Chapter 9
Calculus
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What is Calculus?
Early Results on Areas and Volumes
Maxima, Minima, and Tangents
The “Arithmetica Infinitorum” of Wallis
Newton’s Calculus of Series
The Calculus of Leibniz
Biographical Notes: Wallis, Newton, and Leibniz
9.1
What is Calculus?
• Calculus appeared in 17th century as a system of shortcuts
to results obtained by the method of exhaustion
• Calculus derives rules for calculations
• Problems, solved by calculus include finding areas,
volumes (integral calculus), tangents, normals and
curvatures (differential calculus) and summing of infinite
series
• This makes calculus applicable in a wide variety of areas
inside and outside mathematics
• In traditional approach (method of exhaustions) areas and
volumes were computed using subtle geometric arguments
• In calculus this was replaced by the set of rules for
calculations
17th century calculus
• Differentiation and integration of powers of x (including fractional
powers) and implicit differentiation of polynomials in x and y
• Together with analytic geometry this made possible to find tangents,
maxima and minima of all algebraic curves p (x,y) = 0
• Newton’s calculus of infinite series (1660s) allowed for differentiation
and integration of all functions expressible as power series
• Culmination of 17th century calculus: discovery of the Fundamental
Theorem of Calculus by Newton and Leibniz (independently)
• Features of 17th century calculus:
– the concept of limit was not introduced yet
– use of “indivisibles” or “infinitesimals”
– strong opposition of some well-known philosophers of that time
(e.g. Thomas Hobbes)
– very often new results were conjectured by analogy with previously
discovered formulas and were not rigorously proved
k + (2/n)k + … + (n/n)k](1/n)
• Area ≈ [(1/n)
9.2 Early Results
• → sum 1k + 2k + … + nk
on Areas and Volumes
y = xk
Volume of the solid of
revolution:
area of cross-section is π r2
and therefore it is required to
compute sum
12k + 22k + 32k +… + n2k
1/n
2/n
3/n
(n-1)/n n/n = 1
• First results: Greek mathematicians (method of exhaustion, Archimedes)
• Arab mathematician al-Haytham (10th -11th centuries) summed the series
1k + 2k + … + nk for k = 1, 2, 3, 4 and used the result to find the volume of
the solid obtained by rotating the parabola about its base
• Cavalieri (1635): up to k = 9 and conjectured the formula for positive
integers k
• Another advance made by Cavalieri was introduction of “indivisibles”
which considered areas divided into infinitely thin strips and volumes
divided into infinitely thin slices
• It was preceded by the work of Kepler on the volumes of solids of
revolution (“New Stereometry of wine barrels”, 1615)
• Fermat, Descartes and Roberval (1630s) proved the formula for
integration of xk (even for fractional values of k)
• Torricelly: the solid obtained by rotating y = 1 / x about the x-axis from 1 to
has finite volume!
• Thomas Hobbes (1672): “to understand this [result] for sense, it is not
required that a man should be a geometrician or logician, but that he
should be mad”
9.3 Maxima, Minima, and Tangents
• The idea of differentiation appeared later than that one of
integration
• First result: construction of tangent line to spiral r = aθ by
Archimedes
• No other results until works of Fermat (1629)
f ( x  x)  f ( x)
“modern” approach: lim
x
x 0
Fermat’s approach
(tangent to y = x2)
( x  E ) 2  x 2 2 xE  E 2
slope 

 2x  E
E
E
• E – “small” or “infinitesimal” element which is set equal to
zero at the end of all computations
• Thus at all steps E ≠ 0 and at the end E = 0
• Philosophers of that time did not like such approach
• Fermat’s method worked well with all
polynomials p(x)
• Moreover, Fermat extended this approach to
curves given by p(x,y) = 0
• Completely the latter problem was solved by
Sluse (1655) and Hudde (1657)
• The formula is equivalent
to the use of
implicit differentiation
p ( x, y ) 
m,n
i j
a
x
 ij y
i , j 1

i 1 j
ia
x
 ij y
dy

i j 1
dx
ja
x
 ij y
9.4 The “Arithmetica Infinitorum” of
Wallis (1655)
• An attempt to arithmetize the theory of areas and volumes
1
p
• Wallis found that ∫0 x dx = 1/(p+1) for positive integers p (which
was already known)
1
m/n
• Another achievement: formula for ∫0 x
1
1/2
1
dx
1/3
• Wallis calculated ∫0 x dx, ∫0 x dx,…, using geometric
arguments, and conjectured the general formula for fractional p
• Note: observing a pattern for p = 1,2,3, Wallis claimed a formula
1
for all positive p “by induction” and for fractional p “by
y = x2 (lack of rigour but a great deal of analogy, intuition
interpolation”
and ingenuity)
1 2
∫0 x dx
= 1/3 1
1
1/2
∫0 x dx = 1 - 1/3 = 2/3
• Wallis’ formula:

2 4 4 6 6
     
4 3 3 5 5 7
• Expansion of π as infinite product was known to Viète
(before Wallis’ discovery):
1 1
1 1
1
1 
1 

1 
  
1 
 cos  cos  cos  





4
8
16
2 2
2 2
2
2  


2



• Nevertheless Wallis’ formula relates π to the integers
through a sequence of rational operations
• Moreover, basing on the formula for π Wallis’ found a
sequence of fractions he called “hypergeometric”, which
as it had been found later occur as coefficients in series
expansions of many functions (which led to the class of
hypergeometric functions)
Other formulas for π related to
Wallis’ formula
12
 1
32

2
52
2
72
2
2 
4
Continued fraction
(Brouncker):
Series expansion discovered by 15th century Indian
Euler
mathematicians and rediscovered by Newton,
Gregory and Leibniz:
3
5
7

1
1
1
x
x
x
 1   
tan 1 x  x      sub. x = 1
4
3 5 7
3 5 7
9.5 Newton’s Calculus of Series
• Isaac Newton
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Most important discoveries in 1665/6
Before he studied the works of Descartes, Viète and Wallis
Contributions to differential calculus (e.g. the chain rule)
Most significant contributions are related to the theory of
infinite series
– Newton used term-by-term integration and differentiation to find
power series representation of many of classical functions, such
as tan-1x or log (x+1)
– Moreover, Newton developed a method of inverting infinite
power series to find inverses of functions (e.g ex from log (x+1))
• Unfortunately, Newton’s works were rejected for publication
by Royal Society and Cambridge University Press
9.6 The Calculus of Leibniz
• The first published paper on calculus was by
Gottfried Wilhelm Leibniz (1684)
• Leibniz discovered calculus independently
• He had better notations than Newton’s
• Leibniz was a librarian, a philosopher and a
diplomat
• “Nova methodus” (1864)
– sum, product and quotient rules
– notation dy / dx
• dy / dx was understood by Leibniz literally as a
quotient of infinitesimals dy and dx
• dy and dx were viewed as increments of x and y
The Fundamental Theorem of Calculus
• In “De geometria” (1686) Leibniz introduced
the integral sign ∫
• ∫ f(x) dx meant (for Leibniz) a sum of terms representing
infinitesimal areas of height f(x) and width dx
• If one applies the difference operator d to such sum it
yields the last term f(x) dx
• Dividing by dx we obtain the FTC:
d
f ( x)dx  f ( x)

dx
• Leibniz introduced the word “function”
• He preferred “closed-form” expressions to
infinite series
• Evaluation of integral ∫ f(x) dx was for Leibniz
the problem of finding a known function whose
derivative is f(x)
• The search for closed forms led to
– the problem of factorization of polynomials and
eventually to the Fundamental Theorem of Algebra
(integration of rational functions)
– the theory of elliptic functions
(attempts to integrate 1/√1-x4 )
9.7 Biographical Notes:
Wallis, Newton, and Leibniz
John Wallis
Born: 23 Nov 1616 (Ashford, Kent, England)
Died: 28 Oct 1703 (Oxford, England)
• went to school in Ashford
• Wallis’ academic talent was recognized very early
• 14 years old he was sent to Felsted, Essex to attend the
school
• He became proficient in Latin, Greek and Hebrew
• Mathematics was not considered important in the best
schools
• Wallis learned rules of arithmetic from his brother
• That time mathematics was not consider as a “pure”
science in the Western culture
• In 1632 he entered Emmanuel College in Cambridge
• bachelor of arts degree (topics studied included ethics,
metaphysics, geography, astronomy, medicine and
anatomy)
• Wallis received his Master's Degree in 1640
• Between 1642 and 1644 he was chaplain at Hedingham,
Essex and in London
• Wallis became a fellow of Queens College, Cambridge
• He relinquished the fellowship when he married in 1645
• Wallis was interested in cryptography
• Civil War between the Royalists and Parliamentarians
began in 1642
• Wallis used his skills in cryptography in decoding Royalist
messages for the Parliamentarians
• Since the appointment to the Savilian Chair in Geometry
of Oxford in 1649 by Cromwell Wallis actively worked in
mathematics
Sir Isaac Newton
Born: 4 Jan 1643 (Woolsthorpe, Lincolnshire, England)
Died: 31 March 1727 (London, England)
• A family of farmers
• Newton’s father (also Isaac Newton) was a
wealthy but completely illiterate man who
even could not sign his own name
• He died three months before his son was
born
• Young Newton was abandoned by his
mother at the age of three and was left in
the care of his grandmother
• Newton’s childhood was not happy at all
• Newton entered Trinity College
(Cambridge) in 1661
• Newton entered Trinity College (Cambridge)
in 1661 to pursue a law degree
• Despite the fact that his mother was a
wealthy lady he entered as a sizar
• He studied philosophy of Aristotle
• Newton was impressed by works of
Descartes
• In his notes “Quaestiones quaedam
philosophicae” 1664 (Certain philosophical
questions) Newton recorded his thoughts
related to mechanics, optics, and the
physiology of vision
• The years 1664 – 66 were the most important in
Newton’s mathematical development
• By 1664 he became familiar with mathematical
works of Descartes, Viète and Wallis and began
his own investigations
• He received his bachelor's degree in 1665
• When the University was closed in the summer of
1665 because of the plague in England, Newton
had to return to Lincolnshire
• At that time Newton completely devoted himself to
mathematics
• Newton’s fundamental works on calculus “A
treatise of the methods of series and fluxions”
(1671) (or “De methodis”) and “On analysis by
equations unlimited in their number of terms”
(1669) (or “De analysis”) were rejected for
publication
• Nevertheless some people recognized his genius
• Isaac Barrow resigned the Lucasian Chair
(Cambridge) in 1669 and recommended that
Newton be appointed in his place
• Newton's first work as Lucasian Prof. was on
optics
• In particular, using a glass prism Newton
discovered the spectrum of white light
• 1665: Newton discovered inverse square law of
gravitation
• 1687: “Philosophiae naturalis principia mathematica”
(Mathematical principles of natural philosophy)
• In this work, Newton developed mathematical
foundation of the theory of gravitation
• This book was published by Royal Society (with the
strong support from Edmund Halley)
• In 1693 Newton had a nervous breakdown
• In 1696 he left Cambridge and accepted a government
position in London where he became master of the
Mint in 1699
• In 1703 he was elected president of the Royal Society
and was re-elected each year until his death
• Newton was knighted in 1705 by Queen Anne
Gottfried Wilhelm von Leibniz
Born: 1 July 1646 (Leipzig, Saxony (now Germany)
Died: 14 Nov 1716 (Hannover, Hanover (now Germany)
• An academic family
• From the age of six Leibniz was given free access to
his father’s library
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At the age of seven he entered school in Leipzig
In school he studied Latin
Leibniz had taught himself Latin and Greek by the age of 12
He also studied Aristotle's logic at school
In 1661 Leibniz entered the University of Leipzig
He studied philosophy and mathematics
In 1663 he received a bachelor of law degree for a thesis
“De Principio Individui” (“On the Principle of the Individual”)
• The beginning of the concept of “monad”
• He continued work towards doctorate
• Leibniz received a doctorate degree from University of
Altdorf (1666)
• During his visit to the University of Jena (1663) Leibniz
learned a little of Euclid
• Leibniz idea was to create some “universal logic calculus”
• After receiving his degree Leibniz commenced a legal
career
• From 1672 to 1676 Leibniz developed his ideas related to
calculus and obtained the fundamental theorem
• Leibniz was interested in summation of infinite series by
investigation of the differences between successive terms
• He also used term-by term integration to discover series
representation of π


1
1   1   1 1  1 1 
1
  
 1            


n  1  2   2 3   3 4 
n 1 n( n  1)
n 1  n

1 1 1
1 1
 1 1   11 1    
 1     4  3  5 7     1
 2 2   3 3  4 4 