Differentials, Increments, Infinitesimals

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Differentials, Increments, Infinitesimals All
That

• by
• Costas Efthimiou
• UCF EXCEL Applications of Calculus

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RATES OF CHANGE

• Suppose y is a quantity that depends on another
  quantity x.
• Thus, y is a function of x and we write y f(x).
  
• If x changes from x1 to x2 , then the change in x
  (also called the increment of x) is
• The corresponding change in y is

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RATES OF CHANGE

• The difference quotient
• is called the average rate of change of y with
  respect to x
• over the interval x1,x2 and can be interpreted
  as the slope
• of the secant line PQ in the figure.

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RATES OF CHANGE

• By analogy with velocity, we consider the average
  rate of
• change over smaller and smaller intervals by
  letting x2
• approach x1 and, therefore, letting
  approach 0.
• The limit of these average rates of change is
  called the
• (local) rate of change of y with respect to x at
  x x1.This is interpreted as the slope of the
  tangent to the curve y f(x) at P(x1, f(x1))
• local rate of change
• We recognize this limit as being the derivative
  f(x1).

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LINEAR APPROXIMATIONS

• The approximation
• f(x) f(a) f(a)(x a)
• is called the linear approximation or tangent
  line approximation of f at a.

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THE DIFFERENTIAL AS SMALL QUANTITY
When dxx-a is small, the curve coincides with
the tangent and therefore dy is equal to QS.
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Modeling / Finding Laws
Observe a phenomenon
Derive equations
Use equations to make predictions
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All disciplines of science use calculus
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The Astronomer by Vermeer
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Made by the same painter?
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Hans Van Meegeren (1889-1947)
A story of art, math, and physics
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Hans Van Meegeren
In May 1945, shortly after the liberation of
Holland, two officers arrived at the studio of
van Meegeren, then just a little-known Dutch
painter and art dealer. The officers, from the
Allied Art Commission, were responsible for
repatriating works of art looted by the Nazis.
They had come about a painting discovered among
the collection of Hermann Göring a hitherto
unknown canvas by the great Johannes Vermeer,
entitled The Adultdress.
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Hans Van Meegeren
Since the Nazis had kept detailed records, it had
been easy to trace the sale of the painting back
to van Meegeren. Now, they wanted only the name
of the original owner so that they might return
his priceless masterpiece. When van Meegeren
refused to name the owner, they arrested him and
charged him with treason. If found guilty, he
faced the death penalty. In order to save himself
from serving a long sentence for collaboration
with the Nazis, he pleaded guilty to the lesser
crime of forgery for The Adultress and other
authentic Vermeers.
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Hans Van Meegeren
Van Meegeren invented an ageing process which was
nearly indistinguishable from the real thing.  He
ground his pigments in oil of lilacs and then
mixed them with a special medium.  This was
phenol formaldehyde resin dissolved in either
benzene or turpentine.  The painting was then
baked for several hours at a temperature over
100C.  The result was a paint film which had all
the characteristics of a genuine 17th century
painting.
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Hans Van Meegeren
His best fake paint is Christ and Disciples at
Emmaeus
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Hans Van Meegeren
Prof. Bredius had theorized in print that large
early Vermeers on religious themes might yet turn
up. Accordingly, van Meegeren painted a work
which exactly fulfilled Brediuss scholarly
hypothesis. Other art historians had also
suggested that Vermeer had early in his life
traveled in Italy, and on this count too the
Emmaeus canvas, which showed possible influence
of Carravagio, seemed to confirm an academic
conjecture.
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Hans Van Meegeren
Abraham Bredius was one of the most
authoritative art historians who had dedicated a
great part of his life to the study of Vermeer.
Shortly after having viewed the painting, the 83
year old art historian wrote the Burlington
Magazine, the "art bible" of the times "It is a
wonderful moment in the life of a lover of art
when he finds himself suddenly confronted with a
hitherto unknown painting by a great master,
untouched, on the original canvas, and without
any restoration, just as it left the painter's
studio! And what a picture! we have a ? I am
inclined to say the masterpiece of Johannes
Vermeer of Delft...." No doubts were
advanced since Bredius' opinion was taken as
gospel in the art world so much that he had been
nick-named "the Pope."
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Hans Van Meegeren
In 1947 the trial took place and in order to
demonstrate his case it was arranged that, before
the court under police guard, he would paint
another "Vermeer", Jesus Among the Doctors, using
the materials and techniques he had used for the
other forgeries. During the incredible two year
trial Van Meegeren had confessed that "spurred by
the disappointment of receiving no
acknowledgements from artists and critics....I
determined to prove my worth as a painter by
making a perfect 17th century canvas."
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Hans Van Meegeren
At the end of the trial collaboration charges
were changed to forgery and Van Meegeren was
condemned to one year in confinement. However
many art lovers remained unconvinced that Emmaeus
had been created by the same artist that
created the rest fakes since it was of greater
quality. The final proof was given in 1967 by
scientists by at Carnegie Mellon University
How did they answer this question with
certainty?
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Radioactivity

• Radioactive elements decay. This means that they
  change to something else and therefore they
  decrease in number as time passes by.
• Let N(t) be the number of atoms in time t. We
  write N(t0)N0 for simplicity. The quantity
• is the rate of the change during the interval
  t0,t. In the limit of a zero interval, this
  rate becomes the instantaneous rate of change
  dN/dt

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Radioactivity
In a small interval dt, the number of atoms that
decay dN must be proportional to dt. That is, the
smaller the dt, the smaller the dN. Also, dN
must be proportional to N. That is, the more the
atoms, the more of them decay. Lets call the
proportionality constant ? ?known as the decay
constant. Then dN - ? N dt The
minus is present since dN is negative and dt is
positive.
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Radioactivity
Which function N(t) has the above
differential? Answer
The half-life is also a useful quantity. The
half-life is defined as the time interval during
which half of a given number of radioactive atoms
decay. It relates to ? by
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Dating
Using the formula
we can date objects that contain radioactive
elements
by measuring N(t). Of course this assumes that
we know N0 (hard) and ? (easy) .
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Carbon-14
Carbon-14 is radioactive, with a half-life of
about 5,730 years.
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Carbon-14