Foundations

End Behavior of a Function

The end behavior of a polynomial function is the behavior of the graph of

f (x)

x

approaches positive infinity or negative infinity.

The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.

The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. So, the sign of the leading coefficient is sufficient to predict the end behavior of the function.

Degree	Leading Coefficient	End behavior of the function	Graph of the function
Even	Positive	$\begin{array}{l} f (x) \to + \infty, as x \to - \infty \\ f (x) \to + \infty, as x \to + \infty \end{array}$	Example: $f (x) = x^{2}$
Even	Negative	$\begin{array}{l} f (x) \to - \infty, as x \to - \infty \\ f (x) \to - \infty, as x \to + \infty \end{array}$	Example: $f (x) = - x^{2}$
Odd	Positive	$\begin{array}{l} f (x) \to - \infty, as x \to - \infty \\ f (x) \to + \infty, as x \to + \infty \end{array}$	Example: $f (x) = x^{3}$
Odd	Negative	$\begin{array}{l} f (x) \to + \infty, as x \to - \infty \\ f (x) \to - \infty, as x \to + \infty \end{array}$	Example: $f (x) = - x^{3}$

https://www.google.com/search?ei=rC4KWvhxxIuZAbW5vfgG&q=end+behavior+linear+equation&oq=end+behavior+linear+equation&gs_l=psy-ab.3...7098.16178.0.17174.0.0.0.0.0.0.0.0..0.0....0...1.1.64.psy-ab..0.0.0....0.0WEIVGs1c9U

Anatomy

Mouth
Food begins its journey through the digestive system in the mouth, also known as the oral cavity. Inside the mouth are many accessory organs that aid in the digestion of food—the tongue, teeth, and salivary glands. Teeth chop food into small pieces, which are moistened by saliva before the tongue and other muscles push the food into the pharynx. Oral cavity, cross-section

Teeth. The teeth are 32 small, hard organs found along the anterior and lateral edges of the mouth. Each tooth is made of a bone-like substance called dentin and covered in a layer of enamel—the hardest substance in the body. Teeth are living organs and contain blood vessels and nerves under the dentin in a soft region known as the pulp. The teeth are designed for cutting and grinding food into smaller pieces.
Tongue. The tongue is located on the inferior portion of the mouth just posterior and medial to the teeth. It is a small organ made up of several pairs of muscles covered in a thin, bumpy, skin-like layer. The outside of the tongue contains many rough papillae for gripping food as it is moved by the tongue’s muscles. The taste buds on the surface of the tongue detect taste molecules in food and connect to nerves in the tongue to send taste information to the brain. The tongue also helps to push food toward the posterior part of the mouth for swallowing.
Salivary Glands. Surrounding the mouth are 3 sets of salivary glands. The salivary glands are accessory organs that produce a watery secretion known as saliva. Saliva helps to moisten food and begins the digestion of carbohydrates. The body also uses saliva to lubricate food as it passes through the mouth, pharynx, and esophagus.

Pharynx
The pharynx, or throat, is a funnel-shaped tube connected to the posterior end of the mouth. The pharynx is responsible for the passing of masses of chewed food from the mouth to the esophagus. The pharynx also plays an important role in the respiratory system, as air from the nasal cavity passes through the pharynx on its way to the larynx and eventually the lungs. Because the pharynx serves two different functions, it contains a flap of tissue known as the epiglottis that acts as a switch to route food to the esophagus and air to the larynx.

Esophagus
The esophagus is a muscular tube connecting the pharynx to the stomach that is part of the upper gastrointestinal tract. It carries swallowed masses of chewed food along its length. At the inferior end of the esophagus is a muscular ring called the lower Stomach, gallbladder and pancreas

esophageal sphincter or cardiac sphincter. The function of this sphincter is to close of the end of the esophagus and trap food in the stomach.

Stomach
The stomach is a muscular sac that is located on the left side of the abdominal cavity, just inferior to the diaphragm. In an average person, the stomach is about the size of their two fists placed next to each other. This major organ acts as a storage tank for food so that the body has time to digest large meals properly. The stomach also contains hydrochloric acid and digestive enzymes that continue the digestion of food that began in the mouth.

Small Intestine
The small intestine is a long, thin tube about 1 inch in diameter and about 10 feet long that is part of the lower gastrointestinal tract. It is located just inferior to the stomach and takes up most of the space in the abdominal cavity. The entire small intestine is coiled like a hose and the inside surface is full of many ridges and folds. These folds are used to maximize the digestion of food and absorption of nutrients. By the time food leaves the small intestine, around 90% of all nutrients have been extracted from the food that entered it.
Liver and Gallbladder
The liver is a roughly triangular accessory organ of the digestive system located to the right of the stomach, just inferior to the diaphragm and superior to the small intestine. The liver weighs about 3 pounds and is the second largest organ in the body. The liver has many different functions in the body, but the main function of the liver in digestion is the production of bile and its secretion into the small intestine. The gallbladder is a small, pear-shaped organ located just posterior to the liver. The gallbladder is used to store and recycle excess bile from the small intestine so that it can be reused for the digestion of subsequent meals.

Pancreas
The pancreas is a large gland located just inferior and posterior to the stomach. It is about 6 inches long and shaped like short, lumpy snake with its “head” connected to the duodenum and its “tail” pointing to the left wall of the abdominal cavity. The pancreas secretes digestive enzymes into the small intestine to complete the chemical digestion of foods.

Large Intestine
The large intestine is a long, thick tube about 2 ½ inches in diameter and about 5 feet long. It is located just inferior to the stomach and wraps around the superior and lateral border of the small intestine. The large intestine absorbs water and contains many symbiotic bacteria that aid in the breaking down of wastes to extract some small amounts of nutrients. Feces in the large intestine exit the body through the anal canal.

The digestive system is responsible for taking whole foods and turning them into energy and nutrients to allow the body to function, grow, and repair itself. The six primary processes of the digestive system include:

Ingestion of food
Secretion of fluids and digestive enzymes
Mixing and movement of food and wastes through the body
Digestion of food into smaller pieces
Absorption of nutrients
Excretion of wastes

Ingestion
The first function of the digestive system is ingestion, or the intake of food. The mouth is responsible for this function, as it is the orifice through which all food enters the body. The mouth and stomach are also responsible for the storage of food as it is waiting to be digested. This storage capacity allows the body to eat only a few times each day and to ingest more food than it can process at one time.

Secretion
In the course of a day, the digestive system secretes around 7 liters of fluids. These fluids include saliva, mucus, hydrochloric acid, enzymes, and bile. Saliva moistens dry food and contains salivary amylase, a digestive enzyme that begins the digestion of carbohydrates. Mucus serves as a protective barrier and lubricant inside of the GI tract. Hydrochloric acid helps to digest food chemically and protects the body by killing bacteria present in our food. Enzymes are like tiny biochemical machines that disassemble large macromolecules like proteins, carbohydrates, and lipids into their smaller components. Finally, bile is used to emulsify large masses of lipids into tiny globules for easy digestion.

Mixing and Movement
The digestive system uses 3 main processes to move and mix food:

Swallowing. Swallowing is the process of using smooth and skeletal muscles in the mouth, tongue, and pharynx to push food out of the mouth, through the pharynx, and into the esophagus.
Peristalsis. Peristalsis is a muscular wave that travels the length of the GI tract, moving partially digested food a short distance down the tract. It takes many waves of peristalsis for food to travel from the esophagus, through the stomach and intestines, and reach the end of the GI tract.
Segmentation. Segmentation occurs only in the small intestine as short segments of intestine contract like hands squeezing a toothpaste tube. Segmentation helps to increase the absorption of nutrients by mixing food and increasing its contact with the walls of the intestine.

http://www.innerbody.com/image/digeov.html

Geometry

Independent Events

When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring. An example of two independent events is as follows; say you rolled a die and flipped a coin. The probability of getting any number face on the die in no way influences the probability of getting a head or a tail on the coin.

Dependent Events

When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event.

For example, if you were to draw a two cards from a deck of 52 cards. If on your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is greatly changed because you drew an ace the first time. Let's calculate these different probabilities to see what's going on.

There are 4 Aces in a deck of 52 cards

On your first draw, the probability of getting an ace is given by:

If we don't return this card into the deck, the probability of drawing an ace on the second pick is given by

As you can clearly see, the above two probabilities are different, so we say that the two events are dependent. The likelihood of the second event depends on what happens in the first event.

Conditional Probability

We have already defined dependent and independent events and seen how probability of one event relates to the probability of the other event.

Having those concepts in mind, we can now look at conditional probability.

Conditional probability deals with further defining dependence of events by looking at probability of an event given that some other event first occurs.

Conditional probability is denoted by the following:

The above is read as the probability that B occurs given that A has already occurred.

The above is mathematically defined as:

Set Theory in Probability

A sample space is defined as a universal set of all possible outcomes from a given experiment.

Given two events A and B and given that these events are part of a sample space S. This sample space is represented as a set as in the diagram below.

The entire sample space of S is given by:

Remember the following from set theory:

The different regions of the set S can be explained as using the rules of probability.

Rules of Probability

When dealing with more than one event, there are certain rules that we must follow when studying probability of these events. These rules depend greatly on whether the events we are looking at are Independent or dependent on each other.

First acknowledge that

Multiplication Rule (A∩B)

This region is referred to as 'A intersection B' and in probability; this region refers to the event that both A and B happen. When we use the word and we are referring to multiplication, thus A and B can be thought of as AxB or (using dot notation which is more popular in probability) A•B

If A and B are dependent events, the probability of this event happening can be calculated as shown below:

If A and B are independent events, the probability of this event happening can be calculated as shown below:

Conditional probability for two independent events can be redefined using the relationship above to become:

The above is consistent with the definition of independent events, the occurrence of event A in no way influences the occurrence of event B, and so the probability that event B occurs given that event A has occurred is the same as the probability of event B.

Additive Rule (A∪B)

In probability we refer to the addition operator (+) as or. Thus when we want to we want to define some event such that the event can be A or B, to find the probability of that event:

Thus it follows that:

But remember from set theory that and from the way we defined our sample space above:

and that:

So we can now redefine out event as

The above is sometimes referred to as the subtraction rule.

Mutual Exclusivity

Certain special pairs of events have a unique relationship referred to as mutual exclusivity.

Two events are said to be mutually exclusive if they can't occur at the same time. For a given sample space, its either one or the other but not both. As a consequence, mutually exclusive events have their probability defined as follows:

An example of mutually exclusive events are the outcomes of a fair coin flip. When you flip a fair coin, you either get a head or a tail but not both, we can prove that these events are mutually exclusive by adding their probabilities:

For any given pair of events, if the sum of their probabilities is equal to one, then those two events are mutually exclusive.

Rules of Probability for Mutually Exclusive Events

Multiplication Rule

From the definition of mutually exclusive events, we should quickly conclude the following:
Addition Rule

As we defined above, the addition rule applies to mutually exclusive events as follows:
Subtraction Rule

From the addition rule above, we can conclude that the subtraction rule for mutually exclusive events takes the form;