Get all 5 icanhasmath releases available on Bandcamp and save 55%.
Includes unlimited streaming via the free Bandcamp app, plus highquality downloads of icanhasmatheatre! The Archive., Calculus: The Musical! 3rd Edition., Log: A Rhythm, Be Rational, and CSections.
1. 
5 Sizes of Number
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.


2. 
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!


3. 
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!


4. 
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!


5. 
Ain't Differentiable
02:02


6. 
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.


7. 
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.


8. 
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.


9. 
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!


10. 
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.


11. 
Mean Angst
02:33


Take two points called A and B,
and find the slope so easily:
Rise over Run,
of the line through them.
In between, now there’s a point,
its name is C, and I surely don’t…
Wanna underestimate its importance.
‘Cause what the point proves now
is the Mean Value Theorem,
which holds for continuous curves.
‘Cause the slopes at point C
and the line that I mention,
are equal as can be observed.
I don’t know what the world may need,*
but another theorem’s a good start for me.
Take two points of the same value on a function.
C’s still a point that’s in between,
that Theorem of Value sure is Mean.
think I mean...that it’s time...to extend it.
'Cause what the curve needs now
are some true words of wisdom,
like: Horizontal Tangent at Point C,
yet...
I said I need a point C,
with an instant slope of zero,
like I need a Rolle in my head.
*This line is the same as the original.


12. 
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.


13. 
Maxima & minima
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.


14. 
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.


15. 
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.


16. 
Physicstravaganza
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!


17. 
L'hopital
02:22


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.


18. 

“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.”


19. 
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!


20. 
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.


21. 
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.


22. 


23. 


24. 

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!


25. 
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!

Streaming and Download help
If you like icanhasmath, you may also like:
Bandcamp Daily your guide to the world of Bandcamp