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About Me Literature / Hobbyist Member Darkie's Meeping Moofin17/Male/United States Group group avatar #Random-Fantasy-Cats
 
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What can I say about this? It took my breath away at first sight. Particularly because you took the little detail I gave you and manage...

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:rose: ATTENTION ONCE AGAIN, MY BELOVED AND LOYAL WATCHERS :rose:

---- IS IN DIRE NEED OF A PREMIUM MEMBERSHIP.

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As much as it pains me to say this...

Journal Entry: Wed Aug 24, 2011, 9:27 AM


Journal

This might be my last day on dA for quite a long time. I've decided to take a break so that I can focus on my academics, considering that 4 AP classes will occupy most of my time, making it difficult to both chat to peeps on Skype and work on fanfiction and requests.

I might decide to come back during Winter or Summer break, but it all depends. There'd be nothing worse than being separated from my beloved friends and watchers for such an extensive period of time, so if any of you would like to keep in touch, please send me a request via Skype. My Skype Name is "omegared-19" and my Full Name is "Time Lord Nathan Xylar". If you do not have a Skype, but maybe have an MSN or a Yahoo Messenger, please comment saying so and we can add each other on there.

I hope to see many of you soon. <3



HONORABLE MENTION

:bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred:


To everyone who made my visit here on deviantART such a wonderful experience. I love you all~ <3


:bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred::bulletblue::bulletyellow::bulletblue::bulletred:



:star-empty::star-empty:: Not started
:star-half::star-empty:: Plot and setting have been defined/Art is being drawn
:star::star-empty:: Story is in the process of being typed/Art is sketched
:star::star-half:: Story is being proofread/Art is being shaded and colored
:star::star:: Complete!

:bulletblue:REQUESTS:bulletblue:


:star::star-empty: ~ :iconbloodstainblade: ~ "Overwriting" (Long-term project)


:star::star: ~ :icondarkie-cat: ~ Picnic request


:cake:BIRTHDAY GIFTS:cake:


:star::star: ~ :iconxblackice:


:bulletred::bulletgreen::bulletwhite:MISCELLANEOUS/HOLIDAY PROJECTS:bulletwhite::bulletgreen::bulletred:


:star::star-empty: ~ Birth of a Clan Part II

CSS Layout best viewed in FireFox



Mood::iconadoration-plz:Adoration
Listening to: Cheb Rayan - Rayi Rayi
Playing: Team Fortress 2
Watching: Doctor Who / Torchwood
Eating: Subway sammich
Drinking: Mountain Dew
  • Mood: Adoration
  • Listening to: Cheb Rayan - Rayi Rayi
  • Watching: Doctor Who / Torchwood
  • Playing: Team Fortress 2
  • Eating: Subway sammich
  • Drinking: Mountain Dew

deviantID

~Nathan-Xylar
Darkie's Meeping Moofin
Artist | Hobbyist | Literature
United States
Hallo, everyone! The name's Nathan Xylar! Many of you know me as either the Retaliator, the Knight or Paladin of Chivalry, or the writer to the all-powerful, prominent animator and artist, BLOODSTAINBLADE~! OwO

But enough with the formality~! X3 If you're a person who's viewing my page but isn't on my friends list, then please make an introduction, as I love new colleagues. owo


Likes: Books, Calculus, Chemistry.

Dislikes: Ignorance, assumptions, people whose least favorite subject is either mathematics or science. (This is more of a generalization; if you happen to fall in this category, please don't be afraid that I'll pummel you, because I won't :3)

"Every body remains in a state of rest or uniform motion (constant velocity) unless it is acted upon by an external unbalanced force." ~ Sir Isaac Newton's First Law of Motion



--------------------------------------------------------------------------------



-=[.:CALCULUS AB:.]=-



A derivative is the instantaneous rate of change or the slope of a tangent line to a curve on a graph at any given point.

f ′ (x) = lim h -> 0 [ f ( x + h ) - f (x) ] / h

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Power Rule: ( x ^ n ) ′ = nx ^ ( n - 1 )
d/dx (2x^3) = 6x^2

Product Rule: f ′ (x) = f(x)g(x) ′ + g(x)f(x) ′
or "First D-Second plus Second D-first"
d/dx (2xsinx) = 2x(cosx) + (2)sinx = 2(xcosx + sinx)

Quotient Rule: f ′ (x) = [ g(x)f(x) ′ - f(x)g(x) ′ ] / g(x)^2
or "Low D-High minus High D-Low all over Low squared"
d/dx (2/x^2) = [ x^2(0) - 2(2x) ] / x^4 = -4x/x^4 = -4/x^3

Chain Rule: f(x) = f [ g(x) ] ; f ′ (x) = f ′ [ g(x) ] g(x) ′
d/dx [sin^2(x)] = d/dx (sinx)^2 = 2(sinx)cosx = 2cosxsinx

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Intermediate Value Theorem (IVT):
states that if f(x) on the interval [a,b] is continuous and if f(a) is above/below the x-axis and f(b) is below/above the x-axis, then every y-value between f(a) and f(b) must be hit and therefore, there must be a zero.

Example: Verify that f(x) = x^3 - 3x^2 + 2x - 1 has a zero on [-1,3]

f(-1) = -7
f(3) = 5

IVT confirms that there must be a zero between the two.


Extreme Value Theorem (EVT):
states that if f(x) is continuous on a closed interval, then f(x) has both a minimum and a maximum. However, a function can only have extrema at critical points (see definition underneath MVT) or endpoints.


Mean Value Theorem (MVT):
states that if f(x) is continuous and differentiable on [a,b], then there exists a number c in the closed interval such that:

Derivative at c = Average change of interval

f ′ (c) = [ f(b) - f(a) ] / ( b - a )

Example: Find the x's for which MVT applies for f(x) = 1/x between [-2,2] and g(x) = x^3 - 3x between [-2,2]

f(x) is not continuous or differentiable at x = 0 which lies inside the interval [-2,2], therefore MVT does not apply.

g(x) is a polynomial, therefore it is always continuous and differentiable.

g ′ (x) = 3x^2 - 3
Average change = ( 2 + 2 ) / ( 2 + 2 ) = 1

3x^2 - 3 = 1
3x^2 = 4
x^2 = 4/3
x = (4/3)^(1/2) and -(4/3)^(1/2)

In conclusion, MVT confirms that there are two points, the positive and negative square roots of 4/3, at which the derivative of g(x) equals the average change between the interval.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Critical Point: f ′ (x) = 0
Point of Inflection: f ′′ (x) = 0
Relative Maximum / Concave Down: f ′′ (x) < 0
Relative Minimum / Concave up: f ′′ (x) > 0

Example: Determine all critical points, points of inflection, intervals of increasing and decreasing, intervals of concavity, relative extrema, and indifferentiable values of the position function f(x) = ( x^3 - x ) / ( x^3 - 4x ).

Lim x -> 0 f (x) = 0/0 = ( x^2 - 1 ) / ( x^2 - 4 ) = 1/4

HA: y = 1
VA: x = +/- 2

f ′ (x) ≠ +/- 2

f ′ (x) = [ ( x^2 - 4 )( x^2 - 1 ) ′ - ( x^2 - 1 )( x^2 - 4) ′ ] / ( x^2 - 4 )^2
= [ ( x^2 - 4 ) ( 2x ) - ( x^2 - 1 )( 2x ) ] / ( x^2 - 4 )^2
= ( 2x^3 - 8x - 2x^3 + 2x ) / ( x^2 - 4 )^2
= -6x / ( x^2 - 4 )^2

First derivative equals zero at x = 0; Critical point lies at x = 0.

First derivative is increasing in the intervals between ( -∞ , -2 ) and ( -2 , 0 ) and decreasing in the intervals between ( 0 , 2 ) and ( 2 , ∞ ).

f ′′ (x) = ( x^2 - 4 )( -6x ) ′ - ( -6x )[ ( x^2 - 4)^2 ] ′ / ( x^2 - 4 )^4
= [ ( x^2 - 4 )^2 ]( -6 ) - ( -6x )[ 4x ( x^2 - 4 ) ] / ( x^2 - 4 )^4
= ( x^2 - 4 )( -6 ) - ( -6x )( 4x ) / ( x^2 - 4 )^3
= ( -6x^2 + 24 + 24x^2 ) / ( x^2 - 4 )^3
= ( 18x^2 + 24 ) / ( x^2 - 4 )^3

Second derivative never equals zero on the numerator, but does equal zero/is undefined on the denominator when x = +/- 2; These points, however, are NOT points of inflection since they do not exist within the graph's domain, even though the curvature or concavity of the graph switches.

Second derivative is positive, greater than zero, and concave up at ( -∞, -2 ) and ( 2, ∞ ) and negative, less than zero, and concave down at ( -2, 2 ).

Because the only critical point is at x = 0 and that value just happens to lie between the interval ( -2, 2 ), it is on the part of the graph that concaves down, therefore it is a relative maximum. Because there are no other critical points, there are no relative minima.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Special Trigonometric Derivatives

d/dx sinx = cosx

d/dx cosx = -sinx

d/dx tanx = (secx)^2

d/dx secx = secxtanx

d/dx cscx = -cscxcotx

d/dx cotx = -(cscx)^2


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


Integrals are found through the process of integration which involves reading the derivative table backwards, in a reversed fashion, or inversing the process of finding the derivative. Definite integrals give you a numerical value that represents the area under the curve between the two points on the interval which is the same as the total distance covered if your graph is treated as one of velocity or the increase or decrease in velocity if your graph is treated as one of acceleration.


Integrals are found by taking the limit as n approaches infinity of the (sigma) summation from one to n of f of X sub-i times ΔX (delta-X).



Indefinite (Looking at ALL possible values):
∫ f(x)dx = F(x) + C
F ′ (x) = f(x)
Antiderivative: ∫ (x^n) dx = [ x^( n + 1 ) ] / ( n + 1 )

∫ 2x^3 dx = ( x^4 ) / 2 + C

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Definite integral between two points, a -> b (Seeing as I can't represent that due to limited keyboard function):


FIRST FUNDAMENTAL THEOREM OF CALCULUS

∫ f(x) dx = F(b) - F(a)



Between points 0 and 3:
∫ 2x^3 dx = ( 3^4 ) / 2 - ( 0^4 ) / 2 = 81/2 - 0 = 81/2 = 40.5
The area below the curve is 40.5 units. Therefore, if our graph is one of velocity, the subject went 40.5 feet or miles or whatever distance we want, or if our graph is one of acceleration, our subject's speed is increasing at 40.5 feet or miles per second or minute.


SECOND FUNDAMENTAL THEOREM OF CALCULUS (integral from 0 to x)

d/dx ∫ f(t) dt = f(x) dy/dx



Between points 0 and 3x^2:

d/dx ∫ e^(2t) dt = 6xe^(6x^2)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Mean Value Theorem for Integrals:
states that if f(x) is continuous on a closed interval, then there is a c for which:

∫ f(x) dx = f(c)( b - a )

Therefore, the equation for average value becomes:

[ 1 / ( b - a ) ] ∫ f(x) dx = f(c)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Cross-sectional Area

Find the volume of a solid with a circular base and square cross-sections using x^2 + y^2 = 4 and integration.

f(x) = (4 - x^2)^(1/2)
g(x) = -(4 - x^2)^(1/2)

A(x) = [f(x) - g(x)]^2 = [2(4 - x^2)^(1/2)]^2 = 4(4-x^2) = 16 - 4x^2

(From -2 to 2)
∫ A(x) dx = 16x - (4/3)x^3

16(2) - (4/3)(2)^3 - [16(-2) - (4/3)(-2)^3] =
32 - (32/3) - [-32 + (32/3)] = 64 - (64/3) = 128/3 u^3

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Revolutionary Integration ( all between the interval [a,b] )

Disc Method:

Around the X-axis: π ∫ f(x)^2 dx

Around the Y-axis: π ∫ f(y)^2 dy

Around a moving axis: π ∫ (Top Function - Bottom Function)^2 dx

Washer Method:

Around a perpendicular axis (X) when the intersection points do NOT touch it:
π ∫ [ R(x) ]^2 - [ r(x) ]^2 dx

Shell Method:

Around a parallel axis: 2π ∫ p(x)h(x) dx


Arc Length:

Arc = ∫ [1 + (dy/dx)^2]^(1/2) dx

Surface Revolution:

S = 2π ∫ r(x) [1 + (dy/dx)^2]^(1/2) dx

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Integration by parts

Power Rule: ∫ (x^n) dx = [ x^( n + 1 ) ] / ( n + 1 )

Product Rule: ∫ uv dx -> ∫ v du = uv - ∫ u dv

Quotient Rule: ∫ v/u dx -> ∫ dv/u = v/u + ∫ (v/u^2) du

*Chain Rule: ∫ v(u) dx = [ (∫ (v))(u) ] / du

*ONLY WORKS IF YOU END UP DIVIDING BY A CONSTANT



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


-=[.:CALCULUS BC:.]=- (COMING SOON)
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:iconchainedrei:
-Nudge-
Bark bark! owo

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I believe in Jesus Christ my Savior. If you do too and aren't scared to admit it then copy and paste this in your signature.
Jesus is my life. He's my savior, my rock and my best friend.
The Lord loves you and me. I'm Christian and proud of it!
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:iconcoke13579:
Mood: Joy ~COKE13579 Apr 15, 2012  Student Photographer
wow!!! awesome animater :D

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lovely avatar made by r0se-designs
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:iconpipeachu:
ALSDKJQWEKLAJSDQWEKLJASDMADQWE

HAPPYLATE-BDAY
HAPPYLATE-BDAY

happy late b-daay

amgosh sorry

i had no time to come onnn
and yesterday i was busy sleeping the day off

so HAPPY LATE B-DAY

i owe you a hug *hugs*

--
This signature is so corny, why are you reading it~?
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:iconjayysama:
~Jayysama Mar 28, 2012  Student Writer
Happy Birthday, bro! =D

I know you are away, but when you read this, it isn't too late. I want to to fly to Miami, Florida, and check into the Stillwater Hotel. The people will know your name, and bring you into the correct room. In this room will be a note. This note will contain your directions on saving the world from giant flaming tomato meteors. You are our only hope, because the only way to beat them is writing ridiculous mathematical equations. Plus, these have to be written with one of those fun looptey loop pens. It is mandatory, and you can find them in Disney Land, located in the same state.

I believe in you, soldier.

--
OMG WTF RAAAAAAAGE!
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:iconhawkrise:
~Hawkrise Mar 28, 2012  Hobbyist Traditional Artist
I know youre never on here anymore but Happy birthday! :)
*runs away*

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I currently "believe" in is that NYAN CAT WAS FLYING AROUND IN SPACE AND CREATED EVERYTHING WITH THE COLOURS OF HIS RAINBOW? Now who would like to join the Nyanists?
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:icontharonprinceofdemons:
~TharonPrinceOfDemons Mar 28, 2012  Hobbyist Digital Artist
Happy birthday!

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>.:-Demonn The Time Cat-:.<
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:icondarkie-cat:
Happy Birthday you old turd XD ((not really eue;))

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:iconpipeachu:
HAPPY EARLY-BIRTHDAAAAY
asdjklqwemasd

im going to spam you with comments tomorrow

hope you dont mind :3

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This signature is so corny, why are you reading it~?
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:iconice-or-fire:
Hello!:iconpinkballoonplz:
Happy [early] birthday!
Wishing you miles of smiles in the coming years!


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    Don't press the red button!
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:iconpipeachu:
PROMISE ME YOU'LL COME BACK IN TIME FOR YOUR BIRTHDAAAY

SO I CAN WISH YOU A HAPPY BIRTHDAAAAAY

BECAUSE I MISS YOUUUUU

we all do, i think

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This signature is so corny, why are you reading it~?
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