text:
Ordinary Differential Equations/Applications of Second Order DEQs

There are several uses for second-order differential equations. In this chapter, I will cover the use of second-order differential equations to describe the motion of a mass at the end of a spring.

The chapter is broken up into three sections:
 * 1) ../Simple Harmonic Motion/
 * 2) ../Motion with a Damping Force/
 * 3) ../Motion with an Outside Force/

Chapter Notation
The formulae in this chapter are written with the following notation in mind. If you've learned a different manner of notation, please take note of the differences. I made every attempt to use a standard set of notation.

Important Terms
Terms that I feel deserve your undivided attention will appear like This. You will see this term referred to often in the text that follows, so it's recommended that you fully understand what it means.

Derivatives with respect to time
If a derivation is taken with respect to time (t), then an equivalent symbol is used and is pronounced x double-dot, x triple-dot, etc.
 * Example:$$\frac{d^2x}{dt^2}\equiv\ddot{x}$$

Rendering MATH PNG Images
This chapter was written using the built-in TeX markup language present in MediaWiki. It's recommended that you view the chapter with your preferences set to render all Math in PNG. Check your preferences for this setting.


text:
Ordinary Differential Equations/Simple Harmonic Motion

Simple Harmonic Motion can be used to describe the motion of a mass at the end of a linear spring without a damping force or any other outside forces acting on the mass. It's best thought of as the motion of a vibrating spring.

Laws of Motion
There are generally two laws that help describe the motion of a mass at the end of the spring.
 * 1) Hooke's Law
 * 2) Newton's Second Law

Hooke’s Law
To demonstrate Hooke’s Law, we will use a (massless) spring hung from a ceiling. The ceiling is rigid, and offers no effect on the spring’s motion.

If we leave the spring alone, and don’t attach a mass, the spring will remain in an un-stretched state with a total length $$l$$. Now, if we attach an arbitrary mass ($$m_1>0$$) to the free end of the spring, then the spring will stretch a distance ($$s$$) from the original length ($$l$$). So now $$\mbox{length}_\mbox{total}=s+l$$.

Eventually, the mass will come to rest at this new total length at a position known as the Equilibrium Position. This position is what many of the calculation reference as $$x=0$$.

Here is where Hooke's law comes into play. As the spring stretches, when the mass is attached, a force is exerted on the mass in the direction of the original un-stretched position. This force is expressed using the equation: $$\vec F=\vec ks$$. Where $$\vec F$$ is the force exerted on the mass by the spring, $$\vec k$$ is the spring's Constant of Proportionality, often called the spring constant, and $$s$$ is the distance the spring stretched from its un-stretched position.

Suppose that any movement by the mass in a downward direction is considered positive and upward is negative. If you take the weight ($$\vec W$$) of the mass, it will be equal to the force exerted by the spring when $$x=0$$, or the equilibrium position. Due to this, we now know that $$\vec W=\vec F$$. Since $$\vec W=m\vec g$$ and $$\vec F=\vec ks$$, by substitution, we get $$m\vec g=\vec ks$$. If you set these equal to zero, you'll get $$m\vec g-\vec ks=0$$.

Why is $$\vec ks$$ negative? This is because no matter what the mass does, the spring-with-gravity combination will always exert a force opposite to its motion. For example, as the mass travels downward beyond the equilibrium point, the spring will pull it back upwards in an attempt to regain equilibrium. When the mass gets above the equilibrium point, the spring contribution is less and the net acceleration is then downward toward regaining equilibrium. It behaves the same as if in an idealized example imagined all far removed from any local gravity effect, and that the spring is alternately stretched and compressed about a completely unstressed state. Some examples invent a horizontal version with the mass sliding over a frictionless surface, the better to then introduce a friction component.

Example 1
A mass of $$1/4\mbox{ slug}$$ is attached to a spring and stretches it $$6\mbox{ inches}$$. Calculate the spring's constant of proportionality.

Solution Using the combined equation $$mg-ks=0$$, we know $$m=1/4\mbox{ slug}$$ and $$s=6\mbox{ inches}=1/2\mbox{ ft}$$

(we omit the vector since we're only looking for a magnitude)

By definition, we know $$g=32 ft/s^2$$

Plugging everything in, we get $$(1/4 slug)(32 ft/s^2)-k(1/2 ft)=0$$.

Solving for $$k$$, we find $$8\mbox{ lb}=k(1/2\mbox{ ft})$$ $$16\mbox{ lb}/\mbox{ft}=k$$

Example 2
An object of unknown mass stretches a spring 10 cm from the ceiling. The spring's original length was 7 cm. Calculate the weight of the object if the spring constant is 5 N/m.

Solution First, we need the distance the spring is stretched after the mass is attached. This can be found using $$\mbox{length}_\mbox{total}=s+l$$, where $$\mbox{length}_\mbox{total}=10\mbox{ cm}=.1\mbox{ m}$$ and $$l=7\mbox{ cm}=.07\mbox{ m}$$. Plugging everything in and solving for $$s$$ we find $$s=.03\mbox{ m}$$.

By definition we know $$W=mg$$. With that, and $$mg-ks=0$$, we plug everything in and solve for $$W$$.

$$W-(5\mbox{ N}/\mbox{m})(.03\mbox{ m})=0$$ $$W=.15\mbox{ N}=150\mbox{ g}$$

Newton’s Second Law of Motion
Now, imagine that we have another mass hanging from a spring, which is attached to the ceiling. This mass is then pulled down a distance $$x$$, below the equilibrium point, and released. By Hooke's Law, the spring will be pulled back up, and after reaching it's highest point, start to travel back down.

If the mass continues this motion, without any outside influence, it's known as Free Motion. During this motion, there is acceleration acting on the mass to keep it in motion. This acceleration can be readily found in Newton's Second Law of Motion using $$\vec F=m\vec a$$. Where $$\vec F$$ is the force of the spring acting on the mass $$m$$ and $$\vec a$$ is the acceleration of that mass due to the force acting on it.

From Calculus, we know that $$a=\frac{d^2x}{dt^2}$$ or $$a=\ddot x$$. If we substitute this into our equation from the section on Hooke's Law, we find $$m\ddot x-k(s+x)=0$$. However, since this motion is always in the direction of the spring's force, our equation becomes $$m\ddot x+k(s+x)=mg$$. Expanding this, and solving for $$m\ddot x$$ we get $$m\ddot x=-k(s+x)+mg=-ks-kx+mg$$. Knowing that $$mg-ks=0$$, we can simplify our equation to end with $$m\ddot x=-kx$$.

The Differential Equation of Free Motion or SHM
Finally, if we set the equation above equal to zero, we end up with the following: $$m\ddot x+kx=0$$ Since our leading coeffiecient should be equal to 1, we divide by the mass to get: $$\ddot x+\frac{k}{m}x=0$$ If we set $$\omega^2=\frac{k}{m}$$, we'll have our final form of this equation: $$\ddot x+\omega^2x=0$$ The above equation is known to describe Simple Harmonic Motion or Free Motion.

Initial Conditions
With the free motion equation, there are generally two bits of information one must have to appropriately describe the mass's motion. Generally, one isn't present without the other. For simplicity, we will consider all displacement below the equilibrium point as $$x>0$$ and above as $$x<0$$.
 * 1) The starting position of the mass. $$x_2$$
 * 2) The starting direction and magnitude of motion. $$v$$

For upward motion $$v<0$$, and for downward motion $$v>0$$.

Solution
Multiplying this equation by $$\dot x$$ gives: $$m\ddot x \dot x+k x \dot x=0$$ The first and the second addends are exact derivatives, so this equation may be integrated to obtain the following relation: $$m\frac{{\dot x}^2}{2}+k\frac{x^2}{2}=E$$ The first addend of this relation is known as the kinetic energy of the mass and the second — as the potential energy of the spring. The above integral represents the energy conservation law. This is also a first order separable differential equation. It may be rewritten as $$\frac{dx}{\sqrt{\frac{2 E}{m}-\frac{k}{m}x^2}}=\pm dt$$ The integration of this relation gives

$$\arccos x\sqrt{\frac{k}{2E}} = \pm \sqrt{\frac{k}{m}} t + \varphi$$ Or, finally rearranging the result, substituting $$\omega = \sqrt{k/m}$$, and solving for $$x$$ we obtain $$x = \sqrt{\frac{2E}{k}}\cos (\omega t + \varphi)$$

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


text:
Radiation Oncology/CNS/Introduction

Survival
2-year and 5-year OS of various primary CNS tumors:
 * Astrocytoma, Pilocytic WHO grade I - 90%
 * Astrocytoma WHO grade II - 66%
 * Astrocytoma WHO grade III - 45% and 34%
 * GBM - 9%
 * Oligodendroglioma - 81% and 65%
 * Medulloblastoma - 66% and 60%
 * Ependymoma - 83% and 60%

Tumors that might be irradiated without pathologic diagnosis

 * Optic glioma
 * Diffuse pontine glioma
 * Meningioma
 * Choroidal melanoma

CNS tumors that are most likely to metastasize outside the CNS

 * Glioblastoma
 * Medulloblastoma
 * Ependymoma
 * Hemangiopericytoma
 * Meningioma

Pathologic correlations

 * Rosenthal fibers - JPA
 * Psammoma bodies - meningioma (and papillary tumors such as thyroid or endometrial)
 * Verocay bodies - schwannoma
 * Schiller Duval body - yolk sac tumor
 * Fried egg appearance - oligodendroglioma
 * Pseudorosettes (central lumen not formed by the growth itself, unlike a rosette) - ependymoma, but also PNET, medulloblastoma
 * Homer-Wright rosette - PNET, medulloblastoma, pineoblastoma, neuroblastoma
 * Flexner-Wintersteiner rosette - retinoblastoma

title: Radiation Oncology/CNS/Introduction


text:
Cookbook:Chess Pie

Ingredients

 * 2.5 cups (600 g) white sugar
 * 3 eggs
 * 1 tablespoon vinegar
 * 1 teaspoon vanilla
 * 1 stick (4 oz / 113 g) butter
 * 9-inch (22 cm) pre-made pie crust

Procedure

 * 1) Preheat oven to 375°F (190°C).
 * 2) Mix sugar, butter, and vinegar in saucepan. Bring to a boil.
 * 3) Beat eggs in bowl.
 * 4) Gradually whisk the saucepan contents into the eggs, then add vanilla.
 * 5) Pour mixture into pie crust.
 * 6) Bake pie in 375°F (190°C) oven for 35 minutes.
 * 7) Let cool.

title: Cookbook:Chess Pie


text:
User:Jfingers88

See wikipedia:User:Jfingers88 for more info about me.

 * Cookbook
 * /Templates
 * My contributions

 * Cookbook:Chess Pie
 * Cookbook:Hot Fudge Pudding Cake
 * Cookbook:Blondies

title: User:Jfingers88


text:
User talk:Elfich~enwikibooks

Welcome
Hello and welcome to Wikibooks!

Here are some tips to help you get started:
 * You can sign your name on Talk and vote pages using four tildes, like this: ~~~~.
 * Remember to conduct any editing experiments in the sandbox.
 * Check out our help page if you need additional help.
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Take a look at Naming policy
 * See list of templates which can save your time
 * Remember to maintain a Neutral point of view.
 * Explore, be bold in editing pages, and have fun!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page.

Good luck!--Derbeth talk 20:42, 3 April 2006 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Elfich. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Elfich~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:11, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 04:56, 19 April 2015 (UTC)

title: User talk:Elfich~enwikibooks


text:
Biblical Studies/Christianity/John 15

The Updated Modernity Version (Parallel, Annotated) John 15
  THE NEW TESTAMENT

John 1 − 2  − 3  − 4  − 5  − 6  − 7  − 8  − 9  − 10  − 11  − 12  − 13  − 14  − 15  − 16  − 17  − 18  − 19  − 20  − 21

title: Biblical Studies/Christianity/John 15


text:
User talk:24.106.127.42

Warning
Whenever you delete from a page and save it, other users cannot read it.

This is your last warning. The next time you vandalize a page you will be blocked from editing Wikibooks.  --Derbeth talk 19:45, 7 April 2006 (UTC)

 * I have blocked this IP indefinitely for vandalism on WB for three occasions, having borne in mind that this looks like a static IP and that per WB:VIP, this IP has repeated vandalised other wikimedia projects, Jguk 06:11, 9 April 2006 (UTC)

title: User talk:24.106.127.42


text:
Rueda de Casino/Príncipe Bueno

El Principe Bueno is an Enchufa but at the end before you switch partners the man kisses the woman's hand.

title: Rueda de Casino/Príncipe Bueno


text:
Template talk:Circuit Theory/Page

Maximum Power Transfer/Thevenin Edit
An edit was made on and by: 20:29, 30 March 2006 66.230.82.62. I removed an edit with a revert. I checked the maximum power transfer and thevenin circuit modules and it did not appear appropriate to insert there. Kinglag 23:12, 3 April 2006 (UTC)

title: Template talk:Circuit Theory/Page


text:
Biblical Studies/Christianity/John 16

The Updated Modernity Version (Parallel, Annotated) John 16
  THE NEW TESTAMENT

John 1 − 2  − 3  − 4  − 5  − 6  − 7  − 8  − 9  − 10  − 11  − 12  − 13  − 14  − 15  − 16  − 17  − 18  − 19  − 20  − 21

title: Biblical Studies/Christianity/John 16


text:
User:Jfingers88/Templates

List of easy to use templates.

Cookbook

 * /tsp teaspoon
 * /tbs tablespoon
 * /cup cup
 * /oz ounze
 * /floz fluid ounce
 * /lb pound
 * /in inch

German

 * /dedict German dictionary entry
 * /dedictn Noun entry with spot for gender

title: User:Jfingers88/Templates


text:
User:Jfingers88/Templates/tsp

tsp

title: User:Jfingers88/Templates/tsp


text:
User:Jfingers88/Templates/tbs

tbs

title: User:Jfingers88/Templates/tbs


text:
User:Jfingers88/Templates/cup

cup

title: User:Jfingers88/Templates/cup


text:
User:Jfingers88/Templates/oz

oz

title: User:Jfingers88/Templates/oz


text:
User:Jfingers88/Templates/floz

fl oz

title: User:Jfingers88/Templates/floz


text:
User:Jfingers88/Templates/lb

lb

title: User:Jfingers88/Templates/lb


text:
User:Jfingers88/Templates/in

in

title: User:Jfingers88/Templates/in


text:
User talk:Starchildmom

Hello and welcome to Wikibooks!

Here are some tips to help you get started:
 * You can sign your name on Talk and vote pages using four tildes, like this: ~~~~.
 * Remember to conduct any editing experiments in the sandbox.
 * Check out our help page if you need additional help.
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Take a look at Naming policy
 * See list of templates which can save your time
 * Remember to maintain a Neutral point of view.
 * Explore, be bold in editing pages, and have fun!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page.

Good luck! -- LV (Dark Mark) 00:54, 4 April 2006 (UTC)

 * Thanks for the welcome!--Starchildmom 06:58, 4 April 2006 (UTC)

title: User talk:Starchildmom


text:
Talk:GCSE Science/Acids & Bases, Salts and Electrolysis

we are diligent students!

chemistry
what visible change would be observed when calcium nitrate decomposes under strong heat


text:
User:H0riz0n~enwikibooks

My name is Michael J. Trout. I'm an author, poet, playwright, thinker, living in FUKUi, Japan (I love the name). I have lived all over the world from Bethesda-MD, Palma De Majorca-Spain, West Yorkshire and the Midlands-England, Homosassa FL, Tallahassee FL, Tuscaloosa AL, Montgomery AL, Gaithersburg MD, Arlington MA, Boston MA, Manhattan NY and finally Japan. My focus is the exploration of human truth and I am curious what will happen to humanity in my life-time. Predominately, I muse whether we will foolishly send humanity into a out of control tail spin by our denial of our crude oil addiction, our catastophic impact on global ecology, and the mounting pressures associated with globalization especially as it relates to the awaking of Super-mega-galactic-econo-CHINA monster! It’s going to be very interesting to observe the game unfold, so I picked a great seat to watch it -- Japan.

Contact: mjtrout AT gmail DOT com.

title: User:H0riz0n~enwikibooks


text:
User talk:H0riz0n~enwikibooks

Please dont leave me messages here -- thanks... Check my profile for my email.

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called H0riz0n. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name H0riz0n~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:15, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:01, 19 April 2015 (UTC)

title: User talk:H0riz0n~enwikibooks


text:
Biblical Studies/Christianity/John 17

The Updated Modernity Version (Parallel, Annotated) John 17
  THE NEW TESTAMENT

John 1 − 2  − 3  − 4  − 5  − 6  − 7  − 8  − 9  − 10  − 11  − 12  − 13  − 14  − 15  − 16  − 17  − 18  − 19  − 20  − 21

title: Biblical Studies/Christianity/John 17


text:
General Chemistry/Molar Concepts

A mole is a conversion factor to convert from amu (atomic mass units) to grams and back. Specifically, a mole is 6.02214 x $$10^{23}$$ particles. So, to find the mass of one mole of carbon(12.01 amu) we would use the following equation: $$ amu = (grams)(6.02 * 10^{23}) $$

This number is known as Avogadro's number.

Why moles?
Because atoms and molecules are so small, the molar concept provides us with a way to deal with substances in the laboratory; to measure their properties and observe changes in their characteristics.

For example, a molecule of water is infintesimally small and weighs 18 amu. It consists of two hydrogen atoms and one oxygen atom. However, Avogadro's number of water molecules weighs 18 grams, and has a volume of 18 milliliters. It contains two moles of hydrogen atoms and one mole of oxygen atoms. The molar weight of an element or molecule is also known as the gram formula weight, or simply the atomic weight expressed in grams. Just remember that Avogadro's number is just that -- a number; one can have a mole of anything. A mole of oranges is $$6.02 * 10^{23}$$. (Although this quantity would surely be larger than our entire planet!)

title: General Chemistry/Molar Concepts


text:
Talk:Nissan 240SX Performance Modification/KA24DE and KA24E/Turbocharging

I added some info about the strength of stock internals. Feel free to edit me if I messed anything up :P

thanks for contributing! spread the word.. Sean1978 01:57, 6 April 2006 (UTC)


text:
Talk:Toyota Prius/Driving tricks

May I request that some fact-checking be done here? The 42 mph ICE-run threshold isn't even mentioned, nor is the fact that every action is based on current SOC and driver demand. For more in-depth on B-mode, see [reconstruct URL yourself] techno-fandom.org/~hobbit/cars/b-mode.html [which also tells you who this is, and if you look around a little, how to reach me]. weaky-books isn't handling captcha well enough for me to enter a full URL.

Removed "because diesels don't have throttles", because diesels do have throttles. Original author is confusing fuel injection with carburation possibly.

Diesel engines do not have throttles...

"Approaching Home" -- Not sure about the wisdom of the advice offered. The NiMH cells will deteriorate faster if at low charge during storage.

=
============== Good advice when approaching a red light and you expect to have a long wait. But what if you stop and only then realise it will be a long wait? Easy - just turn the temperature control to cold (and the aircon off of course). Otherwise the NHW-11 will run its ICE to keep the occupants warm.

Another way to encourage ICE shutdown when stopped is a quick flick into reverse (without moving). These tips are from personal experience as I have one. Shannock9 (talk) 00:49, 2 December 2009 (UTC)

title: Talk:Toyota Prius/Driving tricks


text:
Biblical Studies/Christianity/John 18

The Updated Modernity Version (Parallel, Annotated) John 18
  THE NEW TESTAMENT

John 1 − 2  − 3  − 4  − 5  − 6  − 7  − 8  − 9  − 10  − 11  − 12  − 13  − 14  − 15  − 16  − 17  − 18  − 19  − 20  − 21

Return to Christianity contents page

title: Biblical Studies/Christianity/John 18


text:
Biblical Studies/Christianity/John 19

The Updated Modernity Version (Parallel, Annotated) John 19
  THE NEW TESTAMENT

John 1 − 2  − 3  − 4  − 5  − 6  − 7  − 8  − 9  − 10  − 11  − 12  − 13  − 14  − 15  − 16  − 17  − 18  − 19  − 20  − 21

Return to Christianity contents page

title: Biblical Studies/Christianity/John 19


text:
User:Starchildmom/Cell Unit Study

I am using this page to work out the formatting for a potential Wikibook. It is not ready for other's to contribute, I am just working through my ideas.

Please feel free to leave comments and suggestions on the talk page!

=Basic Cells Unit Study=

Introduction:

For this unit we will be using the Usborne Internet-Linked Science Encyclopedia (please check your library before buying) as our initial reference material. I will also be including links to Wikipedia entries for both the English and Simple English (easier reading level) versions when available. The Wikipedia entries can be used as an alternative main source for this unit. The basic structure of this unit is to first do some background research, followed by the activities that will be used to create the unit folder, and finally some laboratory activities which will be recorded and added to the unit folder. The general level for this unit is based on a year 5 (Logic Stage) science program following a classical trivium similar to many popular commercial homeschool curriculum. One big difference in our studies (other than we are Open Source and Creative Commons Licensed, i.e. Free), is that our units embrace technology. To really take advantage of what this unit can offer a good Internet connection is a must. At the present some of the critical information, diagrams, and downloads needed to complete this unit will be in the form of links to those tools on other sites. While most of those sites will have some sort of commercial aspect, all of the resources needed to complete this unit will be part of their no-cost offerings. I hope as time goes on to be able replace those links with resources developed specifically for distribution with this unit. That is part of where the Open Source aspect enters into the big picture. If you, a user, wish to contribute to a unit by creating one of those resources (or even writing your own unit following our format) feel free to do so. You may redistribute it yourself (following the Creative Commons licensing terms) or submit to me for inclusion in the main distribution of this unit (upon approval).

Cell Structure Unit Study

Goals:

To learn the basic structures, types, and processes of cells.

Background Research:

Choose your main resource, or try them all!

Usborne Science Encyclopedia : Read pages 250-251 and 298-299

Wikipedia: Cellular Biology Plant Cells Animal Cells Eukaryotes Prokaryotes Mitosis Meiosis

Introduction to Cellular Biology

Cells are the basic building blocks of life. They are like organic Legos. Each one has a similar basic structure, but the subtle differences (color, number of connectors) make each Lego perform a little differently. Sometimes, even a single block can be enough to form the entire structure. The same is true with cells. There are unicellular organisms and multicellular organisms, simple cells, and complex cells, but the basics of a cell are always there.

The study of cells is called cellular biology, or cytology. A scientist who specializes in cellular biology is called a cytologist. This study includes their physical properties such as their structure and the organelles they contain, their environment and interactions, their life cycle, division and function and eventual death. Knowing the composition of cells and how cells work is important to all of the biological sciences.

Activity1: Vocabulary

Make a 16 word vocabulary book (Big Book of Books p. 97). Add the following words on the top flaps. Look up their definitions in Webster's Student Dictionary online and write them below the flaps. You will add more words later.

 * 1) organic
 * 2) unicellular
 * 3) multicellular
 * 4) organelle
 * 5) organism
 * 6) membrane

Cell History

Cells were discovered by Robert Hooke. He used a microscope to look at organisms in 1665. He named cells after the Latin word cella which means a room. He did this because he thought cells looked like small rooms. Three German biologists - Schleiden, Schwann and Virchov - figured out three rules about all cells: 1.all living things are made of cells 2.the cell is the basic unit of structure and function in all organisms. 3.every cell comes from another cell that lived before it.

Activity 2: Cell Rules & History

Make a Three Pocket Book (Big Book of Science p.20). Make three cards, one for each pocket. On each card write on of the rules above. At the top of the inside write “3 Rules About Cells”. On the bottom flaps, first write “Figured out by German Biologists” near the top edge, below that write the name of one of the biologists on each pocket. On the left outer flap write “Robert Hooke.” On the right outer flap write “discovered cells by using a microscope in 1665.” This should be very neat. Think of it like an art project and make it as decorated (within the theme) as you like. The center-back outer flap will be glued into the final unit study folder, so do not decorate that part.

Basic Cell Structure

In biology, the cell is the basic structure of organisms. All cells only come from other cells. The outside of the cell is a cell membrane. Inside some cells, some parts of the cell stay separate from other parts by membranes. These separate parts are called organelles (like small organs.) They have different functions in the cell (do different work.) Some of these are ribosomes, nucleus (where DNA is), and mitochondria. Kinds of cells

There are two basic kinds of cells: prokaryotic cells and eukaryotic cells.

Prokaryotes are simple cells. Bacteria are types of prokaryotes. They are very simple cells that do not have a nucleus or any other membrane-bound cells found in more complex cells. In order to function, prokaryotic cells must still have certain features. A nucleoid region containing the DNA. This region is not contained in a membrane, and, therefore is not a nucleus. Ribosomes to translate the DNA code into the proteins needed. An outer membrane to hold everything together Some prokaryotes also have cell walls, or flagella (to help them move around)

Eukaryotes are less simple cells. All eukaryotic cells have a nucleus, and may have membrane bound organelles. Most multicellular organisms are made up of eukaryotic cells, although there are unicellular (single-celled) eukaryotes. Plant and animal cells are eukaryote.

Prokaryotic Organisms

All prokaryotic organisms are unicellular.

Most prokaryotic organisms are bacteria. All bacteria (or "one bacterium") are very small organisms (living things). They are so tiny they can only be seen with a microscope. Bacteria are formed with very few cells. Many of them have only one cell. This kind of organism is called a unicellular organism. Bacteria are among the oldest living organisms on Earth. There are more bacteria than any other living organism on the planet. Most bacteria live in the ground or in water, and many live inside of other living organisms. Some bacteria can cause diseases in other organisms, but others are good. A bacterium reproduces (creates more bacteria) by dividing itself into two new "child" bacteria.

Bacteria under the microscope

Kinds of eukaryotic organisms

There are two kinds of eukaryotic organisms: multicellular and unicellular.

Unicellular

A paramecium Unicellular organisms are made of one cell. Examples of unicellular organisms are: Amoeba Paramecium Unicellular organisms live without other cells to help them. So the cell does all things that it needs to live. Each cell: eats moves respires (uses oxygen to make sugar into energy) gets rid of waste reproduce (make more of itself) senses its environment

Multicellular

Multicellular organisms are made from many cells. This can be a small number of cells, or millions of cells. All plants and animals are multicellular organisms. The cells of a multicellular organism are not all the same. They have different shapes and sizes, and do different work in the organism. The cells are specialized. This means they do only some kinds of work. By themselves, they cannot do everything that the organism needs to live. They need other cells to do other work. They live together, but cannot live alone.

Epithelial Cells

Activity 3: Compare and Contrast Prokaryotic and Eukaryotic Cells

Create a three-tab Venn diagram book (Big Book of Science p.17). On the flaps make a Venn diagram comparing and contrasting the two types of cells. Inside, under the appropriate flap, list some types of organisms that have that kind of cell. On the outside sections (will be cover flaps when closed) that corresponds to each cell, draw a colorful diagram of that type of cell.

Plant Cells

Plant cells are quite different from the cells of the other eukaryotic organisms. Their distinctive features are: A large central vacuole (space enclosed by a membrane), which is filled with fluid and helps the cell keep its shape. A cell wall made up mostly of cellulose. The cell wall is thick and rigid, supporting the cell. The cell walls of multiple cells bond together to form the structure of the plant.

Chloroplasts that contain chlorophyll, the pigment that gives plants their green color and allows them to perform photosynthesis.

Plants lack centrioles that are present in animal cells. Like other eukaryotic cells, plant cells typically also have a nucleus, containing most of the cell's DNA, and mitochondria, the 'power plants' of the cell.

Animal Cells

An animal cell is a form of eukaryotic cell. Many tissues in animals are eukaryotic. The animal cell is different from other eukaryotes, most notably plant cells, as they lack cell walls and chloroplasts, and they have smaller vacuoles. Due to the lack of a rigid cell wall, animal cells appear to be circular (though are often deformed by surrounding cells) under microscopes - in three dimensions the cells are normally spherical.

Activity 4: Plant & Animal Cell Diagrams

Color and label the diagrams below. Cut them out and glue onto separate larger pieces of colored paper for a border.

title: User:Starchildmom/Cell Unit Study


text:
User:Tfogarty~enwikibooks

Blended Learning

title: User:Tfogarty~enwikibooks


text:
Talk:Chinese (Mandarin)/Pronunciation of Initials

Untitled
It would be extremely useful if this page listed at least one chinese word for each initial described here along with a recording of a native speaker speaking it. While it is nice to have a description of the sounds, nothing can replace actually hearing it. --128.179.67.72 07:29, 4 April 2006 (UTC)
 * I'm trying to recruit a native speaker to provide these samples. But for now, if you go to Lesson 1 there are recordings of the new vocabulary words, so you should be able to start off pretty well even without looking through the pronunciation section.

 * — Everlong 13:49, 4 April 2006 (UTC)

If you need to find samples, go to the wikipedia page on the IPA representation of the sound and there will be a proper sound sample which you will be able to upload here. Jocke pirat (talk) 17:14, 28 June 2008 (UTC)

I added some sound files, but I don't think any are available for the aspirated sounds. Also, I didn't bother to put in sound files for phonemes which are the same as the English equivalent. TrueDat (discuss • contribs) 05:39, 4 March 2011 (UTC)

clear of the tongue tip
"clear of the tongue tip" is a little vague to me, what specifically is meant by the tongue tip and how do i clear it? YukataNinja 16:31, 14 June 2007 (UTC)

Yeah. The phrase "slowly pass it backwards along the tongue until it is entirely clear of the tongue tip" was used, with slight modifications, to describe j, q and x. But it's not clear what it means. TrueDat (discuss • contribs) 05:06, 4 March 2011 (UTC)

Alternate "c" pronunciation?
There's a site with a handy pronunciation table, apparently of all the standard Mandarin syllables ... but all the syllables beginning "c" are pronounced (to my untrained ear) just like those beginning "k", but with longer vowel sounds afterward. Can someone explain? 70.15.114.2 (talk) 05:44, 1 April 2008 (UTC)

Pronounce
Many pronunciations of this item are not standard.I'm zh-N.这里有很多发音不标准. 范博2004 (discuss • contribs) 07:47, 18 September 2022 (UTC)


text:
Biblical Studies/Christianity/John 20

The Updated Modernity Version (Parallel, Annotated) John 20
  THE NEW TESTAMENT Modernity interprets that: More surely those who receive grace overflowing and the gift of righteousness shall reign in Life through the One, Jesus Christ. RO. 5:17d

John 1 − 2  − 3  − 4  − 5  − 6  − 7  − 8  − 9  − 10  − 11  − 12  − 13  − 14  − 15  − 16  − 17  − 18  − 19  − 20  − 21

Return to Christianity contents page

title: Biblical Studies/Christianity/John 20


text:
Cookbook:Sayur Asem (Indonesian Tamarind Vegetable Soup)

| Cuisine of Indonesia

This vegetable soup is a must-have item on the menu of every West Java (Sundanese) restaurant in Indonesia. This soup has some kind of appetizing effect because of the use of tamarind or (dried) sour starfuit as one of the ingredients. Sayur asem is a compliment to main dish such as ayam Ayam Goreng Kuning, Pepes Ikan or fried salty fish.

Spice paste

 * 1 fresh red chile pepper
 * 2 shallots
 * 2 g terasi
 * ½ tbsp salt

Soup

 * 20 g fresh tamarind or asam jawa tamarind paste, or 5 dried bilimbi fruit
 * 105 g galangal
 * 2 salam leaves
 * 100 g yardlong beans
 * 1 chayote squash
 * 50 g corn
 * 100 g young jackfuit
 * 50 g gnetum (melinjo/belinjau) leaves and fruits
 * 50 g raw peanuts
 * Salt
 * Sugar

Procedure

 * 1) Grind together all the ingredients for the spice paste until smooth.
 * 2) Clean the vegetables, and cut into bite-size pieces.
 * 3) Bring 1.2 liters of water to a boil in a pot. Stir in the spice paste, tamarind, galangal, and salam leaves.
 * 4) Gradually add the vegetables, starting with those that require a longer cooking time.
 * 5) Season with salt and sugar to taste.


text:
PHP and MySQL Programming/XML and PHP

Introduction
XML (eXtensible Markup Language) is used in mainstream development. It might have started off as an attempt at a web standard, but is now used even in more traditional applications as a document standard. For example, the Open Document Format employed by Sun in their StarOffice and OpenOffice suites is based on XML.

Because of its wide-spread use in the IT industry, it is fitting that we as PHP developers know how to make use of XML files in our PHP applications.

XML Structure
Since XML documents are extensible, there are no limits to the tags that you can create to define data with. Here is an example of a simple XML document:

 Isn't this simple! XML is as simple as pie. :-)

The reason that it looks so simple, is because it is so simple! Just as in HTML, elements are enclosed by angled brackets: "<" and ">", where the start element differs from the end element by the exclusion of a forward slash: "/".

Defining the XML Parser
In PHP, you define an XML Parser by using the xml_parser_create function as shown below.

<? $parser = xml_parser_create(ENCODING); ?>

You can think of the $parser variable in terms of a parsing engine for the XML document. Note that the ENCODING can be either:

1. ISO-8859-1 (default)

2. US-ASCII

3. UTF-8

Defining the Element Handlers
Element handlers are defined by means of the xml_set_element_handler function as follows:

<? xml_set_element_handler(XML_PARSER, START_FUNCTION, END_FUNCTION); ?>

The three arguments accepted by the xml_set_element_handler function are:

1. XML_PARSER - The variable that you created when you called the xml_parser_create function.

2. START_FUNCTION - The name of the function to call when the parser encounters a start element.

3. END_FUNCTION - The name of the function to call when the parser encounters an end element.

e.g.:

Defining Character Handlers
Character handlers are created by means of the set_character_handler function as follows:

<? xml_set_character_handler(XML_PARSER, CHARACTER_FUNCTION); ?>

The two arguments accepted by the set_character_handler function are:

1. XML_PARSER - The variable that you created when you called the xml_parser_create function.

2. CHARACTER_FUNCTION - The name of the function to call when the parser encounters character data.

Starting the Parser
To finally start the parser, we call the xml_parse function as follows: <? xml_parse(XML_PARSER, XML); ?>

The two arguments accepted by the xml_parse function are:

1. The variable that you created when you called the xml_parser_create function.

2. The XML that is to be parsed.

e.g.:

Cleaning Up
After parsing an XML document, it is considered good practice to free up the memory that is holding the parser. This is done by calling the xml_parser_free function as follows:

<? xml_parser_free(XML_PARSER); ?>

Parsing XML Documents
We have seen the steps needed to successfully parse a XML document with PHP. Lets take a moment to reflect on how these steps are interconnected.

When a XML parser is initialized, php will go through the XML file. When a starting tag is found, a predefined function created by you, the programmer, is called. The same thing happens when php encounters the text between tags, and the end tags.

Here is a complete example of parsing XML documents. This example is a RSS reader which can be used to display News Articles from any RSS feed which conforms to RSS 1.0 standards.

title: PHP and MySQL Programming/XML and PHP


text:
Teachers' Toolbox

Inspirational introductions to teaching methods.

Teachers%27 Toolbox is an open source Wiki book on teaching methods and ways to improve teaching. The toolbox is intended to give you an overview of methods you can use when teaching in general (applies to all schools, universities...). We hope it will help you to find new ways to activate students and improve their learning. It's *not* intended to be a long theoretical analysis of pedagogical methods, rather emphazises brief inspirational introductions to different ways to teach - and hopefully people will contribute with case stories and examples of teaching material you can develop further. You are very welcome to add to the book!!

If you read in this book please leave us a comment on the discussion page, and that's also the place to write suggestions for new material. Help improving this resource! You are very welcome to add and edit!

=Contents=
 * 1: INTRODUCTION

Why this book? - How YOU can contribute to the book - The Teachers Toolbox History - Other resources

 * 2: TEACHING CLASSES

Zen and the art of keeping students interested - Shorter group work in class sessions

 * 3: LECTURING

Stage performance and large audiences - Helpful Hints - Background Info: The attention of a listener - The plot - Your hands and motion - Your voice and speech - PowerPoint - The black board - Different ways to break up long lectures - Make an audience work in groups

 * 4: PLANNING A COURSE

The goal of teaching: Learning Objectives - Pre-Assesment - Lections - Continuous evaluation - Examination - Post evaluation

 * 5: SUPERVISING AND COACHING STUDENTS

Different types of supervision - Starting a new project for a student - Planing a project step by step - Mindmaps - Strategy -the project plan - Starting supervision of a new student - The supervisor meeting - EXAMPLE: An agenda for the supervisor meeting - Some typical problems and hints to solutions - Where am I heading - The writers block - The disposition - Whats going to happen after my study - The self-imposed high-achiever block - A typical course of a student project - Coaching

 * 6: TEACHING SKILLS TO THE STUDENTS

Group Work - A list of good advice - Feeling stuck - Writing Reports/Papers/Thesis - Report writing - Thesis writing - Writing articles - Making posters - Making presentations - The scientific method - Project Management

 * 7: EVALUATION OF STUDENTS

Written exams - multiple choice - Peer assessment - Reports - Presentations - Project work

 * 8: WAYS TO IMPROVE YOUR OWN TEACHING

Peer coaching - Feedback to other teachers - Student evaluations - Feedback to the supervisor - EXAMPLE: Feedback questionnaire - Teaching portfolio - Video feedback

 * 9: USING WIKI IN TEACHING

Wikibooks can be used in teaching, both as a resource, but also for students to write sections in books to improve them, rather than just writing reports that will not really be used by anyone else.

title: Teachers' Toolbox


text:
User talk:GeeKaa

w:nl:Overleg gebruiker:GeeKaa

title: User talk:GeeKaa


text:
User:GeeKaa

w:nl:Gebruiker:GeeKaa

title: User:GeeKaa


text:
PHP Programming/Commenting and Style

As you write more complex scripts, you'll see that you must make it clear to yourself and to others exactly what you're doing and why you're doing it. Comments and "good" naming can help you make clear and understandable scripts because:
 * When writing a script that takes longer than a week, by the time you're done, you won't remember what you did when you started, and you will most likely need to know.
 * Any script that is commonly used will need rewriting sooner or later. Rewriting is much easier (and in many cases, made possible) when you write down what you did.
 * If you want to show someone your scripts, they should be nice and neat.

The comments
Comments are pieces of code that the PHP parser skips. When the parser spots a comment, it simply keeps going until the end of the comment without doing anything. PHP offers both one line and multi-line comments.

One-Line Comments
One-line comments are comments that start where ever you start them and end at the end of the line. With PHP, you can use both // and # for your one-line comments (# is not commonly used). Those are used mainly to tell the reader what you're doing the next few lines. For example:

It's important to understand that a one-line comment doesn't have to 'black out' the whole line, it starts where ever you start it. So it can also be used to tell the reader what a certain variable does:

The $message = ""; is executed, but the rest of the line is not.

One-line Comment Issues

 * One-line comments end by either:
 * 1) a newline (an actual newline, not the \n newline mark ) OR:
 * 2) a closing PHP tag of the ?> variety
 * If a one-line comment is closed by a closing PHP tag will not be commented. The following will thus print out "2":

Multi-Line Comments
This kind of comment can go over as many lines as you'd like, and can be used to state what a function or a class does, or just to contain comments too big for one line. To mark the beginning of a multiline comment, use /* and to end it, use */. For example:

You can also use this style of comment to skip over part of a line. For example: Although it is recommended that one does not use this coding style, as it can be confusing in some editors.

Multi-line Comment Issues

 * An unfinished multi-line comment will result in an error, unless it is starting (not closing) within an already existing multi-line comment block (i.e., it is non-greedy but only operates first on a complete open and close set)
 * The following will result in an error:
 * The following will not result in an error:
 * Thus, these multi-line comments cannot be nested (the first block up to "ending first comments" will be commented out, but the following */ will not have any opening /*for it). The following will therefore cause an error:

 * One can quickly toggle multiple blocks at a time by combining the styles (though this may not work show as such correctly in a text editor)

Original text with nothing commented out (thus printing "block one block two":

After taking out a / on the first line, the first block is commented out, printing only "block two":

Since the single line // overrides the multi-line /* .. */ two blocks of code can be switched at the same time back and forth into or out of comment mode.

Original text (printing out only "block one")

After taking out a / on the first line, the first block is commented out and the second block is uncommented, printing out only "block two":

 * phpdoc.org/HTMLSmartyConverter/HandS/phpDocumentor/tutorial_tags.pkg.html PHPDocumentor uses multiline comments (albeit with the opening immediately followed by an additional asterisk "/**") and other standardized tags within the comment block to create its automatic documentation. See the website for additional instructions. For example:

Issues with Either Single or Multi-line Comments
When using multi-line comments with the typical "/" as delimiters in a regular expression, it is possible there will be a conflict, as an asterisk at the end of the expression (along with the closing "/" delimiter) could create something that would be parsed as a closer of the comments, thus leaving the succeeding "*/" without an opener:

To avoid the problem, one could:
 * Use other regular expression delimiters--delimiters that may be less frequently encountered (and thus also in need of escaping) in most circumstances anyhow (e.g., "@" besides for email matching may be less frequent than "/")
 * Use an "if" with an impossible conditional that can:

Avoid regular expression conflicts:

and also avoid nesting problems:

One disadvantage of the "if" method, however, is that it will most likely not be recognized by text editors as far as code coloring (though this may be an advantage for debugging, if one wishes to see more clearly what is inside the commented out block while running tests on the code).

Styles
The PHP coding styles are defined in the.

Naming
Naming your variables, functions and classes correctly is very important to understand the program. By convention, it should be done in and without any abbreviation.

If you define the following variables:

They won't say much to anyone. But if you do it like this:

It would be much clearer. But don't go too far. $programmingLanguage, for example is not a good name. It's too long, and will take you a lot of time to type and can affect clarity. A better name may be $progLanguage, because it's shorter but still understandable. A good naming should avoid to write a comment to mark what every variable does.

For instance, these comments are encumbering the script and should be removed:

Magic numbers
When using numbers in a program it is important that they have a clear meaning. For instance it's better to define $menu_items early on instead of using 15 repeatedly without telling people what it stands for. The major exception to this is the number 1; often programmers have to add or subtract 1 from some other number to avoid off-by-one errors, so 1 can be used without definition.

When you define numbers early on in their usage it also makes it easier to adjust the values later. Again if we have 15 menu items and we refer to them ten times, it will be a lot easier to adjust the program when we add a 16th menu item; just correct the variable definition and you have updated the code in 10 places.

Spacing
PHP ignores extra spaces and lines. This means, that even though you could write code like this:

It's better like this, which is the PSR-2 form :

Some programmers prefer this way of writing:

You should also use blank lines between different portions of your script. Instead of

You could write:

And the reader will understand that your script first declares a variable, then welcomes the user, and then checks the variable.

The directives
It's possible to define certain PHP compiling behaviors by some directives written in the command. For example:

title: PHP Programming/Commenting and Style


text:
Carpentry/Power Tools/Drill

A drill is a tool used to bore holes in a material. A drill may either be manually turned by the user or automatically turned by a motor. The latter is an electric drill. The drill bit is the cutting element of the drill. This bit is a cylendrical shaped piece of metal with a spiral cut. Drill bits can be almost any size from fractions of an inch to larger. A drill press is an electric drill fixed to a pedistal.

title: Carpentry/Power Tools/Drill


text:
Category:Cookbook utility categories

These are cookbook-related categories that are meant to assist contributors in improving the cookbook.

title: Category:Cookbook utility categories


text:
Category:Recipes by difficulty

Recipes listed by difficulty.

hy:Կատեգորիա:Բաղադրատոմսեր ըստ դժվարության

title: Category:Recipes by difficulty


text:
User:Paolo Liberatore~enwikibooks

See w:User:Paolo Liberatore

title: User:Paolo Liberatore~enwikibooks


text:
NVIDIA

 NVIDIA  Under development at Wikibooks, a site providing a collection of open-content textbooks that anyone can edit.

The NVIDIA website is www.nvidia.com.

As of now, this is primarily an attempt to restore some of the NVIDIA content under http://www.gmpf.de/index.php/Main_Page, which unfortunately has disappeared. The original content was licensed under the GNU Free Documentation License.

title: NVIDIA


text: Template:NVidia

title: Template:NVidia


text:
User:Richardb

User Richardb from Wiktionary. see my profile there. Just an occasional visitor at Wikibooks.

title: User:Richardb


text:
User:Hapa

= Hello, my name is Kolomona =

I am new to Wiki and Wikibooks

I started a wiki site called www.OpenTutorial.com Please check it out.

There's more info about me on my personal website www.tsickist.com/aboutme

--Hapa 14:30, 4 April 2006 (UTC)

= What am I doing on Wikibooks =
 * Spending my time here Category:Modules_that_need_to_be_wikified

title: User:Hapa


text:
NVIDIA/TV-OUT

Could you imagine a video-night sitting in front of a 17 '' monitor? Wouldn't it be much nicer to go through all your videos using a television? Most video cards don't provide direct SCART-connection to the TV, but some offer a S-Video connection.

This page provides step-by-step documentation on howto setup your X-Server and use Tv-out.

Setting up the X-Server
We need to edit three different parts of your X config file to make the X-server render your desktop or video to a television that is connected to your video card through the S-Video (or Composite) port on your videocard. They are used to associate a Monitor with a Device and define a Screen. If you look through your configfile you will notice that those sections already exist. No wonder, because we need one Device-, one Monitor- and one Screen- Section for each connected monitor/TV. In our case it's the original monitor (CRT) and the television. To make sure your monitor will work later on as well do not touch the existing sections. We will add new sections to support tv-out as well as the original monitor simultaneously. There is also a feature called TwinView which is not covered here.

Please make a backup of your configuration file before you make any changes to it.

Device-Section and its options
This section associates each installed videocard with a driver. As you look through the xorg.conf your device-section section should look like this right now:

Section "Device" Identifier "Videocard0"  # unique identifier for each video card Driver     "nvidia" VendorName "nVidia" BoardName  "nVidia Corporation NV15 [GeForce2 GTS/Pro]"  # see lspci EndSection
 * 1) Options ...

This Device Section controls the DVI/VGA output and ignores the tv-out and all devices connected to it. There are some special nvidia-options to use the tvout.

TVStandard
You should adjust this parameter to your monitor. In general you can take the value from this table:

Usage:

Option   "TVStandard" "NTSC-M"

TVOutFormat
This option is used to force the output to S-Video or Composite. If this option is missing it will be auto-detected. But be careful because this detection might fail. Possible values are SVIDEO and COMPOSITE.

Usage:

Option   "TVOutFormat" "COMPOSITE"

ConnectedMonitor
Usage:

Option   "ConnectedMonitor" "CRT, TV"

A complete device-section
Finally your new device section should look like this:

Section "Device" Identifier "Card_tv" Driver     "nvidia" BusID      "PCI:1:0:0"  # May differ (not needed unless you have two or more cards) Option     "TVOutFormat" "COMPOSITE"  # Or "SVIDEO" Option     "TVStandard" "NTSC-M" Option     "ConnectedMonitor" "TV" EndSection

The Monitor-section
As mentioned we need to setup a monitor device and define the Horizontal Syncronisation rate and the Vertical Refresh rate. You need not search for those numbers, because usually those numbers are the same for most televisions.

Section "Monitor" Identifier  "tv" HorizSync 30-50 VertRefresh 60 EndSection

The Screen-section
To complete the whole configuration we need to add a Screen section that associates the Device and Monitor sections and defines the desired display resolutions.

Section "Screen" Identifier "Screen_tv" Device    "Card_tv" Monitor   "tv" DefaultDepth 16 SubSection "Display" Depth    16 Modes "1280x1024" "1152x864" "1024x768" "800x600" EndSubSection EndSection

The ServerLayout-section
We now have two monitor-sections, two device-sections and two screen-sections. But how does X know what do with with them. Easy answer. There's another section defined with ServerLayout which can be used to switch the used devices. There should already exist a Section for that which controls your current monitor and defines what keyboard and mouse to use. Don't edit it unless you know what your doing. This Tutorial describes how to use several serverlayouts. So we add one more of them using our television Screen section:

Section "ServerLayout" Identifier    "tv" Screen        0  "Screen_tv" 0 0 InputDevice   "Mouse0" "CorePointer" InputDevice   "Keyboard0" "CoreKeyboard" EndSection

Permissions
On a Debian or Ubuntu system you will have to edit allowed_users in /etc/X11/Xwrapper.config to get permission to start the X server.

If xinit reports "user not authorized to run the X server, aborting", edit the file /etc/X11/Xwrapper.config as follows:

Change this line: allowed_users=console to allowed_users=anybody

Additionally, to connect to the server, you may need to launch X with "X :1 -layout tv -auth .Xauthority"

Finally
After putting all that into your configuration file we are ready to test the X-Server and write a little script to launch mplayer on the TV.

Testing Tv-Out
To test the TV out try this from and xterm on your CRT:

X :1 -layout tv

Your monitor should turn blank and might switch into Standby (SignalLost messages could also appear). If your TV is connected and the input selection is set correctly you should see the nVidia logo then the mousepointer should appear. To switch between monitor and TV you can use these key combinations:

''If those keys don't work, try to zap through all: start from F1 and go up to F12. One should be your windowmanager and another is the television. The others are terminal loginprompts.''

(In Ubuntu 7.04 the following has been found with the nvidia-glx driver on a Geforce4 Go 420 on issuing X :1 -layout tv:

(EE) NVIDIA(0): Unable to initialize the X Int10 module; the console may not (EE) NVIDIA(0): be restored correctly on your TV.

This been reported as a bug)

To open a terminal go to your windowmanager, open a prompt and type:

DISPLAY=:1 xterm

If you now switch back to your TV on F8 there will be a terminal popped up in the upper left corner of the screen. Your config passed the test. We can now try to handle all this in a useful way.

When trying the xterm command above, you may get an error similar to:

AUDIT: Tue May 17 14:57:14 2005: 18768 X: client 1 rejected from local host Xlib: connection to ":1.0" refused by server Xlib: No protocol specified

This error occurs if your system is using Xauthentication (xauth) to limit who can access a running X server. You need to tell X that you want to allow access from the local machine to display :1. The following command is one long command that is split across 2 lines. It will configure Xauthentication so that display :1 will accept connections from the same sources as display :0. This command was constructed on and it works on Mandriva 2005 LE, and has been seen to work on Fedora Core 5 and Ubuntu 6.06 (Dapper Drake). To configure Xauthentication for local access, open a prompt and type:

xauth add "$(/bin/hostname)/unix:1" MIT-MAGIC-COOKIE-1 \ $( xauth list | egrep "$(/bin/hostname)/unix:0" | awk '{print $3}' )

Shutting down the TV-SCREEN
It you want to shutdown the TV-screen session, you can easily use the same procedure as if you want to kill your X-Server.

 * 1) Activate the X-Session you want to terminate
 * 2) Press ALT + CTRL + BACKSPACE
 * 3) Switch back to your regular session and continue as if nothing happened

Starting mplayer on TV
To get a command for running mplayer on the TV, create a new file /usr/local/bin/mplayer.tv, with this content:

exec /usr/X11R6/bin/xinit /usr/bin/xterm -ut -e \ /usr/bin/mplayer -stop-xscreensaver -fs -vo sdl "$@" -- /usr/X11R6/bin/X :1 -layout tv
 * 1) !/bin/sh

Use locate or whereis to find the correct path if some commands are located elsewhere. ( FC5: /usr/bin/xinit and /usr/bin/X )

Make the script executeable:

chmod +x /usr/local/bin/mplayer.tv

You can now play all your videos on TV:

mplayer.tv /your/path/to/video.mpeg

mplayerstart: Bash script with resume
mplayerstart is a wrapper script for mplayer which allows the user to resume playback of a previously played file, supports several TV-out methods (including the xinit method above), disables screensavers, restores volume levels, and other features. The most recent version may be obtained here: mplayerstart.

title: NVIDIA/TV-OUT


text:
User:Jnavas2

Hello. My real name is John Navas. My Internet home page page is http://navasgroup.com. To contact me here, please use my discussion/Talk page.

title: User:Jnavas2


text:
NVIDIA/Twin View

TwinView is a feature of modern nVidia graphics cards first introduced on the GeForce2 MX range of cards. TwinView presents a single big screen to the operating system, one advantage of this approach compared to using multiple cards with Xinerama is that hardware accelerated OpenGL applications work on both screens and can be moved from one to other easily.

Buying a TwinView card
If you do not currently have a TwinView card but are interested in buying one, there are few things to look out for:

 * Some cards listed as featuring TwinView have the second display only available through a TV out which is of no real use for dual head system - though this maybe useful for multitasking video with other operations. Popular configurations of TwinView are HD15/TV, DVI/HD15 and DVI/DVI. Many configurations of the later two combinations have the second output which either can be the monitor or a TV out giving the best of both worlds.
 * Most TwinView cards have at least one DVI output which is a DVI-I which can support analog monitors with the aid of an adaptor (such as CRTs and old/cheap LCD displays) in addition to DVI LCD displays, be careful, if this is what you want/need that you don't get a card with DVI-D output which only supports LCDs.
 * The majority of highend NVIDIA cards have a DVI/DVI combination so this is generally easy to find. For a cheaper card you will probably have to settle for a DVI/HD15 card which may require looking at several models before you find one.
 * The resolution of the second display on GeForce2 MX cards is limited by a slow pixel clock (150Mhz) which limits this screen to 1280x1024. GeForce4 (all modern cards) and later cards have both outputs running with the same pixel clock giving the same maxiumum resolution on both screens.

Configuring TwinView with NVIDIA's binary driver
APPENDIX G of the NVIDIA Readme has a comprehensive coverage of the TwinView feature. The configuration of XFree86/Xorg is similar to other setups, the main difference is the "Screen" section, here is an example TwinView setup:

Section "Screen" Identifier	"Screen0" Device		"NVIDIA GeForce2 MX (3123)" Monitor		"Monitor0" DefaultDepth	24 Option "TwinView " "on" Option "TwinViewOrientation" "RightOf" Option "MetaModes" "1280x1024,1280x1024;1280x1024,1024x768;1280x1024,NULL;1280x1024,1280x1024+1200+0" Option "SecondMonitorHorizSync" "30 - 96.0" Option "SecondMonitorVertRefresh" "50 - 120" Subsection "Display" Depth	24 Modes	"1280x1024" "1152x864" "1024x768" EndSubsection EndSection

The options used in this example are specific to TwinView and therefore may need some explanation:

 * "TwinView" needs to be set for TwinView to work
 * "TwinViewOrientation" selects where the second screen is placed in relation to the first one: "RightOf", "LeftOf", and "Clone"
 * "MetaModes" is a list of modes comprised of 1st and 2nd monitor pairs, each pair is separated by a semi-colon.
 * 1280x1024,1280x1024 - both monitors are set to 1280x1024
 * 1280x1024,1024x768 - the first monitor is set to 1280x1024 and second to 1024x768 (on the right in this case)
 * 1280x1024,NULL - the first monitor is set to 1280x1024 the and second is not displaying anything.
 * 1280x1024,1280x1024+1200+0 - the first monitor is set to 1280x1024 and the second is set to 1280x1024 offset by 1200 from the origin of the first screen in the horizonatal direction creating a 80pixel overlap in the middle. This can be used with edge blended projectors (which overlap the image to avoid a noticeable join)
 * SecondMonitorHorizSync The HorizSync option in the Monitor Section is the for the first monitor and this is its equivalent on the second monitor.
 * SecondMonitorVertRefresh The VertRefresh option in the Monitor Section is the for the first monitor and this is its equivalent on the second monitor.
 * Subsection "Display" You may think that you don't need this, however you'd be wrong if you thought that! If only one screen is connected the Nividia driver will say "(WW) NVIDIA(0): Only one display device connected; disabling TwinView." and X will revert to the settings in this section, it is a good idea to check that the default resolution works with both monitors (in case one breaks) so that you can get a graphical login without messing about.

The default meta mode is the left most one. You can switch through the modes in two ways whilst X is running. The first method is to use the Ctrl-Alt-+/- key combination to change native resolution of the screen whilst keeping the virtual screen size the same, this is sometimes called a virtual desktop. The other method only available on modern distributions is to use the xrandr command which changes resolution without creating a virtual desktop, xrandr with no arguments gives a list of all the available modes, to change to mode two use "xrandr -s 2", more details about xrandr can be found on its man page.

Other options

 * NoTwinViewXineramaInfo - Although TwinView presents to the OS as a single screen it also by default uses the Xinerama extension to tell the window manager what physical screens there are. The Window manager will typically use this information to control the behaviour of maximised windows so that they only maximise onto one screen. This option, when set, turns off this feature.
 * TVStandard - For controlling the second monitor on cards that feature it - see the TV-OUT for more details.
 * TVOutFormat - For controlling the second monitor on cards that feature it - see the TV-OUT for more details.
 * TVOverScan - For controlling the second monitor on cards that feature it - see the TV-OUT for more details.
 * ConnectedMonitor - To override the monitor type detected by the Nvidia driver - maybe useful for some old non-DDC monitors.

NB: if you use your flat panels in analog mode, you will need to refer to them as CRTs (e.g., "ConnectedMonitor" "CRT, CRT" (or "CRT-0, CRT-1"))

Alternatives to TwinView
TwinView in the author's opinion is the best way to get a dual-head system due the support of hardware acceleration on both screens:

 * Multiple Similar Nvidia cards using Xinerama, accelerated 3D (via GLX) a new feature introduced in driver version 76.64, see Appendix V in the NVIDIA Readme for more details.
 * in addition to your main AGP card you can use any number of PCI graphics cards and with the aid of the X11R6 Xinerama extension you can span your desktop across the cards and the cards do not have to made by the same manufacturer (though I've never tried this). Xinerama is described by the, fairly outdated, Xinerama-HOWTO. This method does not provide accelerated 3D support the non primary display(s).
 * If you have a system where you cannot get upgraded graphics or extra graphics card (for example in a laptop) you can use VNCwall to create a multihead display using the display output of many networked machines. This option does not provide accelerated 3D support on any displays due to the use of VNC.
 * DMX: provides multi-head support for multiple displays attached to different machines using the X protocol.

TV-out with TwinView
It is possible to use TwinView with a monitor plus a TV if you want everything on your monitor to appear also on the TV. This is particularly useful for watching films and if you don't want to start a separate X display for the TV (as described on TV-OUT). Starting mplayer is also simple, you don't need any special options or scripts to direct output to the TV and while watching the film, your monitor will still be on.

This configuration is taken from the mplayer documentation.

First download the latest nvidia driver from www.nvidia.com, make sure it works and insert the following into your X config file, typically under /etc/X11/xorg.conf if you use xorg:

Section "Device" Identifier	"GeForce" BoardName	"nVidia GeForce2/MX 400" Driver		"nvidia" Option		"NoLogo" Option		"CursorShadow" "on" Option		"TwinView" Option		"TwinViewOrientation" "Clone" Option		"MetaModes" "1024x768,640x480" Option		"ConnectedMonitor" "CRT, TV" Option		"TVStandard" "PAL-B" Option		"TVOutFormat" "SVIDEO" EndSection

The important bits are TwinViewOrientation, MetaModes and TVStandard.

 * 1) TwinViewOrientation: "Clone" means to duplicate the monitor screen to the TV
 * 2) MetaModes: these are pairs of resolutions, first means your monitor, the second the TV
 * 3) TVStandard: see the readme.txt file of the nvidia driver (that you have downloaded from www.nvidia.com) or TV-OUT to figure out what to put here

These setting will clone your monitor and everything will also appear on the TV. To start watching a film just type what you would normally do:

mplayer -fs butch-cassidy-and-the-sundance-kid.avi

title: NVIDIA/Twin View


text:
A-level Mathematics/MEI/C1

This is the first pure module of the MEI A-level Mathematics book. Knowledge of intermediate tier GCSE is assumed. The Exam is an hour and a half long, and is a non-calculator paper. It can be taken in June.

Contents

 * Mathematical Processes
 * Algebra
 * Co-ordinate Geometry
 * Polynomials

title: A-level Mathematics/MEI/C1


text:
A-level Mathematics/MEI/C1/Algebra

Algebra is the branch of mathematics that deals with the relation of quantities. In an equation, both sides are equal, and in an inequality, one side is usually greater than another.

An equation consists of two expressions joined by the equals sign ($$=$$). Everything on the left-hand side is equal to everything on the right-hand side, for example $$2+3=4+1$$. Some equations contain a variable, usually denoted by $$x$$, $$y$$ or $$z$$. An equation with a variable will only hold true for certain values of that variable. For example $$2 + x = 5$$ is only true for $$x=3$$. The values that the variables have when the equation is true are called the solutions of the equation. Therefore $$x=3$$ is the solution of the equation $$2 + x = 5$$.

The language of algebra
There are several terms that you need to be familiar with before you begin to work with algebra.

Variable
A variable is a quantity, whose value is usually unknown. Usually, a variable is given the symbol $$x$$, although any letter can be used.

Constant
A constant is usually a known quantity, which does not involve the variable. Later you will come across unknown constants, which are usually given the symbol $$c$$.

Index
Generally, an index is anything that is written superscript to (slightly above) a symbol. Often indices are used to indicate something raised to a power, such as $$x^3$$ (read as x-cubed), which is the same as $$x \times x \times x$$.

Expression
An expression is a group of symbols which form a mathematical statement. $$2 + 2$$ is an example of an expression.

Term
A term is any variable, constant or a product of variables or constants which are separated by a $$+$$ or a $$-$$ sign. In the expression $$3x+4xy-2y$$, the separate terms are $$3x$$, $$4xy$$, and $$2y$$.

Coefficient
A coefficient is the constant part of a term which multiplies the variable part of the term. For example, in $$2x^3$$, the coefficient of $$x^3$$ is $$2$$.

Equation
An equation is a mathematical statement that two things are equal. There is an equals sign ($$=$$) in between two expressions. $$2+2=4$$ is an example of an equation, as well as $$3+x=7$$.

Identity
An identity is an equation with an unknown, and is true for all values of that unknown. The identity symbol ( ≡ ) is used in place of the equals sign. As an example, $$x - x = 0$$ is always correct no matter what the value of $$x$$ is. It is usual to write $$x - x$$ ≡ $$0$$ when you want to emphasize the fact that it is an identity and not just an equation.

Function
A function relates one input value to one output value. A function of $$x$$ is usually noted as $$f(x)$$. $$f(x)=x^2$$ is an example of a function. It is then possible to write $$f(3)=3^2=9$$ once the function has been defined. $$f(x)= \sqrt x$$ is not a function, because there are two output values (positive and negative) for each input.

Manipulating expressions
Sometimes, expressions will be messier than they need to be, and they can be represented in an easier to understand form. The skills here are essential to the rest of the A-level course, although it is very likely that you have already covered them at GCSE.

Collecting like terms
When collecting like terms, you simply add all the terms in $$x$$ together, all the terms in $$y$$ together, and all the terms in $$z$$ together. The same applies for any other letter that represents a variable.

For example, $$2x+4y+8z-3x-7y-2z+4x$$ becomes:

$$2x-3x+4x = 3x$$

$$4y-7y = -3y$$

$$8z-2z = 6z$$

So, by adding all the answers, $$2x+4y+8z-3x-7y-2z+4x$$ simplified is $$3x-3y+6z$$.

Multiplication
Multiplication of different variables such as $$a \times b$$ becomes $$ab$$. Single variables become indices, so $$ x \times x$$ is $$x^2$$.

Like addition and subtraction, you keep like terms together. So, for example:

$$2x^2z \times 3yz^2 \times 4xy^3$$ becomes:

$$2 \times 3 \times 4 \times x^2 \times x \times y \times y^3 \times z \times z^2$$

which can finally be simplified as:

$$24{x^3}{y^4}{z^3}$$

Expansion of brackets
The skill of expanding brackets is illustrated by the following example.

$$(8x+5y)(3x-6y)$$

$$=(8x+5y)3x-(8x+5y)6y=8x(3x-6y)+5y(3x-6y)$$ use FOIL

F=> first term in each bracket 8x. 3x =24x^2 O=>outside terms(1st in 1st x 2nd in 2nd) 8x. -6y = -48xy I=> inside terms (2nd in 1st x 1st in 2nd) 5y. 3x =15xy L=> last term in each bracket 5y. -6y = -30 y^2

and terms together(usually only the middle 2) 24x^2 -48xy +15xy -30y^2 answer 24x^2-33xy-30y^2

Factorising
Sometimes, expressions can be re-written as the product of their factors. You divide an expression by a factor common to all of the terms in the expression. This is essentially the opposite of expanding brackets, since most of the time the common factor is placed outside of brackets. For example:

To factorise $$10x+15y$$ you must first find a common factor of $$10x$$ and $$15y$$. $$5$$ is easily spotted as a factor. Now you divide the whole expression by $$5$$ leaving you with $$2x+3y$$. Place $$2x+3y$$ within brackets, and then put the $$5$$ outside the bracket. The factorised expression is now $$5(2x+3y)$$, and you can multiply out the expression to make sure you get the original expression.

Another expression, $$x^2-xy^2+3xz$$ has a common factor $$x$$. The factorised form of this expression is $$x(x-y^2+3z)$$.

Fractions
When working with fractions, the rule is to make all of the denominators equal, and then write the expression as one fraction. You need to multiply both the top and bottom by the same amount to keep the meaning of the fraction the same.

For example, for $$\frac {3x}{2} + \frac {2y}{5} - \frac {z}{10}$$, the common denominator is $$10$$.

Multiply both parts by $$5$$: $$\frac {15x}{10}$$

Multiply both parts by $$2$$: $$\frac {4y}{10}$$

Leave this as it is: $$\frac {z}{10}$$

You now have $$\frac {15x}{10} + \frac{ 4y}{10} - \frac {z}{10}$$, which becomes $$\frac {15x+4y-z}{10}$$.

Solving equations
Often, to solve an equation you must rearrange it so that the unknown term is on its own side of the equals sign. By rearranging $$2 + x = 5$$ to $$x = 5 - 2$$, $$x$$ has been made the subject of the equation. Now by simplifying the equation, you can find that the solution is $$x=3$$.

Changing the subject of an equation
You will usually be given equations that are more complex than the example above. To move a term from one side of the equals sign to the other, you have to do the same thing on both sides of the equals sign. For example, to make $$x$$ the subject of $$y = \frac{4 a (x^2 +b)}{3}$$:

Solving quadratic equations
Quadratic equations are equations where the variable is raised to the power of 2 and, unlike linear equations, there are a maximum of two roots. A root is one value of the variable where the equation is true, and to fully solve an equation, you must find all of the roots. For a quadratic equation you can factorise it and then easily find which values make the equation valid. The example above is quite a simple case. You will usually be given a more complicated equation such as $$2x^2 + 5x + 3 = 0$$. If the equation isn't already in the form $$ax^2 + bx + c = 0$$, rearrange it so that it is. The steps needed to factorise $$2x^2 + 5x + 3 = 0$$ are:

So $$(2x + 3)(x + 1) = 0$$ is $$2x^2 + 5x + 3 = 0$$ in factorised form. You can now use the fact that any number multiplied by $$0$$ is $$0$$ to find the roots of the equation. The numbers that make one bracket equal to $$0$$ are the roots of the equation. In the example, the roots are $$-1.5$$ and $$-1$$.

It is also possible to solve a quadratic equation using the quadratic formula or by completing the square.

Simultaneous equations
Simultaneous equations are useful in solving two or more variables at once. Basic simultaneous equations consist of two linear expressions and can be solved by three different methods: elimination, substitution or by plotting the graph.

Elimination method
The basic principle of the elimination method is to manipulate one or more of the expressions in order to cancel out one of the variables, and then solve for the correct solution.

An example of this:

From this, we can see that by multiplying equation (1) by a factor of 2 and then subtracting this new equation from (2), the $$y$$-variable will be eliminated.

(1) $$\times 2 \rightarrow 4x + 6y = 20$$ (1a) (Assigning the number (1a) to this equation)

Now subtracting (2) from (1a):

Now that we have $$2x = 14$$, we can solve for $$x$$, which in this case is $$7$$.

$$x = 7$$.

Substitute the newly found $$x$$ into (1):

$$2 \times 7 + 3y = 10$$

$$14 + 3y = 10$$

And we find that $$y = - 4/3$$

So, the solution to the two equations (1) and (2) are:

$$x = 7$$

$$y = -4/3$$

Substitution method
The substitution method relies on being able to rearrange the expressions to isolate a single variable, in the form variable = expression. From this result this new expression can then be substituted for the variable itself, and the solutions evaluated.

An example of this:

From this expression, it is possible to see that (2) is the most simplistic expression, and thus will be the better choice to rearrange.

Taking (2), and rearranging this into $$x = 6 - y$$. (2a)

Subbing (2a) into (1) we get

$$2(6 - y) + 3y = 12$$

Solving this, we get that $$y = 0$$

Again we can sub this result into one of the original equations to solve for $$x$$. In this case $$x = 6$$.

Note that for situations in which one of the equations is non-linear, you must isolate one variable in the linear equation and substitute it into the non-linear one. Then you can solve the quadratic equation with one of the methods above.

Another form of substitution is if you've got a similar expression in both equations, like in this case:

Here, $$2x$$ is found in both equations, so:

And since $$2x = 2x$$, you could do:

$$10 - 3y = 6 - 6y$$

$$6y - 3y = 6 - 10$$

$$3y = -4$$

$$y = -4/3$$

Now you've got $$y$$, and finding $$x$$ will be the same as above.

Graphical method
By plotting the lines of the two equations, you can solve them by seeing where the lines intersect. If the intersection is at the point (a,b), then the solution is $$x=a$$ and $$y=b$$.

Solving problems with simultaneous equations
Often, you will be given problems which you must be able to write out as a pair of simultaneous equations. You will need to recognise such problems, and write them out correctly before solving them. Most problems will be similar to these examples with some differences.

Example 1

''At a record store, 2 albums and 1 single costs £10. 1 album and 2 singles cost £8. Find the cost of an album and the cost of a single.''

Taking an album as $$a$$ and a single as $$s$$, the two equations would be:

$$2a+s=10$$

$$a+2s=8$$

You can now solve the equations and find the individual costs.

Example 2

''Tom has a budget of £10 to spend on party food. He can buy 5 packets of crisps and 8 bottles of drink, or he can buy 10 packets of crisps and 6 bottles of drink.''

Taking a packet of crisps as $$c$$ and a bottle of drink as $$d$$, the two equations would be:

$$5c+8d=10$$

$$10c+6d=10$$

Now you can solve the equations to find the cost of each item.

Example 3

''At a sweetshop, a gobstopper costs 5p more than a gummi bear. 8 gummi bears and nine gobstoppers cost £1.64.''

Taking a gobstopper as $$g$$ and a gummi bear as $$b$$, the two equations would be:

$$b+5=g$$

$$8b+9g=164$$

The problem can now be solved by using one of the methods above.

Quadratics
A quadratic is a Polynomial of degree 2, in the form $$f(x)=ax^2+bx+c$$.

Graph
A quadratic graph is one that can be written in the form $$y=ax^2+bx+c$$. The graph of $$y=2x^2 + 8x + 2$$ is shown on the right, and as you can see it has the same characteristic "bucket" shape that all quadratics have, called a parabola. The line of symmetry and the vertex (maximum or minimum point) of the graph can be found by plotting the curve point by point, and the root can be found by factorising the quadratic.

However, these properties can be more easily deduced from its completed square form:

$$y=2(x + 2)^2 - 6$$.

Completing the square
Completing the square is the process of changing a quadratic from the form $$ax^2 + bx +c$$ to the equivalent form $$a(x+d)^2 + e$$, where $$a$$, $$d$$ and $$e$$ are constants. For example, the quadratic $$2x^2 + 8x +2$$ would become $$2(x+2)^2 - 6$$.

Changing a quadratic to completed square form makes it easy to find several things, such as the roots of the quadratic and the vertex of the quadratic, without even requiring a sketch.

Here are the steps for completing the square. Don't worry, it's easier than it looks.

So the completed square form of $$y=2x^2 + 8x + 2$$ is $$2(x+2)^2 -6$$. The $$-6$$ tells us that the lowest point of the curve is at $$y=-6$$ and the $$x+2$$ tells us that the line of symmetry is at $$x+2=0$$ or $$x = -2$$. Therefore the vertex is at $$(-2,-6)$$, and if you look at the graph, you can see that is the case.

You can see from the graph that $$y=2x^2 + 8x + 2$$ has one root between $$-4$$ and $$-3$$ and another between $$-1$$ and $$0$$, where it crosses the $$y$$ axis. But how do you find the exact values? Using the completed square form, it can be re-arranged quite easily to find $$x=0$$:

The values of $$x$$ for this specific example are within the expected range, as seen on the graph.

The quadratic formula
The quadratic formula is derived from the general case of completing the square:

$$x =\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

It can be used to find the roots of a quadratic by putting numbers directly into it. For example, for $$y=2x^2 + 8x + 2$$:

$$x =\frac{-8 \pm \sqrt{64-16}}{4} = -2 \pm \sqrt{3}$$

so $$ x = -2 + \sqrt{3}$$ and $$x = -2 - \sqrt{3}$$.

The discriminant
Notice that the quadratic equation contains $$b^2 - 4ac$$ inside a square root sign. This part is called the discriminant and can be considered on its own to determine the number of roots of the equation.

 * If $$b^2 - 4ac<0$$ then you will be unable to find the square roots as you don't know how to square root a negative number. The type of numbers you have encountered so far are known as real numbers and so it is said that the quadratic has no real roots.
 * If $$b^2 - 4ac=0$$ then changing the $$\pm$$ sign in front of the square root won't make any difference, because it is zero either way. You will therefore get the same root twice, so it is said the quadratic has one repeated root.
 * If $$b^2 - 4ac>0$$ then the $$\pm$$ will mean you get two answers, and so you can say the quadratic has two distinct (i.e. different) roots.

Inequalities
An inequality is an expression which compares the relative sizes of points, lines, or curves. Unlike an equation, where both sides of the equals sign are always equal, inequalities can have one side greater than or equal to the other side.

The four signs of inequalities
There are four main basic signs:

 * $$<$$ less than,
 * $$>$$ greater than,
 * $$\le$$ less than or equal to, and
 * $$\ge$$ greater than or equal to.

For example, $$x < 4$$ means that $$x$$ is less than 4, $$x > 4$$ means that $$x$$ is greater than 4, $$x\le 4$$ means that $$x$$ is four or any number less than this, and $$x\ge 4$$ means that $$x$$ is four or any number higher than this.

Note that $$x > y$$ and $$y < x$$ are both essentially the same statement.

If you become confused with which sign means less than and greater than, it is useful to remember that the inequality signs always point to the smaller number.

Combining inequalities
There are some cases where two inequalities can be combined into one. For example, the height of the door was said to be between $$x \ge 1.95$$ and $$x < 2.05$$. The usual way of writing these is $$1.95 \le x < 2.05$$. Notice that the inequality signs are in the same direction. $$2.05 \ge x > 1.95$$ is perfectly acceptable, but it is incorrect to combine opposite facing inequalities and they must be left as two separate inequalities.

Solving linear inequalities
These signs can be used in place of equal signs, and an equation now becomes an inequality (since both sides are not always equal).

For example, instead of $$2x + 4 = 6$$ we could have $$2x + 4 > 6$$.

In this example, $$x$$ may be any number which makes this inequality greater than 6. In this case $$x > 1$$ but $$x \ne 1$$. If the inequality was $$2x + 4 \ge 6$$, then $$x$$ could take the value of 1.

An inequality can be manipulated and therefore solved just like an equation, although there is an extra step you must take when you multiply or divide by a negative number.

Multiplying or dividing by a negative number
When multiplying or dividing by a negative number, you must change the direction of the inequality sign.

For example, look at the inequality $$10 > 5$$. This is correct since 10 is obviously greater than 5. Now if we were to multiply both sides by -1, we would get:

$$-10 > -5$$.

This is incorrect, since -10 is actually less than -5. By reversing the inequality sign, we now have the correct inequality:

$$-10 < -5$$.

Solving quadratic inequalities
To solve a quadratic inequality, you could factorise it, just like a quadratic equation.

Alternatively you could draw its graph as if it was a quadratic equation, and then shade the side that's covered by the inequality. maybe an image here, to demonstrate the method?

Indices
You are probably already familiar with indices, for example $$x^2$$ is just a shorter way of writing $$x \times x$$ and $$x^4$$ is similarly $$x \times x \times x \times x$$. In $$x^5$$, $$x$$ is called the base and $$5$$ is called the power or exponent. $$x^4$$ is pronounced "x to the four", or "x raised to the 4th power" in full. Some powers are so useful that they have special names: $$x^2$$ is referred to as "x squared", $$x^3$$ is "x cubed" and $$x^{-1}$$ (which you will soon learn about if you haven't already encountered it) is called "the reciprocal of x".

Note: The "law of indices" is sometimes also called the "exponent laws" or "power rules". More generally, an index in mathematics is a superscript or subscript to a symbol.

Operations with indices
Using this notation you might notice several patterns.

Multiplication
Firstly, $$x^3 \times x^2$$ is $$\left(x \times x \times x\right) \times \left(x \times x\right)=x \times x \times x \times x \times x$$ which is $$x^5$$. Of course $$3+2=5$$ so you have added the powers together. To clarify, here is an example with numbers: $$2^3 \times 2^5 = 8 \times 32 = 256 = 2^8$$ (like before, $$3+5=8$$)

Division
Secondly $$\frac{x^4}{x^2}$$ is $$\frac{x \times x \times x \times x}{x \times x} = x \times x = x^2$$ (when $$x \neq 0$$). This time $$4-2=2$$ and so you have subtracted the powers.

Here is an example with numbers: $$\frac{10^5}{10^2}=\frac{100000}{100}=1000=10^3$$ and again $$5-2=3$$.

Base raised to two powers
Thirdly $$\left(x^2\right)^3$$ is $$\left(x \times x\right) \times \left(x \times x\right) \times \left(x \times x\right)=x \times x \times x \times x \times x \times x$$ which is $$x^6$$. You can see that $$2 \times 3 = 6$$ and so the powers have been multiplied. Here is another example with numbers: $$\left(3^2\right)^4=9^4=6561=3^8$$ and $$2 \times 4 = 8$$.

Multiple bases
Finally $$\left(xy \right)^3 = xy \times xy \times xy = x \times y \times x \times y \times x \times y = x \times x \times x \times y \times y \times y$$ which is the same as $$x^3y^3$$. Here is an example with numbers: $$\left(2 \times 5\right)^2 = \left(10\right)^2 = 100 = 4 \times 25 = 2^2 \times 5^2$$. There is a similar situation with division: $$\left(\frac{x}{y}\right)^2 = \frac{x}{y} \times \frac{x}{y} = \frac{x \times x}{y \times y} = \frac{x^2}{y^2}$$

The Laws of Indices
The rules that have been suggested above are known as the laws of indices and can be written as:

 * 1) $$x^ax^b = x^{a+b}$$
 * 2) $$\frac{x^a}{x^b} = x^{a-b}$$
 * 3) $$\left(x^a\right)^b = x^{ab}$$
 * 4) $$\left(xy \right)^n = x^n y^n$$
 * 5) $$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$$

Special indices
You may well have realised that $${x^1}=x$$. This can be seen by looking at the pattern of $$x^3=x \times x \times x$$, $$x^2 = x \times x$$ or by doing $$\frac{x^3}{x^2}$$ which is clearly $$x$$ but is also $$x^{3-2}=x^1$$ by law 2.

So far all the examples we have looked at are where the power is a positive integers, but by thinking about the laws it is possible to look at other cases.

Power 0
It is less obvious that $$x^0=1$$ for any $$x$$ (strictly speaking, any $$x \neq 0$$, see the note below.), but this can be shown in a similar way, for example $$\frac{x^4}{x^4}$$ is 1 but must be also be $$x^{4-4}=x^0$$ by law 2.

Negative powers
The next logical step is to ask what $$x^{-1}$$ is. Well by using law 2 "backwards" $$x^{-1} = x^{0-1} = \frac{x^0}{x^1} = \frac{1}{x}$$. A similar argument can be used for any other negative integer, for example $$x^{-3} = x^{0-3} = \frac{x^0}{x^3} = \frac{1}{x^3}$$.

Fractional powers
What if the power isn't even an integer? Suppose you wanted to find $$x^\frac{1}{2}$$, you could say that $${x^\frac{1}{2}} \times {x^\frac{1}{2}} = x^1$$ (by law 3, addition of powers) which means that $$x^\frac{1}{2}$$ must be $$\pm \sqrt x$$. However it is customary to only use the positive root and so $$x^\frac{1}{2}$$ is defined as $$\sqrt x$$. You can use a similar argument for other such fractions, for example $$\left(x^\frac{1}{3}\right)^3=x$$ so $$x^\frac{1}{3} = \sqrt[3]{x}$$.

In summary:
 * $$x^1=x$$
 * $$x^0=1$$
 * $$x^{-n}=\frac{1}{x^n}$$
 * $$x^\frac{1}{n}=\sqrt[n]{x}$$

Sometimes you might have to use the laws to understand what something means, for example $$x^\frac{2}{3} = \left(x^2 \right)^\frac{1}{3}$$ (using law 3), and $$\left(x^2\right)^\frac{1}{3} = \sqrt[3]{x^2}$$ (using the definition above). It's useful to remember the general rule that $$x^\frac{a}{b} = \sqrt[b]{x^a}$$.

Surds
In mathematics, a Surd is an expression containing a root with an irrational solution that can not be expressed exactly – for example, √3 = 1.732050808....

Sometimes it is useful to work in square roots, rather than using an approximate decimal value. Square roots can be manipulated just like algebraic expressions and sometimes it may be possible to eliminate the square root (called rationalising the expression), which may have not been possible if you tried to work with the approximate value. When asked to give the exact value, approximate decimal answers will not do and you will have to manipulate surds in order to give a final answer in simplified surd form.

Simplification of surds
Because surds can be manipulated like algebraic expressions, you can easily multiply out the terms and add the like terms. However, there are also a few rules that will be useful when simplifying surds.

Basic rule of surds
Because $$\sqrt {x} \times \sqrt {x} = x$$, it is useful to know that it can be rearranged to give $$\sqrt {x} = \frac {x} {\sqrt {x}}$$ and $$\frac {1} {\sqrt {x}} = \frac {\sqrt {x}} {x}$$.

Surds as indices
Because $$\sqrt[n]{x} = x^\frac{1}{n}$$ the laws of indices also apply to any n-th root. The most frequently used instances of this are laws 4 and 5 with square roots:

 * $$\left(xy \right)^n = x^n y^n$$ becomes $$\sqrt{xy} = \sqrt{x} \times \sqrt{y}$$
 * $$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$$ becomes $$\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$$

The first of these points is often used to simplify a square root, for example $$\sqrt{200}=\sqrt{100 \times 2}=\sqrt{100} \times \sqrt{2}=10\sqrt{2}$$. In an exam, you will be expected to write all square roots with the smallest possible number inside the square root (i.e. the number inside the root shouldn't have any square factors).

Rationalising the denominator
Another technique to simplify expressions involving square roots is to rationalise the denominator. This means getting rid of square roots from the bottom of a fraction. In the case of a fraction such as $$\frac{5}{\sqrt{3}}$$, both numerator and denominator can be multiplied by $$\sqrt{3}$$ to give $$\frac{5\sqrt{3}}{3}$$.

If the fraction is of the form$$\frac{a}{b+\sqrt{c}}$$ the strategy used in the previous paragraph will only work if it is modified slightly. This time you should multiply the numerator and denominator by $${b-\sqrt{c}}$$. If you are familiar with the standard difference of two squares expansion you should already know what happens next:

$$\frac{a}{b+\sqrt{c}} \times \frac{b-\sqrt{c}}{b-\sqrt{c}} = \frac{ab-a\sqrt{c}}{b^2 + b\sqrt{c} - b\sqrt{c} - \sqrt{c}^2} = \frac{ab - a\sqrt{c}}{b^2-c} $$. As you can see the denominator now does not contain any square roots. For example: $$\frac{2}{\sqrt{3} - 1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1} = \frac{2\sqrt{3}+2}{3-1} = \sqrt{3}+1 $$

Splitting up roots
A common mistake is to split $$\sqrt{x + y}$$ into $$\sqrt{x} + \sqrt{y}$$ or $$\left(x+y\right)^2$$ into $$x^2 + y^2$$, usually whilst moving it to the other side of the equals. Trying a few examples will quickly convince you that this is not possible:  $$\sqrt{25}$$ ≠ $$\sqrt{9} + \sqrt{16}$$ $$\sqrt{64}$$ ≠ $$\sqrt{32} + \sqrt{32}$$  And so on

What is the value of $$0^0$$?
The short answer is that for this course you don't need to know, and you can safely skip this section. If you're still interested then read on:

The question arises because $$x^0=1$$ for any $$x$$ and yet you would expect that $$0^y=0$$ for any $$y$$ as $$0 \times 0 \times 0 \dots = 0$$. It turns out using a value of 1 is quite useful (perhaps even necessary) in various parts of algebra, whereas making it zero doesn't help at all. Almost all mathematicians would therefore either say that $$0^0$$ is 1 or that it is undefined (that is, it can't be given a value). A more technical discussion can be found at http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/

title: A-level Mathematics/MEI/C1/Algebra