MATH1014 Calculus II. A historical review on Calculus

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1 MATH1014 Calculus II A historical review on Calculus Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology September 4, 2015

2 Instantaneous Velocities Newton s paradox Limits

3 Before Newtwon The need to investigate dynamical problems in the 18th century verses static problems in the past was strongly related to the cultural and economics developments at that time. It was Galilei Galileo ( ), called the father of sciences who headed the Scientific Revolution in the 17th century advocating beliefs should be built upon experiments and mathematics and that Philosophy is written in this grand book, the universe... It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures;... (Wiki) He showed that the velocity of a falling body only depends on its mass and has nothing to do with its shape and size He invented telescope and use it to discover the four largest satellites of the planet Jupiter, etc

4 Galilei Galileo ( ) Figure : (Portrait in crayon by Leoni (source Wiki))

5 Galilei Galileo ( ) Italian physicist Advocates heliocentricism instead of geocentrism He was investigated by the Roman Inquisition in 1615, that it could be supported as a possibility, but not as an established fact. Dialogue Concerning the Two Chief World Systems. It s content is about discussions amongst two philosophers and a layman. Under house arrest (1633). Wrote Two New Sciences where he show how mathematics should be used to deal with kinematics and strength of materials. Einstein (1954). Propositions arrived at by purely logical means are completely empty as regards reality. Because Galileo realised this, and particularly because he drummed it into the scientific world, he is the father of modern physics indeed, of modern science altogether.

6 Descartes ( ) Figure : (Wikipedia))

7 Descartes s influences French philosopher, mathematician, and scientist Cogito ergo sum: I think therefore I am. Meditations on First Philosophy is still used as a standard text at most university philosophy departments. Mathematics: Cartesian coordinate system unites Greeks geometry and algebra. Shifting the debate from what is true to of what can I be certain? Descartes shifted the authority of truth from God to humanity.

8 Galileo s philosophy (a) Figure : (Kline Vol. I, p.330))

9 Galileo s philosophy (b) Figure : (Kline Vol. I, p.330))

10 Galileo s philosophy (c) Figure : (M. Kline, Mathematical Thoughts from Ancient to Modern Times, Vol. I, Oxford Univ. Press p.329))

11 Newton s role As M. Kline puts (1972) Great advances in mathematics and science are almost always built on the work of many men who contribute bit by bit over hundreds of years; eventually one man sharp enough to distinguish the valuable ideas of his predecessors from the welter (muddle) of suggestions and pronouncements, imaginative enough to fit the bits into a new account, and audacious (brave) enough to build a master plan takes the culminating and definitive step. In the case of Calculus, this was Newton. In fact, G. W. Leibniz ( ) also invented Calculus. Our calculus notation is also due to Leibniz. Newton s success is built upon Galileo s philosophy and on Kepler s experimental laws.

12 Sir Issac Newton ( ) Figure : (1689 by Sir Godfrey Kneller (Newton Institute))

13 G. W. Leibniz ( ) Figure : Gottfried Wilhelm Leibniz (Wikipedia)

14 Four major types of problems Given the formula of for the distance of an object as a function of time, to find the speed and acceleration of the object, and the converse problem, given the acceleration of the object, to recover the speed and distance. Finding tangents to a curve (e.g.reflections through a lens, direction of a moving body). In fact, even the very definition of a tangent is in question. Finding maximum and minimum values of a function. Finding lengths of curves, areas bounded by curves and volumes bounded by surfaces, gravitational attraction between planets. The Greeks use methods of exhaustion on relatively simple geometric areas and volumes. Their methods work on ad hoc base (with much ingenuity) that lacks generality.

15 Four major types of problems Given the formula of for the distance of an object as a function of time, to find the speed and acceleration of the object, and the converse problem, given the acceleration of the object, to recover the speed and distance. Finding tangents to a curve (e.g.reflections through a lens, direction of a moving body). In fact, even the very definition of a tangent is in question. Finding maximum and minimum values of a function. Finding lengths of curves, areas bounded by curves and volumes bounded by surfaces, gravitational attraction between planets. The Greeks use methods of exhaustion on relatively simple geometric areas and volumes. Their methods work on ad hoc base (with much ingenuity) that lacks generality.

16 Four major types of problems Given the formula of for the distance of an object as a function of time, to find the speed and acceleration of the object, and the converse problem, given the acceleration of the object, to recover the speed and distance. Finding tangents to a curve (e.g.reflections through a lens, direction of a moving body). In fact, even the very definition of a tangent is in question. Finding maximum and minimum values of a function. Finding lengths of curves, areas bounded by curves and volumes bounded by surfaces, gravitational attraction between planets. The Greeks use methods of exhaustion on relatively simple geometric areas and volumes. Their methods work on ad hoc base (with much ingenuity) that lacks generality.

17 Four major types of problems Given the formula of for the distance of an object as a function of time, to find the speed and acceleration of the object, and the converse problem, given the acceleration of the object, to recover the speed and distance. Finding tangents to a curve (e.g.reflections through a lens, direction of a moving body). In fact, even the very definition of a tangent is in question. Finding maximum and minimum values of a function. Finding lengths of curves, areas bounded by curves and volumes bounded by surfaces, gravitational attraction between planets. The Greeks use methods of exhaustion on relatively simple geometric areas and volumes. Their methods work on ad hoc base (with much ingenuity) that lacks generality.

18 Four major types of problems Given the formula of for the distance of an object as a function of time, to find the speed and acceleration of the object, and the converse problem, given the acceleration of the object, to recover the speed and distance. Finding tangents to a curve (e.g.reflections through a lens, direction of a moving body). In fact, even the very definition of a tangent is in question. Finding maximum and minimum values of a function. Finding lengths of curves, areas bounded by curves and volumes bounded by surfaces, gravitational attraction between planets. The Greeks use methods of exhaustion on relatively simple geometric areas and volumes. Their methods work on ad hoc base (with much ingenuity) that lacks generality.

19 Continuity against discreteness Even when a body with distance function S(t) is moving with non-uniform motion between two time S(t 1 ) and S(t 2 ) (t 2 > t 1 ), one can measure average velocity (the common usage speed has no direction): average velocity = distance travelled time taken = S(t 2) S(t 1 ) t 2 t 1 which means distance travelled per unit length. When a body starts from rest to reach certain velocity v 0 later, then its (average) velocity must change at different periods. How to describe an object at every instant? In fact, the very meaning of instantaneous velocity is in question. In fact, one should recognise that mathematics, like any other language, is a language that we use to represent our thoughts (understanding) about the nature. But mathematics (our thoughts) is not the nature itself so there is a limit to this language.

20 Newton s infinitesimals 1. For convenience sake, we can say that Newton invented virtual distance and virtual time to measure virtual velocity. That is, virtual velocity = virtual distance virtual time or just instantaneous velocity which is not a clearly defined term (in any sense). Newton calls these infinitesimal. 2. Let S(t) be the distance function and let dt represents an infinitesimal change in t. So ds(t) = S(t + dt) S(t) is an infinitesimal change in S(t) over the time interval dt 3. So Instantaneous velocity = ds dt = S(t + dt) S(t) dt is essentially what Newton interpreted as the instantaneous velocity although the notation ds, dt was due to Leibniz.

21 So do we have dt = 0? If so, then one would have ds dt = 0 0. That was the question that Newton could not answer satisfactorily during his life time. Newton s trouble Suppose an object moves according to the rule S(t) = t 2 where S measures the distance of the object from the initial position t seconds later. We now compute instantaneous velocity of the object at time t: let dt and ds be the virtual time and virtual distance respectively. Then the change of virtual distance is given by ds = S(t + dt) S(t). So the virtual velocity is ds dt = S(t + dt) S(t) dt = 4(t + dt)2 4t 2 dt Newton then delete the last dt: ds dt = 8 t + 4 /// dt = 8 t. = 8 t + 4 dt.

22 So do we have dt = 0? If so, then one would have ds dt = 0 0. That was the question that Newton could not answer satisfactorily during his life time. Newton s trouble Suppose an object moves according to the rule S(t) = t 2 where S measures the distance of the object from the initial position t seconds later. We now compute instantaneous velocity of the object at time t: let dt and ds be the virtual time and virtual distance respectively. Then the change of virtual distance is given by ds = S(t + dt) S(t). So the virtual velocity is ds dt = S(t + dt) S(t) dt = 4(t + dt)2 4t 2 dt Newton then delete the last dt: ds dt = 8 t + 4 /// dt = 8 t. = 8 t + 4 dt.

23 So do we have dt = 0? If so, then one would have ds dt = 0 0. That was the question that Newton could not answer satisfactorily during his life time. Newton s trouble Suppose an object moves according to the rule S(t) = t 2 where S measures the distance of the object from the initial position t seconds later. We now compute instantaneous velocity of the object at time t: let dt and ds be the virtual time and virtual distance respectively. Then the change of virtual distance is given by ds = S(t + dt) S(t). So the virtual velocity is ds dt = S(t + dt) S(t) dt = 4(t + dt)2 4t 2 dt Newton then delete the last dt: ds dt = 8 t + 4 /// dt = 8 t. = 8 t + 4 dt.

24 Newton s thought So he simply considers that is a virtual distance ds traveled by the object in a virtual time dt. He considers both to be infinitesimal small quantities. So do we have dt = 0? If so, then one would have ds dt = 0 0. That was the question that Newton could not answer satisfactorily during his life time. To put the question differently, is an infinitesimal quantity equal to zero? If dt is infinitely small then it would have to be less than any positive quantity, and we conclude it must be equal to zero. For suppose dt 0 then dt > 0. Hence dt = r > 0 is an actual positive quantity. But then we could find r/2 < dt, contradicting the fact that dt is smaller then any positive quantity. Hence dt = 0. Newton was actually attacked by many people, and among them was the Bishop Berkeley. But he method of calculation of instantaneous velocity has been used by other since then.

25 Newton s thought So he simply considers that is a virtual distance ds traveled by the object in a virtual time dt. He considers both to be infinitesimal small quantities. So do we have dt = 0? If so, then one would have ds dt = 0 0. That was the question that Newton could not answer satisfactorily during his life time. To put the question differently, is an infinitesimal quantity equal to zero? If dt is infinitely small then it would have to be less than any positive quantity, and we conclude it must be equal to zero. For suppose dt 0 then dt > 0. Hence dt = r > 0 is an actual positive quantity. But then we could find r/2 < dt, contradicting the fact that dt is smaller then any positive quantity. Hence dt = 0. Newton was actually attacked by many people, and among them was the Bishop Berkeley. But he method of calculation of instantaneous velocity has been used by other since then.

26 Newton s thought So he simply considers that is a virtual distance ds traveled by the object in a virtual time dt. He considers both to be infinitesimal small quantities. So do we have dt = 0? If so, then one would have ds dt = 0 0. That was the question that Newton could not answer satisfactorily during his life time. To put the question differently, is an infinitesimal quantity equal to zero? If dt is infinitely small then it would have to be less than any positive quantity, and we conclude it must be equal to zero. For suppose dt 0 then dt > 0. Hence dt = r > 0 is an actual positive quantity. But then we could find r/2 < dt, contradicting the fact that dt is smaller then any positive quantity. Hence dt = 0. Newton was actually attacked by many people, and among them was the Bishop Berkeley. But he method of calculation of instantaneous velocity has been used by other since then.

27 Newton s thought So he simply considers that is a virtual distance ds traveled by the object in a virtual time dt. He considers both to be infinitesimal small quantities. So do we have dt = 0? If so, then one would have ds dt = 0 0. That was the question that Newton could not answer satisfactorily during his life time. To put the question differently, is an infinitesimal quantity equal to zero? If dt is infinitely small then it would have to be less than any positive quantity, and we conclude it must be equal to zero. For suppose dt 0 then dt > 0. Hence dt = r > 0 is an actual positive quantity. But then we could find r/2 < dt, contradicting the fact that dt is smaller then any positive quantity. Hence dt = 0. Newton was actually attacked by many people, and among them was the Bishop Berkeley. But he method of calculation of instantaneous velocity has been used by other since then.

28 Newton s Principles of Natural Philosophy (1729) Figure : (M. Kline, Mathematical Thoughts from Ancient to Modern Times, vol. 1)

29 A dynamical problem A rock is launched vertically upward from the ground with a speed of 96 ft/s. Neglecting air resistance, a well-known formula from physics states that the position of the rock after t seconds is given by s(t) = 16t t. The position s is measured in feet with s = 0 corresponding to the ground. Find the average velocity of the rock between 1. t = 1 and t = 3, 2. t = 1 and t = 2.

30 A dynamical problem (figure 2.1)

31 Figure : 2.2a Publisher A dynamical problem

32 A dynamical problem

33 A dynamical problem

34 A dynamical problem

35 A dynamical problem

36 A dynamical problem

37 Revisit: S(t) = t 2 Let s get close but not having dt = 0. Find the average velocity of the object between t = 2 and t = 2.5 ( S(2.5) S(2) ) (2.5) 2 ( (2) 2) = = 18; t = 2 and t = 2.1 ( S(2.1) S(2) ) (2.1) 2 ( (2) 2) = = t = 2 and t = 2.01 ( S(2.01) S(2) ) (2.01) 2 ( (2) 2) = = t = 2 and t = ( S(2.001) S(2) ) (2.001) 2 ( (2) 2) = = t = 2 and t = S(2.0001) S(2) ( ) (2.0001) 2 ( (2) 2) = =

38 Re-assessing the problem Let us begin with the above example about the movement of the object P. Since we are interested to know the magnitude of the average velocity of P near 2, so let us rewrite the expression in the following form: g(x) = S(2 + x) S(2). x This is a function g depends on the variable x, which can be made as close to 16 as we wish by choosing t close to 2. That is, g(x) approaches the value 16 as x approaches 0. On the other hand, we cannot put x = 0 in the function g(x), since both the numerator S(2 + x) S(x) and the denominator x would be zero. We say that the function g has limit equal to 16 as x approaches 0 abbreviated as lim g(x) = 16. x 0

39 Limit definition Note that the above statement is merely an abbreviation for the statement: The function g can get as close to 16 as possible if we let x approach 0 as close as we wish. It is important to note that we are not allowed to put x = 0 above Definition Let a and l be two real numbers. If the value of the function f (x) approaches l as close as we wish as x approaches a, then we say the limit of f is equal to l as x tends to a. The statement is denoted by lim f (x) = l. x a Alternatively, we may also write f (x) l as x a.

40 How to avoid the infinitesimal? Here is the real difficulty: Our thinking process and/or language usage generally does not allow us to describe infinitesimal quantities clearly Mathematicians have found a way to get around describing infinitesimal directly. We say that the function can get as close to a number (limit l) as possible. But we need to pay a heavy price if we want to do so precisely. Here it is. The abbreviation lim x a f (x) = l really means: Given an arbitrary ε > 0, one can find a δ > 0 such that f (x) l < ε, whenever 0 < x a < δ. Both ε and δ represent positive real numbers. Given each/any ε > 0 one can (always) find a δ > 0 such that... holds we refer to this kind of statement as ε δ language interpretation.

41 A linear function example How do we use δ ε to describe lim x 3 = 5? Figure : 2.56 (Publisher)

42 ε = 1 How do we use δ ε to describe lim x 3 = 5?

43 δ = 2 The corresponding δ = 2. That is, 0 < x 3 < 2 guarantees f (x) 5 < 1.

44 ε = 1/2

45 δ = 1 The corresponding δ = 2. That is, 0 < x 3 < 1 guarantees f (x) 5 < 1/2.

46 ε = 1/8, δ = 1/4

47 General ε δ That is, 0 < x 3 < δ guarantees f (x) 5 < ε.

MATH1013 Calculus I. Limits (part I) 1

MATH1013 Calculus I. Limits (part I) 1 MATH1013 Calculus I Limits (part I) 1 Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology March 5, 2014 2013 1 Based on Briggs, Cochran and Gillett: Calculus for