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 Cal-cu-lus,
• The numbers, I love the most.

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 infinitely 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!

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!

Jumps, Pinches, Gaps and Holes,
• Gaps and Holes!

Jumps, Pinches, Gaps and Holes,
• Gaps and Holes!

Vertical Tangents and Asymptotes!

Jumps, Pinches, Gaps and Holes,
• Gaps and Holes!

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!

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.

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.

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
dz
When you take derivatives
• When you take 'em,
• When you take derivatives
  dt
• 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, just a little bit.

• IM-P-L-I-C-I-T find out what it means to me,
• IM-P-L-I-C-I-T take with respect to t.

dx / dt! dz / dt!
• A little respect, just a little bit, just a little bit.

D’riv-ative of Sine X,
is Cosine X.
• Derivative of Secant X is, Amazing!
  Secant X Tan X!
• Driv-ative 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 co-functions are-
  All Negative.

• Ya substitute the functions for the co-functions as implied.

• I said y’know trig don’t choke,
• Derivatives of co-functions are-
  All Negative.

• Ya substitute the functions for the co-functions as implied.

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! Im-
• plicit. 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.

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 Lo-ho-ho-ho-ho-ho!

Lo D Hi! - MINUS! - Hi D Lo! - OVER!
• Lo squared: The Quotient Rule!

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, yeah.

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.

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!
• 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.

Area un-der
• the curve f of x is
• equal to the an-
• ti-driv-a-tive of
• f of x dx.
• It is fundamental

Geometry and
• Analysis tied.

Integral from A to B;
• Anti-derivatae.
• Take the val-ue at B sub
• -tract the val-ue at A.

Under the curve I found,
• some Area.

Under the curve I found,
• some Area.

"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 n-ah.
• ‘Cause width is changing-on-the 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 mul-ti-plication,
• 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.
• Infinitesima-al 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:

An-ti – der-i-va –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 per-fect accuracy.

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!

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 go-ing 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.