The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof
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The Fundamental Theorem of Calculus: History, Intui9on, Pedagogy, Proof V. Frederick Rickey West Point
Transcript of The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof
The Fundamental Theorem of Calculus: History, Intui9on, Pedagogy, Proof
V. Frederick Rickey West Point
Go4ried Wilhelm von Leibniz
(1646 – 1716)
From the Law of IdenGty we obtain
• A – A = 0 • A – A + B – B + C – C + D – D + E – E = 0 • A + (-‐A + B) + (-‐B + C) + (-‐C + D) + (-‐D + E) – E = 0 • Chrisening these differences L, M, N and P, we have:
• L + M + N + P = E – A. – “The sum of the differences is the difference between the first term and the last”
An Easy Example
0 1 4 9 16 25
1 3 5 7 9
So 1+ 3 + 5 + 7 + 9 = 25 – 0.
The sum of the first n odd numbers is n2.
ChrisGaan Huygens asked for the sum of the reciprocals of the
triangular numbers, n (n+1)/2
1 1/2 1/3 1/4 1/5 1/6 1/2 1/6 1/12 1/20 1/30
So 1/2 + 1/6 + 1/12 + 1/20 + 1/30 + . . . = 1 – 0
Leibniz doubled this and got his result.
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Leibniz’s ConcepGon of the Integral
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y∫ dx
The area is a sum of rectangles of height y and infinitesimal width dx.
The integral sign is a long s which stands for sum.
The Sideways Chalk Model
• This is the intuiGon that Leibniz had for the Fundamental Theorem of the Calculus
• It is clear how to differenGate an integral.
The Nova methodus of 1684 – the first paper on the differenGal calculus.
Isaac Newton 1642 -‐ 1727
• 1702 portrait by Kneller
• The original is in the NaGonal Portrait Gallery in London
Took Descartes’s Geometry in hand, tho he had been told it would be very difficult, read some ten pages in it, then stopt, began again, went a liile farther than the first Gme, stopt again, went back again to the beginning, read on Gl by degrees he made himself master of the whole, to that degree that he understood Descartes’s Geometry beier than he had done Euclid.
Descartes’s Geometry, 1637, 1659
Descartes adopted Aristotle’s dictum
The raGos between straight and curved lines are not known, and I believe cannot be discovered by human minds.
La proporGon, qui est entre les droites & les courbes, n’estant pas connuë, & mesme ie croy ne le pouuant estre par les hommes.
La Geometrie (1637), 340.
van Heuraet on Arc Length, 1659
van Heuraet’s recGficaGon, 1659 Define CI, the auxiliary curve, by
Then
Thus arc length is reduced
to area!
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dsdx
=MIk
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k ds∫ = MI dx∫
RecGficaGon Destroyed
• Aristotle’s dictum and • Descartes’ program
• But the story ends well.
The Fundamental Theorem of Calculus
A Method whereby to square such crooked lines as may be squared.
• van Heuraet swapped arc length for area
• Newton swapped area for a tangent
For Newton
• MathemaGcal quanGGes are described by ConGnuous MoGon.
– E.g., Curves are generated by moving points
• In Modern Terms: All variables are funcGons of Gme
• Newton said that quanGGes flow, and so called them fluents.
• How fast they flow – or flex – he called fluxions.
• As Bourbaki would write, “Par abuse de langu,”
d/dt ( fluent ) = fluxion
Intuition: Windshield wipers
Given an equation involving any number of fluent quantities to find the fluxions and vice versa.
In 1700 there were two intuiGons for The Fundamental Theorem of Calculus
Leibniz:
Sideways Chalk Model
Newton:
Windshield Wiper Model
The Isochrone Problem
• Find a curve along which a body will descend equal distances in equal Gmes
• Johann Bernoulli reduces it to the DifferenGal EquaGon √a dx = √y dy.
• Et eorum integralia !
• The curve is a semi-‐cubical parabola,
y3 = 9/4 a x2
Johann Bernoulli in 1743 Son Esprit vit la vérité
Et son coeur connait la jusGce Il a fait l’honneur de la Suisse Et celui de l’humanité
His spirit lives for truth His heart knows jusGce
He is an honor to the Swiss And to all of humanity
• Voltaire
We seek the curve where the square of the ordinate BC is the mean proporGonal between the square of the given length E and the curvilinear figure ABC.
E2 / BC2 = BC2 / Area ABC
Area ABC = y4 / a2
By FTC, y dx = 4 y3 dy / a2
Divide by y and integrate
To get a cubical parabola
y = (stuff) x3
Let BC = y and E = a.
Johann Bernoulli’s definiGon of an integral
We have previously shown how to find the differenGal of a given quanGty. Now we show inversely how to find the integral of a differenGal, i.e., find the quanGty from which the differenGal originates.
E2 / BC2 = BC2 / Area ABC
Area ABC = y4 / a2
By FTC, y dx = 4 y3 dy / a2
Divide by y and integrate To get a cubical parabola
I misread this text. Bernoulli does NOT use FTC but only the notion that an integral is an antiderivative. He uses a definition, not a theorem.
• Johann Bernoulli’s best student !
• Leonhard Euler
• 1707-‐1783
Euler about 1737, age 30
• PainGng by J. Brucker • 1737 mezzoGnt by Sokolov
• Black below and above right eye
• Fluid around eye is infected
• “Eye will shrink and become a raisin”
• Ask your ophthalmologist
• Thanks to Florence Fasanelli
Euler’s Calculus Books
• 1748 IntroducGo in analysin infinitorum 399
402
• 1755 InsGtuGones calculi differenGalis 676
• 1768 InsGtuGones calculi integralis 462
542
508
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2982
• Defines the integral as an anGderivaGve.
• Gives a careful discussion of approximaGng a definite integral with a sum of rectangles.
Read Euler, read Euler, he is our teacher in everything.
Laplace as quoted by Libri, 1846
Lisez Euler, lisez Euler, c'est notre maître à tous.
In the 18th century there was no FTC.
This famous work of 1821 began to introduce rigor into the calculus by defining limits, conGnuity and derivaGves and proving theorems about them.
It was never used as a text.
AugusGn Cauchy 1789 -‐ 1857
AugusGn Cauchy 1823
In his Résumé of 1823, Cauchy
• Gave a careful definiGon as the limit of a sum of areas of rectangles (evaluated at lew endpoints).
• Proved that the integral of a conGnuous funcGon exists.
• Proved the First FTC in a rigorous way.
• Cauchy’s definiGon of the integral is a radical break with the past!
• Euler used lew sums.
• Lacroix and Poisson tried to prove the sums converge.
• Fourier needed to think of the definite integral as an area.
Rosenstein and Temelli, 2001
At the end of the 19th century, authors had two choices regarding the introduction of the integral:
Either one might define the integral as the limit of a certain sum, or, alternatively,
integration is the inverse operation to differentiation.
If the former is chosen, then one must justify some form of the Fundamental Theorem of Calculus;
if the latter, then the use of the integral in applications becomes the sticking point.
The Name “FTC” in Research Monographs.
• Eduard Goursat uses the term “Fundamental Theorem of Calculus“ in his Cours d'analyse mathema;ques (1902).
• Ernest W. Hobson in his Theory of Func;ons of a Real Variable (1907) has a chapter enGtled “The fundamental theorem of the integral calculus.”
• Vallee Poussin in his Cours d'analyse innitesimale (1921) uses the name “relaGon fondamentale pour le calcul des integrals "
William Anthony Granville
• Text used at West Point 1907-‐1948 and 1953-‐1963
Granville 1911
• Granville and Smith define indefinite integraGon as anGdifferentaGon.
• The definite integral is defined as F(b) – F(a), where F’(x) = f(x).
• Thus there is no FTC in our modern sense.
• They use FTC in the sense of du Bois-‐Reymond, 1876, 1880.
The Name “FTC” in Textbooks
• Björling (1877) gives the name “Grundsats” to the second fundamental theorem in a Swedish textbook
• G. H. Hardy, A Course of Pure Mathema;cs (1908), uses the phrase and provides a proof.
• George Thomas uses the phrase in his Calculus (1951).
Randolph and Kac, 1946
First use of the name 2nd FTC, 1958
Why is the FTC “Fundamental”?
It connects the two approaches to the integral:
A summaGon An anG-‐derivaGve
The first is vital for approximaGng (mostly physical) problems.
The second is vital for evaluaGng the resulGng integrals.
Leibniz proves FTC in 1690
