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1. 
All about dx
01:29


Because you know
I'm all about dx 'bout dx differentials.
I'm all about dx 'bout dx differentials.
I'm all about dx 'bout dx differentials.
I'm all about dx 'bout dx
Yeah it's pretty clear I got a lot to do,
but I can work it! work it! and calculate du.
'cause I got that boom boom,
concave or convex
all the right f(all the right x)
I see that g(x) with f(x) on top,
but trend toward zero...whoa whoa
where they gonna stop?
If you've got  Lhopi  Lhopi 
make that rule pop!
'cause we're gonna differentiate
the bottom and the top.
Mama said numbers they come in all kinds of size;
Infinity, Infintesimal, Infinite, Zero, Finite.
You know I don't want no stick figure line
flat and straight with no verve.
I'm gonna calculate slope and the area under the curve.
Because you know
I'm all about dx 'bout dx differentials.
I'm all about dx 'bout dx differentials.
I'm all about dx 'bout dx differentials.
I'm all about dx 'bout dx


2. 
Get Limits
03:15


If you're crying here's a kleenex,
soon you won't be grinning,
the argument I'm winning,
the distance here is thinning.
We're coming to arbitrary value.
So you cry boohoo,
but I'm not touching you.
As delta drops close to none,
downsizing the epsilon,
a shrinking box never gone.
That's how we prove it's a limit.
A space between them so small,
It's like there's nothing at all.
From left to right a close call
and if they meet it's a limit.
There's no point in exploding,
you loose me oscilating,
one side is always waiting
your gap is always gaping.
We're coming to arbitrary value.
So you cry boohoo,
but I'm not touching you.
As delta drops close to none,
downsizing the epsilon,
a shrinking box never gone.
That's how we prove it's a limit.
A space between them so small,
It's like there's nothing at all.
From left to right a close call
and if they meet it's a limit.
and if they meet it's a limit.
From left and right for the limit.
The meeting place is secret,
until we both can see it,
just walk your path to seek it,
we may never reach it.
As delta drops close to none,
downsizing the epsilon,
a shrinking box never gone.
That's how we prove it's a limit.
A space between them so small,
It's like there's nothing at all.
From left to right a close call
and if they meet it's a limit.
and if they meet it's a limit.
From left and right for the limit.


3. 
Feel it Nil
01:50


LEIBNIZ:
Got a curve f of x.
Think I’ll take two points,
distance in between is called h.
Gonna drop the distance down to zero,
Subtract, divide and take the limit.
Ooh woo, ooo ooo, f of x plus h now,
Ooo ooo ooo, minus f of x now,
Got it over h as h drops to nil.
Ooh woo, oo oo f of x plus h now,
Ooo oo oo minus f of x now,
Got it over h as h drops to nil.
NEWTON:
Calculate the Change you need.
Slice of time but maybe slimmer.
Multiply the rate with slivers.
It’s not meant for trolling thieves.
And I’m gonna call it Y DOT!
LEIBNIZ:
Ooh woo oo, dy over dx now,
Is the better way to write the d’rivative now.
Could’ve had your say, but you’re too layzay!
ISAAC:
Fair!
Ooh ooo, sayin' 'fluxion' is slick now.
I came up with that in 1666, now.
LEIBNIZ:
Differential's clutch,
Feelin' jealous much?
BOTH:
Ooh woo, ooo ooo, f of x plus h now,
Ooo ooo ooo, minus f of x now,
Got it over h as h drops to nil.


4. 
The Royal Society
02:34


I've never seen a number like a ghost.
I never had to multiply by infinity.
And I'm not sure how to compare,
one zero to...another zero...
But everyone is like; differential and infinites'mal,
Newton's method, theorems fundamental.
Calculus...It's not as difficult as it seems.
But everybody's like Xeno, limits, epsilon and delta,
Cantor, Liebniz, infinites with sizes.
If you dare...just look to Jean le Rond d'Alembert!
And you'll know the difference, (difference)
Infinite and infinity.
Infinitessimals are small,
as if you had nothing at all.
But they're bigger than zero, (zero)
lacking finite quantity,
say maybe du du du du du.
A differential odyssey.
My friends are eigen vector rows.
I count infinities unless they're uncountable.
And sometimes when you multiplyinfinity by zero
you get real numbers.
But everyone is like; differential and infinites'mal,
Newton's method, theorems fundamental.
Calculus...It's not as difficult as it seems.
But everybody's like Xeno, limits, epsilon and delta,
Cantor, Liebniz, infinites with sizes.
If you dare...just look to Jean le Rond d'Alembert!
And you'll know the difference, (difference)
Infinite and infinity.
Infinitessimals are small,
as if you had nothing at all.
But they're bigger than zero, (zero)
lacking finite quantity,
say maybe du du du du du.
A differential odyssey.


5. 
5 Sizes of Numbers
03:02


There are 5 sizes of numbers,
Big Infinity and small Zero,
And the Finite in the middle,
They’re the ones, I’m sure you know.
But now we look between Finite and Zero.
To numbers so small, they’re nothing at all,
But still a little larger than a Zero.
Their name is Infinitesimal.
On the other side of Finite,
There are numbers too large to say,
Infinites are what we call them,
They are big, in every way.
But they will never quite be Infinity,
They’re not quite as big, not even close.
We’ll use all of these numbers in Calculus,
The numbers, I love the most.


6. 
The Limit's Alright
01:53


I don’t mind if there’s no value at this point,
It’s fine, I’ll find the value anyway.
‘Cause I know, if I come, from the left and the right,
If they meet I will find that the limit’s alright.
The limit’s alright!
Sometimes,
I see that x approaches a,
So x, gets infinitely close to a.
And I know, if I come, from the left and the right,
Close enough, I will find that the limit’s alright.
Each small Epsilon’s gonna have a small Delta for sure!
They’ll get so small, they can’t get any smaller!
Sometimes, I see that x approaches a,
So x, gets arbitrarily close to a.
And I know, if I come, from the left and the right,
If they meet I will find that the limit’s alright.
The limit’s alright!
The limit’s alright!


7. 
Differentiabul
01:43


f of x plus h minus f of x
all over h as h drops to zero
is the formula to find the derivative.
To otherwise state: instantaneous rate.
f of x plus h minus f of x
all over h as h drops to zero
is the formula to find the derivative.
To find the slope at one point.
Infinitesimals dy over dx,
Why he wrote it I can’t say,
Leibniz just liked it better that way!
So,
f of x plus h minus f of x
all over h as h drops to zero
is the formula to find the derivative.
With this I will have to learn to cope!
Leibniz found the limit of the slope!


8. 
Power Rule
01:47


When you have A times an X to the B,
you know you always use:
Power Rule.
Look A B X to the B minus one,
is the derivative.
Power Rule.
Derivatives of constants are always a slope of Zero.
Square Root is the one half power,
you have nothing to fear.
Oh! How can you lose!
For all polynomials you can forget all your troubles,
cause everyone knows you use:
Power Rule!
1 over X is Just,
Power Rule!
X to the minus 1.
Power Rule!
Fo Polynomials, Foo!
La la la, la la la la la la la la la la la la la.
Power Rule!
La la la, la la la la la la la la la la la la la.
Power Rule!
Derivatives of constants are always a slope of Zero.
Square Root is the one half power,
you have nothing to fear.
Oh, Elephant shoes.
For all polynomials you can forget all your troubles,
cause everyone knows you use:
Power Rule!
1 over X is Just,
Power Rule!
X to the minus 1.
Power Rule!
There is no Dana, Just Zool!


9. 

Can’t explain all the theorems that you’re going to reveal!
Irrational, or radical or rational it’s one deal.
When you need the change:
a x to the b
How to change it for:
a x to the b
I believe in a Power Rule!
a b x to the b minus one.
Any all a x to the b
polynomial of any degree.
I believe in a power rule!
Ooh!
Know that square root of x,
is x to the half power.
x to the minus one,
a one over x tower.
When you need the change:
a x to the b
How to change it for:
a x to the b
I believe in a Power Rule!
a b x to the b minus one.
Any all a x to the b
polynomial of any degree.
I believe in a power rule!
Ooh!


10. 

I say Jump or Gap or Cusp or Hole,
Vertical Tangent and Asymptote.
If you see 'em point, it's critical,
'cause they ain't differentiable,
A Jump or Gap think generally
and jump that gap like the General Lee!
A hole discontinuity, that's been removed.
One point removed from the whole's a hole.
and it ain't differentiable,
I say Jump or Gap or Cusp or Hole,
Vertical Tangent and Asymptote.
If you see 'em point, it's critical,
'cause they ain't differentiable,
Join hands, join hands, create a cusp,
when two curves meet continuous.
But don't bestow your misplaced trust.
Now hand to hand and turn it up.
Vertical tangent to fill your cup.
Cobra says, "Smooth curves man, no trouble,"
but they ain't differentiable,
Yo differentiable is the word and girl I got a function for those curves
you may have limits but I ain't got none, my lyrics are continuous,
'cause I'm original not derivative in style,
with analytical critical formulas all the time,
now that I got this interval opened up it's time to end this tangent. What?
I say Jump or Gap or Cusp or Hole,
Vertical Tangent and Asymptote.
If you see 'em point, it's critical,
'cause they ain't differentiable,


11. 
Just a Little Bit
02:11


Change in what? change in time.
change in z...or maybe y.
All I’m asking is for a little respect
when you take derivatives.
dy’s just a little change in height.
dx is a run, but oh so slight.
Ratios of differentials happen with respect!
They’re just a little bit.
Ooh you’re changing and this I mention,
you gotta change in two dimensions.
you change the x I change the y,
it happens with respect!
IM  P  L  I  C  I  T
don’t you solve just let it be.
IM  P  L  I  C  I  T
take with respect to t.


12. 
Triggy Rules
01:03


D’rivative of Sine X,
is Cosine X.
Derivative of Secant X is, Amazing!
Secant X Tan X!
Drivative Tangent X:
Secant Squared X.
Remember the Chain rule, Chain Rule!
Don’t forget the dx, dx!
Triggy rules, triggy rules,
Triggy, triggy, trigg rules,
Triggy rules, triggy rules,
Triggy, triggy, trigg rules,
Y’know trig don’t choke.
Derivatives of cofunctions are
All Negative.
Ya substitute the functions for the cofunctions as implied.
I said y’know trig don’t choke,
Derivatives of cofunctions are
All Negative.
Ya substitute the functions for the cofunctions as implied.


13. 
Product Rule
01:36


First d Last, Last d First,
we must add both terms together.
When our function’s made up of two.
The Product Rule.
First d Last, Last d First,
we must add both terms together.
When our function’s made up of two.
The Product Rule.
If you have X and a Y in your function,
Then use it! Implicit. Remember:
dy, chain rule.
First d Last, Last d First,
We must add both terms together.
When our function’s made up of two.
The Product Rule.


14. 
Quotient Rule
00:32


A quotient of two functions you must differentiate,
Make Hi up high and Lo below and do not hesitate:
Lo D Hi!  MINUS!  Hi D Lo!  OVER!
Over Lo! Over Lohohohohoho!
Lo D Hi!  MINUS!  Hi D Lo!  OVER!
Lo squared: The Quotient Rule!


15. 
Chain Rule
02:33


For compositions,
I thought u substitution.
So change x out now,
save it for the solution.
Oh, you got dy / dx is,
I know It’s just dy / du,
times du / dx.
Oh, respect this Chain Rule.
Chain chain chain…
Chain Rule.
Chain chain chain…
Chain Rule.
Every chain,
has got one more link,
For each composition,
but don’t lose your strength.
Oooh, babe,
You gotta see the function alone!
Don’t matter what’s inside it at all!
Take the derivative, take it easy!
Oh don’t change its insides just you clone
and now apply the:
Chain chain chain…
Chain Rule.
Chain chain chain…
Chain Rule.
Oh, we are not finished!
We still gotta take,
derivative of the inside,
and multiply'all I can babe!
Chain chain chain…
Chain Rule.
Chain chain chain…
Chain Rule.


16. 
Mean Angst
02:33


17. 
A Critical Point
01:26


The first derivative will show you
increase or decrease.
It’s positive or negative
y’know respectively.
If the d’rivative is zero
or it’s undefined,
then that point is critical
and always on your mind!
You know a saddle is a critical point,
that also is an inflection point.
You know, you know a saddle is an inflection point,
that also is a critical point.
The second d’rivative will tell you
concave up or down.
Concave up, positive smile
and down negative frown.
The second d’rivative is zero
in between the sections,
of concave up or down
and they are called points of inflection.
You know a saddle is a critical point,
that also is an inflection point.
You know, you know a saddle is an inflection point,
that also is a critical point.
It’s a critical, it’s a critical, it’s a critical point.


18. 
Poker Trace
02:20


I want to know whatever increase or decrease.
Just take the first derivative and say with me:
 I love it!
Positive or negative means slope is up or down.
And zero shows you where a critical point can be found.
O  zerooo zerooooo...a critical point. A critical point.
O  zerooo zerooooo...a critical point. A critical point.
Do it once, do it twice and then you'll know where my graph's concave!
 Second d'rivative will tell you.
Plus a smile. Minus frown, so you will know where my graph's concave.
 Concave up, or concave down down down.
Second derivative o inflection point!
Second derivative o inflection point!
I can tell you if you're addled  it's a saddle.
Curvy trouble  zero doubled.
Critical point, intersection
with my point inflection
just like a change in entropy
no going back, no way to turn about
it's critical  critical.
Check this curve analytical.
Do it once, do it twice and then you'll know where my graph's concave!
 Second d'rivative will tell you.
Plus a smile. Minus frown, so you will know where my graph's concave.
 Concave up, or concave down down down.
Second derivative o inflection point!
Second derivative o inflection point!


19. 
MAXIMA & mininma
01:26


For Maxima and minima
just take derivitinima!
Happiness, now just assess
the zero, zero, zero, zero!
Don’t forget you must inspect
the endpoints as they are suspect!
Find the values of our function,
look for Highs and Lows!
Local Maxima are on an Interval,
Local Minima are on an Interval!
Global Maxima aren’t on an Interval,
Global Minima aren’t on an Interval!
terval! terval! terval!
Saddle, Peak and Trough and
Saddle, Peak and Trough and
Saddle, Peak and Trough and
Peak and Trough and
Peak and Trough and
Saddle, Peak and Trough and
Saddle, Peak and Trough and
Saddle. Saddle,
Peak and Trough and
Peak and Trough.
and!
Saddle, Peak and Trough and
Saddle, Peak and Trough and
Saddle, Peak and Trough and
Peak and Trough and
Peak and Trough and
Saddle, Peak and Trough and
Saddle, Peak and Trough and
Saddle. Saddle,
Peak and Trough and
Peak and Trough.
Now Maxima and minima
are also called the extrema.
Sometimes they can be absolute
as long as there’s no greater, lesser.
Relative implies a region that the extremum is in.
Don’t confuse a saddle point!
Don’t confuse a saddle point!
DON’t confuse a saddle point!
With an Extremum!
Local Maxima are on an Interval,
Local Minima are on an Interval!
Global Maxima aren’t on an Interval,
Global Minima aren’t on an Interval!
terval! terval! terval!
Saddle, Peak and Trough!
etc.


20. 
Under the Curve
01:58


Area under the curve f of x is
equal to the antidrivative of
f of x dx. It is fundamental
Geometry and Analysis tied.
Integral from A to B;
Antiderivatae.
Take the value at B sub
tract the value at A.
Under the curve I found,
some Area.
Under the curve I found,
some Area.


21. 
Without Riemann
02:59


"G. F. B. Riemann: No gimmicks."
To find the whole area under the curve,
under the curve, under the curve...
To find the whole area under the curve,
under the curve, under the curve...
Interval:
A to B.
f of x.
Stay with me:
f of x,
f of x,
f of x,
f of x,
f of x,
f of x,
f of x....
I’ve a curve at all angles, I’m using rectangles,
to find area so…I use slices, 1 to n.
Well if you want slices this is what I’ll tell ya,
each little width is B minus A over nah.
‘Cause width is changingonthe x its delta –
x, And height: f of x sub i from the middle,
or the right or left cause you’re evaluating,
from the i of the slice that you are calculating.
A! The Area’s from multiplication,
of the width an the height gimme some adulation.
I know that you got Sigma Notation,
when your slices add in a big summation!
So the BCE, won’t disagree,
and let me decree Archimedes.
He had the same idea in ancient Greece.
But it feels exhausting without Rie
mann, Hanover kid summing the bits,
and more cunning and witz, than Brechtian skitz.
And get on it, it’s all working out, you don’t shun it,
I just set up my summation, time to sum it!
And this looks like a job for Rie
mann, add it up from A to B and
the more slices that you see
can increase your accuracy.
And this looks like a job for Rie
mann, add it up from A to B and
the more slices that you see
can increase your accuracy.
Little slices, make area nicest.
The smallest of sizes will be the precisest.
Infinitesimaal bits will entice us,
'till someone comes along with a vision and yells, SWITCH!
A visionary, vision of area,
to start the transformation,
Summation is changing with limits,
‘n’ is getting greater so vast that,
it’s a fact that I know that those little slices outlast,
Attempts to count them!
What a catastrophe!
Cause there can’t be enough
Time to count to infinity!
But step back.
n n n n n n n n n n.
There’s no width to measure,
truly thin, so change the letter,
d for Delta. A differential trend setter.
The new Sigma notation, slimmer is better!
And it’s the best thing,
the S thing’s suggesting.
Divesting x of the sub i, I’m stressing:
Anti – deriva –tive.
When I mention integral of f x d
x. Who made sense, intense from A to B?
Ah Newton? Not him. No not Cauchy.
No! this looks like a job for Rie
mann, add it up from A to B and
the more slices that you see
can increase your accuracy.
And this looks like a job for Rie
mann, add it up from A to B and
the more slices that you see
can increase your accuracy.
And this looks like a job for Rie
mann’s INTEGRAL from A to B and
sliced to infinity,
for your perfect accuracy.


22. 
Physics Extravaganza
00:58


Position is the place you are
at any given time you see.
The instantaneous rate of change
of that is the velocity.
Which is direction and the speed
two parts of information.
Its instantaneous rate of change
is called acceleration.
The total distance traveled
is by no means an atrocity,
the integral of absolute value of the velocity!
Another point of interest know
the integral of force is work.
Accelerations rate of change is
surge or lurch or jolt, or jerk!
Accelerations rate of change is
surge or lurch or jolt, or jerk!
Accelerations rate of change is
surge or lurch or jolt, or jerk!
Accelerations rate of change is
surge or lurch or jolt, or jolt or jerk!
Displacement is how far you are
from your initial starting spot.
Remembering the average value
of a function is a lot:
One over B minus A times the value of the integral
from A to B of f of x dx if on an interval.
One over B minus A times the value of the integral
from A to B of f of x dx if on an interval!


23. 

L’Hôpital
Every now and then I get a little
bit of trouble when I’m taking a limit.
L’Hôpital
Every now and then I get a zero
for the numerator and the denominator.
L’Hôpital
Every now and then I get a limit
that’s confusing in some kind of indeterminate form.
L’Hôpital
Every now and then I get a little
bit terrified but then I think of all your advice.
L’Hôpital
Guillaume François Antoine Marquis de L’Hôpital
Guillaume François Antoine Marquis de L’Hôp!
So we take the rate of change,
of the top and of the bottom.
We don’t need to rearrange.
We’re just going to compare them.
And I know that we’re making this strange,
'Cause we take the limit, again!
If we find that we cannot define
our limit, then we'll have to go repeat one more time!
This almost always works, but if you’re in the dark,
an Oscillating Function may be leaving its mark!
And then this song doesn’t work!
But most of the time, well it does.
For most of the time our song works.
Once upon a time I had trouble with math,
but now they all think that I am smart.
There’s nothing I can’t do,
I have Calculus in the heart.
Once upon a time I was crying all night,
but now I do my math in the dark.
There's nothing I can say,
I have Calculus in the heart.


24. 

“A function can become a sweet series,”
said Taylor as he stepped into the breeze.
“From term zero to term infinity,
you’ll find out every term so easily:
Evaluate the nth derivative at ‘a’,
Times ‘x’ minus ‘a’ to the nth power,
Over n factorial.”
Then Maclurin said, “Hello. I’d prefer it if the ‘a’ was just zero.”
It is the essence of simplicity
factorials and powers: 0, 1, 2, 3, ...
Expansion of the series is the key
relating sine, the cosine, i and e...
Evaluate the nth derivative at ‘a’,
Times ‘x’ minus ‘a’ to the nth power,
Over n factorial.”
Then Maclurin said, “Hello. I’d prefer it if the ‘a’ was just zero.”


25. 
Darcsine
03:31


Tables are turned. Inverted world.
Change will arrive when you derive.
Arcsin implied, now you divide
by the cosine on both sides.
Bait and switch the institution.
Calibrate the substitution.
One over the square root of one
minus x squared and you’re mine.
d arcsine  change in arcsine.
Now you've arrived limitless style,
but you're confined, change is a smile.
Two crocodiles, say what you mean,
infinite heights touch one between.
You and i in love for ages,
can you tell me how it changes?
One over the square root of one
minus x squared and you’re mine.
d arcsine  change in arcsine.
We're changing but your heart is mine!


26. 
Darcsecant
03:05


x in the box in the basement.
x in the box in the basement.
I put an x in the box in the basement.
it was next to a root the encasement.
of the difference, o’ the displacement.
x squared minus one scared,
‘cause there’s someone upstairs,
but her vision’s impaired!
and i’m absolutely certain
that there’s spray paint on the crate!
desecrate  desiccant  she can’t see!
can’t you d  d  x arc secant next
to the x in the box in the basement.


27. 
Darctangent
00:49


One over one plus x squared.
One over one plus x squared.
One over one plus x squared.
That’s the rate of change of arctangent.


28. 
Log: A Rhythm.
02:08


The brightness of a star,
The shaking of the ground,
Computational complexity,
The power of a sound.
The acid in your jar,
The measurements are found
On a logarithmic scale,
The only way I’ll be allowed,
To see the different sizes when the range is just too great,
From super small to super large, is hard to contemplate,
But log base ten will strike and then it’s quite appropriate,
One millionth is negative six, a hundred million’s eight.
See, a logarithmic scale is most concerned with magnitude,
at log base ten the jump is ten times from the one to two.
And two to three and three to four each step is ten times more.
And when we get to sixtythree we’ll fill space universally
with grains of sand the hand of Archimedes’ mind has reckoned
that to find a beach this large without an ocean is a crime,
like every passive second spent in haste!
Desert the massive massive waste of space– time to realign
less focus on the sand, more focus on mankind.
As population rises in an exponential manner,
Like markets do though they seem too prone to constant disaster,
A logarithmic scale will help us see the patterns faster,
at last a graph so prone to change, becomes a graph so rearranged
to lines and planes these simple gains brought to us by a man so strange:
A marvel of a Scotsman and a real necromancer,
kept a spider in a box like any real technodancer.
Named John Napier of Merchistoun, of which he was the laird.
Invented bones. wrote tomes of logarithmic prose.
Promoted points alone in action, Decimals to pwn the fractions!
Simon says, “I miss the circles  but  I think that’s just my first reaction.”
The brightness of a star,
The shaking of the ground,
Computational complexity,
The power of a sound.
The acid in your jar,
The measurements are found
On a logarithmic scale,
An exponential turnaround.


29. 
Crosshairs
03:17


When you say, “I love you,”
you know that I’ll say it back.
When we are escaping,
you know that I’ve got your back.
Peas in pods, and bees in bonnets,
mathematics psych –
Psychoanalytic math of
what I think we’re like:
We’re like a power distribution over multiplication.
Combinations aren’t required, complications won’t arise,
but I fear I see addition symbols floating in your eyes.
Side by side we’re simple;
you are A and I am B.
Raised to any power:
A to P times B to P.
Don’t misunderstand me.
Don’t expect me.
Don’t expand me.
Extra coefficients only serve to make me antsy!
If you and I were added and
then love was multiplied,
love of you plus love of me.
Simplicity.
When operation levels
jump inside to the outside,
they strain they very fabric
of this weak analogy!
We could add another
member to our tiny crew,
Wondertwins can not form Voltron
it takes more than two.
Three would clarify the nature
of association.
You and me are two,
but three would have the same relation!
‘cause we‘re a power distribution over multiplication.
Combinations aren’t required, complications won’t arise,
but I fear I see addition symbols  crosshairs  floating in your eyes.


30. 
e
01:39


I crossed seven bridges to be here to day,
to bring you this important notation.
Not Functions, or Sigma i fear,
though my vision is fading, my voice is still clear.
Not Uler but Euler.
Euler? I barely know how much I’d have if,
I added, reciprocals of all of the factorials,
turns out it’s the number of nature.
Presenting now for all to see
the main event named after me:
it’s
e! the number e, more than 2, less than 3.
e! the number e, a constant transcendentally
One plus 1 over n, all in parentheses,
to the power of n, take n to infinity.
Describe the growth of things, compounded continuously,
or halflife the radiation decays exponentially.
Edges plus two is Faces plus the Vertices.
Off topic, I’m sorry, just came to me, apologies.
The function e to the x  A.K.A. the exp().
Its inverse  the log base e
or natural log naturally.
it’s
e! the number e, more than 2, less than 3.
e! the number e, with beautiful Taylor series.
it’s
e! the number e, more than 2, less than 3.
e! the number e, a constant transcendentally
e to the i x is, cosine x plus i sin x,
it is imperative that you catch what happens next,
you see x  is measured in the radians and not degrees
using my formula it is plain for all to see
that e to the pi times i, plus one makes the quantity of zero.
Abstract I know – but still used to this very day,
when engineers transform the waves, to phasors,
not the kind that blaze, just electricty.
You see it’s
e! the number e, more than 2, less than 3.
e! the number e, as seen on x k c d.
e! the number e, more than 2, less than 3.
e! the number e:
Two point seven one eight two eight one eight two endlessly.
Infinitely continuing streaming irrational
e.


31. 
Rules of Powers
02:07


x to the A times x to the B
is x to the A plus B
Exponent addition is the action.
x to the A over x to the B
Can always be x to the A minus B
Dividing make subtraction.
And x to the A all to the B
Oddly is x to the A times B
And x to the power of One over B
Is really a Root, The Bth root it be!
x to the Zero Always One
Negative exponents are Un
Necessary, if you like the fractions,
Don’t leave I know you hate fractions!
I just thought this time that maybe,
It would be alright if maybe we used fractions.
x to the power of A over B
The Bth root of x to the A, you See?
x to the power of A over B
Can also be x to the A then root of B.
And x to the A all to the B
Oddly is x to the A times B
And x to the power of One over B
Is really a root, The Bth root it be!
x to the Zero always One
Negative exponents are Un
Necessary, if you like the fractions,
Don’t leave I know you hate fractions!
But I thought this time that maybe,
You could really get over your fear of fractions.


32. 
Napier's Curse
01:59


To the power of x,
you never seen it before,
much less solved it,
but you’re like fosho,
‘Cause you have ideas that won’t be denied,
You take the xth root of both sides,
But you’re scratching your head, ‘cause that didn’t help,
If the object is to get the x all by itself!
You see,
the only way to get that exponent down to earth,
for what it’s worth, there’s nothing worse,
And it is called inverse.
In this case the Logarithm’s your hearse,
use Napier’s curse to help you switch it around,
And move that exponent a whole six feet under the ground.
So use it, abuse it, apply it, remove it.
Use it, abuse it, apply it, remove it.
So why is it there  still?
Looking down laughing, deriding while you’re
crying buckets over your spilled poten
Shall you stand this power drilling into your mind,
Kinda like the fleeting feeling that you’re dealing with Pi
rotechnic visions of a exponential monster unwind.
Untie the lies of your lofty tormentor,
resent the disguise and embrace the inventor:
Here’s Johnny with a mantra that will cut to the center
As HE applies the inverse  exponential descenter.
Clear your mind, lift the fog,
the base of the power is the base of the log.
Clear your mind, lift the fog,
The base of the power is the base of the log.
The base of the power is the base of the log.
The base of the power is the base of the log.
The base of the power is the base of the log.
No more exponential messes for you, the essential message to you,
Be the change you want to see, apply Logs to both sides of the
Equation that’s distressing, and oppressing and depressing,
And after all the messing, and the stressing, and the testing.
I suppose logs are a blessing, but they maybe not the best thing.


33. 
Geyser
01:57


The water at the bottom
heated up by molten rock
Is under lots of pressure
from the water at the top.
The water at the bottom
heated up by molten rock
Is under lots of pressure
from cold water at the top.
And So,
And so it doesn’t boil.
And so
it just gets really really hot!
The water at the top is warmed
and slowly turns to steam,
Releasing lots of pressure
from the water underneath.
And when the pressure’s low enough
it happens suddenly.
The superheated water boils,
erupting fervently!
Until...until all the water’s spent!
And then,
and then the cycle starts again!
The water at the bottom
heated up by molten rock
Is under lots of pressure
from the water at the top.
WATER! BOTTOM! HEATED BY THE ROCK!
Is under lots of pressure
from the water at the top.
And So,
And so it doesn’t boil.
It just,
it just gets hotter than normal.
The water at the top is warmed
and slowly turns to steam,
Releasing lots of pressure
from the water underneath.
And when the pressure’s low enough
it happens suddenly.
The superheated water boils,
erupting fervently!
Until...until all the water’s spent!
And then…


34. 
Mantissa
02:28


Now I’m not killing time,
but I gotta find the place I signified,
Not a character, I stick around
don't miss the point I'm on the right
At an estimate, I vaporize.
No ghostly trace. I'm outasight.
But fill that space don't zorak me
Ya need me, I keep things tight.
Cause I’ma Mantissa!
A minor addition
written in the table
With decimal precision
I’ma Mantissa! The accuracy!
Tight as a lock, when precision’s the key...
Six hundred thousand, six, or sixty, point oh oh oh six,
When log base ten is on the scene to me it makes no diff,
exponents build characteristics the big intense,
Then I glance at the significand and chippy my 2 cents
Cause I’ma Mantissa!
A minor addition
written in the table
With decimal precision
I’ma Mantissa! The accuracy!
Tight as a lock, when precision’s the key.
I’ma Mantissa!
A minor addition
written in the table
Sittin’ with Clark Gable.
Mantissa! The decimal part!
Breakin’ your heart,
Breakin’ my heart!


35. 
Bring it Round
01:12


You gotta bring it round,
For the exponent,
You gotta make a Sound,
For that exponent,
You see that log,
With the exponent,
Now free that log,
Of the exponent,
A constant now is found,
It was the exponent,
The constant stands its ground,
The stubborn exponent!
Unless you need,
Another exponent,
So send with speed,
To make an exponent.
You gotta bring it round,
For the exponent,
Around and round, and round
whatever your intent!
You gotta bring it round,
For the exponent,
Around and round, and round...


36. 
Get By My Side
05:04


I got a base…and you got a face,
and I want… to use… my base…to get to your face.
2 to the 4…is multiplied by 2 to the 5,
Result …2 to the nine.
And I find…the same is…true about
Us logs…but inside out.
Plus is to times as Minus to Divide,
I got the base to get you by my side…
so get by my side.
Plus is to times as Minus to Divide,
sharing a base will get you by my side…
now get to my side.
You’re log x, and I’m log y:
And added we will simplify to
Log of x times y. I’m so inclined.
‘Cause I know, your base’s the same as mine,
At least that’s what I hope, so we combine.
Plus is to times as Minus to Divide,
I got the base to get you by my side…
so get by my side.
Plus is to times as Minus to Divide,
sharing a base will get you by my side…
now get to my side.
If perchance we are to be
subtracted then you’re x
up on my y.
Oh so sublime.
Get intertwined.
You’re so fine, that I’ll take either sign,
So please don’t be unkind, time to combine!


37. 
Change of Base
03:09


Every Log must have a base it’s often e or ten,
Input number into place: It's x the Argument.
Awful bases cause a frown,
Shake you up an' slow you down.
But turn the corners of your mouth,
as I make the case
Change of face.
Change of pace.
Change of base.
Change your base.
Mystery Ternary Tree with
Two Hundred FortyThree Leaves
Use logarithm at base three.
How many branches do you see?
But log base three can be replaced!
To hasten matters: Calculate!
Log 243 over Log 3,
at any equal base.
Change of face.
Change of pace.
Change of base.
Change your base.
When change of base has been invoked any base is game.
Just log of x: the argument, over log of b :base.
The bases can be anything as long as they’re the same.
So take your place, the adaptation will not hesitate!
I said the adaptation will not hesitate!
Space. It’s the place.
For your base. All your base.
Change of face.
Change of pace.
Change your base.
Change your base.
Change your.


38. 
The Base 2
05:01


Four ones were being bullied around,
By a mean Five who hung out at
The same play ground.
The four ones were scared
for no matter the weather,
Oh, the Five was very big,
More than all of them together.
Oh four ones, there’s gotta be a way,
to beat a wicked Five at his dirty game
the situations pretty grim,
but there’re four of you,
and one of him.
One day our four heroes
were hiding from the Five.
When they met four Zeroes,
Just glad to be alive.
Together they knew
that they could put him in his place.
When one of the zeroes
got a smile across his face.
His dad had told him this once:
“All you need’s a little space,
For some zeroes and a one
to make a number to amaze.”
So the eight of them united
and decided...to build a base.
They worked out,
If  they built a base ten,
They‘d be over a million!
But that seemed a bit beyond them.
So after careful research planning and review,
they began to build a small base two.
For three Tuesdays, the ones and zeroes stayed up late.
Their base had four seats numbered:
One, two, four and eight.
And so the time had come,
but they could find no trace.
Of that bully five whom they
had plans to utterly disgrace.
They split to look for him,
imagine their surprise,
When two ones saw their base two
bein’ accosted by the Five.
He tried to sit in it,
but in base two Five don’t fit.
The two ones ran back to the base
and in its seats did sit.
They pressed the big blue button
which was labeled activate,
The base it sprang to life,
as they stepped up to the plate,
But five just ran and decked them
knocking ones and the base free
Cause one and one in a base two
still only act as three.
Another one ran to the base
but never made it there,
Cause Five had thrown her
to the ground but she was still aware,
Her actions were distractions,
Now her friends had all arrived,
And they had climbed into the base
to finally challenge five!
Five laughed in their collective face,
“I’m fighting with halfwits!
Three Zeroes and a One,
About to be bashed into bits!”
But what he didn’t realize,
when they hit Activate,
1 0 0 0 in base two’s the same as eight.
Not much more to say,
but that five got it in the face,
He woke up ten times weaker,
feeling somewhat out of place,
He vowed that he’d get even,
at least until he had spied,
The decimal point that the ones
affixed to his left side.


39. 
Scientific Notation
02:13


A toast to numbers great and small,
Like Avogadro’s number holy moley!
It’s six hundred two quintillion
and some change.
The speed of light roughly
let c three hundred million meters
to the second though we think it might be slowing which is strange.
Also of course minute amounts,
Mu–naught magnetic constant counts for this
You thought four π (two τ) divided by ten million small,
But then you walked the Planck’s length blind:
At one point six decillionth of a meter, mind
the gap the edge was quite a way before.
These numbers all are useful, but essentially reducible
to make us able to compute them, keep us up to pace,
for trillion and quintillionth, are both hard to picture when
you know what’s really the important thing is the decimal place.
AND SO…
A necessary explanation, of Scientific Notation.
Big Significand times ten and exponential indication,
Whether minute or immenser, extra Large or quite condenser,
50 cents or 50 grand: a change of five in power land.
C three hundred million transformed into something brilliant,
As the photon glides through space at three times ten to the
eighth. Scientific haven, with an exponential heaven
Magnetic constant: four π (two τ)  times ten to negative seven.
Scientific Notation
It separates computation.
Significand fracas! Look right!
Exponent vs. Exponent
Multiply / divide Miss Siggy
Add / subtract the exponent
Respectively as I have said,
Now simplify the exponent.
Slide around the decimal
Until there is only a single digit to it’s left
There is only one step left.
For each slide left add one,
To the exponential sum,
When sliding right, you take away,
Since Sixteenthirtyseven when
Descartes figured a significant method
Leading to invention of convention,
And you need no invitation, for the mention
Words become undone
Vive! Scientific Notation.


40. 
Dax Axlna (The Loge)
03:34


Dax Axlna Private Eye,
and the case of the loge!
Which is over an x!
Meet Miss Ellen X! 
She has one over ex–tremely
important task, to find the LOGE!
Dax Axlna hired by Miss Ellen X,
Daughter of, The Famous Mr. X!
The Mister X! He’s the one,
who has left us to solve
the Mystery of the Loge!
And what is this Loge?
Well it’s made from a log,
ex perts know it 
to be over an x!
And where is this Loge?
When it’s found they all will know
Up above the glow of  Mr. X’s base.
Mister Saul Nadu,
is here, here to manage the funds,
of the Famous Mr. X
Mister Saul Nadu is here,
here to handle the gold. AKA – Au!
But who is Saul Nadu!
The man who betrayed you!
By you I mean the Famous Mr. X!
He’s cooked all your books,
and the dinner he cooked laced with strychnine,
He told you it was thyme.
Dax Axlna, and Miss Ellen X
race against time,
to discover the loge.
Dax Axlna, and Miss Ellen X
Up on the roof,
which is shaped like an x!
The Loge now is Sighted!
Our heroes are frightened,
By the plight of bullets flying through the night!
But no one is faster than Dax, the Master blaster,
Disaster! for Saul Nadu’s been outclassed!
Dax Axlna Private Eye, has recovered the loge!
Which was over an x!
For Miss Ellen X! She had one over extremely
important task. The case is closed!
Dax Axlna, and Miss Ellen X up on the roof,
in the midst of the night.
Dax Axlna, and Miss Ellen X up on the roof,
in the midst of the night.


41. 

RayleighJeans you must be kidding,
Wilhelm Wien is dead to me.
When you write equations they should work at every frequency.
Measured wavelengths radiating from an ideal blackbody,
At a given temperature show me your inconsistency.
And who the h is he?
A man appearing on the scene.
A man of great discretion sees,
To make the laws complete,
You gotta be discrete.
MAX PLANCK!
Who will save us from the ultraviolet catastrophe? MAX PLANCK!
Who will save us from the ultraviolet catastrophe? MAX PLANCK!
In the middle 1890’s Planck was tryin’ to maximize,
The light emitted by a bulb, to ultrafy efficiency.
So he wrote a small equation based on ideal oscillation.
But to his intense frustration, it still failed for the UV!
In an act of desperation he accepted an assumption,
(for the sake of innovation) that he might not be correct.
Now released from prior notions, he was free to revelations
Contradicting current thinking in ways he could not project.
Statistical mechanics leads,
Him to quantize the energy.
Planck’s constant times the frequency,
Discreet levels of energy,
No more UV catastrophe!
MAX PLANCK!
Who has saved us from the ultraviolet catastrophe? MAX PLANCK!
Who has saved us from the ultraviolet catastrophe? MAX PLANCK!
Planck didn’t stop at energy, no unit has immunity,
when physical relationships can be expressed more succinctly.
Mass, Length, Charge Time, Temperature,
reduced as small as they can be,
Analog is out, the dawning of Quantum Reality.
He has saved us from the ultraviolet catastrophe. MAX PLANCK!
He amazed us with his plan to quantize the energy. MAX PLANCK!
He has saved us from the ultraviolet catastrophe. MAX PLANCK!
Playin’ violin with Einstein in prewar Germany. MAX PLANCK!
Nobel Prize for Physics, in a state of constant hardcore! MAX PLANCK!
Six point six two six times 10 to negative thirty four! MAX PLANCK!
Saved us from the catastrophe. MAX PLANCK! MAX PLANCK!
Saved us from the catastrophe. MAX PLANCK! MAX PLANCK!
Saved us from the catastrophe. MAX PLANCK! MAX PLANCK!
Saved us from the catastrophe.


42. 
GigaBit PicoChip
04:16


Yocto,
Zepto,
Atto,
Femto,
Pico,
Nano,
Micro,
Milli,
Unity. One Thousand Times More.
Kilo,
Mega,
Giga,
Tera,
Peta,
Exa,
Zetta,
Yotta!


43. 
Conic Sections
01:45


Conic Section, Rhythm Section,
Intersection of a Plane and Cones: x 2
The cornerstone of Analytic Geometry!
If you were the Archimedes,
You would burn the enemy ships,
With a parabolic mirror,
pointed out to sea!
Or maybe Johannes Kepler,
Take the data, and it tells ya,
Orbiting around the sun
in perfect ellipses.
Or your name is Alfred Lee
And you invented LORANC
To track location and the speed.
At sea apply Hyperbole!
Or you’re from an ancient clan,
let’s say Mesopotamian.
The Circle spins upon a stand,
A potter’s wheel with clay in hand.
Conic Section, Rhythm Section,
Intersection of a Plane and Cones: x 2
The cornerstone of Analytic Geometry!
Menaechmus, from Alopeconnesus,
Had a Delian dilemma,
Tryin’ to double up
The volume of a cube!
Which is how he stumbled
on the conic sections. Intersections
of two parabolic sections,
Answers they exude.
Given names: Parabola,
Ellipse, and the Hyperbola,
from Apollonius of Perga,
Didn’t name the circle forya.
Circles are a special case:
Ellipses but with added grace.
A horizontal plane embrace,
That’s parallel to the cone base.


44. 
Elliptically
02:49


It’s the end of the line.
I can’t say that we didn’t have a good time.
But failure’s coming. I predict.
Our lives aboard a sinking ship.
Deny it or live in despair.
Why build a road that goes nowhere?
I am set, In my ways.
And there ain’t no future without any change,
But we both fail in compromise,
And so it’s time to say goodbye,
As pain inflames and swells inside,
I look into your starry eyes.
And I remember, I am on a lonely spheroid.
In an elliptical orbit.
Round a slowly burning focus,
with its own elliptical locus.
Though some day it will destroy us,
we’ve got many days before us.
And I say, “It is not the time.”
As we keep on moving oh so quickly through spacetime.
Ambivalent as either point is justified.
Around with no compulsion to decide.
I remembered, I was on a lonely spheroid.
In an elliptical orbit.
Round a slowly burning focus,
with its own elliptical locus.
Though some day it will destroy us,
we’ve got many days before us.
Then I realized they never will look to the end,
No completion, they cycle again and again.
That’s the way that our lives were intended to be,
I will focus on you if you’ll focus on me.
Hold my hand girl and stand on the major axis.
And we’ll live...
And we’ll live...
I remembered, I was on a lonely spheroid.
In an elliptical orbit.
Round a slowly burning focus,
with its own elliptical locus.
Though some day it will destroy us,
we’ve got many days before us.
And I realized they never will look to the end,
No completion, they cycle again and again.
That’s the way that our lives were intended to be,
I will focus on you if you’ll focus on me.
Hold my hand love and stand on the major axis.
And we’ll live...elliptically.


45. 
Hyperboli
01:49


X minus h, Squared.
over rx squared,
Plus y minus k squared.
over ry squareerd.
It all equals one!
Ellipses are one.
Bet they never told ya,
It works for Hyperbola,
The 2 radii,
rx and ry,
normally are Real,
but imagine if you will:
That Rx Squared OR Ry Squared just happen to be Negative,
The only way for this to be, a bumble bee, complexity.
The radius that has an i, don’t get to have no verticii.
But draw both radii to show, the box that holds the asymptotes,
now summoning your mortal strength, you’ll find that pesky focal length.
It’s longer than both radii, when we’re talkin’ Hyperboli!


46. 
Directrix
01:40


Directrix wants you,
An Equidistant to,
A Focus, that you find,
in a lens.
Directrix wants you,
An Equidistant to,
A Focus, that you find,
In A Radar Dish.
You move just to escape,
the space is in the shape
the shape is in your wake
and it flows...
backwards like a wolf
r animal, parabolic.
Directrix wants you,
An Equidistant to,
A Focus, that you find,
On a Suspension bridge.
You move just to escape,
the space is in the shape
the shape is in your wake
and it flows...
backwards like a wolf
r animal, parabolic..
Directrix wants you,
An Equidistant to,
A Focus, that you find,
As you fall back to Earth.


47. 
So it Goes
01:41


So it goes,
and we go with it,
What else,
are we to do?
So it’s time,
Time to admit it,
We both know,
We both know the truth.
So you made,
a couple errors,
with Aristotle,
and Ptolomey.
If they knew now,
All that I found,
Well I think,
They would agree.
So when is the right time to say that you’re wrong?
The longer you wait is the more damage done.
For truth is the truth and eventually,
The truth will remain, for the truth can be seen.
So we’ll look,
up to the heavens,
See what stories,
They have to tell.
You can threaten,
one man to silence.
But the Heavens!
Don’t Scare so well!
So the story,
of our culture.
Advances.
With each new truth.
So it goes,
And we go with it.
What else,
Are we to do.


48. 
Quitting Time
02:00


It’s quitting time, at six o’clock.
A perfect axis, from the bottom to the top.
Major or Minor there is no distinction,
‘cause the focii have merged into one pointy dot.
At the center of the circular clock.
No focal length.
That’s how I feel.
No focal Strength.
Is that a Pigeon.
No focal length.
That’s how I feel.
No focal Strength.
I’m just waiting for the hands to form that sweet diameter.
Dear god it’s only TwentyThree past Four!
I’m feeling Circular!
No focal length.
I’m feeling Circular!
No focal strength.


49. 
Roundabout
01:11


a brave semi major axis,
b the semi minor scared.
For ellipses also circles,
area has been declared:
a times b times pi, for circles
is of course just pi r squared.
The same does not hold for circumference
or perhaps perimeter.
pi d or 2 pi r for circles,
but ellipses don’t transfer.
For Ellipses:
twice the sum of
a squared plus b squared, now go,
take the square root of it all
and multiply by pi but know,
that this is just approximation.
For perfection: Integration.
Read the newest publication
by the good Sir Ivory.
Delivering the information:
Ellipses rectification.
GaussKummer’s Series Summation,
Transforming and reforming
and Transforming
the vast hypergeometry.


50. 
Inscribed Pythagorus
02:23


Equation of the circle is a minor adaptation,
The hypotenuse the radius,
the legs will give you y and x,
and stationary is a point,
the center of the circle,
just two shift transformations,
from the origin to anywhere you’re there.
Inscribed Pythagoras, he’s looking at you, and you must,
Acknowledge him. Appease him. If you’re lucky you will please him,
If you don’t he’ll bring you to a world of hyperbolic grieving,
from which you will never ever go. Though you are always leaving.
The focal lengths, the axes, semimajor, semiminor,
are related to the theorem and there’s no fair theorem finer,
For Ellipses semimajor is the longest of the trio.
The focal length is greatest when pertaining to Hyperbolas...you see....
Inscribed Pythagoras, he’s looking at you, and you must,
Acknowledge him. Appease him. If you’re lucky you will please him,
If you don’t he’ll bring you to a world of hyperbolic grieving,
From which you will never ever go, though you are always leaving.


51. 

A parabolic mirror in the desert.
reflects the waves of solar radiation,
to a tube filled with oil at its focus,
which then runs to a solar power station.
The oil in the tube then heats the water,
which turns a turbine when it turns to steam,
which spins a coil of wire near some magnets.
They’re generating electricity.
It’s AC current!
They’re generating.
It’s alternating,
from the rotating.
Through the dessert!
Along the poles,
and coming in,
to power our homes.
An adapter is then plugged into an outlet.
Electricity flows through a transformer.
Then it passes through a fullwave rectifier,
which is really just four diodes in a square.
The voltage from the wall is still too wavy.
So before it can be used in a machine,
a capacitor assists it with the smoothing,
and now it flows with more consistency.
It’s DC current!
I’ll say directly!
Easy to work with,
more elementary.
An adapter,
is all you need,
to turn the AC
into DC.


52. 
Unit Circle Trigonometry
01:55


Zero, π over 6 and more,
The next one to know is π over 4,
Then π over 3 and π over 2,
Are all of the radians I’ll tell to you!
r is 1, circumference is π times 2
O, thirty, fortyfive, sixty, ninety!
That’s why I love unit circle Trigonometry.
O, thirty, fortyfive, sixty, ninety!
That’s why I love unit circle Trigonometry.
The x’s are 1, square root of 3
Over 2 horizontal they must be.
Square root of 2 over 2 and next are,
One half and zero the stately x.
Secretly they’re cosine but that’s coming next.
O, thirty, fortyfive, sixty, ninety!
That’s why I love unit circle Trigonometry.
O, thirty, fortyfive, sixty, ninety!
That’s why I love unit circle Trigonometry.
Zero, one half, and squareroot of 2
Over 2, that’s "y" I’m telling you,
Squareroot of 3 over 2 and 1.
Sine is the height at any radian,
The squareroot of 4 over 2 is 1!
O, thirty, fortyfive, sixty, ninety!
That’s why I love unit circle Trigonometry.
O, thirty, fortyfive, sixty, ninety!
That’s why I love unit circle Trigonometry.
Now there are three other quadrants to
The unit circle that is in your view,
I’m not going to sing them,
You can work them through,
And while you are doing that I’ll say to you:
r COSINE is X!…And r SINE is Y!
So we all love unit circle Trigonome  tri!
r COSINE is X!…And r SINE is Y!
So we all love unit circle Trigonome  tri!


53. 
Synthetic Division
03:10


Synthetic Division, I love your precision,
You tell me so much, so much more I am wishing...
When my divisor, that's the number up top,
is a positive, and positive answers don't stop,
I know that there won't be factors up higher,
that divisor is an upper bound no greater roots are righter.
When I say remainder, I need a reminder
that's the value of the function when x is the divisor.
Synthetic Division like nuclear fission,
a chain reaction from atomic collision.
For negative divisors, instead of agreement,
my answers' signs differ in consecutive sequence.
This sign alteration, a fine observation,
a lower bound found around a left side location.
When I say remainder, I need a reminder
that the answer of the function when x is the divisor.
Synthetic division like surgery listen
a factor removed with just a tiny incision.
Add it down. Multiply. Bring it up, to the right.
Add it down. Multiply. Bring it up, to the right.
Synthetic Division, the process is easy
multiply by the divisor then you add and repeat it.
So highly efficient, the new coefficient
is one order smaller 'cause it's one root deficient!
When I say remainder, I need a reminder
that the answer of the function when x is the divisor.
Synthetic Division, I'll make my decision,
a factor removed is a zero, envision.


54. 
Complete The Square!
04:10


a x squared plus b x plus c,
equal to a zero,
must we...
suffer from the automatic?
Celebrate the form quadratic!
b  a parabolic figure over me.
c  the y is intercepted by the c.
See the standard form, ignore the c,
factor the a from a and b.
a parenthesis x squared
plus (b over a) times x
plus a little piece!
The missing piece that comes with love from Greece.
The piece, that's missing from the square you must complete!
You must complete. You must complete. You must complete!
Complete the square!
x squared plus b over a times x – not a square.
Complete the square,
Divide b over a times x – into a pair.
Complete the square,
What’s missing (b over 2a) – quantity squared.
Complete the square,
So add the square inside  outside subtract a square.
Balance is important, outside
there's a scalar factor friction.
Feel the gravity on Saturn,
six point four additional percent!
The cancellation should be evident.
Assay the c minus b squared over four a.
Now factor the perfect square,
trinomial to binomial squared.
x plus b over 2 a is squared and now the vertex form is there.
A times (x minus h) squared plus k where...
the vertex is (h comma k) the vertical stretch is the A.
Solve for quadratic formulæ declare!!!
Complete the square!
x squared plus b over a times x – not a square.
Complete the square,
Divide b over a times x – into a pair.
Complete the square,
What’s missing (b over 2a) – quantity squared.
Complete the square,
So add the square inside  outside subtract a square!
Negative b plus or minus,
the square root of all your shyness.
You'll b squared minus four a c.
It's all over 2 a.


55. 
Be Rational
02:14


Though there will always be more,
the set of rationals is small,
Infinity is big you thought
but Cantor says it’s Aleph naught!
Top and bottom,
if you got’em,
you’re my cup of tea!
Some go on forever
with no reason so bizarre!
Don’t append IR!
Just be rational.
πs and es, and sines and bees,
I just can’t comprehend,
No need to transcend!
Just be Rational!
To Be Rational
you gotta be A over B
where B is any integer
except for Zero:
The Exception King.
Radicals aren’t radical,
Unless they terminate.
Please don’t be Irate,
Just be Rational
Five fourths? Quite right.
Phi? If I were ossified!
To Be Rational
you gotta be A over B
where B is any integer
except for Zero:
The Exception King.
Dreamers, Schemers,
Darting is one moment to the next,
Don’t be so complex,
Just Be Rational!
So wear your boots
and end your roots,
at least make them repeat!
Keep it nice and neat,
keep it rational!
Just be Rational.
So fashionable.
Belts look good on you.


56. 
Defining Undefined
01:58


It’s so rational and reasonable
you’ll never ever please ‘em all
descending to the deepest
and then falling from the sky.
Like a shovel beam insisting,
there is really no resisting,
it’s dividing into nothing
lowest low and highest high.
It is not for me, Continuity!
Continuity, it is not for me!
Rising up from deep depression
the increase will always lessen,
'til each day seems less and less
unpleasant from the days of past.
Always upward you are ranging
though it seems that nothing’s changing
and the strangest of the strange thing
is the place you’ll never pass.
Limits wait for me, at infinity!
At infinity, Limits wait for me!
You're so rational and reasonable
you’ll never ever please ‘em all
descending to the deepest
and then falling from the sky.
Asymptotically ambivalent!
No equal or equivalent!
The ultimate in Insolent!
Defining Undefined.


57. 
Asymptote
02:37


We’re closer everyday,
and yet so far away.
I feel the space between,
a barrier unseen.
But I…But I…But I found out something ‘bout you!
You say it’s meant to be,
yet stand there so distantly.
You say you love me more,
I’m not so very sure.
‘Cause I…‘Cause I…‘Cause I found out something ‘bout you!
You’re an Asymptote!
You tease me and you’ll never quit the game.
You’re an Asymptote!
And your behavior’s driving me insane!
Maybe at the end of time,
I will finally know your touch,
and I’m sure it is sublime,
but I’m limiting this crush.
So remove yourself from me, you discontinuity.
But you stand your ground as the upper bound,
and whisper so softly:
A hole I’m not.
Maybe you have got,
the wrong idea my pal.
It’s our destiny, an eternity,
side by side, get comfortable.
And I know I’m not,
very good – so what –
at being rational.
I’ll leave that to you,
maybe while you do,
look at who we are.
I’ll leave that to you,
maybe while you do,
look at who’s creepin’ on who.


58. 
Rash Analysis
01:32


Factor Top and Factor bottom,
Look for zeroes on the top!
Downstairs zeroes are restricted,
Continuity will stop.
Factor Top and Factor bottom,
Cancelation may occur.
Holes appear from points removed!
But cancelling builds character.
The limits at infinity are where you’ll find,
The horizontal asymptotes, but keep in mind:
It’s possible to cross an asymptotic guide,
Specifically the ones found lying on their side.
To find the limits at infinity,
just isolate the highest order terms:
One term on top and one on bottom,
Cancel common factors of them:
If no x is left then you are done.
But if an x is below then the limit must be zero,
x on top grows up to be infinity.
Factor Top and Factor bottom,
What you can’t remove will act,
As a strange mathematic fiction
When it’s zero, that’s a 
Factor Top and Factor bottom,
What you can’t remove, denote:
Two words sitting at the bottom,
Vertical and Asymptote.
Vertical and Asymptote.
Vertical and Asymptote!


59. 
The Ideal
03:29


Robert Boyle packing clothes.
Suitcase bulging at the seams.
An attempt to make it close:
Apply the pressure liberally.
And it seems the idea transferred,
Raise the pressure, drop the volume.
But the ideal is unanswered,
Maybe somewhere it is out there.
Guillaume Amontons,
Boiling water in a pot.
Heavy lid popping up,
Water vapor getting hot.
And it seems the idea transferred,
Hotter gas has higher pressure!
But the ideal is unanswered,
Maybe someone else will get there?
Jacques Charles out in the cold!
Snowpants! He is shivering.
Red balloon on a string,
Gradually is shriveling.
And it seems the idea transferred,
Colder gas has lower volume!
But the ideal still unanswered,
Must be someone else’s problem.
So temperature is related to:
The pressure and the volume,
Also it seems the amount of moles
To count the gas’s molecules.
That occupy the entire space,
From which the gas cannot escape.
Related all by a constant R.
Are you sure? It’s 8.314…
Lorenzo Romano Amedeo Lacquisha Carlo
Avogadro di Quaregna e di Cerreto traded Moles.
And I guess the idea transferred,
Equal moles have equal volume.
But the ideal is unanswered,
Will it stay an unsolved problem.
Monsoir Clapeyron comes on the scene!
Uniting laws with gassy themes:
It’s on the wall
you hear him scream!
P times V is n R T
It’s P for pressure, V for vol.
n the number of the mol.
T’s temperature and it’s in Degrees,
Kelvin says he agrees  he agrees
R you sure it's eight point three...
One four.


60. 
Extraneous
02:56


Extra Answers.
An extra Solution.
Abate the deception,
berate the confusion.
Though all of the answers
you have may make sense,
test to distinguish.
the wise from the dense.
E X T R A N E O U S
Extra answers!
To plague and confound you.
The trouble with scribbles
completely surrounds you.
Not your intention!
But sooner or later:
become their destroyer!
You were their creator.
E X T R A N E O U S
Extraneous solution creeping silent in the night.
A charlatan imposter poseur nogood parasite.
The chocolate with the worm that hides,
the hat that’s full of lice.
Untested is unexposed,
You too will pay the price.
Solving a problem may change the problem.
Solving a problem may change the problem.
You used the power, now choose a door:
Is it sheer luck or Shere Khan?
E X T R A N E O U S
Extraneous solution creeping silent in the night.
A charlatan imposter poseur nogood parasite.
The kitchen where the roaches hide,
The maggots in your rice.
Untested is unexposed,
You too will pay the price.
Solving a problem may change the problem.
E X T R A N E O U S


61. 
Stimulate Emission
03:53


I would like to see light coherently.
As you listen now I will tell you how.
Amplify! Pump it up! Change of Quantum state.
Gaining in the Medium about to radiate!
Pump me up. Bring me down. I can see your phase.
Gaining in the medium, my pulse begins to race!
My electrons are excited and your Photons are invited to
release all inhibition, an orbital transition,
quantum drop, power swap!
We stimulate emission!
I would like a beam frequency in key.
I’ll get you in phase. Waves in step amaze!
Amplify! Pump it up! Change of Quantum state.
Gaining in the Medium about to radiate!
Pump me up. Bring me down. I can see your phase.
Gaining in the medium, my pulse begins to race!
My electrons are excited and your Photons are invited to
release all inhibition, an orbital transition,
quantum drop, power swap!
We stimulate emission!
Invert my population, let's make some radiation.
Back and forth, interfere, come and misbehave,
Filling up your cavity  constructive standing waves.
694 Ruby Red! Violet 416!
I can see you turning red as you increase wavelength.
Amplify! Pump it up! Change of Quantum state.
Gaining in the Medium about to radiate!
Pump me up. Bring me down. I can see your phase.
Gaining in the medium, my pulse begins to race!
My electrons are excited and your Photons are invited to
release all inhibition, an orbital transition,
quantum drop, power swap!
We stimulate emission!
Release all inhibition! An orbital transition!
Quantum drop! Power swap!
We stimulate emission!
We stimulate emission!
We stimulate emission!
Quantum drop! Power swap!
We stimulate emission...


62. 
6.283 (Tau)
02:44


A tower rises from the ground,
a spire shining light.
Who strolls its proud perimeter?
Sir Cumference the knight.
As he circles he'll recite:
Cheers to constant Tau.
Six point two eight three...
Half the tower square is circled in brown hair.
Be wary of this area because there hides a bear.
To tower square we go, to see the spheres on show.
Four times around their surface wound as crust over mantle.
Cheers to constant Tau.
Six point two eight three...
Two thirds of the tower cubes control the singing spheres.
How integral their volume is to anyone who hears.
Take it spheres!
Master Cylinder is there in service to the heir.
A wrapping tight the tower height combines with tower square.
That cylinder just might, contain the dynamite.
Restrain the urge to swiftly purge half the tower square and height.
But if half the square is gone with half the tower height,
the later is collateral the cones will join the fight.
And the volume grows three pointy cones will make the fighting fair.
As the cones battle the cylinder amidst the tower square.
Cheers to constant Tau.
Six point two eight three...

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