EULER's Analytical PROGRAM
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Year 6 Local Linearity and L'hopitals.Notebook December 04, 2018
Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 New Divider! Application of Derivatives: Local Linearity and L'Hopitals Local Linear Approximation Do Now: For each sketch the function, write the equation of the tangent line at x = 0 and include the tangent line in your sketch. 1) 2) In general, if a function f is differentiable at an x, then a sufficiently magnified portion of the graph of f centered at the point P(x, f(x)) takes on the appearance of a ______________ line segment. For this reason, a function that is differentiable at x is sometimes said to be locally linear at x. 1 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 How is this useful? We are pretty good at finding the equations of tangent lines for various functions. Question: Would you rather evaluate linear functions or crazy ridiculous functions such as higher order polynomials, trigonometric, logarithmic, etc functions? Evaluate sec(0.3) The idea is to use the equation of the tangent line to a point on the curve to help us approximate the function values at a specific x. Get it??? Probably not....here is an example of a problem I would like us to be able to approximate by the end of the class. Without the use of a calculator approximate . 2 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Local Linear Approximation General Proof Directions would say, evaluate f(a). If f(x) you find this impossible for some y reason, then that's how you would recognize we need to use local linear approximation! You would: 1) Draw in a tangent line at x = a. -
March 14 Math 1190 Sec. 62 Spring 2017
March 14 Math 1190 sec. 62 Spring 2017 Section 4.5: Indeterminate Forms & L’Hopital’sˆ Rule In this section, we are concerned with indeterminate forms. L’Hopital’sˆ Rule applies directly to the forms 0 ±∞ and : 0 ±∞ Other indeterminate forms we’ll encounter include 1 − 1; 0 · 1; 11; 00; and 10: Indeterminate forms are not defined (as numbers) March 14, 2017 1 / 61 Theorem: l’Hospital’s Rule (part 1) Suppose f and g are differentiable on an open interval I containing c (except possibly at c), and suppose g0(x) 6= 0 on I. If lim f (x) = 0 and lim g(x) = 0 x!c x!c then f (x) f 0(x) lim = lim x!c g(x) x!c g0(x) provided the limit on the right exists (or is 1 or −∞). March 14, 2017 2 / 61 Theorem: l’Hospital’s Rule (part 2) Suppose f and g are differentiable on an open interval I containing c (except possibly at c), and suppose g0(x) 6= 0 on I. If lim f (x) = ±∞ and lim g(x) = ±∞ x!c x!c then f (x) f 0(x) lim = lim x!c g(x) x!c g0(x) provided the limit on the right exists (or is 1 or −∞). March 14, 2017 3 / 61 The form 1 − 1 Evaluate the limit if possible 1 1 lim − x!1+ ln x x − 1 March 14, 2017 4 / 61 March 14, 2017 5 / 61 March 14, 2017 6 / 61 Question March 14, 2017 7 / 61 l’Hospital’s Rule is not a ”Fix-all” cot x Evaluate lim x!0+ csc x March 14, 2017 8 / 61 March 14, 2017 9 / 61 Don’t apply it if it doesn’t apply! x + 4 6 lim = = 6 x!2 x2 − 3 1 BUT d (x + 4) 1 1 lim dx = lim = x!2 d 2 x!2 2x 4 dx (x − 3) March 14, 2017 10 / 61 Remarks: I l’Hopital’s rule only applies directly to the forms 0=0, or (±∞)=(±∞). -
Chapter 4 Differentiation in the Study of Calculus of Functions of One Variable, the Notions of Continuity, Differentiability and Integrability Play a Central Role
Chapter 4 Differentiation In the study of calculus of functions of one variable, the notions of continuity, differentiability and integrability play a central role. The previous chapter was devoted to continuity and its consequences and the next chapter will focus on integrability. In this chapter we will define the derivative of a function of one variable and discuss several important consequences of differentiability. For example, we will show that differentiability implies continuity. We will use the definition of derivative to derive a few differentiation formulas but we assume the formulas for differentiating the most common elementary functions are known from a previous course. Similarly, we assume that the rules for differentiating are already known although the chain rule and some of its corollaries are proved in the solved problems. We shall not emphasize the various geometrical and physical applications of the derivative but will concentrate instead on the mathematical aspects of differentiation. In particular, we present several forms of the mean value theorem for derivatives, including the Cauchy mean value theorem which leads to L’Hôpital’s rule. This latter result is useful in evaluating so called indeterminate limits of various kinds. Finally we will discuss the representation of a function by Taylor polynomials. The Derivative Let fx denote a real valued function with domain D containing an L ? neighborhood of a point x0 5 D; i.e. x0 is an interior point of D since there is an L ; 0 such that NLx0 D. Then for any h such that 0 9 |h| 9 L, we can define the difference quotient for f near x0, fx + h ? fx D fx : 0 0 4.1 h 0 h It is well known from elementary calculus (and easy to see from a sketch of the graph of f near x0 ) that Dhfx0 represents the slope of a secant line through the points x0,fx0 and x0 + h,fx0 + h. -
Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time. -
Indeterminate Forms and Improper Integrals
CHAPTER 8 Indeterminate Forms and Improper Integrals 8.1. L’Hopital’ˆ s Rule ¢ To begin this section, we return to the material of section 2.1, where limits are defined. Suppose f ¡ x is a function defined in an interval around a, but not necessarily at a. Then we write ¢¥¤ (8.1) lim f ¡ x L x £ a ¢ if we can insure that f ¡ x is as close as we please to L just by taking x close enough to a. If f is also defined at a, and ¢¦¤ ¡ ¢ (8.2) lim f ¡ x f a x £ a ¢ we say that f is continuous at a (we urge the reader to review section 2.1). If the expression for f ¡ x is a polynomial, we found limits by just substituting a for x; this works because polynomials are continuous. ¢ But how do we calculate limits when the expression f ¡ x cannot be determined at a? For example, we recall the definition of the derivative: ¢ ¡ ¢ f ¡ x f a ¢©¤ (8.3) f §¨¡ x lim £ x a x a The value cannot be determined by simply evaluating at x ¤ a, because both numerator and denominator ¢ ¡ ¢ are 0 at a. This is an example of an indeterminate form of type 0/0: an expression f ¡ x g x , where both ¢ ¡ ¢ ¡ ¢ f ¡ a and g a are zero. As for 8.3, in case f x is a polynomial, we found the limit by long division, and then evaluating the quotient at a (see Theorem 1.1). For trigonometric functions, we devised a geometric argument to calculate the limit (see Proposition 2.7). -
Lecture 29 Prep
Review What does tell us about ? on an interval is concave up on on an interval is concave down on Second Derivative Test for local maxima and minima: Suppose is in the domain of and is continuous near if and then has a local min at then has a local max at Q1: Consider the functions and Which statement is true? A) Both functions have a local minimum at B) Both functions have a local maximum at C) For these functions, the Second Derivative Test does not help me decide whether the functions have a local maximum or minimum at D) For these functions there is no method (except for drawing the graphs) to decide whether there is a local maximum of minimum at . Answer: and Note so both functions have a critical number In these two examples, we can decide what happens at by using the First Derivative test: when and also when , so is increasing near both on the left and right sides of . has no local max or min at . for and for , so, by the First Derivative Test, has a local minimum at . But both The moral of the example is that the Second Derivative Test doesn't always work: it never gives you a wrong answer, but when the second derivative is at a critical point, the test leaves you undecided. C) is the correct answer. Q2: If represents the average of all SAT scores across the country, then Homer is saying A) and B) and C) and D) and Solution: The graph of as a function of time should be decreasing and concave up, resembling This corresponds to answer C) Draw the graph of a function for which: Exercise done in class. -
Mathematical Language and Mathematical Progress by Eberhard Knobloch
Mathematical language and mathematical progress By Eberhard Knobloch In 1972 the space probe Pioneer 10 was launched, carrying a plaque which contains the first message of mankind to leave our solar system1: The space probe is represented by a circular segment and a rectangle designed on the same scale as that of the man and the woman. This is not true of the solar system: sun and planets are represented by small circles and points. We do not know whether there are other intelligent beings living on stars in the universe, or whether they will even understand the message. But the non-verbal, the symbolical language of geometry seemed to be more appropriate than any other language. The grammar of any ordinary language is so complicated that still every computer breaks down with regards to it. 1 Karl Märker, "Sind wir allein im Weltall? Kosmos und Leben", in: Astronomie im Deutschen Museum, Planeten - Sterne - Welteninseln, hrsg. von Gerhard Hartl, Karl Märker, Jürgen Teichmann, Gudrun Wolfschmidt (München: Deutsches Museum, 1993), p. 217. [We reproduce the image of the plaque from: http://en.wikipedia.org/wiki/Pioneer_plaque ; Karine Chemla] 1 The famous Nicholas Bourbaki wrote in 19482: "It is the external form which the mathematician gives to his thought, the vehicle which makes it accessible to others, in short, the language suited to mathematics; this is all, no further significance should be attached to it". Bourbaki added: "To lay down the rules of this language, to set up its vocabulary and to clarify its syntax, all that is indeed extremely useful." But it was only the least interesting aspect of the axiomatic method for him. -
Lagrange's Theory of Analytical Functions and His Ideal of Purity of Method
Lagrange’s Theory of Analytical Functions and his Ideal of Purity of Method Giovanni Ferraro, Marco Panza To cite this version: Giovanni Ferraro, Marco Panza. Lagrange’s Theory of Analytical Functions and his Ideal of Purity of Method. Archive for History of Exact Sciences, Springer Verlag, 2012, 66, pp.95-197. hal-00614606 HAL Id: hal-00614606 https://hal.archives-ouvertes.fr/hal-00614606 Submitted on 12 Aug 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LAGRANGE’S THEORY OF ANALYTICAL FUNCTIONS AND HIS IDEAL OF PURITY OF METHOD GIOVANNI FERRARO AND MARCO PANZA ABSTRACT. We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline. -
God, King, and Geometry: Revisiting the Introduction to Cauchy's Cours D'analyse
God, King, and Geometry: Revisiting the Introduction to Cauchy's Cours d'Analyse Michael J. Barany Program in History of Science, Princeton University, 129 Dickinson Hall, Princeton, New Jersey 08544, United States Abstract This article offers a systematic reading of the introduction to Augustin-Louis Cauchy's landmark 1821 mathematical textbook, the Cours d'analyse. Despite its emblematic status in the history of mathematical analysis and, indeed, of modern mathematics as a whole, Cauchy's introduction has been more a source for suggestive quotations than an object of study in its own right. Cauchy's short mathematical metatext offers a rich snapshot of a scholarly paradigm in transition. A close reading of Cauchy's writing reveals the complex modalities of the author's epistemic positioning, particularly with respect to the geometric study of quantities in space, as he struggles to refound the discipline on which he has staked his young career. Keywords: Augustin-Louis Cauchy, Cours d'analyse, History of Analysis, Mathematics and Politics 2000 MSC: 01-02, 65-03, 97-03 1. Introduction Despite its emblematic status in the history of mathematical analysis and, indeed, of modern mathematics as a whole, the introduction to Augustin-Louis Cauchy's 1821 Cours d'analyse has been, in the historical literature, more a source for suggestive quotations than an object of study in its own right. In his introduction, Cauchy definitively outlines what were to be the foundations of his new rigorous mathematics, invoking both specific mathematical practices and their underlying philosophical principles. His text is thus a fecund encapsulation of the mathematical and epistemological work which would make him \the man who taught rigorous analysis to all of Europe" (Grabiner, 1981, p. -
A Neutrosophic Binomial Factorial Theorem with Their Refrains
Neutrosophic Sets and Systems, Vol. 14, 2016 7 University of New Mexico A Neutrosophic Binomial Factorial Theorem with their Refrains Huda E. Khalid1 Florentin Smarandache2 Ahmed K. Essa3 1 University of Telafer, Head of Math. Depart., College of Basic Education, Telafer, Mosul, Iraq. E-mail: [email protected] 2 University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA. E-mail: [email protected] 3 University of Telafer, Math. Depart., College of Basic Education, Telafer, Mosul, Iraq. E-mail: [email protected] Abstract. The Neutrosophic Precalculus and the Two other important theorems were proven with their Neutrosophic Calculus can be developed in many corollaries, and numerical examples as well. As a ways, depending on the types of indeterminacy one conjecture, we use ten (indeterminate) forms in has and on the method used to deal with such neutrosophic calculus taking an important role in indeterminacy. This article is innovative since the limits. To serve article's aim, some important form of neutrosophic binomial factorial theorem was questions had been answered. constructed in addition to its refrains. Keyword: Neutrosophic Calculus, Binomial Factorial Theorem, Neutrosophic Limits, Indeterminate forms in Neutrosophic Logic, Indeterminate forms in Classical Logic. 1 Introduction (Important questions) 2. Let 3 Q 1 What are the types of indeterminacy? √퐼 = 푥 + 푦 퐼 3 2 2 2 3 3 There exist two types of indeterminacy 0 + 퐼 = 푥 + 3푥 푦 퐼 + 3푥푦 퐼 + 푦 퐼 3 2 2 3 a. Literal indeterminacy (I). 0 + 퐼 = 푥 + (3푥 푦 + 3푥푦 + 푦 )퐼 3 As example: 푥 = 0, 푦 = 1 → √퐼 = 퐼. -
Calculus 1, Chapter 4 “Applications of Derivatives” Study Guide Prepared by Dr
Calculus 1, Chapter 4 “Applications of Derivatives” Study Guide Prepared by Dr. Robert Gardner The following is a brief list of topics covered in Chapter 4 of Thomas’ Calculus. 4.1 Extreme Values of Functions on Closed Intervals. Absolute minimum and maximum, extreme values, Extreme-Value Theorem for Continuous Functions (Theorem 4.1), local maximum and minimum (relative extrema), Local Ex- treme Values (Theorem 4.2), critical point, finding extrema of continuous func- tion on a closed and bounded interval. 4.2 Mean Value Theorem. Rolle’s Theorem (Theorem 4.3), Mean Value Theo- rem (Theorem 4.4), Functions with Zero Derivatives are Constant Functions (Corollary 4.1), Functions with the Same Derivative Differ by a Constant (Corollary 4.2), position/velocity/acceleration, Algebraic Properties of Natu- ral Logarithms (Theorem 1.6.1/Theorem 4.2.A), Properties of Natural Expo- nentials (Theorem 4.2.B). 4.3 Monotone Functions and the First Derivative Test. increasing/decreasing function on an interval, monotonic function, First Deriva- tive Text for Increasing and Decreasing (Corollary 4.3), First Derivative Test for Local Extrema (Theorem 4.3.A), sign tests of f 0. 4.4 Concavity and Curve Sketching. Increasing/decreasing y0, concave up and concave down functions on open intervals, convex function, Second Derivative Test for Concavity (Theorem 4.4.A), concavity emojis, point of inflection, sign tests of f 00, Second Derivative Test for Local Extrema (Theorem 4.5), “Proce- dure for Graphing y = f(x)” (seven steps). 4.5 Indeterminate Forms and L’Hˆopital’s Rule 0/0 and ∞/∞ indeterminate forms of limits, L’Hˆopital’s Rule (Theorem 4.6), L’Hˆopital’s Rule for ∞/∞ Indeterminate Forms (Theorem 4.5.A), ∞ − ∞ indeterminate form, 0 · ∞ indeterminate form, 1∞ indeterminate form, 00 indeterminate form, ∞0 in- determinate form, limits and exponentials (Theorem 4.5.B), Cauchy’s Mean Value Theorem (Theorem 4.7), proof of L’Hˆopital’s Rule. -
History of Mathematics in the Higher Education Curriculum
History of Mathematics in the Higher Education Curriculum Mathematical Sciences HE Curriculum Innovation Project Innovation Curriculum HE Sciences Mathematical Edited by Mark McCartney History of Mathematics in the Higher Education Curriculum Edited by Mark McCartney A report by the working group on History of Mathematics in the Higher Education Curriculum, May 2012. Supported by the Maths, Stats and OR Network, as part of the Mathematical Sciences Strand of the National HE STEM Programme, and the British Society for the History of Mathematics (BSHM). Working group members: Noel-Ann Bradshaw (University of Greenwich; BSHM Treasurer); Snezana Lawrence (Bath Spa University; BSHM Education Officer); Mark McCartney (University of Ulster; BSHM Publicity Officer); Tony Mann (University of Greenwich; BSHM Immediate Past President); Robin Wilson (Pembroke College, Oxford; BSHM President). History of Mathematics in the Higher Education Curriculum Contents Contents Introduction 5 Teaching the history of mathematics at the University of St Andrews 9 History in the undergraduate mathematics curriculum – a case study from Greenwich 13 Teaching History of Mathematics at King’s College London 15 History for learning Analysis 19 History of Mathematics in a College of Education Context 23 Teaching the history of mathematics at the Open University 27 Suggested Resources 31 History of Mathematics in the Higher Education Curriculum Introduction Introduction Mathematics is usually, and of course correctly, presented ‘ready-made’ to students, with techniques and applications presented systematically and in logical order. However, like any other academic subject, mathematics has a history which is rich in astonishing breakthroughs, false starts, misattributions, confusions and dead-ends. This history gives a narrative and human context which adds colour and context to the discipline.