So You Think You

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So You Think Youre Educated, But You Dont Know
Calculus

• A brief introduction to one of humanitys
  greatest inventions

Michael Z. Spivey Department of Mathematics and
Computer Science Samford University December 1,
2004
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What is Calculus?

• Calculus is the mathematics of change.
• It has two main branches
• Differential calculus
• Involves calculating rates of change from
  functions
• Integral calculus
• Involves determining a function given information
  about its rate of change

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Outline of Talk

• The intellectual progression from arithmetic to
  algebra to calculus
• Ideas that led to the development of calculus
• The tangent line problem
• The area problem
• The Cartesian coordinate system
• Historical side note
• What, exactly, is calculus?
• Differentiation
• Integration
• The Fundamental Theorem of Calculus
• The impact of calculus
• Immediate applications
• Impact on Western thought and contribution to the
  Enlightenment
• Calculus today

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Arithmetic

• With arithmetic, the unknown always occurs at the
  end of the problem.
• Example 357 982 ?

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Algebra

• With algebra, the unknown can be incorporated at
  the beginning of the problem.
• Example. We know that x satisfies the following
  relationship x2 3x 4 0. Find x.

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Calculus

• With calculus, the unknown can be incorporated at
  the beginning of the problem, and it can be
  allowed to change.
• Example. We know that x changes according to the
  following rule

Find a formula giving x at any time t.
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Ideas That Led to the Development of Calculus,
Part I The Tangent Line Problem

• It takes two points to determine a line.
• So, if we have two points on a curve, then we can
  determine the line through those points.
• In particular, we can determine the slope of the
  line.

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Ideas That Led to the Development of Calculus,
Part I The Tangent Line Problem

• But how do we determine the slope of the tangent
  line?
• The problem is that theres only one point.

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Ideas That Led to the Development of Calculus,
Part I The Tangent Line Problem

• The Greeks had solved the tangent line problem
  for a whole host of geometrical shapes, including
  the circle, the ellipse, and various spirals.
• But the problem remained Is there a general
  method for finding the slope of a tangent line
  i.e., a method that will work on any curve?

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Ideas That Led to the Development of Calculus,
Part II The Area Problem

• How do you find the area enclosed by a curve?
• Again, the Greeks had solved the area problem for
  a whole host of geometrical shapes, including the
  circle and the ellipse.
• Example The area of a circle is given by A
  pr2, where r is the circles radius.

r
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Ideas That Led to the Development of Calculus,
Part II The Area Problem

• Is there a general method for finding the area
  enclosed by curves?

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Ideas That Led to the Development of Calculus,
Part III The Invention of the Cartesian
Coordinate System

• Until the 17th century, algebra and geometry were
  considered two separate branches of mathematics.
• The use of a coordinate system shows how the two
  are related, though
• The set of all points that satisfy an algebraic
  equation determines a curve.
• And any curve determines an algebraic expression.
• The invention of the Cartesian coordinate system
  is credited to Descartes and Fermat and is named
  after Descartes.

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Ideas That Led to the Development of Calculus,
Part III The Invention of the Cartesian
Coordinate System

• The Cartesian coordinate system also helps show
  why the tangent line problem is so important.
• The slope of the tangent line measures the rate
  of change of the curve.
• In other words, the slope of the tangent line
  measures how much y changes as x changes.
• In particular, if x represents time, then the
  slope of the tangent line tells us how fast y is
  moving.

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The Invention (Discovery?) of Calculus

• Using the Cartesian coordinate system as a tool,
  and building on the work of others, Newton and
  Leibniz separately solved both the tangent line
  problem and the area problem.
• The tools to solve the tangent line problem and
  related problems are the differential calculus.
• The tools to solve the area problem and related
  problems are the integral calculus.
• Their great accomplishment, though, was to show
  that the tangent line problem and the area
  problem are, in some sense, actually inverses of
  each other!

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Historical Side Note The Controversy

• There was a lot of fighting among the scientific
  community in the late 1600s and early 1700s over
  who invented calculus first Newton or Leibniz.
• We now know that Newton invented calculus in 1665
  but didnt publish his results until 1704, in
  the appendix to his Optiks.
• Interestingly enough, calculus is not in the
  Principia Mathematica, published in 1687.
• Newton used calculus to achieve his scientific
  results published in the Principia, but in the
  book itself he used geometrical arguments, not
  calculus, to justify his mathematical claims.
• Leibniz invented calculus in 1673 but didnt
  publish his results until 1684.
• So Newton invented it first, but Leibniz
  published first.
• Newtons supporters won the fight, and so today
  we give Newton the credit for inventing calculus.
• Historical ironies
• We use Leibnizs notation today when we teach
  calculus, not Newtons.
• Leibnizs superior notation and treatment of the
  subject led to a flourishing of scientific
  applications of calculus on the European
  continent, while British science after Newton
  languished.

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Differentiation

• Remember that the slope of the tangent line to a
  graph of y versus x is just a way of expressing
  how much y changes as x changes.
• So the tangent line problem is simply a prototype
  for the more general problem of finding rates of
  change.
• The process of finding rates of change is called
  differentiation.

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Differentiation Notation

• The notation for the rate of change of a quantity
  as x changes is
• So, to express how the quantity y changes as x
  changes we write
• This is actually Leibnizs notation.

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Differentiation

• There is a very nice technique for finding the
  rates of change of all of the most
  commonly-encountered functions, such as
• Polynomial functions
• Rational functions
• Trigonometric functions
• Exponential functions
• Logarithmic functions
• Most of Calculus I involves learning this
  technique and how to apply it to different kinds
  of problems.

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Integration

• Solving the area problem involves the process of
  integration.
• The integration notation is as follows.
• Let f(x) be the height of the curve forming the
  top boundary of the enclosed area to the right.
• Then the area under the curve from a to b is
  denoted

f(x)
a
b
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The Fundamental Theorem of Calculus

• Unfortunately, there are not any nice, direct
  integration techniques like the one for
  differentiation.
• The great accomplishment of Newton and Leibniz
  was to realize the truth of what we call the
  Fundamental Theorem of Calculus
• Differentiation and integration are essentially
  inverse processes.
• This means that the area problem can be solved by
  using the differentiation technique backwards.

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The Fundamental Theorem of Calculus

• Let the upper bound on the region be variable
  call it x.
• If I make a tiny increase in x, Im adding a thin
  rectangle to the area of the region.
• The area of that rectangle is height of the
  rectangle, f(x), times the change in x.
• Mathematically, we write
• What this means, then, is that the rate at which
  the area of the region changes as x changes is
  f(x).
• In mathematical notation, this last statement is
  expressed as

f(x)
a
x
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The Fundamental Theorem of Calculus

• Looking more closely at what we have here, we see
  that if we integrate a function from a to x and
  then differentiate it with respect to x, we get
  back the original function.
• So differentiation and integration are inverse
  processes!
• And so the area problem can be solved by doing
  differentiation backwards!

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Immediate Applications of Calculus

• The tangent line problem is, as weve said, a
  prototype problem for anything involving a rate
  of change, including
• Velocity
• Acceleration
• The area problem is also a prototype problem for
  a whole host of other problems, including
• Volume
• Mass and center of mass
• Work

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Calculus and Mechanics

• With calculus as his tool, Newton was able to use
  his theory of gravity to solve and place into one
  theoretical framework nearly all of the
  outstanding problems in terrestrial and celestial
  mechanics.
• Basically, he was able to explain why nearly
  everything on earth and in space moved the way it
  did.
• For example, he was able to
• Give a theoretical justification for Keplers
  prediction of the elliptical orbits of the
  planets
• Explain the movements of the comets
• Explain why tides occur
• Describe the motion of pendulums
• Newton singlehandedly completed the scientific
  revolution.
• http//www.phy.hr/dpaar/fizicari/xnewton.html

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• Nature and Natures laws
• Lay hid in night
• God said, Let Newton be!
• And all was light.
• - Alexander Pope

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The Impact on Western Thought

• Newton and Leibniz had invented a new
  mathematical tool, based on a relationship that
  no one had noticed before.
• Newton then used that tool and his theory of
  gravity to explain the motion of nearly
  everything on earth and in the heavens.
• What are the implications of this?

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Implication 1 We dont need God to explain why
things move.

• If gravity can explain why the planets move the
  way that they do, we dont have to posit a God
  that keeps them in motion.
• However, gravity doesnt explain why the planets
  started moving in the first place only why they
  stay in motion.
• So we need to presuppose God in order to explain
  why the planets first started moving.
• Possible conclusion
• Maybe thats all that God does.
• He doesnt interact with His creation.
• He creates, starts things moving, and then steps
  back to watch.
• This is Deism the idea of a Watchmaker God.

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Implication 2 We can use mathematics to model
the universe.

• While many scientists before Newton had used
  mathematics in their work, much of the science up
  to Newtons time was descriptive, not
  quantitative.
• For example, Aristotle said that every object has
  a natural resting place. Fire naturally wants to
  be in the heavens, and stones naturally want to
  be near the center of the earth.
• Theres no mathematics in this.
• Calculus was so successful as a tool for solving
  physical problems that it revolutionized the way
  science was done.
• Since Newton, mathematics has been the language
  of science since Newton, we have used
  mathematics to model the universe.

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Implication 3 Why should these discoveries stop?

• Calculus was extremely powerful at solving
  problems in mechanics.
• Scientists who came after Newton were able to use
  calculus on many other problems as well, such as
• Motion of a spring
• Fluid force and fluid flow
• Discoveries were being made in other areas of
  science, too.
• If we can solve so many problems now, why cant
  we continue to solve scientific problems?

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Implication 4 Maybe there is a calculus for
other fields of knowledge.

• Maybe other areas of human life have their own
  calculus, too.
• Maybe each field has its own fundamental
  principle that explains everything in it.
• If this is true, and we can figure out what these
  principles are, then we can solve all of
  societys problems in areas such as
• Ethics
• Economics
• Government
• History
• Philosophy

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Calculus Today

• Calculus isnt starting any more intellectual
  revolutions today.
• But its still being taught.
• Its still the most powerful mathematical tool in
  existence.
• Its still the basis for much of modern science.

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Some Modern Applications of Calculus

• Population modeling (ecology)
• If we can describe the growth rate of a
  population, we can use calculus to find a formula
  for the population at any time.
• Marginal analysis (economics)
• The additional cost of making one more item is
  the rate of change of the cost with respect to
  the number of items made.
• Weather forecasting (meteorology)
• Historically, this is the origin of the study of
  chaos.
• Determining spaceship orbits and re-entry (space
  exploration)