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Calculus II Honors: Limits, Infinity, and Applications

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Calculus II - Honors: MATH 1921-800
Goals for this class
Explore calculus using a wider lens
E.g., examine the notion of a limit from a wider perspective
“Infinite Powers: How calculus reveals the secrets of the universe”
by Steven Strogatz
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Infinity Principle
Infinity Principle centers around 2 themes - mysteries &
methodology
Methodology: Divide-&-Conquer strategy - Cutting & Rebuilding
Cutting - infinitely fine subtraction which is used to quantify the
differences between the parts - called Differential Calculus
Rebuilding - integrates parts back into the original whole - called
Integral Calculus
Big challenge - Cope with infinity
Mysteries: mystery of curves, mystery of motion, mystery of change
Resolution of these mysteries had far reaching consequences on
civilization and our everyday lives
Figures like squares, cubes, planes proved to be staighthforward
Round things were brutal!
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Mystery of curves
How much area and volume a sphere can hold
Circumference and area of a circle were unsurmountable problems
Calculus grew out of geometers’ curiosity and frustration with
roundness
Breakthrough - insisting that curves were actually made of straight
pieces
Issue - pieces have to be infinitesimally small & infinitely numerious
Zoom closely enough on a circle - the portion of it under the
microscope begins to look straight & flat
Curves arose naturally in parabolic arc of a ball in flight, elliptical
orbit of Mars, convex shape of a lens (development of microscopes
& telescopes in late Renaissance Europe)
Led to fascinations with mysteries of motion on Earth and in the
solar system
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Mystery of motion
Experiments - measured motion of a pendulum, clocked accelerating
distance of a ball rolling down a ramp
Kepler was in a state of “sacred frenzy” when he found his laws of
planetary motion
However, math & geometry could not adequately explain the
patterns
Motion was often not steady.
E.g., ball rolling down a ramp, direction of travel kept changing,
planets moved faster when they got close to the sun & slowed down
when they receded from it
Infinity Principle came to the rescue just like it had for curves
Changing speed was made up of infinitely many & infinitely similar
brief motions at a constant speed. Even the jerkiest driver cannot
make the speedometer needle move by much
Over an infinitesimal time interval - needle was not moving at all
Criticized for playing fast and loose with infinity in mid-1600s
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Isaac Newton
Found a way to combine symbols of algebra with infinity represented any curve as a sum of infinitely many simpler curves
described by powers of x, like x2 , x3 , x4 , etc.
Motion of any kind always unfolds one infinitesimal step at a time
Handful of differential equations (laws of motion & gravity)
explained everything from the arc of a cannonball to orbits of
planets
Work launched Enlightment - impacted philosophers, poets,
underpinned space program & work of African-American
mathematician Katherine Johnson (Hidden Figures)
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Mystery of change
Are the laws of change similar to laws of motion?
With experiment & observation, scientists worked out the laws of
change and used calculus to solve them & make predictions
Examples - Einstein applied calculus to a model of atomic transitions
to predict an effect called stimulated emission [ s & e in laser - Light
Amplification by Stimulated Emission]
Applied medicine to shape 3-drug therapy for HIV patients
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Infinity
Shaped by everyday concerns such as redrawing boundaries of
farmers’ fields after summer flooding of the Nile washed boundaries
away
Geometers found circles to present challenges - much harder to
analyze than triangles, rectangles, etc.
Calculus began as an outgrowth of geometry
Goal - use infinity to build a bridge between the curved and the
straight
Infinity helped to find the area of a circle
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Area of a circle using a pizza analogy
Pizza = perfectly flat and round with an infinitesimally thin crust
Want to measure circumference and radius
Cut into four quarters and rearrange
Cut into 8 slices and rearrange
Cut one of the slanted endpieces in half and move that half to the
other side
Keep increasing number of slices
Creating a sequence of shapes that keep getting closer to a rectangle
Call rectangle limiting rectangle
Area of circle = rC
2 , proved by Archimedes
Limit of infinitely many slices gave us the rectangular shape =
things became simpler at infinity
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Map of Tennessee
Get Printable Maps From:
Waterproof Paper.com
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Lecture 2: Limits & the Riddle of the Wall
A limit is like unattainable goal
E.g. Riddle of the Wall
Walk halfway to a wall
Then, walk half remaining distance
Then, half of that, and on and on
Will you finally get to the wall?
Get arbitrarily close to the wall
Wall plays the role of a limit
Parable of .333 . . .
Convert 1/3 into an equivalent decimal
1/3 can be written as 0.333 . . . where the dot-dot-dots mean than
the threes repeat infinitely
dot-dot-dots = represent the infinitely many threes that we cannot
possibly write = completed infinity
dot-dot-dots = represent a limit = limit of successive decimals
generated during long division on the fraction 1/3 = potential infinity
Pretending that process actually terminates & somehow reaches the
nirvana of infinity can get us in trouble!
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Limits & the Parable of the Infinite Polygon
Take a circle and put three dots on the circle and connect them
Board
More dots we use =⇒ rounder polygon becomes =⇒ sides get
shorter and more numerous
As we add more dots, polygons approach the original circle as a limit
Infinity is bridging two worlds. That is, taking us from the rectilinear
to the round
In pizza proof, infinity took us from round to a rectangle
Polygon gets closer & closer to being a circle but never truly gets
there =⇒ potential infinity
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Allure & Peril of Infinity
Limiting shape is simpler & more symmetrical
Circle simpler that any of the thorny polygons that approach it
Pizza proof, rectangle was simpler than the scalloped shapes
This is the allure of infinity. That is, everything becomes better
there
Should we take the plunge and say that a circle is truly a polygon
with infinitely many infinitesimal sides?
Might be condemned to logical hell (sin of completed infinity)!
If circle = infinite polygon with infinitesimal sides, then how long are
sides of polygon? Zero length?
If so, infinity × zero = circumference of circle! Absolute garbage!
Don’t know what infinity × zero is!
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Sin of Dividing by Zero
Imagine dividing 6 by 0.1. Answer: 60
Imagine dividing 6 by 0.01. Answer: 600
Imagine dividing 6 by 0.0000001. Answer: 60 000 000
The smaller the divisor, the bigger the answer
As the divisor approaches zero, the answer approaches infinity
That is why we cannot divide by zero. The truth is the answer
would be infinite.
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Sin of Completed Infinity
We thought we could actually reach the limit - that we could treat
infinity like any number
Greek philosopher Aristotle (4th century BCE) warned that being
imprecise with infinity in this way could lead to all sorts of logical
trouble
Thinking of our pizza proof, potential infinity would mean that the
pizza can be sliced into more and more slices, as many as desired
but still a finite number of slices and all of nonzero length
This leads to no logical difficulties
Completed infinity =⇒ infinite number of slices of zero length was
forbidden by Aristotle
His edict followed for the next 2200 years
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Zeno & his Paradox of the Arrow
Zeno argued that space and time are discrete, that is, they are
composed of tiny indivisible units, say pixels of space and time
Paradox: If space and time are discrete, an arrow in flight can never
move, because at each instant (pixel in time) the arrow is at some
definite place (a specific set of pixels in space). Hence, at any given
instant, the arrow is not moving!
Not moving between instants because, by assumptions, there’s no
time between instants
We know from watching movies & videos on our digital devices,
motion is very much continuous even when it’s discretized
Owing to our perceptual limitations, the motion of the video looks
smooth
Sometimes our senses do really deceive us
If the chopping into pixels is too choppy, we can tell the difference
between continuous & discrete
For many practical purposes, the discrete can stand in for the
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Zeno meets the Quantum
Do infinitesimally small things exist in the real world?
Realm of Quantum Mechanics = Nature on smallest scales
Riddle of the Wall from quantum perspective
If walker is an electron, there’s a chance it might walk right through
the wall.
Effect known as quantum tunneling
Hard to make sense of this in classical terms
Quantum explanation is that electrons are described by probability
waves
Waves obey an equation formulated by Austrian physicist, Erwin
Schrodinger (1925). Solution to Schrodinger’s eqn show that a small
portion of the electron probability wave exists on the far side of the
impenetrable barrier
=⇒ there is some small nonzero probability that the electron will
be detected on the far side of the barrier, as if it had tunneled
through the wall
With calculus, can calculate the rate at which tunneling events occur
& experiments have confirmed predictions
Alpha particles tunnel out of uranium nuclei at the predicted rate to
produce effect known as radioactivity
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Nature on the smallest scale = Quantum Mechanics
Tunneling also critical in nuclear-fusion processes that make the sun
shine
Hard for us to think of a world on an atomic level
Calculus has taken the place of intuition
Quantum mechanics radical in in many ways but retains the
assumption that space & time are continuous
Maxwell made same assumption in his theory of electricity &
magnetism
Newton in his theory of gravity & Einstein in his theory of relativity
All rely on assumption of continuity
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Does time lose its continuous character?
We don’t know what it’s like on such a small scale
No consensus on how to visualize space & time on these small scales
Agreement on how small the scales may be
Forced upon us by three fundamental constants of nature
G = gravitational constant
~ = reflects strength of quantum effects (appears in Schrödinger’s
wave equation of quantum mechanics)
c = speed of light (speed limit for the universe)
Planck - father of quantum theory - combine these constants to
produce a scale of length
p
Planck length = ~G/c3
When we plug in the values of the constants, Planck length = 10−35
metres
Corresponding time to travel this distance = 10−43 seconds
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What kind of numbers are we talking about?
Numbers put a bound on how fine we can slice space or time
Take largest distance possible = estimated diameter of the universe
Divide by Planck’s length
This extreme ratio is a number with only 60 digits in it
That is the most we need to express distance in terms of time
Using way more digits than that would be colossal overkill!
Yet, in calculus we use infinitely many digits all the time
In the idealized world of calculus, we pretend that everything can be
split finer & finer without end
Whole story of calculus built on that
With infinity, things are made simpler
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The Man Who Harnessed Infinity = Archimedes
Archimedes
His use of Infinity Principle & his approach of blending mathematics
with physics
Estimation of π
Let’s think of a circle as track
Say we walked around the track in x steps - the steps would be
represented by small straight lines around the circle
Can estimate length of track by counting the number of steps &
multiplying by the length of a step
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Archimedes - Estimation of π
Archimedes proceeded similarly
Began with taking 6 straight steps along the circle
Divide hexagon into 6 equilateral triangles
C > 6r
Define π as the ratio of the circumference to the diameter
π = C/d = C/2r > 6r/2r = 3 ← lower bound for π
Went from 6 steps → 12 steps → 24 steps → 48 steps → . . . → 96
steps
Got progressively harder to calculate step length
Applied Pythagorean theorem and computed square roots by hand
Ultimately, with his 96-gon, he showed that
3 + 10/71 < π < 3 + 10/70
3 + 10/70 reduces to 22/7 ← famous approximation
No mention of π in Euclid’s Elements (a generation or two before)
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Tao of Pi
π gave rise to the notion of irrational numbers
Around the same time, the Greeks realized that the diagonal of a
unit square was also not a whole number!
As of now, twenty-two trillion digits of π have been computers by
some of the fastest computers
Yet twenty-two trillion digits is nothing compared to the infinitude of
digits that define the actual pi
Where are these digits?? They don’t exist in the material world, as
of now
π is defined as the unattainable limit of a never-ending process.
Unlike the sequence of polygons approaching a circle, there is no end
in sight for π, no limit we can ever know
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Archimedes & the Quadrature of the Parabola
Arc of a three-point shot in basketball approximates a parabola
To Archimedes, a true parabola is obtained by slicing through a cone
with a plane
Steeper cut produces an ellipse
A cut that has the same slope as the cone itself produces a parabola
Archimedes wanted to compute the quadrature of a parabolic
segment
Find quadrature = express area between parabola and line segment
in terms of the area of a square/rectangle/triangle/rectilinear figure
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Quadrature of the Parabola
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Quadrature of the Parabola
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Quadrature of the Parabola
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Achimedes - The Method
The two slats balance each other
Same is true for every vertical slice
All the ribs from the parabola end up at S
They balance all the slats from the huge outer triangle ACD
Since those slats have not moved, this means all the parabolic mass
shifted to S balances the huge triangle right where it is
Next, Archimedes replaces the infinitely many slats of the huge
outer triangle with an equivalent point of their own, called the
triangle’s center of gravity
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Achimedes - The Method
Huge triangle acts as if its entire mass were concentrated at that
single center of gravity
Archimedes shows that this point lies on the line F C at a point
precisely three times closer to the fulcrum F than S is
Since entire mass of the triangle sits three times closer to the pivot
point, the parabolic segment must weigh a third as much as the
huge triangle in order for them to balance
=⇒ area of parabolic segment = 1/3 huge outer triangle ACD
Outer triangle has four times of inner triangle ABC
Archimedes deduces that the parabolic segment must have 4/3 area
of triangle ABC inside it!
Problem with argument!
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Completed infinity used in “The Method”
Archimedes argument deals with Completed Infinity!
Outer triangle = made up of all parallel lines inside itself
Outer triangle = Completed infinity of slats
Likewise, parabolic segment = made up of all the parallel lines
drawn inside the curve
Archimedes went on to apply “The Method” to many other
problems about curved shapes
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Surface area & volume of sphere
Figure: Sphere in a cylindrical hatbox
Using the Method, Archimedes found that the sphere had 2/3 the
volume of the hatbox & 2/3 of its surface area (assuming tob and
bottom lids are also counted).
Archimedes Palimpsest first came to light in 1899 in a Greek
Orthodox library in Constantinople
Now lives in the Walters Art Museum in Baltimore
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The Method - From Computer Animation to Facial Surgery
Archimedes legacy lives on today
Characters seem lifelike due to Archimedes insight - any smooth
surface can be convincingly approximated by triangles
The more triangles used and the smaller we make them, the better
the approximation becomes
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From Computer Animation to Facial Surgery
What’s true for mannequins is equally true for ogres, clownfish & toy
cowboys
Like Archimedes, modern-day animators at DreamWorks created
Shrek’s round belly and his trumpet-like ears out of tens of
thousands of polygons
Even more such polygons were required for a tournament scene in
which Shrek battled local thugs - each frame of that scene took over
forty-five million polygons
No trace of the polygons anywhere in the finished movie!
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Archimedes - The Method - in Avatar
Figure: Avatar by DreamWorks
Infinity Principle teaches us that the straight and the jagged can
impersonate the curved and the smooth
In Avatar, level of polygonal detail became more extravagant
Director James Cameron insisted that animators use about 1 million
polygon to render each plant in the imaginary world of Pandora
It cost 300 million dollars to produce!
First movie to use polygons by the billions
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The Method in computer-generated movies
Toy Story used far fewer polygons
Took an animator a week to sync an eight-second shot
Whole film took 4 years and 800 000 hours of computer time to
complete
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Geri’s Game
Figure: Geri’s Game
First computer-animated film with a human main character
Geri was built from angular shapes
Animators at Pixar fashioned Geri’s head from a complex
polyhedron, a three-dimensional gem-like shape that consisted of
about 4500 corners with flat facets in between them
Animators subdivided those facets repeatedly to create an
increasingly detailed depiction
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Geri’s Game
Subdivision process took up much less memory than earlier methods
and allowed for much faster animations
Revolutionary advance in computer animation
However, it channeled Archimedes
To estimate π, Archimedes walked around a circle in 3 steps
(triangle)
He increased his number of steps until he got to a 96-gon
Animators approximated Geri’s wrinkly forehead, nose and folds of
skin in his neck by repeatedly subdividing a polyhedron
(Geri’s Game)
DreamWorks took the next steps forward in realism and emotional
expressiveness in Shrek
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Facial Surgery
Surgeries for patients with overbites and malformations from birth
Applied mathematicians Deuflhard, Weiser, & Zachow used calculus
& computer modeling to predict the outcomes of complex facial
surgeries
Built an accurate map of a patient’s facial-bone structure
Biomechanically accurate model of the material properties of skin
and soft tissues such as fat, muscle, tendons, blood vessels
With computer model, surgeons could perform operations on virtual
patients
Computer calculated how the virtual soft tissue behind the face
would move and reconfigure itself in response to stresses produced
by the face’s new bone structure
Soft tissues posed its own geometrical challenges. The team
represented the three dimensional volume between the skull and
behind the face by hundreds of thousands of tetrahedrons
Their model successfully predicted the position of 70% of the
patient’s facial skin
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Archimedes & Mystery of Motion
Figure: Archimedes
Legacy - first principled use of infinite processes to quantify the
geometry of curved shapes
Did not address challenges having to do with motion
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Upcoming Presentations
Chapter 3 - Discovering the Laws of Motion: Group 1
Chapter 4 - Dawn of Differential Calculus: Group 2
Chapter 5 - Crossroads: Group 3
Chapter 6 - Vocabulary of Change: Group 4
Chapter 7 - Secret Fountain: Group 3
Chapter 8 - Fictions of the Mind: Group 4
Chapter 9 - Logical Universe: Group 2
Chapter 10 - Making Waves: Group 1
Group 1 - Phoebe, Renie, Hallie, Christian
Group 2 - Allison, Callie, Chao, Jacob W.
Group 3 - Destin Harris, Alex, Jackson Hanchek, Dillon Williams
Group 4 - Luke, Blaine, Jacob D., Jackson Hughes
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Chapter 3 - Important Concepts
Galileo & retrograde motion
Misconception about sun rotating around the earth
Issues with Catholic Church
First practitioner of the scientific method - ball on inclined plane
Differential calculus & Galileo
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