The Tree of Calculus
by Susan Cantey © 2008
The complete words to
Calculus Memory Book Songs
(The Tree of Calculus CD)
Susan Cantey © 2008
Climbing the Tree of Calculus
Gonna tell you the story of a boy named Zach,
Minding his business on hid granddad’s ranch,
Walking through a field of contented cows, he bumped into a tree,
Well Zach was a strong athletic youth,
He did what any other boy would do,
Took a deep breath and began to climb,
Didn’t know that he would be….
Climbing,
Climbing,
Climbing the tree of Calculus!
The trunk was slick, he was losing his grip,
Cried out for help as he started to slip,
Then below he felt some hands and voices yelling “Don’t stop!”
John and Maggie, Alex and Alyssa,
Evan and Matthew, Sam and Amina,
Nirmal, Michael, Lei and Zach,
Were on their way to the top!
Climbing,
Climbing,
Climbing the tree of Calculus!
(instrumental bridge – this is for you Joel, you wacky programming kid!)
Climbing,
Climbing,
Climbing the tree of Calculus!
Climbing,
Climbing,
Climbing the tree of Calculus!
Original music and lyric by Susan Cantey
The Limit
The LIMIT is the basis of all calculus, though early on they didn’t know it was,
Cauchy’s definition should be memorized, ‘cause that’s what it’s all about.
If y goes to L whenever x goes to a, from either side and in any old way,
Then L is the LIMIT of f(x) as x approaches a.
How close? How close? Gotta find a really small delta of course,
Given tiny epsilon, delta’s gotta be, small proportionally.
The absolute value of x minus a less than delta’s gotta make
The absolute value of y minus L less than epsilon,
Hey! That’s the LIMIT! That’s what it’s all about!
“Hokey Pokey” 1949 © by Larry LaPrise – true origin of melody (and lyric) is debated
Continuity
The definition of continuity at a point where x equals a,
Is the limit as x approaches a of f of x which equals f of a,
In layman’s terms this means the function value
Is the same as the value being approached,
No steps, no holes, no vertical asymptotes,
Just one smooth curve to speed you on your way.
Traditional Irish melody – Londonderry Air – O’ Danny Boy
First Derivative
The definition of derivative, must be memorized,
And it comes in two versions; you’ll need both to survive,
f of x minus f of t is the one you’ll like the best,
Divided by x minus t now let t go to x,
Instantaneous rate of change, velocity,
Slope of the curve, f prime or dy dx,
Use it to find any min or max,
….first derivative.
For the second definition, h goes to zero,
f of x plus h minus f of x over h and now you know,
The definition of derivative, which must be memorized
And when you know both versions; maybe you’ll get a five.
f prime of x is the limit as t goes to zero,
f of t minus f of x, over t minus x
original music by Susan Cantey – One Man Woman
Simple Rules
How do you take the derivative of
The product of f times g?
Well it’s g times f prime plus f times g prime,
That’s all you need.
Simple rules, simple rules,
That’s all you need, just a few special tools,
Then you can take the derivative
Of anything, with just a few simple rules.
How do you take the derivative of
The quotient of f over g?
Well it’s g times f prime minus f times g prime,
All over geeeee squared,
What about composites? Well you’ll need the chain rule,
Take f prime of the outer, copy the inner, oh yeah, and don’t forget your baby!
Original music by Susan Cantey – Over the Hill
Derivatives You Need to Know
Some derivatives you need to know: I’m gonna sing them now,
First the function and then f prime, please try not to frown,
I’m only gonna do the hard ones; I’m hoping that you know,
How to take the derivative of any polynomial,
Sine x goes to cosine x; tangent, secant squared,
Cosine goes to negative sine, secant goes to secant tangent.
Cotan x negative cosecant squared,
Cosecant x negative cosecant cotan,
Log x goes to one over x; e to the x stays the same.
Sine inverse goes to one over the square root of one minus x squared,
Tan inverse goes to one over one plus x squared,
Cosine inverse is the opposite of the derivative of sine inverse…
You’ll have to look up secant inverse, ‘cause to sing it would be a big mess,
A to the x goes to a to the x times log a,
Log base a of x goes to one over x log a
Original music by Susan Cantey – No More Sad Songs
Higher Derivatives
First derivative, first derivative, take the derivative one more time, second derivative.
Second derivative, second derivative, take the derivative one more time, third derivative.
Third derivative, third derivative, take the derivative one more time, fourth derivative.
Fourth derivative, fourth derivative, take the derivative one more time, fifth derivative.
Fifth derivative, fifth derivative, take the derivative one more time, sixth derivative.
Sixth derivative, sixth derivative, take the derivative one more time, seventh derivative.
Seventh derivative…
Traditional children’s tune – Row, Row, Row Your Boat
Implicit Differentiation
Implicit differentiation isn’t really new,
Implicit differentiation, application of the chain rule,
Here’s exactly what to do…
Treat x the same as you always have,
He hasn’t changed, oh, don’t treat him bad,
It’s y’s derivatives that you multiply
Oh let those babies live!
CHORUS
Now technically, x has a baby too,
But dx / dx has no effect on you,
So don’t you worry at all
dy / dx is proper protocol.
CHORUS
Now you know what you should do…
Original music by Susan Cantey
Related Rates
For some strange reason students find related rates problems make them lose their minds.
The key is in the picture labeled the right way; variables vary; constants stay the same,
Just relate the variables!
An equation or two is all that you need.
Then take the derivative
With respect to t!
The thing that makes these problems so slow is the convoluted geometry you must know,
Similar triangles, Pythagoras and a myriad of volume formulas,
CHORUS
Please don’t forget that you gotta answer in English,
And always include correct and proper units.
CHORUS
Please don’t flush the product rule!
Original music by Susan Cantey – The Day the Water Stopped
Mins and Maxes
(and concavity)
Don’t let mins and maxes get you down; just turn your frown upside down,
Up and down that’s what derivatives do; always the same; ain’t nothin’ new.
When f prime is negative, f decreases; when f prime is positive, f increases,
And this can’t change unless f prime is zero or undefined.
Extrema I insist, might exist, if f prime is zero or doesn’t exist,
But they must be verified by a change of sign; so do it right, draw a number line.
f double prime negative, concave down,
f double prime is positive, concave up,
Inflection points, my oh my; f double prime must change signs.
Beethoven’s melody – Fur Elise
Newton’s Method
Newton’s Method is quite good at approximating real roots,
Maybe that’s how calculators do the clever things they do,
Drop the tangent from a point then compute its intercept,
Use that for the next calculation, then repeat it all again.
The iterative formula is x sub n plus one
Equals x sub n minus f over f prime of x sub n,
Just make sure to start near the zero past anywhere extrema might be,
Keep on using Newton’s formula till you get the answer that you need.
Beethoven’s melody – 9th Symphony
Mean Value Theorem
On an interval from a to b, the average rate of change
Is f of b minus f of a all over b minus a,
Somewhere in between, there will usually be
The very same value for some f prime of c.
If you go up one level there’s a formula that gives
The average value of a function by subtracting its anti-derives,
Then one more time, you must divide
By the change in the x values to get the answer right.
Mean Value Theorem guarantees,
Given continuity and differentiability,
Somewhere in between,
The derivative is the same,
As the average rate of change.
Any time the average value is what you need,
Here is the strategy: integrate that quantity,
Now divide by b minus a, yes, that’s the way
To apply the Mean Value Theorem, average value to obtain.
Instrumental break
The Man Value Theorem, average value will obtain.
Traditional Irish melody - L. Scott – Annie Laurie
L’Hopital
If you like L’Hopital, clap your hands. If you like L’Hopital, clap your hands.
He makes LIMITS easy even when your stomach’s queasy,
If you like L’Hopital, clap your hands.
There are seven indeterminate forms, seven indeterminate forms,
To use L’Hopital, ya gotta put them all
In the following fractional form.
Zero over zero is nice; infinity over infinity will do,
It’s entirely up to you to find, the algebra you need to use,
How do you change zero times infinity? How do you change zero times infinity?
Turn one factor on its head; divide by it instead,
That’s how you change zero times infinity?
Infinity minus infinity’s not too bad. Infinity minus infinity’s not too bad.
You’ll have to factor something out, but there isn’t any doubt,
Infinity minus infinity’s not too bad.
Worst are the exponential ones. Cause first you take the log and that’s no fun.
Then bring down the power, now do it like the others,
Don’t forget to anti-log when you’re done!
Traditional children’s tune – origin disputed – If You’re Happy and You Know It
Euler’s Method
Euler’s Method is the one used
To estimate an unknown function,
Employing tiny tangent lines,
Approximations you will find,
But if the curve is concave up,
Your answer won’t be big enough.
Initial values specified,
To get dy you must multiply
dy / dx times delta x;
Add that to y that’s what comes next,
Now you’ve got the new y-value,
Just a little more I need to tell you.
Repeat the process with the new numbers,
Not so hard if you’ll remember
To put your work inside a table,
Please write neatly if you’re able,
If f is concave down, then you’ll find
Your estimate is a little bit high.
Traditional tune – Twinkle Twinkle Little Star
Riemann Sums
Let’s all calculate a Riemann Sum, everyone, here’s how it’s done,
Rectangles underneath a curve, two dimensional phenomenon,
Delta x for the widths, y values for the heights,
When these areas are combined, the Riemann Sum is what we find.
A lower or an upper sum, minimum or maximum,
Maybe left or right or middle point, we can use any one we want,
The more delta x is reduced, the more rectangles we produce,
Closer to the actual quantity, the Riemann Sum will tend to be.
Bach’s tune – Minuet in G
The Fundamental Theorem
To Find the Derivative of an integral,
From a to x of little f in some other variable,
Backwards and forwards, I bet you can guess,
The answer will be little f of x.
The fundamental theorem of Calculus is this:
Little f in the integrand, big F its anti-deriv,
Big F of b, big F of a, now find the difference,
The total change in big F is what it represents!
It’s the fundamental theorem.
When you take the derivative after integrating f
With complicated limits like a to g of x,
The answer is f of g times
g’s little baby…g prime…
original music by Susan Cantey – I Don’t Wanna Want You
Area
To find the area under the curve of a rectangular function, this is your unction,
Set up an integral from a to b and if f’s positive the area twill be,
You’ll get the displacement, even if f’s negative,
But if you want the total area, take the absolute value before you integrate.
To find the area in parametric form, you will still integrate y times dx,
But be clever in your endeavor; change everything to “t”; that’s what comes next,
In polar form it’s quite different; one half r squared is the integrand,
The pi inside the circle is in d theta, but it’s disguised
Traditional Italian tunes – Santa Lucia and O’ Sole Mio
U-Substitution
Other than algebraic manipulation,
U-substitution is your most powerful weapon,
You need it in your arsenal,
To fight the mighty integral,
In order to use u-substitution
You need to see an inside function with a baby,
Let u equal the inside function then compute du,
Then change the integrand into u-expressions completely,
Don’t forget to change the limits of integration
If the integral is a definite one,
Or else change back to x
When you’re done…
Original music by Susan Cantey
Parts and Partial Fractions
Backwards product rule, integration by parts,
Two functions unrelated, no way to integrate it,
One factor is f prime; the other factor is g,
The trick is to choose each one of them right as you will shortly see.
g prime should be simpler or at least no worse than before,
f prime’s anti-derivative shouldn’t be a terrible chore,
The original problem is equal to g times f minus integral new,
Integrate f g prime, that’s what you do, though occasionally there is more.
Repeat the process if you need to
Boil it down; boil it down,
Try not to go in circles,
That defeats the purpose,
Unless you change signs,
Add it to the other side.
Backwards quotient rule, partial fractions,
The denominator is the key; you need to factor it completely,
Set the problem equal to separate fractions, oh just a few,
Solve for A, B, C and D, integrate individually.
Now we’ve reviewed two methods in your arsenal,
You can integrate lots of things when you know these methods well,
Practice makes you better; give it your best effort,
These problems won’t take forever; just count to ten and begin…oh yeah.
Traditional children’s tune – Three Blind Mice
Volumes of Rotation
Susan Cantey © 2007
We will rotate round the axes yes we will,
We will rotate round the axes yes we will,
We will rotate round the axes, rotate round the axes,
We will rotate round the axes, yes we will.
Disks around x-axis with one function,
Integrate pi y squared times dx,
Disks around x-axis with one function,
Integrate pi y squared times dx,
Disks around y-axis with one function,
Integrate pi x squared times dy,
Disks around y-axis with one function,
Integrate pi x squared times dy,
If you’re doing disks with two functions
Use pi times the following quantity,
Big R squared minus little r squared and then use the same letter
As the rotational x or y-axes.
If you rotate around any other line,
You will need to change your radii,
Bring the constant over to the other side,
Now r is x or y plus or minus.
For shells, the variables get reversed,
And the formula is pi r h of course,
Use dx with the y-axis; dy with the x-axis,
With shells, opposites are attracted.
Traditional Melody – She’ll Be Coming Round the Mountain
Work
I’ve been working on my calculus all the live long day,
And I found the hardest problems to ever come my way
Involve the work of pumping liquids and stretching springs out too,
But quitters never win and winners never quit, so I figured out what to do.
Work equals force times distance but force and distance vary,
That’s what makes the integral an absolute necessity,
Force needed to stretch a spring from its natural length,
Equals k times x and when you integrate that you get work,
k is a special constant and x is the length you stretch it,
The limits are the incremental differences which the start and stop positions suggest.
Short instrumental riff
REPEAT CHORUS
Pumping liquids is complex; you need weight per unit times the area
Times the distance times the thickness which is just dx,
In the metric system, instead of weight you’ll need,
The mass per unit, called density, times the constant of gravity,
There are several other applications like buckets and cables and ropes,
But the fundamental thing you need to know is that work is the integral of force.
REPEAT CHORUS
(Now this song is over…hope you know what to do!)
Traditional American melody – I’ve Been Working on the Railroad
Diff E Q
First order diff E Q contain dy / dx,
But only one kind is in your calculus text,
The type where students try
To separate x and y,
Putting x stuff on the right
Y stuff on the left,
Then integrate…both sides.
Diff E Q, easy to do,
Separate and integrate,
Shouldn’t need to hesitate,
Diff E Q
With initial conditions,
You’ll need to solve for C,
But do that before solving for y
Or the algebra will drive you crazy.
Original music by Susan Cantey
Arc Length
Arc length in any form is purely plug and chug
Formulas and integrals make the problems fun
The basic part of the formula is the square root of one plus f prime squared,
All the rest, follow from this, you only have to integrate…
Arc length, no big deal, you just have to memorize,
Arc length, easier than you realize,
Regular, polar, parametric, no one can deny,
By the book, hook or crook, just take a look; the formula identify
Arc length in parametric form, instead of one plus f prime squared,
Underneath that square root symbol, a different sum is made,
dy / dt squared, plus dx / dt squared too,
Given three dimensions, oooh, I hate to mention, just means another term for you…
CHORUS
Polar form is easy too; here’s all you’ve got to do,
r squared plus its derivative squared, that’s all that’s new…
CHORUS
Original music by Susan Cantey – Matched Up
Series
Series are serious business; the terms are the first thing you check,
If they don’t go to zero, divergence is a sure bet,
Alternating series are the next best, ‘cause if the terms go to zero, they will converge,
Moreover the error in S sub n is less than the very next term.
Positive series are tougher; you’ll need a bevy of tests,
The terms need to shrink so much faster sometimes it’s hard to know which test is best,
P-series, p-series, one over n to the power of p,
They will converge if p’s greater than one; if not it’s a divergent sum.
If last comes to last, you can always compare,
Your series to one from the past,
If the limit of their ratio has a positive value,
What the old one must do, the new one will too!
Factorials necessitate the ratio test; exponents require the root test,
One is the cut off for both of them; convergence when the limit is less,
Or you can make a sub n into a function and its integral examine,
If you get real number positive result
Then your series convergences to some unknown sum.
Traditional Scottish tune – My Bonnie Lies Over the Ocean
Taylor Series
Who’s afraid of Maclaurin? No, no, no, not I,
Who’s afraid of Taylor Series? No, no, no, not I,
e to the x is equal to one plus x plus x squared over two factorial
Plus x cubed over three factorial plus x to the fourth over four factorial,
Plus dot dot dot fo-o-o-ore-e-ve-er; Plus dot dot dot fo-o-o-ore-e-ve-er;
That’s the way it goes with series: patterns that are easy to remember.
Cosine of x is equal to one minus x squared over two factorial
Plus x to the fourth over four factorial minus x to the sixth over six factorial,
Plus dot dot dot fo-o-o-ore-e-ve-er; Plus dot dot dot fo-o-o-ore-e-ve-er;
That’s the way it goes with series: patterns that are easy to remember.
Sine of x is the same as cosine except instead of even powers we use the odds instead,
Tangent inverse is the same as sine except you leave out all the exclamation points,
One over one minus x is the same as e to the x
But without any denominators at all.
Oh, woe is me if I ever must derive a series from scratch, it’s a formula I lack,
Sigma is scary all by itself, let alone all that other stuff,
nth prime of a..., over n factorial times x minus a to the nth,
Don’t forget to start at zero…it’s a formula you must know...
I know sigma’s scary all by itself, let alone all that other stuff,
nth prime of a..., over n factorial times x minus a to the nth,
Don’t forget to start at zero…
Sine of x equals x minus x cubed over three factorial,
Plus x to the fifth over five factorial minus x to the seventh over seven factorial,
Plus x to the ninth over nine factorial minus x to the eleventh over eleven facto…rial…..
Melodies - Sonatina by Muzio Clementi
Toreador Song from Carmen by Georges Bizet
The Tree of Calculus
Last night, I had a dream that I was standing underneath the shade of a mighty tree,
And its roots went down into the ground, deep in the soil below,
Oh tree of calculus,
How great thy branches,
Differential and integral calculus,
Rising high above the earth,
Holding secrets of great worth,
Your benefits are limitless.
I was transfixed, looking up at a heart carved on the trunk,
Someone had written: “I love calculus.”
Lost in time, I was mesmerized by the leaves dancing in the sun.
Oh tree of calculus,
How great thy branches,
Differential and integral calculus,
Rising high above the earth,
Holding secrets of great worth,
Your benefits are limitless.
Two thousand years you tarried,
With great minds you parried,
Today you reach for the sky,
Your fruit satisfies!
Oh tree of calculus,
How great thy branches,
Differential and integral calculus,
Rising high above the earth,
Holding secrets of great worth,
Your benefits are limitless.
Oh tree of calculus….
Original music by Susan Cantey