450°F (230°C) for 12 minutes.

Notes, tips, and variations

 * Vegetable shortening can have a bland flavor. You can substitute the same volume of butter for the shortening in this recipe if you want the butter flavor.

title: Cookbook:Biscuits I


text:
Talk:Algebra/Equations

Proposal for deltion
I propose that this page be merged with the solving equations page. Nfgdayton (talk) 17:18, 26 July 2010 (UTC)

The previous chapter Arithmetic has become quite long and this one, "Simple symbolic manipulation," is shorter. My suggestion is to cover the commutative, associative, distributive, and other fundamental laws of real numbers at the beginning of this chapter, especially since there are thoughts of covering a few extra topics (percentages, rounding, etc.) in "Arithmetic" as mentioned in comments at the bottom of its edit page. Also, it would be a good idea to introduce sets briefly and discuss solution sets to single-variable equations equations in this chapter.

There is a comment at the bottom of this chapter in the edit page about adding simultaneous equations to this chapter. It will become long enough as it is. I have started a new chapter on Simultaneous Equations in the second part of the book, "Equations and Functions." It comes right after the "Function Graphing" chapter, which I wrote. Therefore, simultaneous equations will be able to be discussed using graphs in the new chapter.

H Padleckas 13:31, 11 Jul 2004 (UTC)

Incorrect expansion?
"3x − 2 > 2(x − 3). First, you may want to expand the right hand side: 3x − 2 > 2x − 6"

Quoted from the Solving Inequalities section of the article. Is that expansion true? What if x = 4, for example? That expansion assumed what, exactly, is applicable here?

title: Talk:Algebra/Equations


text:
Cookbook:French Fries

| Side dishes

French fries (or chips) are a well-known element of fast food.

Ingredients

 * Potatoes (those with a high dry matter content such as Russet or Maris Piper work best)
 * Cooking oil
 * Salt

Procedure

 * 1) Peel potatoes, and cut into thin straw-like pieces.
 * 2) Wash potatoes well with water to remove excess starch.
 * 3) Place fries in a saucepan with water, and simmer or parboil on the stove for about 15 minutes.
 * 4) Drain the fries, and dry them well.
 * 5) Deep fry in 175°C vegetable oil, lard, or drippings for 4–5 minutes until light golden and crisp.
 * 6) Remove from oil, and place on plate with paper towels to absorb the leftover cooking oil. Transfer to another plate if necessary.
 * 7) Add salt and shake to ensure all French fries are evenly salted.

Notes, tips, and variations

 * You can vary the size of the potato slices. Thick wedge shapes are home-style fries or English-style chips, while thinly julienned potatoes make for shoestring potatoes. Cooking times will need to be adjusted accordingly.
 * Try adding other spices after cooking. Minced garlic is popular in California cuisine.
 * Instead of ketchup, serve with mayonnaise, gravy, barbecue sauce, or salt, hummus, and vinegar.
 * For poutine (a French-Canadian treat), toss cooked fries with cheese curds and smother with gravy.
 * Some cook the fries a single time in a generous amount of oil pre-heated to a temperature around 375 °F (190 °C) until they are golden and slightly crisp.
 * The method recommended by most cookbooks and used by many restaurants, is to cook them in two stages: first at around 350 °F (177 °C) until the fries are nearly cooked but limp and pale; after they have been removed from the oil and allowed to cool, they are briefly fried at a higher temperature of around 375 °F (190 °C) until golden and crisp.
 * A method attributed to the celebrated French chef Joël Robuchon for the home cook says to put the sliced potatoes into a saucepan with just enough cold oil in it to cover the potatoes, then to cook them over high heat until golden, stirring occasionally.

Warnings

 * Do not attempt to extinguish fat fires with water! If the oil becomes too hot, it may catch fire. Do not attempt to extinguish it with water, as this will cause an explosive reaction: oil floats on water, so water will only help the oil move up and get more oxygen. Instead, turn off the stove if possible, try to cover fire with a lid or any other piece of solid, non-flammable material, and use a class B fire extinguisher if the fire is still not under control. If that fails to work, evacuate and call the emergency service.

Recipes featuring French fries

 * Poutine
 * Fish and Chips
 * Chips, Cheese and Gravy

title: Cookbook:French Fries


text:
User:Offray~enwikibooks

Esta es mi páginar personal en los Wikibooks.

Por lo pronto he empezado a usar los wikibooks para un libro de cálculo del curso de vacaciones que me encuentro dictando y un solucionario. Espero crear más contenido usando esta infraestructura proximamente (y mejorar esta página ;-) ).

title: User:Offray~enwikibooks


text:
Cryptography/Polyalphabetic substitution

A Polyalphabetic substitution cipher is simply a substitution cipher with an alphabet that changes. For example one could have two alphabets:

Plain Alphabet:    A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Cipher Alphabet #1: B D F H J L N P R T V X Z A C E G I K M O Q S U W Y Cipher Alphabet #2: Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

Now to encrypt the message ``The quick brown fox jumped over the lazy dogs" we would alternate between the two cipher alphabets, using #1 for every first letter and #2 for every second, to get: ``Msj joxfp dicda ucu tfzkjw ceji msj xzyb hln".

Polyalphabetic substitution ciphers are useful because the are less easily broken by frequency analysis, however if an attacker knows for instance that the message has a period n, then he simply can individually frequency analyze each cipher alphabet.

The number of letters encrypted before a polyalphabetic substitution cipher returns to its first cipher alphabet is called its period. The larger the period, the stronger the cipher. Of course, this method of encryption is certainly not secure by any definition and should not be applied to any real-life scenarios.


text:
User:Cky

 * 1) REDIRECT User:Cky

title: User:Cky


text:
Objective-C Programming/concepts

There are a number of key Objective-C concepts that have close relations to the practice of object-oriented programming in general. For the moment, we won't look at exact syntax until later.

 * If you have previous experience in object-oriented programming, you may wish to skip these sections and examine the Summary section

Objective-C is just C (a superset of C), with Lisp like object-oriented syntax
Object-orientation in objective-C has some similarities with C++, and is a way of organising memory like c-style structs, except that it is expected that the structures have pointers to functions that can reference any offset defined as an instance variable or function pointer , so the object is a memory location from which the offsets to instance variables can be found , and where there is a pointer back to the class definition that gives the addresses of functions of the class.

In C++, to call a member function from a member function of a class, the syntax is this -> func ( a, b );

or

(*this).func (a, b);

this represents a pointer to "this" object, the instance of a class which defined the function where "this" is being used.

In objective-C, 'self' is the keyword to replace 'this', and the invocation would be [self func: a, b ]

So here, the parameters to the function are not enclosed in parentheses like most other c based languages, but

Constructors in java are called like this in a constructor of a child class, when the parent class's constructor needs to be invoked.

Child( a, b, c ) {  super(a); this.b = b; this.c = c;  }

Similarly, in objective-C, for a class interface @interface MyClass : MyParentClass { MyClassB * b;   int  c;  } @property (copy, nonatomic) NSInteger d;

-(void) func1; -(void) func2: (int)param1, (id) param2; @end

there is an elided init constructor function, and

the constructor might be

- (id) init: (id)a, (id) b, (int) c, (NSInteger) d { self = [super init:a ]; // -(2) if (self) { _b = [[B alloc] initByCopy:b] ;    _c = c;      self.d = d;    }    return self;   }

and there exists a constructor for MyParentClass, which is

- (id) init: (id) a // - (2a) { .. }

The above example shows:
 * 1) id is a placeholder for a class type.
 * 2) In a function declaration or definition, parentheses are conventionally put around the return type, and before each parameter to give their types.
 * 3) self is a supplied instance variable that references an object of the class, but can be initially an object of the supertype , as in (2) and (2a).
 * 4) the convention is to call alloc which is a class function for a class to obtain an object, and then to initialize the object by calling an object function with a function conventionally being 'init' or beginning with 'init..' , so the pattern is my_object = [[A_Class alloc] init:a_param1, a_param2], which means "allocate the memory for an object templated by A_Class, and then call init on the object with a_param1 and a_param2".
 * 5) some simple C language types such as int, double, char*, can be represented by more complex classes beginning with NS.
 * 6) In the above example, _b and _c don't have self preceding it, but d does. d is defined as a property in the interface declaration part of the class. A property has elided set/get functions for the instance variable, and can have property attributes like ( copy, nonatomic ) - copy means "self.d = d" is short for "self.d = [ [NSInteger alloc] initByCopy:d ] , where NSInteger is the class for d.
 * 7) it's helpful to remember that objective-C is an extension of c, and variable names usually refer to a short sequence of 4, 8 or 16 bytes, which either represent numbers (int, long, float, double, char) or addresses ( char* , class instance pointers ). This is a bit unlike java, where variables are references to objects, and it is elided or hidden how the objects are referenced, and how their memory is managed. Objective-c frameworks added in reference-counting automatic deallocating memory management later to the NS.. classes family.

C and Standard Unix system interface underlie Objective C
Any of the standard Unix libraries which are available in C are available in Objective C, so this includes standard system interfaces to services such as file input/output, dynamic ( malloc/free ) allocation, string functions, maths functions, process creation, synchronization, and interprocess communication via file locks, thread locks, thread condition variables, semaphores, pipes, sockets, shared memory management, signals. However, objective C allows grouping of data structures with methods, and gives rise to shortcut organizing structured programming for domains such as GUI construction, using object-oriented patterns such as polymorphism and delegation ( via objective-c "protocol" feature ). Additional features also include automated references counting so that dynamic allocated objects are automatically deallocated without garbage collection. This gives rise to replacement of standard operating system interface functions (malloc/free), with standard objective c. object oriented idioms from NSObject e.g. NS * p = [[NS alloc] init] ; for those familiar to dot only syntax eg java , means same as   MYClass p = (MyClass.alloc).init; "Call static factory class method alloc, and invoke init on object."

In c, there is only MyStruct * p = malloc ( size of(MyStruct)); later on, the pointer p may share the memory address with another pointer, say q, and there would be 2 references, then if later, q = z, and even later p= z, but wait, there is a memory leak, so free must be called on p because it is the last reference to the memory allocation produced by the call to malloc above. free(p); In older objective c frameworks, [p dealloc]; However, automatic reference counting or ARC, means that when p = z, in the example above, the reference count for the NS object pointed to by p falls to zero, and free is called on the hidden pointer held by the ARC support framework.

Example 2, instead of char * charbuf[maxlength]; snprintf ( charbuf, maxlength-1, "hello %s, %d times!", "world", 2);

there is

NSString* mystrbuf = [[NSString alloc] initWithFormat: "hello %s, %d times!", "world", 2];

What is object-oriented programming?
Object-oriented programming (often just shortened to OO or OOP), is a programming paradigm where code is written in units called classes, where functions associated with data structures are declared and defined together. For example, ANSI C has data in structs, and similarly, an OO language stores data in classes or objects, but also declares the functions that operate on instances of structured data (objects). A class itself can be an object, with associated data and class functions, but there is only one instance per process space. (in general, operating systems have one process per running program, but allow multiprocess programs which are programs that start off other programs ).

For example, if you are creating a word processor, you might represent the document by an object that has text as its data, and associate with it different tasks to perform upon the document's data, such as "spell check".

The word processor then would be a small program that might draw the window and show the document's text. A spell check button would perform the "spell check" task upon that document's object.

However, if we were editing two documents at once, they would differ probably in the contents of the documents- the data. But all the documents would be more or less the same: Each document contains a space for its text, and has a couple of defined tasks that operate on the text. Each document object has essentially the same form. We call this form the class of the object, and we often give it a name. Let's call the document class Document (Style note: we often capitalize names of classes).

Now, each document object has the form of the Document. We say that each document object is an instance of Document. So, for example, our word processor operates on instances of the Document class. Sometimes we may call an instance of the Document class just "a Document".

Remember that each object has some data. For example, with the Document class, each <tt>Document</tt> has data to hold the text of the document. We call the data that each object holds instance variables. So, a <tt>Document</tt> may specify an instance variable to hold the text, or an instance variable to hold the word count, or the author, or any other data we may want to specify. Each <tt>Document</tt> has instance variables that are separate from the instance variables of any other <tt>Document</tt> that may exist.

Each object also has a number of tasks that it can perform on its instance variables. In Objective-C, we call these tasks methods. The object's methods operate on the object's instance variables.

However, there's more to objects than just this. With Objective-C, we can do a lot of interesting things with objects.

Working with objects
Let's stick with the example of our word processor. Say we want to create a new instance of <tt>Document</tt> so we can work with it. This process is called instantiation.

Now that we have an active instance of an object, we can work with it, by using the object's methods. We'll look at defining and working with methods later. Each method merely acts like a function in C, and in Objective-C, there is only a minor syntax change.

When we're completely done with the object and want to get rid of it (so it stops taking up memory), we do what is known as destruction or deallocation of the object. This is like <tt>free</tt>'ing a <tt>malloc</tt>'d array, for example, in C. However, if, in the process of using the object's methods, we allocate some extra memory for its instance variables, we must free that memory too.

Object techniques
When we're creating new classes, we may want to base the new object's behaviour on a class, which we have previously created. For example, if we want to create a document class that has details of style information, we may wish to base that class on the `Document` class, which we have previously created.

This is a common enough feature that it is included in object-oriented systems, and it is given the name of inheritance. When a class inherits from another, the new class takes on the form of the old class and can add extra instance variables and methods.

So, if we wanted to create a `StyledDocument` class, we would inherit from the `Document` class. In Objective-C, a class only inherits from one other class. This is known as single inheritance.

Interfaces and implementation
As in the real world, there is a separation between how objects are used, and how objects work. For example, with our word processor, there is a consistent interface to the spellchecker (how it is used), such as buttons to jump to the next misspelt word, to go back, and to correct. But, behind the scenes, the way the spellchecker works (the implementation) can be done in many different ways.

In Objective-C, the interface and implementation are kept separate. This allows different objects to work and link together in a well-defined manner. To change channels on your television, for example, you use the interface of the buttons on the television set instead of opening it and fiddling around with the electronics.

Keeping the implementation separate from the interface means that we can change anything in the implementation (even substituting it with another implementation) and everything should still work, because objects (buttons etc) interact with each other only through their interfaces.

Summary
So, before we get to learning Objective-C syntax, let's recap:
 * An object is an instance of a class
 * which has defined upon it tasks known as methods
 * and data known as instance variables
 * A class can inherit from one (and only one) other class (single inheritance)

title: Objective-C Programming/concepts


text:
User:"Pablo Quintero"

=Axiomas=

Los siguientes ejercicios fueron tomados del Libro "Calculus de Thomas Apostol".

 * Demuestre:

1. $$a.0=0.a=0$$

 * Demostración:

Por hipótesis decimos que,

 * $$a\in \mathbb R$$

Por existencia de neutros, podemos sumar un $$0$$ a $$a.0$$ y sigue siendo $$a.0$$

 * $$a.0+0=a.0$$

Con el fin de aplicar propiedad transitiva, buscamos una ecuación que sea igual a $$a.0$$,(esta será demostrada mas adelante),

 * $$a.0+a.0=a.0$$ (*)

Al conocer que ambas ecuaciones son iguales a $$a.0$$,se igualan por propiedad transitiva

 * $$a.0+0=a.0+a.0$$

Por el teorema: si a+c=a+b, entonces c=b; podemos cancelar $$a.0$$ en ambos lados.

 * $$a.0=0$$

 * (*) $$a.0+a.0=a.0$$

Por existencia de neutros para la suma, decimos que

 * $$a.0+a.0=a.(0+0)$$

Por existencia neutros, decimos que $$0+0=0$$

 * $$a.0+a.0=a.0$$

 * QED

 * "Pablo Quintero"

2. $$(-a)b=-(ab)$$

 * Demostración:

Por hipótesis decimos que,

 * $$a,b\in \mathbb R$$

Por existencia de inversos para la suma podemos decir que

 * $$ab+(-ab)=0$$

Con el fin de aplicar propiedad transitiva, buscamos una ecuación que sea igual a $$0$$, (esta será demostrada mas adelante),

 * $$ab+(-a)b=0$$ (*)

Al conocer que ambas ecuaciones son iguales a $$0$$, se igualan por propiedad transitiva

 * $$ab+(-ab)=ab+(-a)b$$

Por el teorema: si a+c=a+b, entonces c=b; podemos cancelar $$ab$$ en ambos lados.

 * $$-(ab)=(-a)b$$

 * (*) $$ab+(-a)b=0$$

Por propiedad distributiva podemos expresar la igualdad como

 * $$ab+(-a)b=[a+(-a)]b$$

Por existencia de inverso para la suma decimos que,

 * $$ab+(-a)b=[0]b$$

Por Teorema $$a.0=0.a=0$$, tenemos que

 * $$ab+(-a)b=0$$

 * QED

 * "Pablo Quintero"

3. $$(a/b)+(c/d)=[(ad+bc)/bd]$$

 * Demostración:

Por hipótesis decimos que,

 * $$a,c\in \mathbb R$$

y

 * $$b,d\in \mathbb R - (0)$$

Como solo se trabaja con suma y multiplicación, por notación cambiamos la expresión de la igualdad

 * $$(a/b)+(c/d)=(ab^-1)+(cd^-1)$$

Por existencia de neutros para la multiplicación, podemos multiplicar  la expresión por $$1$$. Este $$1$$ se puede expresar de la forma (bd)(bd)^-1.

 * $$(ab^-1)+(cd^-1)=[(ab^-1)+(cd^-1)](bd)(bd)^-1$$

Por propiedad distributiva tenemos que

 * $$(ab^-1)+(cd^-1)=[(ab^-1)(bd)+(cd^-1)(bd)](bd)^-1$$

Por las propiedades: conmutativa, asociativa y existencia del reciproco, tenemos que

 * $$(a/b)+(c/d)=[(ad)+(bc)](ba^-1)$$

Para que la igualdad sea como la presentada en el enunciado, por notación le cambiamos la expresión a

 * $$(a/b)+(c/d)=[(ad+bc)/bd]$$

 * QED

 * "Pablo Quintero"

4. $$-0=0/(bc)$$

 * Demostración:

Por hipótesis decimos que,

 * $$0\in \mathbb R$$

y

 * $$-0\in \mathbb R$$

Por existencia de inverso para la suma tenemos que

 * $$0+(-0)=0$$

Por existencia de neutros para la suma tenemos que

 * $$0+0=0$$

Al conocer que ambas ecuaciones son iguales a $$0$$, se igualan por propiedad transitiva

 * $$0+(-0)=0+0$$

Por el teorema: si a+c=a+b, entonces c=b, cancelamos $$0$$ en ambos lados

 * $$(-0)=0$$

 * QED

 * "Pablo Quintero"

5. $$(a-b)+(b-c)=(a-c)$$

 * Demostración:

Por hipótesis decimos que,

 * $$a,b,c\in \mathbb R$$

Como solo se trabaja con suma y multiplicación, por notación cambiamos la expresión de la igualdad

 * $$(a-b)+(b-c)=[a+(-b)]+[b+(-c)]$$

Por propiedad asociativa tenemos que

 * $$(a-b)+(b-c)=[a+[(-b)+b]]+(-c)$$

Por existencia de inverso para la suma tenemos que

 * (a-b)+(b-c)=[a+0]+(-c)</math

Por existencia de neutros para la suma tenemos que

 * $$(a-b)+(b-c)=a+(-c)$$

Para que la igualdad sea como la presentada en el enunciado, por notación le cambiamos la expresión a

 * $$(a-b)+(b-c)=(a-c)$$

 * QED

 * "Pablo Quintero"

6. $$(-a/b)=(-a)/b$$

 * Demostración:

Por hipótesis decimos que,

 * $$a\in \mathbb R$$

y

 * $$b\in \mathbb R - (0)$$

Como solo se trabaja con suma y multiplicación, por notación cambiamos la expresión de la igualdad

 * $$(-a/b)=(-ab^-1)$$

Por existencia de neutros para la multiplicación, podemos multiplicar  la expresión por $$1$$. Este $$1$$ se puede expresar de la forma (b)(b^-1).

 * $$(-a/b)=(-ab^-1)(b)(b^-1)$$

Por propiedad asociativa tenemos que

 * $$(-a/b)=[(-ab^-1)(b)](b^-1)$$

Por propiedad asociativa tenemos que

 * $$(-a/b)=[(-a)[(b^-1)b]](b^-1)$$

Por existencia del reciproco tenemos que

 * $$(-a/b)=[(-a)(1)](b^-1)$$

Por existencia de neutros para la multiplicación tenemos que

 * $$(-a/b)=(-a)(b^-1)$$

Para que la igualdad sea como la presentada en el enunciado, por notación le cambiamos la expresión a

 * $$(-a/b)=(-a)/b$$

 * QED

 * "Pablo Quintero"

=Desigualdades=

Los siguientes ejercicios fueron tomados del libro "Calculo de Edwin J. Purcell, sexta edición".

 * Resuelva:

1. $$(-6)<(2x+3)<(-1)$$

 * Respuesta:

Se suma $$ -3$$ a cada parte de desigualdad, con el fin de eliminar el $$3$$ y despejar la variable sola

 * $$(-6-3)<(2x+3-3)<(-1-3)$$

Operando se obtiene

 * $$(-9)<(2x)<(-4)$$

Se divide por $$2$$ a cada parte de desigualdad, con el fin de eliminar el $$2$$ y despejar la variable sola

 * $$(-9/2)<(2x/2)<(-4/2)$$

Operando se obtiene

 * $$(-9/2)<(x)<(-2)$$

Se da el resultado en términos de intervalos

 * $$(-9/2,-2)$$

 * "Pablo Quintero"

2. $$(10x+1)>(8x+5)$$

 * Respuesta:

Se pasa $$(8x+5)$$ al otro lado con el fin de dejar la variable a un solo lado

 * $$10x+1-8x-5>0$$

Operando se obtiene

 * $$2x-4>0$$

Se pasa el $$4$$ al otro lado con el fin de despejar la variable

 * $$2x>4$$

Se pasa el $$2$$ a dividir al otro lado con el fin de despejar la variable

 * $$x>4/2$$

Operando se obtiene

 * $$x>2$$

Se da el resultado en términos de intervalos

 * $$(2,\infty)$$

 * "Pablo Quintero"

3. $$(3x+5)>(7x+17)$$

 * Respuesta:

Se pasa $$(7x+17)$$ al otro lado con el fin de dejar la variable a un solo lado

 * $$3x+5-7x-17>0$$

Operando se obtiene

 * $$-4x-12>0$$

Se pasa el $$-12$$ al otro lado con el fin de despejar la variable

 * $$-4x>12$$

Se pasa el $$-4$$ a dividir al otro lado con el fin de despejar la variable

 * $$x<-12/4$$

Operando se obtiene

 * $$x<-3$$

Se da el resultado en términos de intervalos

 * $$(-\infty,-3)$$

 * "Pablo Quintero"

4. $$(2+3x)<(5x+1)<(16)$$

 * Respuesta:

Se separa la desigualdad con el fin de operar con mayor facilidad

 * $$(2+3x)<(5x+1)$$, $$(5x+1)<(16)$$

Se pasa $$(5x+1)$$ al otro lado con el fin de dejar la variable a un solo lado. Se pasa el $$1$$ al otro lado con el fin de despejar la variable

 * $$2+3x-5x-1<0$$,  $$5x<16-1$$

Operando se obtiene

 * $$1-2x<0$$,       $$5x<15$$

Se pasa el $$1$$ al otro lado con el fin de despejar la variable. Se pasa a dividir el $$5$$ al otro lado con el fin de despejar la variable

 * $$-2x<-1$$,       $$x<15/5$$

Se pasa a dividir el $$-2$$ al otro lado con el fin de despejar la variable
 * $$x>-1/-2$$,      $$x<15/5$$

Operando se obtiene

 * $$x>1/2$$,        $$x<3$$

Ordenando la desigualdad se obtiene

 * $$1/2<x<3$$

Se da el resultado en términos de intervalos

 * $$(1/2,3)$$

 * "Pablo Quintero"

5. $$(2x-4)<(6-7x)<(3x+6)$$

 * Respuesta:

Se separa la desigualdad con el fin de operar con mayor facilidad

 * $$(2x-4)<(6-7x)$$, $$(6-7x)<(3x+6)$$

Se pasa $$(6-7x)$$ al otro lado con el fin de dejar la variable a un solo lado. Se pasa $$(3x+6)$$ al otro lado con el fin de dejar la variable a un solo lado

 * $$2x-4-6+7x<0$$,  $$6-7x-3x-6<0$$

Operando se obtiene

 * $$-10+9x<0$$,     $$-10x<0$$

Se pasa el $$-10$$ al otro lado con el fin de despejar la variable. Se pasa a dividir el $$-10$$ al otro lado con el fin de despejar la variable

 * $$9x<10$$,        $$x>-1/10$$

Se pasa a dividir el $$9$$ al otro lado con el fin de despejar la variable

 * $$x<10/9$$,       $$x>-1/10$$

Ordenando la desigualdad se obtiene

 * $$-1/10<x<10/9$$

Se da el resultado en términos de intervalos

 * $$(-1/10,10/9)$$

 * "Pablo Quintero"

6. $$|x-5|<|x+1|$$

 * Respuesta:

Se eleva al cuadrado cada valor absoluto con el fin de quitar el mismo

 * $$(x-5)^2<(x+1)^2$$

Factorizando se obtiene

 * $$x^2-10x+25-24/-12$$

Operando se obtiene

 * $$x>2$$

Se da el resultado en términos de intervalos

 * $$(2,\infty)$$

 * "Pablo Quintero"

7. $$|3x+3/x+1|<1$$

 * Respuesta:

Por propiedades del valor absoluto, se puede expresar la desigualdad como

 * $$|3x+3|/|x+1|<1$$

Se pasa a multiplicar $$|x+1|$$ al otro lado con el fin de separar los valores absolutos

 * $$|3x+3|<|x+1|$$

Se eleva al cuadrado cada valor absoluto con el fin de quitar el mismo

 * $$(3x+3)^2<(x+1)^2$$

Factorizando se obtiene

 * $$x^2+6x+9<x^2+2x+1$$

Se pasa $$( x^2+2x+1)$$ al otro lado con el fin de dejar la variable a un solo lado

 * $$x^2+6x+9-x^2-2x-1<0$$

Operando se obtiene

 * $$4x+8<0$$

Se pasa el $$8$$ al otro lado con el fin de despejar la variable

 * $$4x<-8$$

Se pasa a dividir el $$4$$ al otro lado con el fin de despejar la variable

 * $$x<-8/4$$

Operando se obtiene

 * $$x<-2$$

Se da el resultado en términos de intervalos

 * $$(-\infty,-2)$$

 * "Pablo Quintero"

=Derivadas=

Los siguientes ejercicios fueron tomados del libro "Ejercicios de Calculo 1 y Calculo Diferencial de Takeuchi, Yu".

De las siguientes funciones, encuentre sus derivadas:

1.$$f(x)=\sqrt[2]{1-x^2}$$
 * Respuesta:

$$f'(x)=\frac {-x} {\sqrt[2]{1-x^2}}$$
 * "Pablo Quintero"

2.$$f(x)=\frac {x^2+x+1} {x^2-x+1}$$
 * Respuesta:

$$f'(x)=\frac {(2x+1)(x^2-x+1)-(2x-1)(x^2+x+1)} {(x^2-x+1)^2}$$ $$f'(x)=\frac {-2(x^2-1)} {(x^2-x+1)^2}$$
 * "Pablo Quintero"

3.$$f(x)=\frac {xa} {x^2-a^2} + \frac {xa} {x^2+a^2}$$
 * Respuesta:

$$f'(x)=\frac {a(x^2-a^2)-xa(2x)} {(x^2-a^2)^2} + \frac {a(x^2+a^2)-xa(2x)} {(x^2+a^2)^2}$$ $$f'(x)=\frac {-a(x^2-a^2)} {(x^2-a^2)^2} + \frac {a(x^2+a^2)} {(x^2+a^2)^2}$$
 * "Pablo Quintero"

4.$$f(x)=\frac {x-\sqrt[2]{x}} {x+\sqrt[2]{x}} $$
 * Respuesta:

$$f'(x)=\frac {(1-1/2\sqrt[2]{x})(x+\sqrt[2]{x})-(x-\sqrt[2]{x})(1-1/2\sqrt[2]{x})} {(x+\sqrt[2]{x})^2}$$ $$f'(x)=\frac {2\sqrt[2]{x}+1} {(x+\sqrt[2]{x})^2}$$
 * "Pablo Quintero"

5.$$f(x)=\frac {2x^2+x-8} {3x+2}$$
 * Respuesta:

$$f'(x)=\frac{(4x+1)(3x+2)-3(2x^2+x-8)} {(3x+2)^2}$$ $$f'(x)=\frac{6x^2-8x+26} {(3x+2)^2}$$
 * "Pablo Quintero"

6.$$f(x)=\frac {x^2*a^2} {x^3+a^3}$$
 * Respuesta:

$$f'(x)=\frac {2xa^2(x^3+a^3)-3x^2(x^2*a^2)} {(x^3+a^3)^2}$$ $$f'(x)=\frac {xa^2(2a^3-x^3)} {(x^3+a^3)^2}$$
 * "Pablo Quintero"

7.$$f(x)=\sqrt[2]{(\frac {x+1}{x-1})^3}$$
 * Respuesta:

$$f'(x)=(\frac {x+1}{x-1})^3/2$$ $$f'(x)=3/2(\frac {x+1}{x-1})^1/2*\frac {(x-1)-(x+1)}{(x-1)^2}$$ $$f'(x)=\frac {-3}{(x-1)^2}*\sqrt[2]{\frac {x+1}{x-1}}$$
 * "Pablo Quintero"

=Máximos y Mínimos=

Los siguientes ejercicios fueron tomados del libro "Ejercicios de Calculo 1 y Calculo Diferencial de Takeuchi, Yu".

1.Para enviar cierto tipo de paquetes por correo la empresa transportadora exige que sean de base cuadrado y que la suma de sus lados no supere 150 cm. Halle el volumen máximo que estos paquetes pueden encerrar.

Sean x, x, y las dimensiones del paquete, entonces

 * $$x+x+y=150$$

El volumen del paquete $$V = x^2y$$

 * $$V =x^2y=x^2(150-2x)$$	 (0 ≤ x ≤ 75)

 * $$dV/dx=300x-6x^2=0$$

entonces $$ x=0$$, $$x=50$$

tenemos que cuando $$ x=0$$, $$V=0$$	y cuando $$x=50$$, $$V=50^3$$

Si $$ x=75$$ entonces $$V=0$$, por lo tanto, el volumen máximo es

 * $$50^3=125000cm^3$$
 * "Pablo Quintero"

2.Una recta paralela al eje OY corta en P y Q las curvas

 * $$y=3x2+7x$$, $$y=3x$$

Halle la longitud máxima del segmento PQ.

PQ=f(x)=|$$(3x^2+7x)$$–$$(3x)$$|=$$|3x^2+4x|$$

 * PQ=$$3x^2+4x$$    sí   $$x \in (-\infty, -4/3) U (0, \infty)$$

 * PQ=$$-3x^2$$–4x    sí   $$x \in (-4/3,0)$$

 * f’(x)=$$6x+4x$$    sí    $$x\in (- \infty, -4/3) U (0, \infty)$$

 * f’(x)=$$-6x$$–4x    sí   $$x\in (-4/3, 0)$$

f es decreciente en ($$-\infty$$, -4/3), creciente en (-4/3, -2/3), decreciente (-2/3, 0), creciente en (0, $$\infty$$)

Luego f toma un máximo local en

 * $$x=-2/3$$
 * "Pablo Quintero"

3.Hallar el máximo y el mínimo de x2 + y2 cuando x + y = 1, x ≥ 0, y ≥ 0.

Sea S=x2+y2  donde x+y=1       (x≥0, y≥0) entonces S=$$x^2+(1-x)^2$$=$$2x^2$$–$$2x+1$$

Los puntos críticos de S:

 * dS/dx=$$4x$$–$$2$$=$$0$$

 * $$x=1/2$$, $$S=1/2$$

Los valores de S en los extremos son:
 * S(0)=1, S(1)=1

Entonces:
 * S(0)=S(1)=1      (máximo de S)

 * S(1/2)=1/2       (mínimo de S)
 * "Pablo Quintero"

=Razón de Cambio=

Los siguientes ejercicios fueron tomados del libro "Ejercicios de Calculo 1 y Calculo Diferencial de Takeuchi, Yu".

1.Un radio de una esfera crece a la velocidad de 1mm/seg, a qué velocidad están creciendo superficie y su volumen cuando el radio es 10 cm?

Sea V, S el volumen y la superficie de la esfera de radio r, entonces

 * V=$$4/3 \pi r^3$$, 	 S=$$4 \pi r^2$$

entonces

 * dV/dt =$$4 \pi r^2 $$(dr/dt),	 dS/dt=$$8 \pi r$$ dr/dt

Si dr/dt =$$1$$(mm/seg),
 * = $$0.1$$(cm/seg), r = $$10$$cm

entonces

 * dV/dt=$$4 \pi 10^2 (0.1)=40 \pi$$(cc./seg)

 * dS/dt=$$8 \pi 10(0.1)=8 \pi$$($$cm^2$$/seg)
 * "Pablo Quintero"

2.Para gases ideales se sabe que PV=constante, siendo P la presión del gas y V el volumen del recipiente que lo contiene. Cómo varía la presión de un gas contenido en un recipiente que disminuye su volumen a la razón de 10c.c./seg., cuando V=$$500c.c. y P=15kg./ cm^2?

 * PV=c	 (c = constante)

Entonces

 * (dV/dtV+PdV/dt)=0

Pero
 * dV/dt=-10(cm^3/seg)

 * V=500 (cm^3), P=15(kg/cm^2)

entonces
 * dP/dt=-(P/V)dV/dt==0.3 (kg/cm^2$$seg)
 * "Pablo Quintero"

3.Los ejes mayor y menor de una elipse aumentan sus longitudes a las velocidades respectivas de 1cm/seg, y 2cm/seg. A qué rata crece su área cuando el eje mayor mide 10 cm y el eje menor 6 cm?. (Área de una elipse: S=$$\pi$$ab siendo a y b las semi-longitudes de sus ejes).

Sean a,b los ejes mayor y menor de la elipse, entonces el área de la elipse S es:

 * S=$$\pi$$ab

 * dS/dt=$$\pi$$b(da/dt)+$$\pi$$a(db/dt)

Pero

 * da/dt =$$1$$, db/dt =$$2$$.

Entonces

 * dS/dt=pb+$$2\pi$$a=$$\pi$$(6)+$$2\pi(10) $$ =$$26\pi$$ ($$cm^2$$/seg)
 * "Pablo Quintero"

title: User:"Pablo Quintero"


text:
Wikibooks:Computer science bookshelf

Category — Subject —  Dewey Decimal — Library of Congress

This bookshelf covers books about computer science - that is, books on software design, computer programming, and the theory of computation. See also Category:Shelf:Computer science

To be moved to Shelf:Computer science

 * A-level Computing
 * The Design and Organization of Data Centers
 * History of Computers
 * IB Computer Science
 * Medical Informatics
 * Open Source
 * Concepts of Computer Graphics
 * Shelf:Software reverse engineering
 * Systems Theory
 * Internet Technologies [[image:50%.svg]]
 * The World of Peer-to-Peer (P2P)
 * Communication Systems
 * Sensor Networks
 * Video Game Design
 * CORBA [[image:25%.svg]]
 * Video Game Design
 * Genetic Programming and Algorithms
 * Java Programming/Struts
 * The Complete Programming FAQ
 * Robotics
 * VLF-reception with the PC
 * Theory of Polygon and Mesh Modeling
 * Computer Programming/Linux Programming [[image:25%.svg]]

Programming

 * Console Game Programming
 * Classic Systems:
 * Nintendo
 * NES[[image:25%.svg]]
 * Super NES[[image:50%.svg]]
 * Game Boy (classic)[[File:00%.svg]]
 * Nintendo 64[[File:00%.svg]]
 * Game Boy Advance[[File:00%.svg]]
 * Sega
 * SMS[[File:00%.svg]]
 * Sega Genesis[[image:25%.svg]]
 * Sega Game Gear[[File:00%.svg]]
 * Sega Saturn[[File:00%.svg]]
 * Sega Dreamcast[[File:00%.svg]]
 * Modern Systems
 * PlayStation Portable [[File:00%.svg]]
 * XGameStation
 * Related:
 * Emulation

de:Regal:EDV

title: Wikibooks:Computer science bookshelf


text:
User:せくそろじすと

sqares

title: User:せくそろじすと


text:
User:The Saitou~enwikibooks

MetaWiki Wikipedia Nipponese Wikipedia English Wiktionary English Wikibooks English Wikicities MediaWiki other account

I am Nipponese.

I was born in 1979, a self-proclaimed student of Bachelor of Engineering (Electrical Engineering, March 2004), unemployed.

Links

 * The_Saitouの日記（MainBlog）
 * The_Saitouの日記2（SubBlog）

ja:User:せくそろじすと big

title: User:The Saitou~enwikibooks


text:
Talk:The Voynich Manuscript/F1v

Meta-discussion: (about the wiki page itself, not the content).

This is my first attempt at moving content from the interlinear to this wikibook. I have some experience with wikis but little with this particular variety. Some of the links gave me trouble, in particular the image links. From my reading of the help document it shouldn't be treating these image links specially, but obviously it is. The one inserts the image directly into the page which may have copyright issues, the other two links don't seem to work and so the wiki replaces what would in theory by the image with simply the filename. I don't know how to protect these so that they are treated as mere links like the others.

KenCausey

title: Talk:The Voynich Manuscript/F1v


text:
The Voynich Manuscript/F2r

Transcription, comments, theories, links to do with VMs page f2r to be added here...

Description:
One plant centered on page.

 * Root: left-right symmetrical, planar, looking sort of like headless stick man with outstretched arms and gripping hands. Overpainted with dark color.

 * Stem: short and thin.

 * Branches: two main ones, almost upright, slightly bent. Irregular thickness, awkward shape (cf. top right leaf). Light-clored with a dark vein grossly painted along the middle.

 * Leaves: Dark overpaint, shaped like a triangular fan with five lanceolated fingers.

 * Stalk: medium length, looks like a branch.

 * Flowers: three, on tip of main branch.

 * Stalk: longish, looks like a branch.

 * Chalyx: an oblong oval swelling at end of stalk, partial dark overpaint.

 * Petals: many, narrow and long; the flower looks like a shaving brush.

 * Core: hidden by petals.

Two paragraphs (with 6.4 and 5.4 lines), left- and right-justified: one at the top, slightly narrower, interrupted by three flowers; and one near the bottom, interrupted by two branches.

Comments:
The root looks rather strange, the stems and leaves are somewhat awkward, but the flowers look normal.

Petersen tentative identification is "cyanus segetum, cornflower (caeruleus) (cf 153)". That must be

Centaurea_cyanus (cornflower,fiodaliso;bluet,kornblume)

http://www.runeberg.org/nordflor/7.html

The flower shape and branching pattern seem to match fairly well. However, those features are not very specific. [What about color?] Also, the leaves of Centaurea cyanus are long and thin, with smooth edges or a few very shallow teeth, attached directly to the stem; quite unlike those of f2r.

Infusions of Centaurea cyanus were once used as a febrifugue (whole plant) and eye wash (flowers only).

References:
[1] Claude Martin http://www.voynich.info/3_deciphered_text_selection/3_1_folio_f2r/3_1_1_f2r_line_p1.html

title: The Voynich Manuscript/F2r


text:
File:Bubble01.png

bubble sort fig. 1

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text:
File:Bubble02.png

bubble sort fig. 2

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text:
File:Bubble03.png

bubble sort fig. 3

title: File:Bubble03.png


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File:Bubble04.png

bubble sort fig. 4

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File:Bubble05.png

bubble sort fig. 5

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File:Bubble06.png

bubble sort fig. 6

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text:
Algorithm Implementation/Sorting/Bubble sort

Bubble Sort
The bubble sort is also known as the ripple sort. The bubble sort is probably the first, reasonably complex module that any beginning programmer has to write. It is a very simple construct which introduces the student to the fundamentals of how sorting works.

A bubble sort makes use of an array and some sort of "swapping" mechanism. Most programming languages have a built-in function to swap elements of an array. Even if a swapping function does not exist, only a couple of extra lines of code are required to store one array element in a temporary field in order to swap a second element into its place. Then the first element is moved out of the temporary field and back into the array at the second element's position.

Here is a simple example of how a bubble sort works: Suppose you have a row of children's toy blocks with letters on them. They are in random order and you wish to arrange them in alphabetical order from left to right.

 * Step 1. Begin with the first block. In this case, the letter G. (Fig. 1.)

Fig. 1

 * Step 2. Look at the block just to the right of it.

 * Step 3. If the block to the right should come before the block on the left, swap them so that they are in order (Fig. 2.)

Fig. 2

 * If you were doing this by hand, you might just pick the blocks to be moved with one in each hand and cross your arms to swap them. Or you might move the first one out of its position temporarily, move the second one in its place, then move the first one to the now empty position (this is the difference between having a single function to do the swap, or writing some code to do it).

 * Step 4. Compare the next block in line with the first, and repeat step 3. Do this until you run out of blocks. Then begin step one again with the second block. (Fig. 3,4,5,6)

Fig. 3 - Pass #2 Fig. 4 - Pass #3 Fig. 5 - Pass #4 Fig. 6 - Completed Sort

Why is it called a bubble sort?
The bubble sort gets its name because elements tend to move up into the correct order like bubbles rising to the surface.

C++
C++ can use the C bubble sort above, or use this for random-access iterators of generic containers and a generic comparison operator:

Ada
Sorts an array of integers.

AutoIt
Sorts an array of variant data types

Go
Go implementation with package sort

Haskell
bubbleSort []=[] bubbleSort x=   (iterate swapPass x) !! ((length x)-1) where swapPass [x]=[x] swapPass (x:y:zs) | x>y = y:swapPass (x:zs) | otherwise = x:swapPass (y:zs)

OCaml
This version checks for changes in a separate step for simplicity.

Pascal
A simplified version:

Rust
Rust function for generic types that implement Copy and PartialOrd traits (e.g. numeric types or strings).

VBA
see discussion for possible mistake

COBOL
If you are dealing with a large volume of data, use the COBOL SORT using sequential files. For sorting a WORKING STORAGE table, the following example assumes that the table is already loaded. The literals "a" indicates the size of the row, and "b" how many rows in the table.

Implementazioni di algoritmi/Bubble sort


text:
User talk:Jaime.rincon

Limites
Evalue los siguientes límites:

1. $$lim_{x\to 4}\frac {x-4} {x^2-x-12}$$
 * Respuesta:
 * $$lim_{x\to 4}\frac {x-4} {(x-4)(x+3)}$$
 * $$lim_{x\to 4}\frac {x-4} {x+3}$$
 * $$= 1/7$$

Jaime.rincon

2. $$lim_{x\to 3}\frac {x^3-27} {x^2-9}$$
 * Respuesta:
 * $$lim_{x\to 3}\frac {(x-3)(x^2+3x+9)} {(x-3)(x+3)}$$
 * $$lim_{x\to 3}\frac {x^2+3x+9} {x+3}$$
 * $$= 9/2$$

Jaime.rincon

3. $$lim_{h\to 0}\frac {(x+h)^2-x^2} {h}$$
 * Respuesta:
 * $$lim_{h\to 0}\frac {x^2+2hx+h^2-x^2} {h}$$
 * $$lim_{h\to 0}{2x+h}$$
 * $$= 2x$$

Jaime.rincon

4. $$lim_{x\to \infin}\frac {3x-2} {8x+7}$$
 * Respuesta:

$$lim_{x\to \infin}\frac {3-(2/x)} {8+(7/x)}$$ $$= 3/8$$ Jaime.rincon

5. $$lim_{x\to \infin}\frac {6x^2+2x+1} {5x^2-3x-4}$$
 * Respuesta:

$$lim_{x\to \infin}\frac {6+(2/x)+(1/x^2)} {5-(3/x)-(4/x^2)}$$ $$= 6/5$$ Jaime.rincon

6.$$lim_{x\to \infin}\frac {x^2+x-2} {4x^3-1}$$
 * Respuesta:

$$lim_{x\to \infin}\frac {(1/x)+(1/x^2)-(2/x^3)} {4-(1/x^3)}$$ $$=0$$ Jaime.rincon 7.$$lim_{x\to \infin}\frac {2x^3} {x^2+1}$$
 * Respuesta:

$$lim_{x\to \infin}\frac {2} {(1/x)+(1/x^3)}$$ $$= n.e$$ Jaime.rincon

8.$$lim_{x\to 3}\frac {x^2-9} {x^2-6x+9}$$
 * Respuesta:

$$lim_{x\to 3}\frac {(x-3)(x+3)} {(x-3)(x-3)}$$ $$lim_{x\to 3}\frac {x+3} {(x-3)}$$ $$= 6/0 = n.e$$ Jaime.rincon

9.$$lim_{x\to 2}\frac {|x^2-4|} {x-2}$$
 * Respuesta:

$$lim_{x\to 2}\frac {(x+2)(x-2)} {x-2}$$ $$= -4$$ $$= 4$$ Jaime.rincon
 * si x<2, $$lim_{x\to 2}\frac {|x^2-4|} {x-2}=lim_{x\to 2}\frac {-x^2+4} {x-2}$$
 * si x>2, $$lim_{x\to 2}\frac {|x^2-4|} {x-2}=lim_{x\to 2}\frac {x^2-4} {x-2}$$

10.$$lim_{x\to 0}\frac {\sqrt[2]{9+x}-3} {x}$$
 * Respuesta:

=$$lim_{x\to 0}[\frac {\sqrt[2]{9+x}-3} {x}][\frac {\sqrt[2]{9+x}+3} {\sqrt[2]{9+x}+3}]$$ =$$lim_{x\to 0}\frac {x} {x[\sqrt[2]{9+x}+3]}$$ =$$lim_{x\to 0}\frac {1} {\sqrt[2]{9}+3}$$ =$$1/6$$ Jaime.rincon

Derivadas
De las siguientes funciones, encuentre sus derivadas:

1.$$f(x)=\sqrt[2]{1-x^2}$$
 * Respuesta:

$$f'(x)=\frac {-x} {\sqrt[2]{1-x^2}}$$ Jaime.rincon

2.$$f(x)=\frac {x^2+x+1} {x^2-x+1}$$
 * Respuesta:

$$f'(x)=\frac {(2x+1)(x^2-x+1)-(2x-1)(x^2+x+1)} {(x^2-x+1)^2}$$ $$f'(x)=\frac {-2(x^2-1)} {(x^2-x+1)^2}$$ Jaime.rincon 3.$$f(x)=\frac {xa} {x^2-a^2} + \frac {xa} {x^2+a^2}$$
 * Respuesta:

$$f'(x)=\frac {a(x^2-a^2)-xa(2x)} {(x^2-a^2)^2} + \frac {a(x^2+a^2)-xa(2x)} {(x^2+a^2)^2}$$ $$f'(x)=\frac {-a(x^2-a^2)} {(x^2-a^2)^2} + \frac {a(x^2+a^2)} {(x^2+a^2)^2}$$ Jaime.rincon

4.$$f(x)=\frac {x-\sqrt[2]{x}} {x+\sqrt[2]{x}} $$
 * Respuesta:

$$f'(x)=\frac {(1-1/2\sqrt[2]{x})(x+\sqrt[2]{x})-(x-\sqrt[2]{x})(1-1/2\sqrt[2]{x})} {(x+\sqrt[2]{x})^2}$$ $$f'(x)=\frac {2\sqrt[2]{x}+1} {(x+\sqrt[2]{x})^2}$$ Jaime.rincon

5.$$f(x)=\frac {2x^2+x-8} {3x+2}$$
 * Respuesta:

$$f'(x)=\frac{(4x+1)(3x+2)-3(2x^2+x-8)} {(3x+2)^2}$$ $$f'(x)=\frac{6x^2-8x+26} {(3x+2)^2}$$ Jaime.rincon

6.$$f(x)=\frac {x^2*a^2} {x^3+a^3}$$
 * Respuesta:

$$f'(x)=\frac {2xa^2(x^3+a^3)-3x^2(x^2*a^2)} {(x^3+a^3)^2}$$ $$f'(x)=\frac {xa^2(2a^3-x^3)} {(x^3+a^3)^2}$$ Jaime.rincon

7.$$f(x)=\sqrt[2]{(\frac {x+1}{x-1})^3}$$
 * Respuesta:

$$f'(x)=(\frac {x+1}{x-1})^3/2$$ $$f'(x)=3/2(\frac {x+1}{x-1})^1/2*\frac {(x-1)-(x+1)}{(x-1)^2}$$ $$f'(x)=\frac {-3}{(x-1)^2}*\sqrt[2]{\frac {x+1}{x-1}}$$ Jaime.rincon

8.$$f(x)= \frac{\sqrt[2]{x^2+x+1}} {\sqrt[2]{x^2-x+1}}$$
 * Respuesta:

$$f'(x)= (\sqrt[2]{x^2+1})-(\sqrt[2]{x^2-1})$$ $$f'(x)= 1/2(\frac{2x} {\sqrt[2]{x^2+1}})-1/2(\frac{2x} {\sqrt[2]{x^2-1}})$$ $$f'(x)= x({\frac{1} {\sqrt[2]{x^2-1}}}-{\frac{1} {\sqrt[2]{x^2-1}}})$$ Jaime.rincon

9.$$f(x)= \frac{\sqrt[3]{x+1}+1} {\sqrt[3]{x+1}-1}$$
 * Respuesta:

$$f'(x)= 1+ \frac {2} {\sqrt[3]{x+1}-1}$$ Jaime.rincon

10.$$F(x)= x\sqrt{5+x^2}$$
 * Respuesta:

$$F'(x)= \sqrt{5+x^2}+x\cdot \frac {1}{2\sqrt{5+x^2}}\cdot 2x$$ $$F'(x)= \sqrt{5+x^2}+\frac {2x^2}{2\sqrt{5+x^2}}$$ $$F'(x)= \frac {\sqrt{5+x^2}+x^2}{\sqrt{5+x^2}}$$ $$F'(x)= \frac {5+x^2+x^2}{\sqrt{5+x^2}}$$ $$F'(x)= \frac {5+2x^2}{\sqrt{5+x^2}}$$ Jaime.rincon

11.$$ f(x)=\frac {1-senx}{1+cosx}$$
 * Respuesta:

$$ f'(x)=\frac {-cosx(1+cosx)-(1-senx)-senx}{1+cosx}$$ $$ f'(x)=\frac {-cosx-cos^2x+senx-sen^2x}{1+cos^2x}$$ $$ f'(x)=\frac {-cosx-1+sen^2x+senx-sen^2x}{(1+cosx)^2}$$ $$ f'(x)=\frac {-cosx-1+senx}{(1+cosx)^2}$$ Jaime.rincon

12.$$\ f(x)= (sen(x)+tan(x)) (sec(x)-tan(x))$$
 * Respuesta:

$$\ f'(x)=(tan(x)sec(x)+sec(x)^2)(sec(x)-tan(x))+(tan(x)sec(x)-sec(x)^2)(sec(x)tan(x))$$ $$\ f'(x)=(tan(x)sec(x)+sec(x)^2)(sec(x)-tan(x))+(tan(x)sec(x)-sec(x)^2)(sec(x)+tan(x))$$ $$\ f'(x)=(tan(x)sec(x)^2-tan(x)^2sec(x)+sec(x)^3-sec(x)^2tan(x))+(tan(x)sec(x)^2+tan(x)^2sec(x)-sec(x)^3-sec(x)^2tan(x))$$ $$\ f'(x)= -tan(x)^2sec(x)+sec(x)^3+tan(x)^2sec(x)-sec(x)^3$$ $$\ f'(x)= 0 $$ Jaime.rincon

13.Hallar f''(x) si f(x)=$$\frac {x^2}{x^2-9}$$ f(x)=$$\frac {x^2}{x^2-9}$$
 * Respuesta:

f'(x)=$$\frac {2x(x^2-9)-x^2(2x)}{(x^2-9)^2}$$ f'(x)=$$\frac {2x^3-18x-2x^3}{(x^2-9)^2}$$ f'(x)=$$\frac {-18x}{(x^2-9)^2}$$ f''(x)=$$\frac {-18(x^2-9)^2+18x(4x)(x^2-9)}{(x^2-9)^4}$$ f''(x)=$$\frac {(x^2-9)[-18(x^2-9)+72x^2]}{(x^2-9)^4}$$ f''(x)=$$\frac {54x^2+162}{(x^2-9)^3}$$ Jaime.rincon 14.$$ f(x)=\frac {(x-1)^2}{x^2}$$
 * Respuesta:

$$ f'(x)=\frac {2(x-1)(x^2)-(x-1)^22x}{x^4}$$ $$ f'(x)=\frac {(2x-2)(x^2)-(x^2-2x+1)2x}{x^4}$$ $$ f'(x)=\frac {2x^3-^2x^2-2x^3+4x^2-2x}{x^4}$$ $$ f'(x)=\frac {2x^2-2x}{x^4}$$ $$ f'(x)=\frac {2x(x-1)}{x^4}$$ $$ f'(x)=\frac {2(x-1)}{x^3}$$ Jaime.rincon

15.$$ f(x)=x+(\frac {1}{x})^2$$
 * Respuesta:

$$ f'(x)=2(x+\frac {1}{x})(1-\frac {-1}{x^2})$$ $$ f'(x)=(2x+\frac {2}{x})(1-(\frac {1}{x^2})$$ $$ f'(x)=(\frac {2x^2+2}{x})(\frac {x^2-1}{x^2})$$ $$ f'(x)=\frac {2x^2-2x^2+2x^2-2}{x^3}$$ $$ f'(x)=\frac {2x^4-2}{x^3}$$ Jaime.rincon

Maximos y minimos
1.La función y = f(x)/x tiene valor extremo en Saa. Demostrar que la recta tangente a la curva y = f(x)en x = a pasa por el origen.

 * Y = f(x)/x,

 * Dy/dx = f’ (x) x – f’ (x)/x2

Entonces tenemos

 * f’(a)a=f(a)            (1)

La ecuación de la recta tangente a la curva  y=f(x) en (a, f(a)) es:

 * y–f(a)=f’(a)(x-a)            (2)

Reemplazando la condición (1) en (2):

 * y–f(a)=f’(a)[x–f(a)]

O sea

 * y = f’(a)(x),

ésta pasa por el origen.

2. Hallar el máximo y el mínimo de 3x + 4y cuando x2 + y2 = 1,

Sea S = 3x + 4y donde x2 + y2 = 1      (x ³ 0, y ³ 0)

En los puntos críticos tenemos:

 * dS/dx=3+4(y’)=0

 * 2x+2y(y’)=0

 * $$4x=3y$$

entonces:

 * $$x = 3/5$$, $$y = 4/5$$, $$S = 5$$

En los extremos los valores de S son:

 * $$x = 0$$, $$y = 1$$; $$S = 4$$
 * $$x = 1$$, $$y = 0$$; $$S = 3$$

Por lo tanto:

Máximo de S = 5, Mínimo de S = 3

3.Pruebe que la distancia mínima al eje OX del punto máximo o mínimo de la curva y = ax2+bx+c se obtiene sí b2–4ac=0.

El punto máximo (o mínimo) de la curva y=ax2 +bx+c esta en

 * $$x=-b/2a$$,  :$$y=-b+4ac/4a$$

La distancia del punto al eje OX es:

 * D=|-b^2+4ac/4a|

Entonces D es mínimo cuando b–4ac=0

 * y = f’(a)(x),

ésta pasa por el origen.

Razón de cambio
1.Sean OA, OB y OC tres aristas mutuamente perpendiculares de un paralelepípedo. Si OA crece $$1cm$$/seg, OB crece $$2$$cm/seg y OC decrece 3 cm/seg. Diga con qué rata varía el volumen del sólido cuando OA=$$10$$cm, OB=$$8$$cm, OC=$$12$$cm?

Sean x=OA,y=OB,z=OC,entonces por hipótesis:

 * dx/dt=$$1$$(cm/seg),dy/dt=$$2$$(cm/seg),dz/dt=$$

-3$$(cm/seg),

Sea V el volumen del paralelepípedo entonces

 * dV/dt=(dx/dt)(yz)+(xz)(dy/dt)+(x)(y)(dz/dt)

Si x=$$10$$,y=$$8$$,z=$$12$$ entonces

 * dV/dt=$$(1)(8)(12)$$+$$2(10)(12)$$+$$(-3)(10)(8) $$

 * = $$96$$ ($$cm^3$$/seg).

2.La altura y el radio de un cilindro circular recto aumentan a las velocidades de a cm/seg y b cm/seg. ¿Cuáles son las ratas de crecimiento de su volumen y su superficie total? Sean V, S el volumen y la superficie del cilindro de radio r y altura h, entonces

 * V =$$\pi r2 $$h S =$$2 \pi r2+2 \pi$$r h

entonces

 * dV/dt=$$2 \pi$$r h(dr/dt)+$$\pi r^2$$h(dh/dt)

 * dS/dt =$$4 \pi$$r(dr/dt)+$$2 \pi$$h(dr/dt)+$$2 \pi$$r(dh/dt)

Pero como dh/dt=a, dr/dt=b entonces tenemos

 * dV/dt=$$\pi$$r($$2$$h b+r a)

 * dS/dt=$$2 \pi$$($$2$$r b+h b+r a)

3.Sea f(x) derivable en $$x >0$$, y f(0)= $$0$$, si f ’(x) es creciente en $$x>0$$ entonces demuestre que g(x)=F(x)/x es también creciente en $$x >0$$.

 * g’(x)=[f’(x)(x)–f(x)]/$$x^2$$

 * f‘(x)(x)–f(x)=f‘(x)(x)–[f(x)–f(0)]

 * =f‘(x)x–f(c)(x–$$0$$)	 ($$0$$$$0$$

title: User talk:Jaime.rincon


text:
Template:Objective-C Programming

The Objective-C programming language: Objective-C concepts - Objective-C syntax - Objective-C in depth - Objective-C advanced features

title: Template:Objective-C Programming


text:
User:Ado thetechy

Ado thetechy is a school student on the Central Coast of New South Wales, Australia. He lives at home with his father and two brothers. His interests include computing, mathematics and graphic design.

title: User:Ado thetechy


text:
Cookbook:Fried Eggs

| Breakfast

Fried eggs, a popular breakfast dish in the United States, Canada, and Western Europe, take about 5 minutes to prepare. They are often served with toast or with other fried dishes.

Ingredients

 * 2 teaspoons butter or olive oil
 * 2–3 large eggs, depending on appetite
 * Salt to taste
 * Pepper to taste (optional)

Equipment

 * Small (10-inch) frying pan
 * Spatula
 * Gas ring, at medium heat

Procedure

 * 1) Melt the butter in the pan over low to medium heat.
 * 2) Crack open the eggs into the pan and let fry until the yolks begin to harden at the edges (indicated by a lightening in the yolk color).
 * 3) Using the spatula, flip the eggs over and allow to cook 10 seconds for over-easy, 30 seconds for over-medium, or up to 1 minute for over-hard.
 * 4) Add salt and pepper to taste, and serve.

Notes, tips, and variations

 * [[Image:Fried egg, sunny side up.jpg|thumb|right|Sunny side-up fried egg]]It often considered a cardinal sin to allow the yolk of an egg to break (although some prefer fried eggs with broken yolks). If the cook ensures that the eggs are fresh the yolks will be more 'pert', and less liable to break.
 * For sunny side-up eggs, do not flip the eggs. Use a slightly lower heat, and cover the pan with a lid for some of the cooking period. An ideal sunny side-up has a runny yolk, but the white should be cooked.
 * Uncooked egg white (often called jelly) is normally unwanted and less-preferred than a partially cooked yolk, so err on the side of overcooking rather than undercooking the eggs.
 * For a richer taste, try sprinkling the egg with grated Parmesan or freshly-chopped basil and chives.
 * When frying eggs for sandwiches, rolls or wraps, it is often a good idea to break the yolks. This increases the coverage of yolk so that every bite gets some yolk, and it prevents uncooked yolk dripping out.

Warnings

 * Many people prefer their eggs only partially cooked. Please see the egg ingredient page for possible health concerns of eating raw eggs.

달걀 프라이

title: Cookbook:Fried Eggs


text:
Intelligence Intensification/Information Grasping

Vocabulary
Learn to be cognizant of words that you do not yet understand. Cognizant means to be consciously informed.

It is generally known that it is easier to learn something which you want to learn. When do we want to learn? One situation when we usually want to learn is when we have to in order to solve a concrete problem. This leads to the concept of information availability. If you have access to information when you need it, you will learn more. For instance, if you need to understand a word and have a handheld computer with a dictionary, you will probably look it up. If you haven't any dictionary available, and the need of knowing the word cease to exist, you never learn the word.

We also learn when we are curious. If you can access the information you need, again maybe using a handheld computer, you can learn while you are curious. Later in the day you might feel that other things are more important.

Metaphor
According to some cognitive scientists, notably George Lakoff, most abstract concepts are founded on metaphor. The number line represents quantity as distance, for example, and the Cartesian coordinate system generalizes that same metaphor to function in two dimensions. René Descartes did not invent this metaphor: people from many cultures say "the price went up" (or the equivalent), regardless of their native language. The metaphor "more IS up" (in Lakoff's notation) is quite old, but Descartes did formalize and extend it, allowing it to be applied to a new array of intricate concepts.

But a metaphor can only be extended by a certain amount, after which it loses its meaning. Much of the power of mathematics comes from systematic and well-documented limits to the application of its metaphors. As soon as a new metaphor is added to the field, mathematicians thoroughly test exactly where it does and does not apply, and record their results explicitly in theorems that guide future work.

The convention in other fields is to give metaphors much less explicit limits, forcing the reader or listener to find them out by trial and error. Returning to the "more IS up" example, some quantities tend to go up only with difficulty but go down quite easily, or vice-versa. In these cases, your intuitive understanding of gravity can be added to the metaphor, if you imagine that some quantities sink, while others float. If this doesn't apply, then the metaphorical "weight" of a quantity won't help your understanding.

Some teachers are cognizant of the role that metaphor plays in learning, but in many courses, the metaphors behind the concepts presented are only vaguely hinted at. Actively searching for them and thoroughly testing their limits as early as possible can be quite helpful.


text:
Talk:Guide to Social Activity/Dating ideas

Couldn't dating be a seperate book? (And have a link in the "getting a girl" book to the dating book.)

Because dating is a much broader category than getting a girl.

For example, my situation. I have a girl, we met thru coincidence and it clicked, neither of us were searching for a boy/girl-friend. So I have no need for the "getting a girl" book. However info on dating in general is handy in my situation. And there are dozens of other people that could benefit from good dating ideas that aren't intersted in "getting a girl". I also have some good dating idea's, but I won't add them to the "getting a girl book" because I am opposed to the ethics behind this book. However I would love to share my idea's with other people who are searching for a good idea for a date.

My proposel is a Dating book, or more precisely a "Going out for two book". With Dating idea's and little tips. Interesting idea's to have a good time for lovers.

 * I agree. Please sign you comments. Noldoaran 18:45, 6 Aug 2004 (UTC)

maybe you could meet a girl in the office of the psychiatrist where you should be going to book yourself a long series of appointments

I like the ideas on this topic; especially the 'don't be discouraged' part, where it states that you'll either succeed or get good practice experience. Come to think of it, certain parts of daily living can be approached this way.


text:
Talk:Designing a Training Manual

This page has been copied from Wikipedia, specifically ; because its style was instructive rather than encyclopedic. Duncharris 12:59, 13 Jul 2004 (UTC)
 * When you want content added from Wikipedia, please request an import so we don't lose the edit histoy. It's more work to import, then history-merge. Please ask at WB:RFI or WB:AN for imports in the future. – Mike.lifeguard  | talk 23:20, 10 October 2007 (UTC)

title: Talk:Designing a Training Manual


text:
User:Duncharris~enwikibooks

Hi. Please see my page on Wikipedia, http://en.wikipedia.org/wiki/User:Duncharris for stuff about me.

Duncharris 13:00, 13 Jul 2004 (UTC)

title: User:Duncharris~enwikibooks


text:
User:Mikegr~enwikibooks

Nachrichten auf meiner Diskussionsseite auf Wikipedia hinterlassen.

title: User:Mikegr~enwikibooks


text:
The Voynich Manuscript/F2v

Transcription, comments, theories, links to do with VMs page f2v to be added here... :-)

Description:
One plant centered on page.

 * Root: long straight rhizome, covered with scales or round stalk scars.
 * Stem: single, vertical, slightly bent, shooting out from middle of rhizome; dark painted.
 * Branches: there is the stub of a lost branch at mid-stalk.
 * Leaves: a single large leaf, horizontal, shaped like a very fat C, with smooth rim. Painted solid dark.
 * Stalk: short.
 * Flowers: one white flower, pointing almost straight up.
 * Stalk: long, light colored, with a darker vein; part of it obscured by leaf which should be behind it.
 * Chalyx: well-drawn, with thick tongue-shaped sepals, light colored.
 * Petals: well-drawn, flaring, lily shaped, serrated edges, white underpaint.
 * Core: not visible.
 * Stamen: long, with a few short hairs.

Two paragraphs (with lines, respectively), left- and right-justified: one at the top, interrupted by the single flower, and one just below mid-page, interrupted by the main stem.

Comments:
The plant looks quite natural and well-drawn.

Presumably, the flower stem was in front of the leaf in the original drawing; but the dark overpaint covered it. The original outline of the leaf may have been obscured, too.

The leaf shape suggests it is a water plant. With that assumption, the choices are quite few. Rene suggested

Nymphaea candida (a.k.a. Nymphaea alba; water lily, seerose)

http://runeberg.org/nordflor/pics/181.jpg

title: The Voynich Manuscript/F2v


text:
The Voynich Manuscript/F3r

Transcription, comments, theories, links to do with VMs page f3r to be added here...

Description:
One plant, flush against the right margin.

 * Roots: a bunch of unbranched tentacle-like roots radiating from the stem's base. Light colored, partial dark overpaint.
 * Stem: rather thick, bent. Light colored, dark overpainted stripe (shadow) along its left side.
 * Branches: none.
 * Leaves: lance-shaped with an S bend, very dense, growing straight off the stem on both sides, each partially covering the previous one. Stem ends with one unpaired leaf. Light colored, with a dark-painted shadow on the bottom of each leaf, and a dotted line just inside the leaf's rim.
 * Flowers: none.

Four paragraphs (with 9.5, 3.5, 2.7, and 2.8 lines): a longer one at the top, and three short ones just below mid-page. The first two are left-justified and follow the plant's outline on the right. The last two are left- and right-justified, and interrupted by the main stem.

Comments:
The plant looks strange because of the dense leaves. Perhaps they are actually the sheaths of leafstalks, badly drawn? Perhaps the dots along the leaf rim are short spines?

title: The Voynich Manuscript/F3r


text:
The Voynich Manuscript/F3v

Transcription, comments, theories, links to do with VMs page f3v to be added here... :-)

Description:
One plant, flush against the bottom and right margins of the page.

 * Root: single long root, consisting of six "telescoping" conical segments, dark painted, with flat light-painted top. Tip of root is bent into a hook and ends with a small round knob.
 * Stem: long, straight, vertical, with a dark "shadow" stripe crudely painted along its length.
 * Branches: none.
 * Leaves: paired, disk shaped with two "tails", two "arms", and a swallowtail-shaped "head", with long outward-curved tips.
 * Stalks: medium length, thin.
 * Flowers: two, very unequal size, growing out of the tip of the stem.
 * Stalks: short, thin, like leaf stalks.
 * Calyx: heavy, swollen, shaped like a sweet potato cut in half, dark painted.
 * Petals: single row, light, semicircular, very small.
 * Core: flat, light-colored, filled with dots; looks like a pepper shaker.

Two paragraphs (with 6.0 and 7.5 lines), both in the top half of the page. The text is left-justified and follows the plant outline on the right.

Comments:
The plant looks quite strange. The telescoping root is hard to explain. The leaf shape is unusual but not impossible. The bloated flower chalices are very strange; they may have been drawn from pressed and dried specimens.

title: The Voynich Manuscript/F3v


text:
The Voynich Manuscript/F4r

Transcription, comments, theories, links to do with VMs page f4r to be added here... :-)

Description:
One plant, mostly in the right half of the page.

Two paragraphs (with 3.8 and 8.6 lines), both above mid-page. The first is left- and right-justified, and interrupted once by the flowers; the second is left-justified and follows the plant's outline on the right.

Comments:
Plant appearance: very normal, exceptionally well-drawn. The layout of leaves and flowers is very naturalistic. The "dark painter" apparently spared it.

title: The Voynich Manuscript/F4r


text:
The Voynich Manuscript/F4v

Transcription, comments, theories, links to do with VMs page f4v to be added here...

Description:
One plant, mostly in the right half of the page.

Two paragraphs, both above mid-page. The first is left- and right-justified; the second is left-justified, and follows the plant's profile on the right.

Comments:
Plant appearance: somewhat bizarre, and badly drawn. It seems to have two types of leaves. The lower ones are too small, badly spaced and badly proportioned, and in unnatural positions. The upper ones look like pompons. The root has a strange, ungainly shape (but that may be the work of the Dark Painter). The stems have uneven and unnatural thickness.

This drawing is almost certainly an expanded version of plant f101r1[1,5]. Stolfi suggested that the person who drew this page thought that the star-shaped things in f101r1[1,5] were flowers, so he supplied fantasy leaves; and he only noticed the mistake halfway through the plant.

title: The Voynich Manuscript/F4v


text:
The Voynich Manuscript/F5r

Transcription, comments, theories, links to do with VMs page f5r to be added here... :-)

Description:
One plant, horizontally centered, reaching only 3/4 up the page.

 * Root: five unbranched roots, mostly horizontal. Each root looks like an earthworm, light-colored with a few dark dots, with a  short and fat swelling about 4/5 of the way to the tip.

 * Stalk: straight, vertical, slightly tapered, uncolored(?).

 * Leaves: a single ring of large, lance shaped leaves, closely spaced, attached mid-stem with no stalks. The leaves curl down in a semicircle, like an umbrella, until they almost touch the stem again; their thin tips curl further into hooks. Dark overpaint.

 * Flowers: one, at tip of stem, pointing up.

 * Stalk: medium, thin, uncolored(?); extension of stem.

 * Chalix: thimble-shaped, partial dark overpaint.

 * Petals: long, hair like, drooping.

 * Core: hemispherical, like a bubble.

Two paragraphs (with 3.6 and 2.5 lines), both above the plant, left- and right-justified.

Comments:
The plant looks very strange but not impossible. The oddest details are the position of the tubers, the shape and disposition of the leaves, their curled tips, the "shaggy dog" flower.

title: The Voynich Manuscript/F5r


text:
The Voynich Manuscript/F5v

Transcription, comments, theories, links to do with VMs page f5v to be added here...

Description:
One branching plant that takes up the lower 3/4 of the page.

 * Root: two dark-painted potato-shaped tubers next to the stem, with a few medium-length "mousetail" roots shooting from them.

 * Stem: straight and vertical, light under(?)color, with a fantastic loop at the top. The stem's base is wrapped with a short uncolored sheath.

 * Branches: several, diagonal, mostly straight.

 * Leaves: four lobes with serrated edges, naturalistically curved. Each lobe has a central vein.

 * Stalks: medium-short, thin, look like branches. Light coulored, many of them with crudely overpainted dark splotches.

 * Flowers: several, medium-small, mostly upright, some drooping.

 * Stalk: short, sometimes bent by weight.

 * Chalyx: rounded, light colored, like a rose's but on the fat side.

 * Sepals: short and pointed, light colored.

 * Petals: half a dozen, small, tongue-shaped, spread out; dark overpaint almost but not quite fills their outline.

One paragraph (5.5 lines) above the plant, left- and right-justified.

Comments:
The plant looks fairly normal, except for the fantastic loop in the stalk. Leaves and flowers are very naturalistic, but some are spoiled by the crude dark overpaint.

title: The Voynich Manuscript/F5v


text:
Electrodynamics/Coulombs Law

Coulomb's Law for Point Charges
The repulsive or attractive electrostatic force between two point charges is determined by an equation called Coulomb's law. Consider the situation where we have two charges, labeled for convenience q, and Q ("q" or "Q" are common variables to describe a point charge, and we will use these letters to describe charges throughtout this book). The following picture shows charge q at a certain point with charge Q at a distance of r away from it. The presence of Q causes an electrostatic force to be exerted on q. The distance vector between Q and q is r.

Using Coulomb's Law, we can find the strength of the electric force between these two charges:

 * $$F = k_\mathrm{e} \frac {q Q}{r^2}$$

Where: $$k_\mathrm{e}$$ = Coulomb's constant = 8.9875 × 109  N·m2/C2 in free space.

The magnitude of the electrostatic force F, on charge q, due to charge Q, equals Coulomb's constant times the product of the two charges (in coulombs) divided by the square of the distance r, between the charges q and Q.

Dielectrics
The value of Coulomb's constant given here is such that the preceding Coulomb's Law equation will work if both q and Q are given in units of coulombs, r in meters, and F in newtons and there is no dielectric material between the charges. A dielectric material is one that reduces the electrostatic force when placed between charges. Furthermore, Coulomb's constant can be given by:

 * $$k = \frac {1}{4 \pi \epsilon }$$

where ε = permittivity. When there is no dielectric material between the charges (for example, in free space or a vacuum),

 * $$\epsilon _0 = 8.85419 \times 10^{-12} \frac{C^2}{(N \dot m^2)}$$

Air is only very weakly dielectric and the value above for ε0 will work well enough with air between the charges. If a dielectric material is present, then:

 * $$\epsilon = \kappa \epsilon _0\,$$

where κ is the dielectric constant which depends on the dielectric material. In a vacuum (free space), κ = 1 and thus ε = ε0. For air, κ = 1.0006. Typically, solid insulating materials have values of κ > 1 and will reduce electric force between charges. The dielectric constant can also be called relative permittivity, symbolized as εr in Wikipedia.

This shows that the electric force between the charges decreases as the charges are located further from each other by the square of the distance between them. As the charges become located further enough apart, their effect on each other becomes negligible.

Force Vectors
Any force on an object is a vector quantity. In a force vector, the direction is the one in which the force pulls the object. The symbol F is used here for the electric force vector. If charges q and Q are either both positive or both negative, then they will repel each other. This means the direction of the electric force F on q due to Q is away from Q in exactly the opposite direction, as shown by the red arrow in the preceding diagram. If one of the charges is positive and the other negative, then they will attract each other. This means that the direction of F on q due to Q is exactly in the direction towards Q, as shown by the blue arrow in the preceding diagram. The Coulomb's equation shown above will give a magnitude for a repulsive force away from the Q charge. If the magnitude given by the above equation is negative due to opposite charges, the direction of the resulting force will be towards Q, an attractive force. In other sources, different variations of Coulombs' Law are given, including vector formulas in some cases (see Wikipedia link and reference(s) below).

Vector Form
We can vectorize Coulomb's Law to use position and force vectors instead of scalar quantities. We can express the location of charge q as rq, and the location of charge Q as rQ. In this way we can know both how strong the electric force is on a charge, but also what direction that force is directed in. Coulomb's Law using vectors can be written as:

 * $$\mathbf{F} = \frac{kqQ(\mathbf{r}_q - \mathbf{r}_Q)}{|\mathbf{r}_q - \mathbf{r}_Q|^3}$$

If we have r = rq - rQ, and r = |r|, then we can rewrite this equation as:

 * $$\mathbf{F} = \frac{kqQ\mathbf{r}}{r^3}$$

The vector F is a force vector that shows the direction and the magnitude of the force.

n Charges
In many situations, there may be many charges, Q1, Q2, Q3, through Qn, on the charge q in question. Each of the Q1 through Qn charges will exert an electric force on q. The direction of the force depends on the location of the surrounding charges. A Coulomb's Law calculation between q and a corresponding Qi charge would give the magnitude of the electric force exerted by each of the Qi charges for i = 1 through n, but the direction of each of the component forces must also be used to determine the individual force vectors, F1, F2, F3, ...., Fn. To determine the total electric force on q, the electric force contributions from each of these charges add up as vector quantities, not just like ordinary (or scalar) numbers.

 * $$\mathbf{F}_{q} = \mathbf{F}_{1} + \mathbf{F}_{2} + \mathbf{F}_{3} + ..... + \mathbf{F}_{n}$$

The total electric force on q can be added to any other forces affecting it as a vector quantity to obtain the total force vector on the charged object q. In many cases, there are billions of electrons or other charges present, so that geometrical distributions of charges are used with equations stemming from Coulomb's Law.

We can re-write Coulomb's law as a sum of n charges:

 * $$\mathbf{F_n} = \sum_{i \ne n} \frac{q_n q_i(\mathbf{r}_n - \mathbf{r}_i)}{4 \pi \epsilon_0|\mathbf{r}_n - \mathbf{r}_i|^3}$$

title: Electrodynamics/Coulombs Law


text:
User:Wheat~enwikibooks

I've been contributing to wikipedia for a while but just recently discovered the wikibooks project. I hope to contribute some useful things to the PHP section. I've also started a project for ASP.NET

title: User:Wheat~enwikibooks


text:
Scrapebook Linguistics

Linguistics is the study of human language(s) by scientific method(s) in the spoken, written and preconscious form.

Languages that cannot be reliably classified into any family are known as language isolates.

In the scientific practice of linguistics, several distinct areas of study are recognized. Each represents a different aspect or level of abstraction. These range from Phonetics--the study of the acoustic, anatomical, and other such aspects of the physical production or qualities of vocal sounds; to syntax--the study of the rules and organization of words and the relationships existing between them.

 * Phonetics is the study and systematic classification of the sounds made in spoken utterance.
 * Phonology is the study and description of the sound changes in a language.
 * Morphology is the study of the internal structure of words.
 * Syntax is the study of the structure of phrases, sentences and language.
 * Semantics is the study of the relationship between words and their meanings.
 * Pragmatics is the study of how meaning changes according to context.
 * Stylistics is the study of why a particular choice of language is used in a particular context.

Computational Linguistics
An interdisciplinary field dealing with the statistical and logical modeling of natural language from a computational perspective. This modeling is not limited to any particular field of linguistics.

There are two major purposes of computational linguistics:
 * 1) To help linguists study natural languages more easily
 * 2) To make it possible for ordinary people to use natural language when using a computer

Computational linguists often use large bodies of digitized text or speech called corpora as a basis for teaching computer programs the proper use of a language, or to compare the use of a language in one context to it's use in another context. This is also called corpus linguistics.

Some of the practical uses of computational linguistics include:
 * Speech recognition
 * Speech production
 * Machine translation
 * Improvements to data mining, including  search engine design.

See Also: computational linguistics

title: Scrapebook Linguistics


text:
User:Jansegers

My name is Pieter Jansegers.

I'm a teacher of French in the Dutch speaking part of Belgium.

I did study French and Italian (both litterature and linguistics) at the University of Ghent.

I'm an ICT enthusiast and my contribution on this system will be concentrated on three points:

1) French

2) linguistics

3) ICT use in general

some of my sites:

http://jansegers.blogspot.com

http://vocabulaire.eduinfo.com

http://ticerama.eduinfo.com

other info needed, see:

http://dmoz.org/profiles/jansegers.html

title: User:Jansegers


text:
User:Emilia~enwikibooks

Demostraciones de límites
Teniendo en cuenta la definición de límite demuestre que los límites pedidos existen

1)$$\lim_{x\to 4} 3x-7=5$$
 * Análisis Preliminar
 * Si el límite existe se debe cumplir:
 * 0 < |x-4| < δ → |(3x-7)-5| < ε
 * Consideremos la parte derecha
 * (3x-7)-5| < ε
 * 3x-12| < ε
 * 3(x-4)| < ε
 * 3| |x-4| < ε
 * x-4| < $$\frac {\epsilon} {3}$$
 * Sea δ >0, se elige un δ = $$\frac {\epsilon} {3}$$
 * 0 < |x-4| < δ → |x-4| < $$\frac {\epsilon} {3}$$
 * →|x-4| |3| < ε
 * →|3x-12| < ε
 * →|(3x-7)-5| < ε

Emilia

2)$$\lim_{x \ to 4} \frac {1} {2} x +3 = 5$$
 * Analisis Preliminar
 * Si el límite existe, se debe cumplir:
 * 0 < |x-4| < δ → |$$\frac {1} {2} x$$ +3 -5| < ε
 * Consideremos la parte derecha
 * \$$frac {1} {2} x$$ +3 -5| < ε
 * |$$\frac {1} {2} x$$-2| < ε
 * 2|$$\frac {1} {2} x$$ -2| < 2ε
 * |x-4|< 2ε
 * Sea δ >0, se elige un δ = 2ε
 * 0 < |x-4| < δ →  |x-4|< 2ε
 * → |$$\frac {|x-4|} {|2|}|$$ < ε
 * → |$$\frac {1} {2}$$ x-2|< ε
 * → |($$\frac {1} {2}$$x+3)-5| < ε

Emilia

3)$$\lim_{x \ to1} \frac {2x^2+2x-4} {x-1} = 6$$
 * Análisis Preliminar
 * Si el límite existe se debe cumplir:
 * 0 < |x-1| < δ → |$$\frac {2x^2+2x-4} {x-1}$$ - 6| < ε
 * Consideremos la parte derecha
 * $$\frac {2x^2+2x-4} {x-1}$$ - 6| < ε
 * 6 - ε < $$\frac {2x^2+2x -4} {x-1}$$ < ε +6
 * 6 - ε < $$\frac {2(x^2+x -2)} {x-1}$$ < ε +6
 * 6 - ε < $$\frac {2(x-1)(x+2)} {x-1}$$ < ε +6
 * 6 - ε < 2(x+2) < ε+6
 * 3-$$\frac {\epsilon} {2}$$ < x+2 < $$\frac {\epsilon} {2}$$ +3
 * -$$\frac {\epsilon} {2}$$ < x+1 <$$\frac {\epsilon} {2}$$
 * x-1| < $$\frac {\epsilon} {2}$$
 * Sea δ >0, se elige un δ = $$\frac {\epsilon} {2}$$
 * 0 < |x-1| < δ → |x-1| < $$\frac {\epsilon} {2}$$
 * →|x-1|< $$\frac {\epsilon} {2}$$
 * → -$$\frac{\epsilon}{2}$$< x-1 < $$\frac{\epsilon}{2}$$
 * →3-$$\frac{\epsilon}{2}$$< x-1+3<$$\frac{\epsilon}{2}$$+3
 * →3-$$\frac{\epsilon}{2}$$< x+2 <$$\frac{\epsilon}{2}$$+3
 * →$$\frac{6-\epsilon}{2}$$< x+2<$$\frac{6+\epsilon}{2}$$
 * →6-ε < 2(x+2) < 6+ε
 * → -ε < 2(x+2)+6 < ε
 * →|2(x+2)+6| <ε
 * →|$$\frac {x-1} {x-1}$$| |2(x+2)-6| < ε
 * →|$$\frac {2(x+2)(x-1)}{x-1}$$ - $$\frac {6(x-1)} {x-1}$$| < ε
 * →|$$\frac {2(x^2-x+2x+2)} {x-1}$$ -6| < ε
 * →|$$\frac {2x^2+2x+4} {x-1}$$ -6 | < ε

Emilia

4)$$ \lim_{x \ to 3} \sqrt{3x-5} = 2$$
 * Análisis Preliminar
 * Si el límite existe, se debe cumplir:
 * 0 < |x-3| < δ → |$$\sqrt {3x-5}$$ - 2| < ε
 * Consideremos la parte derecha
 * $$\sqrt {3x-5}$$ - 2| < ε
 * $$\sqrt {3x-5} - 2|^2$$ < $$\epsilon^2$$
 * $$(\sqrt {3x-5})^2 - (2)^2$$| < $$\epsilon^2$$
 * 3x-5-4|< $$\epsilon^2$$
 * 3x-9|< $$\epsilon^2$$
 * 3(x-3)| < $$\epsilon^2$$
 * 3| |x-3| < $$\epsilon^2$$
 * x-3| < $$\frac {\epsilon^2} {3}$$
 * Sea δ >0, se elige un δ = $$\frac {\epsilon^2} {3}$$
 * 0 < |x-3| < δ → |x-3| < $$\frac {\epsilon^2} {3}$$
 * →|3x-9| < $$\epsilon^2$$
 * →|(3x-5)-4| < $$\epsilon ^2$$
 * →|$$\sqrt{3x-5} - \sqrt{4}$$| < $$\sqrt{\epsilon^2}$$
 * →|$$\sqrt{3x-5}$$ - 2| < ε

Emilia

Derivadas
Encontrar las derivadas pedidas

1)Derivar:$$\ y=x^2+2x+17 $$
 * $$\ y'=2x+2 $$

Emilia

2)Derivar:$$\ y=(x+4)(2x+3)$$
 * $$\ y'=x(2x+3)+ 2(x+4)$$
 * $$\ 2x^2+3x+2x+4$$
 * $$\ 2x^2+5x+4$$

Emilia

3)Derivar:$$y= \frac{2x+1}{x+6}$$
 * $$y'=\frac{2(x+6)-1(2x+1)}{(x+6)^2}$$
 * $$y'=\frac {2x+12-2x+1}{(x+6)^2}$$
 * $$y'=\frac{11}{(x+6)^2}$$

Emilia

4)Derivar:$$\ y^3+7y=x^3$$
 * $$\ 3y^2y'+7y=3x^2$$
 * $$\ y'= \frac{3x^2}{3y^2+7}$$

Emilia

5)Derivar:$$y= \frac{e^(2x+3)}{(x+1)^2}$$
 * $$y'= \frac {e^(2x+3) 2(x+1)^2-2(x+1)e^2x+3}{(x+1)^4}$$
 * $$y'= \frac {(x+1)[e^(2x+3)2(x+1)-e^2x+3]}{(x+1)^4} $$
 * $$y'= \frac {e^(2x+3) 2(x+1)-2e^2x+3}{(x+1)^3}$$
 * $$y'= \frac {2e^(2x+3) [(x+1)-1]}{(x+1)^3}$$

Emilia

6)Derivar:$$\ y=3x^4x^3+2x-5$$
 * $$y'= 3x^4x^3+2x-5 [\frac{3 (4x^3+2x-5}{3x}+12x^2+2(ln3x)]$$
 * $$y'= 3x^4x^3+2x-5 [\frac{ (4x^3+2x-5}{x}+12x^2+2(ln3x)]$$
 * $$\ y'= 3x^4x^3+2x-5 [\frac{ (4x^3+2x-5+12x^2+2)(ln3x)}{x}]$$

Emilia

7)Derivar: $$\ y^2+x^2=16$$
 * $$\ 2yy'+2x=0$$
 * $$\ 2yy'= -2x$$
 * $$\ y'= \frac{-2x}{2y}$$
 * $$\ y' = \frac {-x}{y}$$

Emilia

8)Derivar:$$f(x)= \frac {1}{x+1}$$
 * $$\ f'(x)= \frac {-1}{(x+1)^2}$$

Emilia

Derivadas Trigonometricas
1)Derivar:$$\ y=Cos^-1x$$
 * $$\ Cosy=x$$
 * $$\ -Senyy'=1$$
 * $$\ y'=\frac{1}{-seny}$$
 * $$\ y'=\frac{-1}{\sqrt{1-cos^2y}}$$
 * $$\ y'=\frac{-1}{\sqrt{1-x^2}}$$

Emilia

2)Derivar:$$\ y=tan^-1x$$
 * $$\ tany=x$$
 * $$\ sec^2yy'=1$$
 * $$\ y'=\frac{1}{sec^2y}$$
 * $$\ y'=\frac{1}{1+tan^2y}$$
 * $$\ y'=\frac{1}{1+x^2}$$

Emilia

3)Derivar$$\ y=sec^-1x$$
 * $$\ secy=x$$
 * $$\ (tany secy)y'=1$$
 * $$y'=\frac{1}{tany secy}$$
 * $$y'=\frac{1}{secy\sqrt{sec^2-1}}$$
 * $$y'=\frac{1}{x\sqrt{x^2-1}}$$

Emilia

4)Derivar:$$\ y=csc^-1x$$
 * $$\ cscy=x$$
 * $$\ (-cscy ctgy)y'=1$$
 * $$\ y'=\frac{1}{-cscy ctgy}$$
 * $$\ y'=\frac{-1}{cscy \sqrt{csc^2y-1}}$$
 * $$\ y'=\frac{-1}{x\sqrt{x^2-1}}$$

Emilia

5)Derivar:$$\ y= (sen(x)+tan(x)) (sec(x)-tan(x))$$
 * $$\ y'=(tan(x)sec(x)+sec(x)^2)(sec(x)-tan(x))+(tan(x)sec(x)-sec(x)^2)(sec(x)tan(x))$$
 * $$\ y'=(tan(x)sec(x)+sec(x)^2)(sec(x)-tan(x))+(tan(x)sec(x)-sec(x)^2)(sec(x)+tan(x))$$
 * $$\ y'=(tan(x)sec(x)^2-tan(x)^2sec(x)+sec(x)^3-sec(x)^2tan(x))+(tan(x)sec(x)^2+tan(x)^2sec(x)-sec(x)^3-sec(x)^2tan(x))$$
 * $$\ y'= -tan(x)^2sec(x)+sec(x)^3+tan(x)^2sec(x)-sec(x)^3$$
 * $$\ y'= 0 $$

Emilia

6)Derivar:$$\ y=e^(2x) arccos(3x+1)$$
 * $$y'= e^(2x) 2x arccos(3x+1)+e^2x\frac{3}{\sqrt{1-(3x+1)}}$$
 * $$y'= e^(2x) 2x arccos(3x+1)+e^2x\frac{3}{\sqrt{-3x}}$$
 * $$y'= e^(2x) (2xarccos(3x+1)+\frac{3}{\sqrt{-3x}}$$

Emilia

7)Derivar: $$\ y= 6^(tan^2(3x))$$
 * $$\ y'= 6^(tan^2(3x)) ln6 2tan(3x) sec^2(3x) 3$$
 * $$\ y'= 6^(tan^2(3x)) ln6 6tan(3x) sec^2(3x) 3$$

Emilia

8)Derivar:$$f(x)= \frac {1}{x^2+1}+x^5cos(x)$$
 * $$f'(x)= \frac {-2x}{(x^2+1)^2}+5x^4cos(x)-sen(x)x^5$$

Emilia

luz Emilia

2)Erica tiene 200 pies de valla con la cual planea encerrar un patio rectangular para su perro. Si desea encerrar el area máxima, cuáles serian las dimensiones de esta?

 * $$200=2y+2x *$$
 * $$A=xy$$
 * despejamos y en *     $$y=100-x$$
 * reemplazando          $$A=x(100-x)$$
 * $$A=100x-x^2$$
 * derivando            $$A'=100-2x$$
 * igualando a cero     $$ 0=100-2x$$
 * despejando          $$-2x=-100$$
 * $$x=\frac{-100}{-2}$$
 * $$x=50$$
 * reemplazando x en * $$200=2y+2(50)$$
 * despejando y         $$ 2y=200-100$$
 * $$2y=100$$
 * $$y=50$$

Las dimensiones para encerrar el area máxima serian x=50; y=50 (Tomado del Cálculo de Purcell) Emilia

Razón de Cambio
1) Una escalera de 2m está apoyada contra una pared, la base de la escalera resbala alejandose de la pared, a razón de 0.3m/s. ¿Con qué rapidez desciende el extremo superior de la escalera cuando se encuentra a 0.8m del piso?

 * $$\frac{dx}{dt}=0.3m/s$$
 * $$\frac{dy}{dt}=? cuando x=0.8m$$
 * $$y^2+x^2=4$$
 * $$2yy'+2xx'=0*$$
 * $$Si x=0.8$$
 * $$y^2+0.64=4$$
 * $$y^2=4-0.64$$
 * $$y=\sqrt{3.36}$$
 * $$y=1.83$$
 * reemplazando en *
 * $$2(1.83)y'+2(0.3)(0.8)=0$$
 * $$3.66y'=-0.48$$
 * $$y'=-0.13$$

La escalera desciende con una rapidez de 0.13m/s Emilia

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