Transcript So: You Think You’re Educated, but

So You Think You’re Educated,
But You Don’t Know Calculus
A brief introduction to one of
humanity’s greatest inventions
Michael Z. Spivey
Department of Mathematics and Computer Science
Samford University
December 1, 2004
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
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
Arithmetic
• With arithmetic, the unknown always
occurs at the end of the problem.
• Example: 357 + 982 = ?
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.
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:
dx
 5x
dt
Find a formula giving x at any time t.
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.
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
there’s only one point.
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?
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 = πr2, where r
is the circle’s radius.
r
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?
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.
x2  y 2  1
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.
y  x3  3
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!
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 didn’t 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 didn’t publish his results until
1684.
– So Newton invented it first, but Leibniz published first.
• Newton’s supporters won the fight, and so today we give Newton the
credit for inventing calculus.
• Historical ironies
– We use Leibniz’s notation today when we teach calculus, not Newton’s.
– Leibniz’s 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.
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.
Differentiation Notation
• The notation for the rate of change of a
quantity as x changes is d
dx
• So, to express how the quantity y changes as x
changes we write
d
dy
( y ), or
dx
dx
• This is actually Leibniz’s notation.
Differentiation
• There is a very nice technique for finding the
rates of change of all of the most commonlyencountered 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.
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: b
 f ( x )dx
a
f(x)
a
b
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.
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,
I’m 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:
Area  f ( x)x
• 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:
x

d 
 f ( x)dx   f ( x)
dx  a

f(x)
a
x
The Fundamental Theorem of
Calculus
x


d
  f ( x)dx   f ( x)
dx  a

• 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!
Immediate Applications of Calculus
• The tangent line problem is, as we’ve 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
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 Kepler’s 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
Nature and Nature’s laws
Lay hid in night;
God said, “Let Newton be!”
And all was light.
- Alexander Pope
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?
Implication #1: We don’t need God to
explain why things move.
• If gravity can explain why the planets move the way that
they do, we don’t have to posit a God that keeps them in
motion.
• However, gravity doesn’t 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 that’s all that God does.
He doesn’t interact with His creation.
He creates, starts things moving, and then steps back to watch.
This is Deism – the idea of a Watchmaker God.
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
Newton’s 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.
– There’s 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.
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
can’t we continue to solve scientific problems?
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 society’s problems in areas
such as
–
–
–
–
–
Ethics
Economics
Government
History
Philosophy
Calculus Today
• Calculus isn’t starting any more intellectual
revolutions today.
• But it’s still being taught.
– It’s still the most powerful mathematical tool in
existence.
– It’s still the basis for much of modern science.
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)