Biyernes, Marso 9, 2012

Tablet PC


Tablet PC
A tablet personal computer or tablet PC is tablet-sized computer that also has the key features of a full-size personal computer. A tablet PC is essentially a small laptop computer, equipped with a rotatable touchscreen as an additional input device, and running a standard (or lightly adapted) PC operating system like Windows or Linux.
The term was made popular with the Microsoft Tablet PC concept presented by Microsoft in 2001. Today, the term tablet is also used to refer to computer-like devices operated primarily by a touch screen but not intended to run general PC operating systems or applications.

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What is a Tablet PC?
Tablet PC is a lightweight version of a PC in the shape of a flat panel. User interact with it by tapping the screen and hand-writing with a penlike device. Tablet PCs will also let users wirelessly connect to the Internet and other PCs. In a word, a Tablet PC combines the mobility of a handheld computer with the computing power of a notebook.
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What are the main characteristics of a Tablet PC?
  • Powerful -- Tablet PC can run almost all PC applications compatiable with Windows XP, plus the applications which specifically target Tablet PC. On connectivity side, besides the built-in wireless support, Tablet PCs can also be placed into a docking station at a desk to support a large monitor, network connection, keyboard, mouse, and other peripherals.
  • Mobility -- The Tablet PC comes in two basic forms, the "convertible" with an integrated keyboard, and the ultra-mobile "pure tablet". All Tablet PCs will be very light — ranging from 2 to 3 pounds. Combined with wireless network support, Tablet PC gives you greater mobility and immediate access to the full power of your PC.
  • Natural interaction with computer -- Using a digital pen and the Input Panel utility, you can write directly on the screen and save your notes in your own handwriting or convert them to typed text for input into other applications. Speech recognition is an integral component of every Tablet PC, allowing users to dictate content or control applications with their voice.
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Is Tablet PC right for me?
Tablet PC's special features make it best choice for people who have jobs requiring them to be on the move, whether attending meetings in the office or conducting business on the road. You will also find Tablet PC very useful if you like to have the full power of a notebook at a greater mobility.
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What are differences between two versions of tablet PC?
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There are two versions of the tablet PC -- the slate style and the convertible style.
Slate-style tablet appears similar to a large PDA. The approximate size of the device is 8.5 inches by 12 inches, with a color screen measuring around 10 inches diagonally with control buttons on the edge of the screen. The tablet PC connects to a docking station and peripherals including keyboard and mouse, but can be carried around as a mobile device, with a stylus for input. The slate design has the advantage of being simple and sleek without hinges or moving parts, but it has no keyboard or mouse when the device is mobile, away from the docking station.
The second style, the convertible, is essentially a laptop PC that converts to a tablet PC. The convertible tablet PC has a clamshell design similar to the laptop PC; but when opened, the screen side of the PC swivels 180 degrees, and the backside of the screen folds down on top of the keyboard with the screen side facing up. The size of the convertible design is similar to the slate, although slightly thicker to accommodate for the hinged keyboard component. The main advantage of the convertible style is its similarity to the laptop PC--a familiar device for the consumer. The convertible style also has the advantage of the attached keyboard, but the hinged clamshell design creates a bulkier device that may be susceptible to damage or breaking.
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What operating system is Tablet PC running?
The Tablet PC runs a specialized version of Microsoft's Windows XP operating system. This version offers the features and capabilities of Windows XP Professional, plus additional features specific to the Tablet PC, such as handwritting support and voice recognision. All existing Windows software applications would still be able to run on Tablet PCs.
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What CPU does Tablet PC use?
Most hardware manufactures are currently using Intel Pentium III (above 800MHz) ultra low voltage CPU for mobile computer.
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Which companies make Tablet PCs?
So far Acer, Compaq, ElectroVaya, FIC, Fujitsu, Tatung, VIA Technologies and ViewSonic etc. have announced their Tablet PC products. More companies are expected to join the list in the near future.
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What types of Tablet PC are currently available and where can I get one?
Tablet PCs from several companies, e.g. Acer, Compaq, Fujitsu, Motion Computing, Toshiba, ViewSonic etc., are currently avaialable in the market. Please refer to the Products Center page for detailed product specifications and their availability at some of the online retailers.
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Where can I find more information about Tablet PC?
Microsoft's Tablet PC home page is a good starting point to learn more about Tablet PC, e.g. Tablet PC system overviewfeatures, and demos etc.  Or you may refer to Related Links page for other tablet PC related websites

Features of Tablet PC
Tablet personal computer or Tablet PC is increasingly becoming a replacement for laptops and desktops and the youth love to get hold of this new technological change in the form of Tablet PC. Given below is the list of some of the features or characteristics of tablet PC which makes it special among youth –
1. The first and foremost feature of tablet PC is that this is the new technology and new technology whether you like it or not attract people towards it.
2. Tablet PC is much smaller than a laptop and therefore it is easy to carry as it is very thin and people who travel frequently are attracted to tablet PC.
3. Tablet PC have on-screen virtual keyboard and this is why they are different from laptop in terms of size and dimensions.
4.Tablet PC are sleek in design making it attractive to the people, and it is one of the reason for its growing popularity as humans have the tendency to go towards things which are attractive.
5. The tablet PC can be laid flat and rotated easily which is not possible in case of laptops or desktops.

Application
Microsoft released the Tablet PC interface several years ago. This platform allows users to more naturally interact with a computer by using a pen in addition to other input devices. It also allows for the use of digital ink as a datatype within applications. This additional pen interaction and digital ink can improve the interaction of both students and faculty with the PC. Some potential applications of Tablet PCs in education include:

•Classroom presentations which incorporate handwritten notes on PowerPoint or Classroom Presenter slides;
•Electronic grading of papers with digital ink;
•Digital note taking and sharing of handwritten notes;
•Improved development of digital media through the use of the pen interface with drawing or CAD applications;
•Development of rapid eLearning applications which can incorporate handwritten examples; and
•In class collaboration between faculty and students with portable computers.
This study is designed to investigate the potential benefits and challenges encountered by students and faculty members when using Tablet PCs. In Fall 2005, we will focus on the use of Tablet PCs by faculty members in Architectural Engineering along with several graduate students to determine the applications and functions that improve their teaching, learning, and research initiatives. Faculty and students will complete a survey before and after the semester. They will also report any issues encountered using the Table PCs throughout the semester.

In Spring 2006, we will focus on the development of a collaborative classroom within the ICon Lab using Tablet PCs as an integral tool within one graduate level course (AE 597F - Virtual Facility Prototyping). The Access Grid will be used to organize the faculty and student collaboration during the class and VNC will be used so that any student in the classroom will be able to share their screen with the entire class. The value of this collaborative learning environment will be assessed.

Throughout these two semesters, the applications used, functions performed, benefits gained, and challenges faced by the users will be documented and explored so that future applications of Tablet PCs within educational environments can be better designed.

The software installed on the Gateway M275 Tablet PCs (installed on the Microsoft XP Tablet PC Edition 2005) includes :
•Microsoft Office for Tablet PC - Adds the ability to use digital ink in MS Office applications.
•Microsoft OneNote 2003- An note taking program that allows you to capture, organize and save either handwritten or typed notes.
•Microsoft Experience Pack - Includes the Ink Desktop, Snipping Tool, Ink Art, Media Transfer and other applications.
•Microsoft Education Pack - Includes Equation Writer, Ink Flash Cards, Send to OneNote, and a trial version of GoBinder planning software.
•Microsoft PowerToys for Tablet - Includes many small applications designed for the Tablet interface.
•Microsoft Journal - Allows for the markup and distribution of documents with ink as well as note taking.
•Adobe Photoshop and Illustrator - Allows for the easy modification and development of images using the pen interface.
•Classroom Presenter 2.0 - Presentation software designed for collaborative views
•Google Sketchup 6 - A 3D CAD application that is designed around the tablet interface. It is very easy to learn and use. Faculty members can request a free license for use in their class.
Additional software installed on the Tablet PCs which are not specifically designed to use the tablet interface includes:
•Autodesk Revit - 3D Building Information Modeling application.
•Autodesk AutoCAD - CAD applications.
•Navisworks Jetstream - 3D Model Review Application
•UltraVNC - Screen and file sharing software that can be used for collaboration. This software can also allow remote connection of a tablet PC to a computer with a projector.
Why use a Tablet PC?
·       Tablet PCs allow teachers and students greater flexibility in learning environments. Students and teachers no longer have to be in computer labs or classrooms.
·       They allow for mobility in the room to enable you to assist in student collaborations.
·       Tablet PCs can be used anywhere in or out of the school building.
·       Stay mobile in or out of the classroom but still have access to the information you normally have on your desktop or laptop PC.
·       Tablet PCs remove the barriers that are created from the screens of laptops and desktops to allow you to maintain eye contact and interaction with your students.
·       Saves schools money on paper.
·       Greater communication between students and teachers.
·       Add enthusiasm to a lesson.


How are educators using Tablet PCs?

Teacher use:
·       To grade papers. Tablet PCs provide teachers with the ability to insert comments right into a students work in Microsoft Office.
·       As a whiteboard or chalkboard connected to a projector. Teachers are able to have students brainstorm in class by writing their thoughts on the Tablet PC and save their work without having to give up chalkboard space or worry about someone erasing it. Tablet PCs also allow teachers to provide notes to students typed so they are easy for students to read.
·       Create integrated assignments for the students use.
·       Create worksheets students can fill in and submit electronically.

Student use:
·       Create and use integrated assignments. Students or teachers can combine reading assignments, class notes and other learning material into one document or multiple documents at hand without the hassle of fumbling through books for the materials.
·       The ability to take notes in their own handwriting and also search through those notes.
·       Collect and share web sites and information for research.
·       Allows students to submit assignments but also retain a copy.


Huwebes, Marso 8, 2012

Philosophy

On Definition: A Lecture in Logic

DEFINITION IS A STATEMENT that gives the meaning of a term or explains what a term means. As it clarifies the limits by which a word or term should be used and understood, definition helps eliminate confusion and ambiguity in the use of terms, thereby minimizing, if not totally eradicating, misunderstanding and misconception in communication.
          Definition comprises two elements: the definiendum or the term to be defined; and the definiens or the defining term. In the definition, “Mathematics is the study of the relationships among numbers, shapes, and quantities,” Mathematics is the definiendum and the study of the relationships among numbers, shapes, and quantities is the definiens.

4 Main Types of Definition
1. Nominal definition or ‘definition by name’ simply conveys what the term means, not what the thing is.
Kinds of Nominal Definition
a. Etymological definition gives the root word, derivation, or origin of a term.
Examples:
Philosophy (philia, love; Sophia, wisdom) is love of wisdom.
Agoraphobia (“agora”, open space; “phobia”, fear) is the fear of open or public spaces.
Transport (“trans”, across; “portare”, to carry) means, “to carry across”.
Pantheism (“pan”, all; “theos”, God; “ismos”, system of belief) is the belief that God is everything and everything is God.

b. Synonymous definition gives the equivalent word (which is usually the better known or more common) of the term being defined.
Example:
Novice means “beginner.”
Aficionado means “enthusiast”.
Crimson means “red”.

c. Biverbal definition gives the translation of the term in another language.
Example:
Vox populi means “voice of the people”.
Caveat means “warning or proviso”.
A bundok is a “mountain”.

d. Definition by full form gives the word or group of words represented by abbreviation (shortenings, contractions, initialisms, and acronyms.)
Examples: Shortenings                                                                                                                               cont. means continued; km. means kilometer; hippo means hippopotamus; limo means limousine; bus means omnibus; taxi means taxicab; deli means delicatessen; zoo means zoological garden; bike means bicycle.
Examples: Contractions
Dr. means doctor, St. means saint or street; couldn’t means could not; didn’t means did not, we’ve means we have.
Examples: Initialisms
CSI is Crime Scene Investigation; LA is Los Angeles; A.W.O.L. is absent without official leave; a.m. is ante meridiem
Examples: Acronyms
AIDS is Acquired Immune Deficiency Syndrome; laser is light amplification by stimulated emission of radiation; scuba is self-contained underwater breathing apparatus; UNESCO is United Nations Educational, Scientific, and Cultural Organization.

2. Denotative definition or definition by example elucidates the meaning of a term by mentioning or showing its referent/s.
Kinds of Denotative Definition
a. Demonstrative definition or ostensive definition explicates the meaning of a term by showing or pointing at the object.
For instance, a teacher may explain the meaning of a “laptop” by bringing one and showing it to the class. A boxing trainer may try to answer the question, “What does ‘left hook’ mean?” by executing it. Proper names, like “Pres. Obama” may also be defined ostensively.

b. Enumerative definition is done by naming the referents of the term.
Ex: If someone asks, “What is a fruit?” and we answer by saying, “The apple is a fruit. The orange is a fruit. The mango is a fruit. The guava is a fruit.”

3. Descriptive definition elucidates the meaning of a term by citing the essential or accidental characteristics of the referent/s of the term.
Kinds of Descriptive definition
a. Distinctive definition mentions the property of a thing that emanates from its essence or unique trait.
Ex: Man can have sense of humor. (Ability to see something as funny is a property of man, which arises from his being rational).
H20 can take the solid, liquid, and gas state of matter.

b. Accidental definition explains a thing by stating its contingent or unessential characteristics.
Examples:
The house has three bathrooms.
The car is yellow.
Kobe Bryant has tattoos.

c. Genetic definition explains how a thing originated or depicts how something is produced.   
Example:
Yeast breads are made by mixing, kneading, and rising the dough, then shaping and baking the bread.
Pepsi-Cola (Pepsi) is a mixture of carbonated water, cane-sugar syrup, and an extract from tropical kola nuts.
A lunar eclipse occurs when the earth is between the sun and the moon and its shadow darkens the moon.
A solar eclipse happens when the moon is between the sun and the earth and its shadow moves across the face of the earth.

d. Definition by efficient cause gives the maker or producer of a thing.
Examples:
Nike, Inc. was founded by former track athlete Philip Knight and his college coach William Bowerman.
The Chronicles of Narnia was written by C. S. Lewis.
Candies are made by a confectioner.
The Bible was written by God’s selected servants through His commissioning and inspiration.
                 
e. Definition by final cause declares the purpose or end of which a thing is produced.
Examples:
School is an institution for teaching students.
Exercise is an activity undertaken to enhance general physical health.
A calculator is a device used to compute arithmetic operations.

4. Connotative definition or essential definition provides the nature or essential characteristics of the referents denoted by a term. It has two elements: the genus or the property that a thing has in common with its co-members in the same class, and the specific difference or the thing’s essential property that distinguishes it from other members of the same class.
Example:
Octagon is a polygon with eight sides.
Dictionary is a reference book of word meanings or equivalents.
Tsunami is a large ocean wave caused by an underwater earthquake or another movement of the Earth's surface.
Blizzard is a storm characterized by extreme cold, strong winds, and a heavy snowfall.

Rules of a Good Connotative Definition
Connotative definition is very much useful in logic because it is a definition, which alone gives the essential meaning of a term. It helps eliminate ambiguity and it reduces the vagueness of the term in its essential attributes. The following are some rules in making a connotative definition:
1. The defining term must be equivalent to the term being defined. The definition must neither be too broad nor too narrow.
The definition, A typhoon is a violent storm is too broad since hurricane and blizzard, which are different from typhoon, are also violent storms. A guitar is a stringed musical instrument is also too broad definition because many musical instruments other than guitar are also stringed.
On the other hand, the definition Martial Arts are Korean Tae Kwon Do is too narrow as it excludes Aikido, Judo, Karate, and other forms of martial arts. Same is the case of A basketball player is a point guard for it excludes basketball players who are playing other positions.

2. Definition should not be circular. The term being defined should not be contained in the defining term; otherwise, the definition gives little or no information at all and thus fails in its purpose to explain the meaning of a term.
Examples:
A player is one who plays.(Circularity can be avoided by saying A player is somebody taking part in a sport game.)
A cookbook is a book for cooking. (Better: A cookbook is a book containing recipes and directions for preparing food.)
An answering machine is a machine that answers. (An answering machine is a phone recording device that plays a message to callers and records messages from them.)
A green card is a card that is green. (In the United States, a green card is an identity card and work permit issued to nationals of other countries.)
A blue book is a book that is blue. (It is a blank notebook used in schools for writing examination answers.)

3. The definition must be stated in univocal and simple terms. The definition should not be expressed in obscure, ambiguous, or figurative language. In defining terms, perplexing and highly theoretical explanations, as well as subjective judgments, must be avoided.
Examples:
Figurative definitions:
Marriage is a bribe to make the housekeeper think she's a householder.--Thornton Wilder (1897 - 1975)
Science is a first-rate piece of furniture for a man's upper chamber, if he has common sense on the ground floor.--Oliver Wendell Holmes (1809 - 1894)
Dreams are cages within which we observe the cages without.--William Wantling (1933 - 1974)
Beauty is a shell from the sea where she rules triumphant till love has had its way with her.--William Carlos Williams (1883 - 1963)
Friendship is a disinterested commerce between equals; love, an abject intercourse between tyrants and slaves. – Oliver Goldsmith (1730 - 1774)

Obscure definitions:
Life is our reaction to the basic insecurity which constitutes its substance.--José Ortega y Gasset (1883 - 1955)
Reason is…a wonderful and unintelligible instinct in our souls, which carries us along a certain train of ideas, and endows them with particular qualities, according to their particular situations and relations.--David Hume (1711 - 1776)
Consciousness… is of a teeming multiplicity of objects and relations, and what we call simple sensations are results of discriminative attention, pushed often to a very high degree.--William James (1842 - 1910)
Consciousness is thoroughgoing dialectical restlessness, this melee of presentations derived from sense and thought, whose differences collapse into oneness.--G. W. F. Hegel (1770 - 1831)
Reason is…an act of faith to assert that our thoughts have any relation to reality at all.--G. K. Chesterton (1874 - 1936)

Subjective definitions:
Death is my neighbour now.--Edith Evans (1888 - 1976) (British actor.
Said a week before her death)
Happiness [is] a good bank account, a good cook, a good digestion.--attributed to Jean-Jacques Rousseau (1712 - 1778)
Philosophy [is] living voluntarily among ice and high mountains—seeking out everything strange and questionable in existence, everything so far placed under a ban by morality. -- Friedrich Wilhelm Nietzsche (1844 - 1900)
Love is the difficult realization that something other than oneself is real.--Iris Murdoch (1919 - 1999)
Opera is when a guy gets stabbed in the back and instead of bleeding he sings.
--Ed Gardner (1904 - 1963)

4. The definition should not be needlessly negative.  As much as possible, the definition must be in the affirmative form for it must explain what a term means rather than what it does not.
          To define basketball as “a ballgame that is neither volleyball nor football” is obviously to fail to explain its meaning. Notice too that there are many other kinds of ballgame, aside from volleyball and football, which are not meant by the term basketball.
Other examples:
A male is not a female.
Long is the opposite of short.
A ball pen is neither a pencil nor a chalk.
Moral is that which is not immoral.
Negative definition, nonetheless, are accepted for terms whose meaning are necessarily negative (e.g. Infinity is a state of being which is without limits) and when the terms refer to a privation (e.g. A deaf is someone who cannot hear.)

Calculus problems


PROBLEM: An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume ?

SOLUTION  : Let variable x be the length of one edge of the square base and variable y the height of the box.

The total surface area of the box is given to be
48 = (area of base) + 4 (area of one side) = x2 + 4 (xy) ,
so that
4xy = 48 - x2
or
$ y = \displaystyle{ 48 - x^2 \over 4x } $
$ = \displaystyle{ 48 \over 4x } - \displaystyle{ x^2 \over 4x } $
$ = \displaystyle{ 12 \over x } - (1/4)x $ .
We wish to MAXIMIZE the total VOLUME of the box
V = (length) (width) (height) = (x) (x) (y) = x2 y .
However, before we differentiate the right-hand side, we will write it as a function of x only. Substitute for y getting
V = x2 y
$ = x^2 \big( \displaystyle{ 12 \over x } - (1/4)x \big) $
= 12x - (1/4)x3 .
Now differentiate this equation, getting
V' = 12 - (1/4)3x2
= 12 - (3/4)x2
= (3/4)(16 - x2 )
= (3/4)(4 - x)(4 + x)
= 0
for
x=4 or x=-4 .
But $ x \ne -4 $ since variable x measures a distance and x > 0 . Since the base of the box is square and there are 48 ft.2 of material, it follows that$ 0 < x \le \sqrt{ 48 } $ . See the adjoining sign chart for V' .
http://www.math.ucdavis.edu/~kouba/MAXMINCHARTSdirectory/MaxMin3der1.gif
If
x=4 ft. and y=2 ft. ,
then V = 32 ft.3 is the largest possible volume of the box.

Sample problems


Example 1.
            Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.
Solution: Let variables x and y represent two nonnegative numbers. The sum of the two numbers is given to be 9 = x + y
so that
y = 9 - x
We wish to MAXIMIZE the PRODUCT
P = x y2 .
However, before we differentiate the right-hand side, we will write it as a function of x only. Substitute for y getting
P = x y2
= x ( 9-x)2
Now differentiate this equation using the product rule and chain rule, getting
P' = x (2) ( 9-x)(-1) + (1) ( 9-x)2
= ( 9-x) [ -2x + ( 9-x) ]
= ( 9-x) [ 9-3x ]
= ( 9-x) (3)[ 3-x ]
= 0
for
x=9 or x=3 .
Note that since both x and y are nonnegative numbers and their sum is 9, it follows that 0 ≤ x ≤ 9.
If
and
then
x=0
y=9
P=0
x=3
y=6
P=108
x=9
y=0
P=0

If x=3 and y=6, then P=108 is the largest possible product. (substitute to P=xy2)
Example 2.
            A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to be 1  inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that least amount of paper is used?
Solution: Let A be the area to be minimized
A = (x+3)(y+2)            primary equation
The printed area inside the margins is given by
24 = xy                                    secondary equation
Solving this equation for y produces y= . Substituting into the primary equation produces
A = (x+3)(+2)
    =30+2x+              function of one variable
Because x must be positive, you are interested only in values of A for x>0. To find the critical numbers, differentiate with respect to x.
 = 2- = 0 or x2=36

So the critical numbers are x=±6. You do not have to consider x=-6 because it is outside the domain. The First Derivative Test confirms that A is a minimum when x=6. So, y= =4 and the dimensions of the page should be x+3=9 inches by y+2=6 inches.

References:
Calculus Concepts and Contexts Second Edition by James Stewart

Optimization Absolute Maxima and Minima


OPTIMIZATION ABSOLUTE MAXIMA AND MINIMA


A function f has an absolute maximum at c if f(c) ≥ f(x), for all x in the domain of f. At x=c, the graph reaches its highest point. The number f(c) is called the maximum value of f.
A function f has an absolute minimum at c if f(c) ≤ f(x), for all x in the domain of f. At x=c, the graph reaches its lowest point. The number f(c) is called the minimum value of f.
Together, the maximum and minimum values are called the extreme values of the function f.

GUIDELINES FOR SOLVING MAX./MIN. PROBLEMS
1. Read each problem slowly and carefully. Read the problem at least three times before trying to solve it. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking. If you misread the problem or hurry through it, you have NO chance of solving it correctly.
2. If appropriate, draw a sketch or diagram of the problem to be solved. Pictures are a great help in organizing and sorting out your thoughts.
3. Define variables to be used and carefully label your picture or diagram with these variables. This step is very important because it leads directly or indirectly to the creation of mathematical equations.
4. Write down all equations which are related to your problem or diagram. Clearly denote that equation which you are asked to maximize or minimize. Experience will show you that MOST optimization problems will begin with two equations. One equation is a "constraint" equation and the other is the "optimization" equation. The "constraint" equation is used to solve for one of the variables. This is then substituted into the "optimization" equation before differentiation occurs. Some problems may have NO constraint equation. Some problems may have two or more constraint equations.
5. Before differentiating, make sure that the optimization equation is a function of only one variable. Then differentiate using the well-known rules of differentiation.
6. Verify that your result is a maximum or minimum value using the first or second derivative test for extreme values.
Rounded Rectangle: First Derivative Test for Absolute Extreme Values

Suppose that c is a critical number of a continuous function f on an interval

(a) If f’(x) > 0 for all x<c and f’(x) < 0 for all x>c, then f(c) is the absolute maximum value of f.
(b) If f’(x) < 0 for all x<c and f’(x) > 0 for all x>c, then f(c) is the absolute minimum value of f.
Text Box: Second Derivative Test for Absolute Extreme Values
Let l be the range of all possible optimal values of    and further suppose that   is continuous on I, except possibly at the endpoints. Finally suppose that   is a critical point of    and that c is in the interval I.  Then,
1. If    for all x in I  then    will be the absolute minimum value of    on the interval I.
2. If    for all x in I  then    will be the absolute maximum value of    on the interval I.