OpenLLM-France/wikimedia huggingface dataset:
# Caro-Kann Defence
## 4...Qd5

The dramatic, and probably unsound, 4...Qd5 was played by Thomas Johansson against Nora Medvegy in the EU Championship and Cork Chess Congress in 2005. White was unable to take advantage, and the game continued as follows:
1. e4 c6 2. d4 d5 3. Nc3 dxe4 4. Nxe4 Qd5 5. Nc3 Qd6 6. Bc4 Nf6 7. Nge2 Bf5 8. Bf4 Qd8 9. d5 Nxd5 10. Nxd5 cxd5 11. Qxd5 Qxd5 12. Bxd5 Nc6 13. O-O-O f6 14. Nd4 Nxd4 15. Rxd4 e5 16. Bxb7 Rb8 17. Bc6+ Kf7 18. Bd5+ Kg6 19. g4 exf4 20. gxf5+ Kxf5 21. Re1 g5 22. Bf7 Rb5 23. Bd5 Bc5 24. Be6+ Kg6 25. Rd2 Rhb8 26. c3 Bb6 27. Bd7 Re5 28. Rxe5 fxe5 29. Rd6+ Kf7 30. f3 Ke7 31. Rd1 h5 32. Bf5 Kf6 33. Be4 g4 34. Rd6+ Ke7 35. Rh6 Rg8 36. Rh7+ Kf6 37. fxg4 hxg4 38. Rh6+ Kg5 39. Re6 f3 40. Rxe5+ Kf4 41. Re7 Rh8 42. Bh7 Bg1 43. Rf7+ Kg5 44. Be4 Rh3 45. Rf5+ Kh4 46. c4 Bxh2 47. c5 Kg3 48. c6 Kf2 49. Kd2 Rh8 50. Kd3 Rd8+ 51. Bd5 Bc7 52. Kc4 Ke2 53. Rf7 Rxd5 54. Kxd5 f2 55. Rxc7 f1=Q 56. Re7+ Kd3 57. c7 Qf5+ 58. Kc6 Qc8 59. Rd7+ Kc4 60. b3+ Kb4 61. Rd4+ Kc3 62. Rd8 Qa6+ 63. Kd7 Qb5+ 64. Kd6 Qb6+ 1/2-1/2
When contributing to this Wikibook, please follow the Conventions for organization.

## References

* Kasparov, Garry, & Keene, Raymond 1989 Batsford chess openings 2. ISBN 0-8050-3409-9.
# Caro-Kann Defence - Advance Variation

This line is the most common response to the 3. e5, the Advance Variation of the Caro-Kann, which has gained popularity after having previously been widely regarded as inferior for many years, owing chiefly to the strategic demolition that Aron Nimzowitsch (playing as White) suffered at the hands of José Capablanca in one of their games at the New York 1927 tournament:
The Advance Variation has since been revitalized by aggressive lines such as the Bayonet Attack (4.Nc3 e6 5.g4) favored by Latvian Grandmaster Alexei Shirov or the less ambitious variation (4.Nf3 e6 5.Be2 c5 6.Be3) popularized by English Grandmaster Nigel Short.

## Theory table

For explanation of theory tables, see theory table and for notation, see algebraic notation. .
1.e4 c6 2.d4 d5 3.e5 Bf5
* 4
* Nc3 e6
* Nf3 e6
* h4
* Be3
* c3
* Nd2
* c4
* Bd3 Bxd3 -=
* Ne2
* g4
* f4
* Be2
* a3
* h3
* Bd2
When contributing to this Wikibook, please follow the Conventions for organization.

## References

* Nunn's Chess Openings. 1999. John Nunn (Editor), Graham Burgess, John Emms, Joe Gallagher. ISBN 1-8574-4221-0.
* Modern Chess Openings: MCO-14. 1999. Nick de Firmian, Walter Korn. ISBN 0-8129-3084-3.
* Kasparov, Garry, & Keene, Raymond 1989 Batsford chess openings 2. ISBN 0-8050-3409-9.
* This Chess related article is a stub. Please assist Wikibooks by expanding it.
# Chess Opening Theory/1. e4/1...c6/2. d4/2...d5/3. Nc3/3...dxe4/4. Nxe4/4...Nf6/5. Qd3/5...e5
## 5...e5?

This move appears to completely free Black's position and equalize the game, but it really loses time in a wide open position and falls behind in development. This puts Black is a critical position. This also gives black an opportunity to fall for a brilliant mating attack that was played in the game Reti-Tartakower, 1910. It is rarely a good idea to open up the position if you are behind in development, and Black is doing just this here.
Better is to trade knights on e4 and then move the other knight to d7. He can then play for the freeing...c5, ...e5, or attack the queen with ...Nf6 After this mistake, black loses time and development.

## Reti-Tartakower, 1910

The game continued: 1. e4 c6 2. d4 d5 3. Nc3 dxe4 4. Nxe4 Nf6 5.Qd3 e5? 6.dxe5 Qa5+ 7.Bd2 Qxe5 8.0-0-0! (Sacrificing a piece to get at the unprotected black king). Nxe4?? (This gets mated. However, he was so far behind in development that he already had a poor position. 9..Qxe4 loses the queen to 10.Re1) 9.Qd8+!! Kxd8 10.Bg5+ Kc7 (10...Ke8 Rd8#) 11.Bd8# A remarkable finish!
The final position is given in the second diagram below. According to chessgames.com, the game Reti-Tartakower, 1910 was the "Game of the day" on Jan 16 2005 and the "Sucker Punch".

## References

Hodgson, Julias. Quick Chess Knockouts. Globe Print Press. p. 108-109. ISBN 1-85744-045-5. {{cite book}}: Cite has empty unknown parameter: |1= (help)
* ↑ http://www.chessgames.com/perl/chessgame?gid=1250654
# Caro-Kann Defence: Finnish Variation

With move 4. h6 Black creates escape square for light squared bishop's later use. It is usually consired as an inaccuracy and it is therefore rarely seen in higher levels.
White's most usual respond is 5. Nf3. Natural development move 5. Bc4 is also played often and usually with an aim to sacrifice bishop in f7.
When contributing to this Wikibook, please follow the Conventions for organization.
# Caro-Kann Defense: Modern Variation

4. Nxe4 transposes into the main line Caro-Kann Defense.

## Theory table

For explanation of theory tables, see theory table and for notation, see algebraic notation.
1.e4 c6 2.d4 d5 3.Nd2 dxe4
| | 4 | 5 | 6 | 7 | |
| Main Line | Nxe4 Bf5 | Ng3 Bg6 | h4 h6 | Nf3 Nd7 | = |
When contributing to this Wikibook, please follow the Conventions for organization.
# Caro-Kann f6

This has been tried quite a few times, but chances of success are low.

## Epilogue

The f6 Caro-Kann is a rare line that weakens the kingside and hinders the development of the king's knight, but prepares an e5 push.
# Caro-Kann f6

This refutes the f6 line.

## Epilogue

Ng3 is an interesting but important move that allows the e5 pawn push, but gains a huge lead in development after 5...e5 6. Nf3.
# Christian Hymns/Lo! Star-Led Chiefs
## Background
### Text

* Author: Reginald Heber
* Source:

### Music

Hymn Tune(s):
* Lo! Star-Led Chiefs

## Lyrics

Lo! star-led chiefs, Assyrian odours bring,
And bending Magi
and bending Magi seek their infant King!
Lo! star-led chiefs, Assyrian odours bring,
And bending Magi
bending Magi seek their infant King!
bending Magi seek their infant King!
Mark'd ye, mark'd ye where hov'ring o'er his head,
hov'ring o'er his head, o'er his radiant head,
The dove's white wings celestial, celestial glory shed,
celestial glory shed, celestial glory shed?
Lo! star-led chiefs, Assyrian odours bring,
And bending Magi seek their infant King!
Mark'd ye, mark'd ye where, hov'ring o'er his head,
where, hov'ring o'er his head, o'er his radiant head.
The dove's white wings celestial,
celestial glory shed? o'er his radiant head,
Mark'd ye, where, hov'ring o'er his radiant head,
The dove's white wings celestial glory shed,
The dove's white wings celestial glory shed?
# Christian Hymns/O Heavenly Word, Eternal Light
## Background
### Text

* Author:
* Translator: John Mason Neale, alt.
* Source:

### Music

Hymn Tune(s):
* O Heiland, Reiss
* Verbum Supernum Prodiens

## Lyrics

O heavenly Word, Eternal Light,
Begotten of the Father’s Might,
Who, in these latter days, art born
For succor to a world forlorn.
Our hearts enlighten from above,
And kindle with Thine own true love,
That we, who hear Thy call today,
May cast earth’s vanities away.
And when as Judge Thou drawest nigh,
The secrets of all hearts to try;
When sinners meet their awful doom,
And Saints attain their heavenly home;
O let us not, for evil past,
Be driven from Thy Face at last,
But with the blessed evermore
Behold Thee, love Thee, and adore.
To God the Father, God the Son,
And God the Spirit, Three in One,
Praise, honor, might and glory be
From age to age eternally.
# Christian Hymns/O Come, O Come, Emmanuel
## Background
### Text

* Author:
* Translator: John Mason Neale, alt.
* Source:

### Music

Hymn Tune(s):
* Veni, Emmanuel

## Lyrics

O come, O come, Emmanuel,
And ransom captive Israel,
That mourns in lonely exile here
Until the Son of God appear.
Rejoice! Rejoice!
Emmanuel shall come to thee, O Israel.
O come, Thou Wisdom from on high,
Who orderest all things mightily;
To us the path of knowledge show,
And teach us in her ways to go.
Rejoice! Rejoice!
Emmanuel shall come to thee, O Israel.
O come, Thou Rod of Jesse, free
Thine own from Satan’s tyranny;
From depths of hell Thy people save,
And give them victory over the grave.
Rejoice! Rejoice!
Emmanuel shall come to thee, O Israel.
O come, Thou Day-spring, come and cheer
Our spirits by Thine advent here;
Disperse the gloomy clouds of night,
And death’s dark shadows put to flight.
Rejoice! Rejoice!
Emmanuel shall come to thee, O Israel.
O come, Thou Key of David, come,
And open wide our heavenly home;
Make safe the way that leads on high,
And close the path to misery.
Rejoice! Rejoice!
Emmanuel shall come to thee, O Israel.
O come, O come, great Lord of might,
Who to Thy tribes on Sinai’s height
In ancient times once gave the law
In cloud and majesty and awe.
Rejoice! Rejoice!
Emmanuel shall come to thee, O Israel.
O come, Thou Root of Jesse’s tree,
An ensign of Thy people be;
Before Thee rulers silent fall;
All peoples on Thy mercy call.
Rejoice! Rejoice!
Emmanuel shall come to thee, O Israel.
O come, Desire of nations, bind
In one the hearts of all mankind;
Bid Thou our sad divisions cease,
And be Thyself our King of Peace.
Rejoice! Rejoice!
Emmanuel shall come to thee, O Israel.
# Christian Hymns/O Little Town of Bethlehem
## Background
### Text

* Author: Phillips Brooks
* Source: Christmas Songs and Easter Carols
* Source: The Church Porch

### Music

Hymn Tune(s):
* Forest Green
* St. Louis

## Lyrics

O little town of Bethlehem,
How still we see thee lie!
Above thy deep and dreamless sleep
The silent stars go by.
Yet in thy dark streets shineth
The everlasting Light;
The hopes and fears of all the years
Are met in thee to-night.
For Christ is born of Mary,
And gathered all above,
While mortals sleep, the Angels keep
Their watch of wondering love.
O morning stars, together,
Proclaim the holy birth!
And praises sing to God the King,
And peace to men on earth!
How silently, how silently,
The wondrous gift is given;
So God imparts to human hearts
The blessings of His Heaven.
No ear may hear His coming,
But in this world of sin,
Where meek souls will receive Him still,
The dear Christ enters in.
Where children pure and happy
Pray to the blessed Child,
Where misery cries out to Thee,
Son of the Mother mild;
Where Charity stands watching
And Faith holds wide the door,
The dark night wakes, the glory breaks,
And Christmas comes once more.
O holy Child of Bethlehem,
Descend to us, we pray!
Cast out our sin and enter in,
Be born in us to-day.
We hear the Christmas angels,
The great glad tidings tell;
O come to us, abide with us,
Our Lord Emmanuel!
# Christian Hymns/Lo! He Comes with Clouds Descending
## Background
### Text

* Author: Charles Wesley
* Source: Hymns of Intercession for All Mankind

### Music

Hymn Tune(s):
* Cum Nubibus
* Helmsley
* Kensington New
* Picardy
* Regent Square
* St. Thomas
* Störl
* Westminster Abbey

## Lyrics

Lo! He comes with clouds descending,
Once for favour'd sinners slain!
Thousand, thousand saints attending,
Swell the triumph of his train:
Hallelujah,
God appears, on earth to reign!
Every eye shall now behold him
Rob'd in dreadful majesty,
Those who set at nought and sold him,
Pierc'd, and nail'd him to the tree,
Deeply wailing
Shall the true Messiah see.
The dear tokens of his passion
Still his dazzling body bears,
Cause of endless exultation
To his ransom'd worshippers;
With what rapture
Gaze we on those glorious scars!
Yea, amen! let all adore Thee,
High on thine eternal throne!
Saviour, take the power and glory,
Claim the kingdom for thine own,
Jah, Jehovah,
Everlasting God, come down.
# Christian Hymns/Lo! He Cometh, Countless Trumpets
## Background
### Text

* Author: John Cennick
* Source: Collection of Sacred Hymns

### Music

Hymn Tune(s):
* Benediction
* Haydn
* Judgment
* Temple Boro
* Watchman

## Lyrics

Lo! he cometh, countless trumpets
Blow before the bloody sign;
'Midst ten thousand saints and angels,
See the Crucified shine.
Hallelujah! hallelujah!
Welcome, welcome, bleeding Lamb!
Now his merit by the harpers
Thro' th' eternal deep resounds;
Now resplendent shine his nail-prints,
Ev'ry eye shall see his wounds:
They who pierc'd him, they who pierc'd him,
Shall at his appearance wail.
Ev'ry island, sea, and mountain,
Heav'n and earth shall flee away;
All who hate him, must, ashamed,
Hear the trump proclaim the day:
Come to judgment, come to judgment,
Stand before the Son of man.
Saints who love him, view his glory,
Shining in his bruised face,
His dear person on the rainbow.
Now his peoples head shall raise:
Happy mourners, happy mourners,
Lo! in clouds, he comes, he comes !
Now redemption, long expected,
See in solemn pomp appear;
All his people, once rejected,
Now shall meet him in the air:
Hallelujah! hallelujah!
Now the promis'd kingdom's come.
View him smiling, now determin'd
Ev'ry evil to destroy;
All the nations now shall sing him
Songs of everlasting joy:
O come quickly, O come quickly,
Hallelujah! come, Lord, come.
# Christian Hymns/Lord Christ, When First Thou Cam'st To Earth
## Background
### Text

* Author: Walter Russell Bowie
* Source:

### Music

Hymn Tune(s):
* Allein Gott In Der Höh
* Kirken Den Er Et Gammelt Hus
* Mit Freuden Zart

## Lyrics

Lord Christ, when first thou cam’st to earth,
upon a cross they bound thee,
and mocked thy saving kingship then
by thorns with which they crowned thee:
and still our wrongs may weave thee now
new thorns to pierce that steady brow,
and robe of sorrow round thee.
O aweful Love, which found no room
in life where sin denied thee,
and, doomed to death, must bring to doom
the powers which crucified thee,
till not a stone was left on stone,
and all those nations’ pride, o’erthrown,
went down to dust beside thee!
New advent of the love of Christ,
shall we again refuse thee,
till in the night of hate and war
we perish as we lose thee?
From old unfaith our souls release
to seek the kingdom of thy peace,
by which alone we choose thee.
O wounded hands of Jesus, build
in us thy new creation;
our pride is dust, our vaunt is stilled,
we wait thy revelation:
O love that triumphs over loss,
we bring our hearts before thy cross,
to finish thy salvation.
# Christian Hymns/On Jordan's Bank, the Baptist's Cry
## Background
### Text

* Author: Charles Coffin
* Translator: John Chandler
* Source: Hymns of the Primitive Church

### Music

Hymn Tune(s):
* Alstone
* Puer Nobis
* Winchester New

## Lyrics

On Jordan’s bank, the Baptist’s cry
Announces that the Lord is nigh:
Come then and hearken, for he brings
Glad tidings from the King of kings.
E'en now, the air, the sea, the land
Feel that their Maker is at hand;
The very elements rejoice,
And welcome Him with cheerful voice.
Then cleansed be every Christian breast;
And furnished for so great a Guest.
Yea! let us each our hearts prepare
For Christ to come and enter there.
For Thou art our Salvation, Lord,
Our refuge and our great Reward.
Without Thy grace our souls must fade,
And wither like a flower decayed.
Stretch forth thine hand, to heal our sore,
And make us rise to fall no more;
Once more upon thy people shine,
And fill the world with love divine.
To Him, who left the throne of heaven,
To save mankind, all praise be given:
Like praise be to the Father done,
And Holy Spirit, Three in One.
# Christian Hymns/Lift Up Your Heads, Ye Mighty Gates
## Background
### Text

* Author: Georg Weissel
* Translator: Catherine Winkworth
* Source: Lyra Germanica

### Music

Hymn Tune(s):
* Macht Hoch Die Tür
* Truro

## Lyrics

Lift up your heads, ye mighty gates,
Behold the King of glory waits;
The King of kings is drawing near,
The Saviour of the world is here;
Life and salvation doth He bring,
Wherefore rejoice and gladly sing:
We praise Thee, Father, now!
Creator, wise art Thou!
The Lord is just, a Helper tried,
Mercy is ever at His ride,
His kingly crown is holiness,
His sceptre, pity in distress,
The end of all our woe He brings;
Wherefore the earth is glad and sings:
We praise Thee, Saviour, now,
Mighty in deed art Thou!
Oh blest the land, the city blest,
Where Christ the Ruler is confest!
Oh happy hearts and happy homes
To whom this King in triumph comes!
The cloudless Sun of joy He is,
Who bringeth pure delight and bliss:
O Comforter Divine,
What boundless grace is Thine!
Fling wide the portals of your heart,
Make it a temple set apart
From earthly use for Heaven's employ,
Adorn'd with prayer and love and joy;
So shall your Sovereign enter in,
And new and nobler life begin:
To Thee, O God, be praise,
For word and deed and grace!
Redeemer, come! I open wide
My heart to Thee,--here, Lord, abide!
Let me Thy inner presence feel,
Thy grace and love in me reveal,
Thy Holy Spirit guide us on
Until our glorious goal is won!
Eternal praise and fame
We offer to Thy name.
# Christian Hymns/Once He Came In Blessing
## Background
### Text

* Author: Johann Roh
* Translator: Catherine Winkworth
* Source: Chorale Book for England

### Music

Hymn Tune(s):
* Advent
* Gottes Sohn Ist Kommen

## Lyrics

Once he came in blessing,
All our ills redressing,
Came in likeness lowly,
Son of God most holy,
Bore the cross to save us,
Hope and freedom gave us.
Still He comes within us,
Still His voice would win us
From the sins that hurt us;
Would to Truth convert us
From our foolish errors,
Ere He comes in terrors.
Thus if thou hast known Him,
Not ashamed to own Him,
Nor dost love Him coldly,
But wilt trust Him boldly,
He will now receive thee,
Heal thee, and forgive thee.
But through many a trial,
Deepest self-denial,
Long and brave endurance,
Must thou win assurance
That His own He makes thee,
And no more forsakes thee.
He who thus endureth
Bright reward secureth;
Come then, O Lord Jesus,
From our sins release us.
Let us here confess Thee,
Till in heaven we bless Thee.
# Embedded Systems/Freescale Microcontrollers

Freescale Semiconductor (formally Motorola Semiconductor Products Sector) spun-off from Motorola in July 2004. Freescale makes many microcontrollers (MCU's) and also a whole host of other devices such as sensors, DSP's and memory, to name a few.
The Freescale Microcontrollers come in 5 families.
* 6800 descendents: 8 bit or 16 bit
* 68000 descendents: 32 bit
* MCORE: 32 bit
* PowerPC family: 32 bit
* ARM family: 32 bit. We discuss ARM core Freescale microcontrollers in another chapter, Embedded Systems/ARM Microprocessors.

## 8-bit MCUs
### Freescale HC08

There are many variations on the HC08 CPU core; The 68HC908JL8 is one example. the HC908Jl3 offer 256 bytes of RAM (random access memory) and 4K bytes of Flash ROM (Read only memory). The HC08 cores offer a maximum bus speed of 8 MHz, a 20 MHz crystal may be used as the external clock source(as the oscillator is internally divided by 4 to give 8 MHz bus speed). Typical peripheral components of the microcontroller include:
* Two 16 bit, free running timers.
* SCI (serial communications interface,(RS232))
* 12 channel 8-bit Analogue to digital converters (A/D)
The HC08 microcontrollers are usually supplied in 28 pin or 32 pin DIL packages, but can also be obtained in serface-mount SOIC footprints

### Freescale HC11 (Legacy)
## 16-bit MCUs
### Freescale HC16
## 32-bit Embedded Processors
### 68k/ColdFire

The 68k family and the nearly-binary-compatible ColdFire family are 32 bit processors capable of running Linux.
There is a Debian Linux port to 68k processors with a MMU. A Debian Linux port to ColdFire processors with a MMU is "in progress".
There are several ColdFire chips that, as of 2008, are available for under $5 (in qty 1). Those low-cost chips do not include a MMU, and so cannot run a full version of Linux. w:uClinux runs on chips without a MMU, and has been ported to some ColdFire chips on platforms with at least 1 MB of RAM.
Yes, but does uClinux actually run on a chip that costs less than $5 ?
Most (all?) currently manufactured ColdFire and 68k chips are available only in surface mount packages, not in any DIP package.

### PowerPC

* First Generation: G1 (601)
* Second Generation: G2 (603, 603e, 604)
* Third Generation: G3 (750, 750CX, 750CX3, 750FX, 750GX)
* Fourth Generation: G4 (7400,7450)

## Further reading

* w:Freescale 68HC12
* w:Freescale ColdFire
* w:PowerPC
* The 68HC12 discussion forum at EmbeddedRelated is still pretty active, apparently because 68HC12 dev boards (such as those from EVBplus) are typically lower-cost than dev boards with most other microcontrollers.
* EE Compendium: resources for using Freescale's HC12 family
* w:Motorola 68000 family
# Calculus/Solutions

* Precalculus/Solutions
* Infinity is not a number/Solutions
* Limits/Solutions
* Differentiation/Differentiation Defined/Solutions
* Chain Rule/Solutions
* Some Important Theorems/Solutions
* Differentiation/Basics of Differentiation/Solutions
* L'Hôpital's rule/Solutions
* Related Rates/Solutions
* Differentiation/Applications of Derivatives/Solutions
* Integration/Solutions
Navigation:
Main Page · Precalculus · Limits · Differentiation · Integration · Parametric and Polar Equations · Sequences and Series · Multivariable Calculus · Extensions · References
# Calculus/Summation notation

Summation notation allows an expression that contains a sum to be expressed in a simple, compact manner. The uppercase Greek letter sigma, Σ, is used to denote the sum of a set of numbers.

## Example

    $\sum _{i=3}^{7}i^{2}=3^{2}+4^{2}+5^{2}+6^{2}+7^{2}$
   Let $f$ be a function and $M,N$ are integers with $N<M$ . Then
       $\sum _{i=N}^{M}f(i)=f(N)+f(N+1)+f(N+2)+\cdots +f(M)$ .
   We say $N$ is the lower limit and $M$ is the upper limit of the sum.
We can replace the letter $i$ with any other variable. For this reason $i$ is referred to as a dummy variable. So...
    $\sum _{i=1}^{4}i=\sum _{j=1}^{4}j=\sum _{\alpha =1}^{4}\alpha =1+2+3+4$
Conventionally we use the letters $i$ , $j$ , $k$ , $m$ for dummy variables.

## Example

    $\sum _{i=1}^{5}i=1+2+3+4+5$
Here, the dummy variable is $i$ , the lower limit of summation is 1, and the upper limit is 5.

## Example

Sometimes, you will see summation signs with no dummy variable specified, e.g.,
    $\sum _{1}^{4}i^{3}=100$
In such cases the correct dummy variable should be clear from the context.
You may also see cases where the limits are unspecified. Here too, they must be deduced from the context.

### Common summations

$\sum _{i=1}^{n}c=c+c+\cdots +c=nc\ ,\ c\in \mathbb {R}$
$\sum _{i=1}^{n}i=1+2+3+\cdots +n={\frac {n(n+1)}{2}}$
$\sum _{i=1}^{n}i^{2}=1^{2}+2^{2}+3^{2}+\cdots +n^{2}={\frac {n(n+1)(2n+1)}{6}}$
$\sum _{i=1}^{n}i^{3}=1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(\sum _{i=1}^{n}i\right)^{2}=\left({\frac {n(n+1)}{2}}\right)^{2}$
$\sum _{i=1}^{\infty }a_{i}=\lim _{t\to \infty }\left[\sum _{i=1}^{t}a_{i}\right]$
# Calculus/Rolle's Theorem

If a function, $f(x)\$, is continuous on the closed interval $[a,b]\$, is differentiable on the open interval $(a,b)\$, and $f(a)=f(b)\$, then there exists at least one number c, in the interval $(a,b)\$ such that $f'(c)=0\ .$
Rolle's Theorem is important in proving the Mean Value Theorem.

## Examples

Example:
$f(x)=x^{2}-3x$. Show that Rolle's Theorem holds true somewhere within this function. To do so, evaluate the x-intercepts and use those points as your interval.
Solution:
1: The question wishes for us to use the x-intercepts as the endpoints of our interval.
Factor the expression to obtain $x(x-3)=0$. x = 0 and x = 3 are our two endpoints. We know that f(0) and f(3) are the same, thus that satisfies the first part of Rolle's theorem (f(a) = f(b)).
2: Now by Rolle's Theorem, we know that somewhere between these points, the slope will be zero. Where? Easy: Take the derivative.
    $dy \over dx$ $=2x-3$
Thus, at $x=3/2$, we have a spot with a slope of zero. We know that $3/2$ (or 1.5) is between 0 and 3. Thus, Rolle's Theorem is true for this (as it is for all cases). This was merely a demonstration.
Navigation:
Main Page · Precalculus · Limits · Differentiation · Integration · Parametric and Polar Equations · Sequences and Series · Multivariable Calculus · Extensions · References