Euler and Lagrange on Pell's Equation
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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. -
Rationale of the Chakravala Process of Jayadeva and Bhaskara Ii
HISTORIA MATHEMATICA 2 (1975) , 167-184 RATIONALE OF THE CHAKRAVALA PROCESS OF JAYADEVA AND BHASKARA II BY CLAS-OLOF SELENIUS UNIVERSITY OF UPPSALA SUMMARIES The old Indian chakravala method for solving the Bhaskara-Pell equation or varga-prakrti x 2- Dy 2 = 1 is investigated and explained in detail. Previous mis- conceptions are corrected, for example that chakravgla, the short cut method bhavana included, corresponds to the quick-method of Fermat. The chakravala process corresponds to a half-regular best approximating algorithm of minimal length which has several deep minimization properties. The periodically appearing quantities (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, etc.) are correctly understood only with the new theory. Den fornindiska metoden cakravala att l~sa Bhaskara- Pell-ekvationen eller varga-prakrti x 2 - Dy 2 = 1 detaljunders~ks och f~rklaras h~r. Tidigare missuppfatt- 0 ningar r~ttas, sasom att cakravala, genv~gsmetoden bhavana inbegripen, motsvarade Fermats snabbmetod. Cakravalaprocessen motsvarar en halvregelbunden b~st- approximerande algoritm av minimal l~ngd med flera djupt liggande minimeringsegenskaper. De periodvis upptr~dande storheterna (jyestha-m~la, kanistha-mula, ksepaka, kuttakara, os~) blir forstaellga0. 0 . f~rst genom den nya teorin. Die alte indische Methode cakrav~la zur Lbsung der Bhaskara-Pell-Gleichung oder varga-prakrti x 2 - Dy 2 = 1 wird hier im einzelnen untersucht und erkl~rt. Fr~here Missverst~ndnisse werden aufgekl~rt, z.B. dass cakrav~la, einschliesslich der Richtwegmethode bhavana, der Fermat- schen Schnellmethode entspreche. Der cakravala-Prozess entspricht einem halbregelm~ssigen bestapproximierenden Algorithmus von minimaler L~nge und mit mehreren tief- liegenden Minimierungseigenschaften. Die periodisch auftretenden Quantit~ten (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, usw.) werden erst durch die neue Theorie verst~ndlich. -
IJR-1, Mathematics for All ... Syed Samsul Alam
January 31, 2015 [IISRR-International Journal of Research ] MATHEMATICS FOR ALL AND FOREVER Prof. Syed Samsul Alam Former Vice-Chancellor Alaih University, Kolkata, India; Former Professor & Head, Department of Mathematics, IIT Kharagpur; Ch. Md Koya chair Professor, Mahatma Gandhi University, Kottayam, Kerala , Dr. S. N. Alam Assistant Professor, Department of Metallurgical and Materials Engineering, National Institute of Technology Rourkela, Rourkela, India This article briefly summarizes the journey of mathematics. The subject is expanding at a fast rate Abstract and it sometimes makes it essential to look back into the history of this marvelous subject. The pillars of this subject and their contributions have been briefly studied here. Since early civilization, mathematics has helped mankind solve very complicated problems. Mathematics has been a common language which has united mankind. Mathematics has been the heart of our education system right from the school level. Creating interest in this subject and making it friendlier to students’ right from early ages is essential. Understanding the subject as well as its history are both equally important. This article briefly discusses the ancient, the medieval, and the present age of mathematics and some notable mathematicians who belonged to these periods. Mathematics is the abstract study of different areas that include, but not limited to, numbers, 1.Introduction quantity, space, structure, and change. In other words, it is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Mathematicians seek out patterns and formulate new conjectures. They resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity. -
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. -
Quantum Algorithm for Solving Pell's Equation
CS682 Project Report Hallgren's Efficient Quantum Algorithm for Solving Pell's Equation by Ashish Dwivedi (17111261) under the guidance of Prof Rajat Mittal November 15, 2017 Abstract In this project we aim to study an excellent result on the quantum computation model. Hallgren in 2002 [Hal07] showed that a seemingly difficult problem in classical computa- tion model, solving Pell's equation, is efficiently solvable in quantum computation model. This project explains the idea of Hallgren's quantum algorithm to solve Pell's equation. Contents 0.1 Introduction..................................1 0.2 Background..................................1 0.3 Hallgren's periodic function.........................3 0.4 Quantum Algorithm to find irrational period on R .............4 0.4.1 Discretization of the periodic function...............4 0.4.2 Quantum Algorithm.........................5 0.4.3 Classical Post Processing.......................7 0.5 Summary...................................8 0.1 Introduction In this project we study one of those problems which have no known efficient solution on the classical computation model and are assumed hard to be solved efficiently, but in recent years efficient quantum algorithms have been found for them. One of these problems is the problem of finding solution to Pell's equation. Pell's equation are equations of the form x2 − dy2 = 1, where d is some given non- square positive integer and x and y are indeterminate. Clearly, (1; 0) is a solution of this equation which are called trivial solutions. We want some non-trivial integer solutions satisfying these equations. There is no efficient algorithm known for solving such equations on classical compu- tation model. But in 2002, Sean Hallgren [Hal07] gave an algorithm on quantum compu- tation model which takes time only polynomial in input size (log d). -
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. -
Equation Solving in Indian Mathematics
U.U.D.M. Project Report 2018:27 Equation Solving in Indian Mathematics Rania Al Homsi Examensarbete i matematik, 15 hp Handledare: Veronica Crispin Quinonez Examinator: Martin Herschend Juni 2018 Department of Mathematics Uppsala University Equation Solving in Indian Mathematics Rania Al Homsi “We owe a lot to the ancient Indians teaching us how to count. Without which most modern scientific discoveries would have been impossible” Albert Einstein Sammanfattning Matematik i antika och medeltida Indien har påverkat utvecklingen av modern matematik signifi- kant. Vissa människor vet de matematiska prestationer som har sitt urspring i Indien och har haft djupgående inverkan på matematiska världen, medan andra gör det inte. Ekvationer var ett av de områden som indiska lärda var mycket intresserade av. Vad är de viktigaste indiska bidrag i mate- matik? Hur kunde de indiska matematikerna lösa matematiska problem samt ekvationer? Indiska matematiker uppfann geniala metoder för att hitta lösningar för ekvationer av första graden med en eller flera okända. De studerade också ekvationer av andra graden och hittade heltalslösningar för dem. Denna uppsats presenterar en litteraturstudie om indisk matematik. Den ger en kort översyn om ma- tematikens historia i Indien under många hundra år och handlar om de olika indiska metoderna för att lösa olika typer av ekvationer. Uppsatsen kommer att delas in i fyra avsnitt: 1) Kvadratisk och kubisk extraktion av Aryabhata 2) Kuttaka av Aryabhata för att lösa den linjära ekvationen på formen 푐 = 푎푥 + 푏푦 3) Bhavana-metoden av Brahmagupta för att lösa kvadratisk ekvation på formen 퐷푥2 + 1 = 푦2 4) Chakravala-metoden som är en annan metod av Bhaskara och Jayadeva för att lösa kvadratisk ekvation 퐷푥2 + 1 = 푦2. -
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. -
International Conference on the 900Th Birth Anniversary of Bhāskarācārya
International Conference on the 900th Birth Anniversary of Bhāskarācārya 19, 20, 21 September 2014 ABSTRACTS Vidya Prasarak Mandai Vishnu Nagar, Naupada, Thane 400602 2 I BHĀSKARĀCĀRYA’S LIFE AND TIMES A. P. JAMKHEDKAR, Mumbai. ‘Learning and Patronage in 12th -13th Century A.D.: Bhaskarācarya and the Śāndilya Family, A Case Study’ … … … … … 5 II BHASKARĀCĀRYA’S POETIC GENIUS Pierre-Sylvain FILLIOZAT, Paris. ‘The poetical face of the mathematical and astronomical works of Bhāskarācārya’ … … … … … … … … 7 K. S. BALASUBRAMANIAN, Chennai. ‘Bhāskarācārya as a Poet’ … … … … 8 GEETHAKUMARI, K. K., Calicut. ‘Contribution of Līlāvatī to Prosody’ … … … 9 Poonam GHAI, Dhampur. ‘Līlāvatī men Kāvya-saundarya’ … … … … … … 10 III THE LĪLĀVATĪ K. RAMASUBRAMANIAN, Mumbai. ‘The Līlā of the Līlāvatī: A Beautiful Blend of Arithmetic, Geometry and Poetry’ … … … … … … … … … 11 PADMAVATAMMA, Mysore. ‘A Comparative Study of Bhāskarācārya’s Līlāvatī and Mahāvīrācārya’s Gaṇitasārasaṁgraha’… … … … … … … … 11 K. RAMAKALYANI, Chennai. ‘Gaṇeśa Daivajña’s Upapattis on Līlāvatī’ … … 12 Anil NARAYANAN, N., Calicut. ‘Parameswara’s Unpublished Commentary on Lilavati: Perceptions and Problems’ … … … … … … … … … … … 13 P. RAJASEKHAR & LAKSHMI, V. Calicut. ‘Līlāvatī and Kerala School of Mathematics’ 14 N. K. Sundareswaran & P. M. Vrinda, Calicut. ‘Malayalam Commentaries on Līlāvatī – A Survey of Manuscripts in Kerala’ … … … … … … 15 Shrikrishna G. DANI, Mumbai. ‘Mensuration of Quadrilaterals in Lilavati’ … … 16 Takanori KUSUBA, Osaka. ‘Aṅkapāśa in the Līlāvatī’ … … … … … … … 17 Medha Srikanth LIMAYE, Mumbai. Use of Bhūta-Saṅkhyās (Object Numerals) in Līlāvatī of Bhāskarācārya’ … … … … … … … … … … 17 Sreeramula Rajeswara SARMA, Düsseldorf. ‘The Legend of Līlāvatī’ … … … 17 IV THE BĪJAGAṆITA Sita Sundar RAM, Chennai. ‘Bījagaṇita of Bhāskara’ … … … … … … … 19 K. VIDYUTA, Chennai. ‘Sūryaprakāśa of Sūryadāsa – a Review’ … … … … 20 Veena SHINDE-DEORE & Manisha M. ACHARYA, Mumbai. ‘Bhaskaracharya and Varga Prakriti: the equations of the type ax2 + b = cy2’ … … … … … 20 K. -
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.