DOCSLIB.ORG

Differential and Integral Calculus

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Differential and Integral Calculus DIFFERENTIAL AND INTEGRAL CALCULUS WITH APPLICATIONS BY E. W. NICHOLS Superintendent Virginia Military Institute, and Author of Nichols's Analytic Geometry REVISED D. C. HEATH & CO., PUBLISHERS BOSTON NEW YORK CHICAGO w' ^Kt„ Copyright, 1900 and 1918, By D. C. Heath & Co. 1 a8 v m 2B 1918 ©CI.A492720 & PREFACE. This text-book is based upon the methods of " limits " and ''rates/' and is limited in its scope to the requirements in the undergraduate courses of our best universities, colleges, and technical schools. In its preparation the author has embodied the results of twenty years' experience in the class-room, ten of which have been devoted to applied mathematics and ten to pure mathematics. It has been his aim to prepare a teachable work for begimiers, removing as far as the nature of the subject would admit all obscurities and mysteries, and endeavoring by the introduction of a great variety of practical exercises to stimulate the student's interest and appetite. Among the more marked peculiarities of the work the follow- ing may be enumerated : — i. A large amount of explanation. 2. Clear and simple demonstiations of principles. 3. Geometric, mechanical, and engineering applications. 4. Historical notes at the heads of chapters giving a brief account of the discovery and development of the subject of which it treats. 5. Footnotes calling attention to topics of special historic interest. iii iv Preface 6. A chapter on Differential Equations for students in mathematical physics and for the benefit of those desiring an elementary knowledge of this interesting extension of the calculus. 7. An arrangement of topics admitting of extensive elimina- tions without destroying the continuity of the subject. 8. A clear, open page. The author desires to express here his acknowledgments to the friends who have aided him in his work. To Chas. M. Snelling, A.M., University of Georgia, and to T. H. Taliaferro, Ph.D., State College of Pennsylvania, the author's obligations are peculiarly great. Not only have they given valuable counsel, but they have been largely instrumental in freeing the work from typographical errors. PREFACE TO REVISED EDITION In presenting the revised edition of this work to the public, I wish to express my acknowledgment to L. W. Smith, A.M., Ph.D. Professor of Mathematics, Washington and Lee University, and to Colonel C. W. Watts, C.E., Professor of Mathematics, Virginia Military Institute. To these gentlemen the care of the revision oi the work was submitted and to them is due full credit for all im- provements. E. W. NICHOLS. Lexington, Va., July 16, 1917. CONTENTS. PART I. DIFFERENTIAL CALCULUS. CHAPTER I. QUANTITIES. FUNCTIONS. ARTS. PAGES 1-2. Quantity. Classes of 3 3-4. Constants. Variables 3 5. Illustrations 3-4 6-8. Functions. Classes of 4-7 9. Notation. Examples 7-9 CHAPTER II. FUNDAMENTAL PRINCIPLES. io-ii. Increment. Uniform and Varied Change ' 10 12. Uniform and Varied Motion 11 13-14. Differential. Illustrations 11-12 15-16. Rate. Relation between a Differential and a Rate . 12-13 17-18. Velocity. Component Velocities 14 dy 19-20. Significations of — • Remark 15—17 CHAPTER III. DIFFERENTIATION. 21-22: History. Differential Calculus. Differentiation .... 18 23-29. Differentiation of Algebraic Functions. Examples . 19-28 30-32. Differentiation of the Logarithmic Functions 29-3 1 33-34. Differentiation of the Exponential Functions. Examples . 31—35 vi Contents ARTS. PAGES 35-43. Differentiation of the Trigonometric Functions. Examples 35-40 44-52. Differentiation of the Circular Functions. Examples . 40-46 % CHAPTER IV. LIMITS. 53-57. History. Limit. Principles . 47-49 58-59. First Derivative. Examples 49-52 60-67. Differentiation by Method of Limits 52—57 68. First Derivative as a Fraction 57 CHAPTER V. ANALYTICAL APPLICATIONS. Analytical Applications 59—71 CHAPTER VI. GEOMETRIC APPLICATIONS. Cartesian Curves. 69-70. Tangent. Normal. Examples 72—75 71-72. Subtangent. Subnormal. Examples 75—78 73-75. Asymptotes. Examples 78-84 Polar Curves. 76-77. Tangent. Subtangent 84-86 78. Normal. Subnormal. Examples 86-88 79. Asymptotes. Examples 88-90 CHAPTER VII. SUCCESSIVE DIFFERENTIATION. 80-81. Successive Differentials and Derivatives. Examples . 91-94 82. Applications • 94~96 83. Leibnitz's Theorem. Examples . 96-99 84. Non-equicrescent Variables. Examples 99-104 Contents vn CHAPTER VIII. SERIES. ARTS. PAGES 85-91 History. Varieties of. Methods of Development . 105-108 92-93 Maclaurin's Theorem. Examples . , 108-113 94 Euler's Exponential Values of Sine and Cosine ... 113 95-96 Taylor's Theorem. Examples '. 113-117 97 Bernouilli's Series 117 99-102 Lagrange's Theorem. Tests for Development . 11 8-1 24 CHAPTER IX. ILLUSORY FORMS. 103-105. History. Forms — > — , -• Examples 127-131 _ a 00 00 106-107. Forms > — > — . Examples ' — r ..,„ , OH 00 a 00 132-135 o. 00, 00— 00. 108. Forms Examples I 35~ I 3o 00 00 co°, i 00 °°, 109-110. Forms o°, , , etc. Examples .... 136-138 CHAPTER X. MAXIMA AND MINIMA. 1-113. History. Conditions for. Illustration 139-142 :4-n6. Methods of Investigation. Suggestions. Examples . 142-151 117. Formulae. Problems 151-158 CHAPTER XI. PARTIAL AND TOTAL DIFFERENTIATION. 118-119. Partial Differentials and Derivatives. Examples . 159- 161 120. -Euler's Theorem. Examples 161-162 21-122. Total Differentials and Derivatives. Examples . 162-168 23-124. Successive Partial Differentiation 168-171 125. Successive Total Differentiation 172 viii Contents CHAPTER XII. DIRECTION OF CURVATURE. POINTS OF INFLEXION. Cartesian Curves. ARTS. PAGES 126-127. Investigation for Direction of Curvature 1 73-1 74 128. Point of Inflexion. Examples 174-176 Polar Curves. 129-130. Investigation for Direction of Curvature . * . 176—1 78 131. Point of Inflexion. Examples 178-179 CHAPTER XIII. CURVATURE. EVOLUTE AND INVOLUTE. 132-133. History. Measure. Circle and Radius of Curvature 180-183 134-135. Expressions for Radius. Maximum Curvature. Ex- amples 183-187 136-140. Evolute. Involute. Examples 188-193 CHAPTER XIV. CONTACT OF CURVES. ENVELOPES. 141 -144. Orders of Contact. Examples 194-200 145-146. Families of Curves. Envelope 200-201 147-148. Equation of Envelope. Examples 201-206 CHAPTER XV. SINGULAR POINTS. 149-152. Multiple Points. Isolated Points. Point d'Arret . 207-200 153. Methods of Investigation. Examples 209-217 CHAPTER XVI. LOCI. 1 54—1 55. Algebraic Equations. Suggestions. Examples . 21S- 156. Polar Equations. Suggestions. Examples .... 22u -2 Contents IX PART II. INTEGRAL CALCULUS. CHAPTER I. TYPE FORMS. PAGES I57-I59 Integrals. Integration. Notation 225-228 160 Indefinite Integrals. Constant of Integration . 228 161 Elementary Principles 229 162-164 Type Formulae. Examples 229-243 165 Integration by Parts. Examples 243-246 CHAPTER II. RATIONAL FRACTIONS. 166-168 Fractional Differential and Cases . 247-248 169 Factors Real and Unequal. Examples . 248-251 - . 2 2 170 Factors Real and Equal. Examples . 2 5 55 171 Factors Imaginary and Unequal. Examples 255-259 172 Factors Imaginary and Equal. Examples 259-262 IRRATIONAL FRACTIONS. 173-174. Methods of Rationalization 262 175—177. Monomial, Binomial and Quadratic Surds. Examples 262-268 178. Methods in Special Cases. Examples 268-272 CHAPTER III. BINOMIAL DIFFERENTIALS. 179-180. Reduction to Form, xm {a + bxn)Pdx 273-274 181-183. Rationalization. Examples 274-279 184-187. Reduction Formulae. Examples 279-288 CHAPTER IV. TRIGONOMETRIC INTEGRALS. 188-189. Trigonometric Formulae. General Rule 289-290 m 190. fta.n xdx. I* coX™ xdx. Examples 290-292 Contents PAGES n re fsec xdx. f csc xdx. Examples 292-293 m 1 m w 192 f ta.n x see xdx. fcot x csc xdx. Examples . 294-295 m m xdx. cos xdx. Examples 193-194 fsm J 296-298 m n 195 J sir\ x cos xdx. Examples 298-301 196-198 Reduction Formulae 301-305 199 Reduction to Algebraic Forms. Examples .... 306-311 CHAPTER V. DEFINITE INTEGRALS. 200-202. Method of Determining the Constant 312-313 203. Method of Eliminating the Constant 314 204. Notation. Examples 3 J 4-3i5 Applications 316-318 CHAPTER VI. GEOMETRIC APPLICATIONS. 205-206. Quadrature. Examples 319-324 207-208. Rectification. Examples 324-329 209. Surfaces and Volumes of Revolution. Examples . 329-333 210. Surfaces and Volumes in General. Examples . 333~33S CHAPTER VII. SUCCESSIVE INTEGRATION. 211. Successive Integration 336~337 212-213. Double and Triple Integration . 337-33% 214. Definite Double and Triple Integration. Examples . 33%~339 CHAPTER VIII. GEOMETRIC APPLICATIONS. 215. Quadrature by Double Integration. Examples . 340-341 216. Surfaces and Volumes by Double and Triple Integra- ". tion. Examples . 342-346 Contents XI CHAPTER IX. DIFFERENTIAL EQUATIONS. PAGES 217--2l8. Definitions. Orders. Degrees 347-348 Form, (x) dx-\-<j> (x) o. dy Examples . 219. / ft (y) <f> t ( y) — 348-350 (x,y) . 220-221. Form, f dy + <£ (x,y) dx = o. Examples. 350-353 222--223. Form, dy + Pydx = Qdx 353~354 n 224. Form, dy -{- Pydx — Qy dx. Examples 354~35 225--226. Exact Differential Equations. Examples 35°-358 227--228. Integrating Factor. Examples 358-362 229 Equations of First Order and nth Degree. Examples . 362-364 23O Equations of «th Order. Examples 364-367 CHAPTER X. MECHANICAL APPLICATIONS. 8I-237 Rectilinear Motion 368-374 238-240 Curvilinear Motion 374~379 241-249 Centers of Gravity 379-384 '5^255 Moments of Inertia 384-386 256-261 Deflection and Slope of Beams 386-393 _ 262. Strongest Rectangular Beam 393 394 PART I. DIFFERENTIAL CALCULUS. DIFFERENTIAL CALCULUS, CHAPTER I. QUANTITIES. FUNCTIONS. 1. Quantity. That which can be increased, diminished, measured, or in general, anything to which mathematical pro- cesses are applicable is called Quantity. Time, space, motion, velocity, force, and mass are examples, and with these, as with other quantities, we shall have more or less to do in illustrating and applying the principles which are to follow. 2. Classes of Quantity. In the abstract science of the Cal- culus, as in Analytic Geometry, quantities are divided into two general classes, viz., Constants and Variables.
Load more

Recommended publications
  • Trigonometry and Spherical Astronomy 1. Sines and Chords
    CHAPTER TWO TRIGONOMETRY AND SPHERICAL ASTRONOMY 1. Sines and Chords In Almagest I.11 Ptolemy displayed a table of chords, the construction of which is explained in the previous chapter. The chord of a central angle α in a circle is the segment subtended by that angle α. If the radius of the circle (the “norm” of the table) is taken as 60 units, the chord can be expressed by means of the modern sine function: crd α = 120 · sin (α/2). In Ptolemy’s table (Toomer 1984, pp. 57–60), the argument, α, is given at intervals of ½° from ½° to 180°. Columns II and III display the chord, in units, minutes, and seconds of a unit, and the increase, in the same units, in crd α corresponding to an increase of one minute in α, computed by taking 1/30 of the difference between successive entries in column II (Aaboe 1964, pp. 101–125). For an insightful account on Ptolemy’s chord table, see Van Brummelen 2009, where many issues underlying the tables for trigonometry and spherical astronomy are addressed. By contrast, al-Khwārizmī’s zij originally had a table of sines for integer degrees with a norm of 150 (Neugebauer 1962a, p. 104) that has uniquely survived in manuscripts associated with the Toledan Tables (Toomer 1968, p. 27; F. S. Pedersen 2002, pp. 946–952). The corresponding table in al-Battānī’s zij displays the sine of the argu- ment at intervals of ½° with a norm of 60 (Nallino 1903–1907, 2:55– 56). This table is reproduced, essentially unchanged, in the Toledan Tables (Toomer 1968, p.
    [Show full text]
  • Notes on Curves for Railways
    NOTES ON CURVES FOR RAILWAYS BY V B SOOD PROFESSOR BRIDGES INDIAN RAILWAYS INSTITUTE OF CIVIL ENGINEERING PUNE- 411001 Notes on —Curves“ Dated 040809 1 COMMONLY USED TERMS IN THE BOOK BG Broad Gauge track, 1676 mm gauge MG Meter Gauge track, 1000 mm gauge NG Narrow Gauge track, 762 mm or 610 mm gauge G Dynamic Gauge or center to center of the running rails, 1750 mm for BG and 1080 mm for MG g Acceleration due to gravity, 9.81 m/sec2 KMPH Speed in Kilometers Per Hour m/sec Speed in metres per second m/sec2 Acceleration in metre per second square m Length or distance in metres cm Length or distance in centimetres mm Length or distance in millimetres D Degree of curve R Radius of curve Ca Actual Cant or superelevation provided Cd Cant Deficiency Cex Cant Excess Camax Maximum actual Cant or superelevation permissible Cdmax Maximum Cant Deficiency permissible Cexmax Maximum Cant Excess permissible Veq Equilibrium Speed Vg Booked speed of goods trains Vmax Maximum speed permissible on the curve BG SOD Indian Railways Schedule of Dimensions 1676 mm Gauge, Revised 2004 IR Indian Railways IRPWM Indian Railways Permanent Way Manual second reprint 2004 IRTMM Indian railways Track Machines Manual , March 2000 LWR Manual Manual of Instructions on Long Welded Rails, 1996 Notes on —Curves“ Dated 040809 2 PWI Permanent Way Inspector, Refers to Senior Section Engineer, Section Engineer or Junior Engineer looking after the Permanent Way or Track on Indian railways. The term may also include the Permanent Way Supervisor/ Gang Mate etc who might look after the maintenance work in the track.
    [Show full text]
  • Coverrailway Curves Book.Cdr
    RAILWAY CURVES March 2010 (Corrected & Reprinted : November 2018) INDIAN RAILWAYS INSTITUTE OF CIVIL ENGINEERING PUNE - 411 001 i ii Foreword to the corrected and updated version The book on Railway Curves was originally published in March 2010 by Shri V B Sood, the then professor, IRICEN and reprinted in September 2013. The book has been again now corrected and updated as per latest correction slips on various provisions of IRPWM and IRTMM by Shri V B Sood, Chief General Manager (Civil) IRSDC, Delhi, Shri R K Bajpai, Sr Professor, Track-2, and Shri Anil Choudhary, Sr Professor, Track, IRICEN. I hope that the book will be found useful by the field engineers involved in laying and maintenance of curves. Pune Ajay Goyal November 2018 Director IRICEN, Pune iii PREFACE In an attempt to reach out to all the railway engineers including supervisors, IRICEN has been endeavouring to bring out technical books and monograms. This book “Railway Curves” is an attempt in that direction. The earlier two books on this subject, viz. “Speed on Curves” and “Improving Running on Curves” were very well received and several editions of the same have been published. The “Railway Curves” compiles updated material of the above two publications and additional new topics on Setting out of Curves, Computer Program for Realignment of Curves, Curves with Obligatory Points and Turnouts on Curves, with several solved examples to make the book much more useful to the field and design engineer. It is hoped that all the P.way men will find this book a useful source of design, laying out, maintenance, upgradation of the railway curves and tackling various problems of general and specific nature.
    [Show full text]
  • Development of Trigonometry Medieval Times Serge G
    Development of trigonometry Medieval times Serge G. Kruk/Laszl´ o´ Liptak´ Oakland University Development of trigonometry Medieval times – p.1/27 Indian Trigonometry • Work based on Hipparchus, not Ptolemy • Tables of “sines” (half chords of twice the angle) • Still depends on radius 3◦ • Smallest angle in table is 3 4 . Why? R sine( α ) α Development of trigonometry Medieval times – p.2/27 Indian Trigonometry 3◦ • Increment in table is h = 3 4 • The “first” sine is 3◦ 3◦ s = sine 3 = 3438 sin 3 = 225 1 4 · 4 1◦ • Other sines s2 = sine(72 ), . Sine differences D = s s • 1 2 − 1 • Second-order approximation techniques sine(xi + θ) = θ θ2 sine(x ) + (D +D ) (D D ) i 2h i i+1 − 2h2 i − i+1 Development of trigonometry Medieval times – p.3/27 Etymology of “Sine” How words get invented! • Sanskrit jya-ardha (Half-chord) • Abbreviated as jya or jiva • Translated phonetically into arabic jiba • Written (without vowels) as jb • Mis-interpreted later as jaib (bosom or breast) • Translated into latin as sinus (think sinuous) • Into English as sine Development of trigonometry Medieval times – p.4/27 Arabic Trigonometry • Based on Ptolemy • Used both crd and sine, eventually only sine • The “sine of the complement” (clearly cosine) • No negative numbers; only for arcs up to 90◦ • For larger arcs, the “versine”: (According to Katz) versine α = R + R sine(α 90◦) − Development of trigonometry Medieval times – p.5/27 Other trig “functions” Al-B¯ırun¯ ¯ı: Exhaustive Treatise on Shadows • The shadow of a gnomon (cotangent) • The hypothenuse of the shadow
    [Show full text]
  • Mathematics Tables
    MATHEMATICS TABLES Department of Applied Mathematics Naval Postgraduate School TABLE OF CONTENTS Derivatives and Differentials.............................................. 1 Integrals of Elementary Forms............................................ 2 Integrals Involving au + b ................................................ 3 Integrals Involving u2 a2 ............................................... 4 Integrals Involving √u±2 a2, a> 0 ....................................... 4 Integrals Involving √a2 ± u2, a> 0 ....................................... 5 Integrals Involving Trigonometric− Functions .............................. 6 Integrals Involving Exponential Functions ................................ 7 Miscellaneous Integrals ................................................... 7 Wallis’ Formulas ................................................... ...... 8 Gamma Function ................................................... ..... 8 Laplace Transforms ................................................... 9 Probability and Statistics ............................................... 11 Discrete Probability Functions .......................................... 12 Standard Normal CDF and Table ....................................... 12 Continuous Probability Functions ....................................... 13 Fourier Series ................................................... ........ 13 Separation of Variables .................................................. 15 Bessel Functions ................................................... ..... 16 Legendre
    [Show full text]
  • Tables of Trigonometric Functions in Non-Sexagesimal Arguments
    Tables of Trigonometric Functions in Non-Sexagesimal Arguments Excluding the ordinary tables of trigonometric functions in sexagesimal arguments the two principal groups of such tables are those with arguments in A. Radians,—tables of this type have been already listed in RMT 81; and B. Grades. But we shall also consider tables of the trigonometric functions with arguments in C. Mils; Dl. Gones; D2. Cirs; and E. Time. B. Grades. The first stage in the evolution of the centesimal division of the quadrant was by the division of each sexagesimal degree into 100 minutes and each of these minutes into 100 seconds. This was conceived already about 1450 by Theodericus Ruffi, in a Latin codex in Munich.1 The centesimal division of the degree was also presented by Vieta in his Calendarij Gregoriani, 1600, p. "29"(Opera M athematica, 1646, p. 487). It was this source which sug- gested to Henry Briggs the choice of every hundredth of a degree as the unit in the preparation of his wonderful "sine canon";2 see RMT 79. Late in the eighteenth century the centesimal division of the quadrant was definitely in the thought of scholars then in Berlin and especially of Johann Carl Schulze (1749-1790), author of Neue und erweiterte Sammlung logarithmischer, trigonometrischer und anderer zum Gebrauch der Mathematik unentbehrlicher Tafeln [also French t.p.], Berlin, 2v., 1778. He had studied under J. H. Lambert (1728-1778) and became a member of the Academy of Sciences at Berlin, in which Lagrange (following Euler) was director of its mathematical section for a score of years before he moved to Paris in 1787.
    [Show full text]
  • Before Calculators Uavs and Traffic Karakoram Glaciers a Look at Computations Batteries No Issue Monitoring Movement SURVEYORS’ CALCULATION METHODS USED BC
    JUNE 2016 HOLDOUTS! Before Calculators UAVs and Traffic Karakoram Glaciers A look at computations Batteries no issue Monitoring movement SURVEYORS’ CALCULATION METHODS USED BC n the dark days BC (Before Calculator), surveyors invented and adapted methods which would lessen their calculations. Here are some interesting relationships, known only to I old-guy surveyors. To Find D on a Railroad Curve “Stringlining” in railroad surveying refers to a method to realign the tracks. Due to unbalanced forces, the tracks in curves tend to creep and shift, making the middle ordinates (m) for equal-length chords (c) along the curve not equal, as they should be for a circular curve. (See the sketch.) So, the tracks can be “stringlined” or literally moved to make the curve circular, by forcing the m’s to be equal for the same c. To see the amazing process of how this is physically done, go to YouTube and search for “Gandy Dancer.” (There’s a similar process for the spiral-curve portion of a track, but that’s for another day.) For the chord definition of Degree of Curvature used by the railroads, a well-known and very close approximation is: For a 62-foot chord, 1° Degree of Curvature yields 1 inch of Middle Ordinate (distance, m, from the chord midpoint to the curve midpoint). You remember, m = (c2D)/45,840 (m and c in feet). And, m and D are proportional. So, using this relationship, should you need to measure the D of a railroad curve, here’s the procedure: On the inside of the rail, mea- sure and mark two points (A and B in the sketch) 61.80 feet apart (a chord).
    [Show full text]
  • The Forgotten Trigonometric Functions, Or How Trigonometry Was Used in the Ancient Art of Navigation (Before GPS!)
    The Forgotten Trigonometric Functions, or How Trigonometry was used in the Ancient Art of Navigation (Before GPS!) Recently, as I was exploring some mathematical concept I came across some terms that were suspiciously similar to some familiar trigonometric functions. What is an excosecant? How about a Havercosine? As it turns out, these belong to a family of trigonometric functions that once enjoyed very standard usage, in very notable and practical fields, but in recent times have become nearly obsolete. ( ) Versine !"#$%& ! = 1 − cos ! !(!!!"# !) = The most basic of this family of functions is the ! versine, or versed sine. The versine, abbreviated ! 1 − cos ! versin of some angle x is actually calculated by = 2 !! ! 2 subtracting the cosine of x from 1; that is versin(x) = 1 - cos (x). Through some manipulations with half- 1 = 2 !"#! ! !! angle identities, we can see another equivalent 2 expression: Notice, we do not need the ± as the versine will always represent non-negative value (since cosine ranges from negative one to positive one, one minus these values will range from zero to two). This was especially helpful as navigators used logarithms in their calculations, so they could always take the logarithm of the versine. It is interesting that the Latin name for versine is either sinus versus (flipped sine, to distinguish from sinus rectus, the vertical sine) or sagitta, meaning arrow. It is easy to see, when looking at a drawing of the sine, cosine and versine why the reference to arrow occurs. Envision a bow and arrow; on the sketch, the arc AB is the bow, segment AB is the bowstring, and segment CD is the drawn arrow shaft, thus, the reference to an arrow.
    [Show full text]
  • Product Sheet
    MEASURING DEVICES We calibrate your measuring devices! Contact your ROBEL partner. ROMEAS Versine Versine Measuring Device 82.05 For measuring track curves • Not obstructionless • Two angle supports in aluminium with level • ~148 ft chord • Ruler • Complete in canvas bag • Weight ~5.7 lbs ROMEAS Versine Versine Measuring Device 82.06 For easy checking and determination of versines in track curves • Can be used for all common rail profiles • Measuring chord adjustable to rail height and from side to side • Easy attachment to rail foot • Obstructionless • Consisting of 2 tensions clamps with rope drum and 70 m nylon rope • Weight ~9.5 lbs ROMEAS Level Sighting and Levelling Unit 82.07 For determining the longitudinal level (with reference to the top of rail) and cross level tasks • Scales every ~0.4", alternating in ~0.4"& 0.8" graduation • Minus zone of level slide marked with a red bar • Telescope magnification 24-fold • Lens opening ~1.5" • Portrait image • Shortest adjusting distance ~15.8" • Adjusting distance ~0.05 to 1.2" • Rotation range with micro adjustment 360° • System height above top of rail ~51.2" • Tilting device used as sighting facility • Tilting range ± 40 ‰ • Levelling rod used for levelling at fixed points • Scale graduation 0,04" • Measuring range of basic scale ± ~31.5" • Measuring range of adjustable level slide + ~6" – ~4" • Total weight ~29 lbs ACCESSORIES 9800820010 Target rod lighting 9800820001 Push-on part 0 – 150 (Standard = 0 - 50) consisting of: 1 cold cathode neon lamp, 12 V, 9800820008 Transport and storage box of water-resistant, for side lighting of adjustable level slide, glued plywood with carrying handles including one rechargeable battery.
    [Show full text]
  • The Development of Calculus in the Kerala School
    The Mathematics Enthusiast Volume 11 Number 3 Number 3 Article 5 12-2014 The development of Calculus in the Kerala School Phoebe Webb Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Webb, Phoebe (2014) "The development of Calculus in the Kerala School," The Mathematics Enthusiast: Vol. 11 : No. 3 , Article 5. Available at: https://scholarworks.umt.edu/tme/vol11/iss3/5 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME, vol. 11, no. 3, p. 495 The Development of Calculus in the Kerala School Phoebe Webb1 University of Montana Abstract: The Kerala School of mathematics, founded by Madhava in Southern India, produced many great works in the area of trigonometry during the fifteenth through eighteenth centuries. This paper focuses on Madhava's derivation of the power series for sine and cosine, as well as a series similar to the well-known Taylor Series. The derivations use many calculus related concepts such as summation, rate of change, and interpolation, which suggests that Indian mathematicians had a solid understanding of the basics of calculus long before it was developed in Europe. Other evidence from Indian mathematics up to this point such as interest in infinite series and the use of a base ten decimal system also suggest that it was possible for calculus to have developed in India almost 300 years before its recognized birth in Europe.
    [Show full text]
  • Working Group Report: a Brief History of Trigonometry for Mathematics Educators
    Adams. G. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 36(1) February 2016 Working group report: a brief history of trigonometry for mathematics educators Leo Rogers and Sue Pope British Society for the History of Mathematics (BSHM); Manchester Metropolitan University Despite the words: ‘Mathematics is a creative and highly inter-connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems.’ in the Purpose of Study section, the new English mathematics curriculum alludes only to Roman Numerals in the primary programme of study, and there is no further mention of historical or cultural roots of mathematics in the aims, or in the programmes of study. In contrast, the increased expectations for lower and middle attainers in the new curriculum challenge teachers to make more mathematics accessible and memorable to more learners. The history of mathematics can provide an engaging way to do this. There are also many opportunities in post-16 mathematics. Further to our recent article on quadratic equations, we use trigonometry to illustrate some of the ways that history of mathematics can enrich teaching of this topic. Keywords: history of mathematics; trigonometry; BSHM Introduction In the 2014 National Curriculum for mathematics (DfE, 2014) all KS4 (ages 14-16) students are expected to make links to similarity including trigonometric ratios, and apply trigonometric ratios to finding lengths in right angled triangles and 2D figures. Higher attainers are expected to recognise, sketch and interpret graphs of trigonometric functions for angles of any size (in degrees), and extend application of trigonometric ratios to include 3D figures, the sine and cosine rules, and the area of a triangle .
    [Show full text]
  • Earth-Satellite Geometry
    EARTH-REFERENCED AIRCRAFT NAVIGATION AND SURVEILLANCE ANALYSIS Michael Geyer S π/2 - α - θ d h α N U Re Re θ O Michael Geyer Frederick de Wit Navigation Vertical Plane Model Peter Mercator Federico Rostagno Spherical Surface Model Surveillance Project Memorandum — June 2016 DOT-VNTSC-FAA-16-12 Prepared for: Federal Aviation Administration Wake Turbulence Research Office DOT Volpe Center FOREWORD This document addresses a basic function of aircraft (and other vehicle) surveillance and navi- gation systems analyses — quantifying the geometric relationship of two or more locations relative to each other and to the earth. Here, geometry means distances and angles, including their projections in a defined coordinate frame. Applications that fit well with these methods include (a) planning a vehicle’s route; (b) determining the coverage region of a radar or radio installation; and (c) calculating a vehicle’s latitude and longitude from measurements (e.g., of slant- and spherical-ranges or range differences, azimuth and elevation angles, and altitudes). The approach advocated is that the three-dimensional problems inherent in navigation/surveil- lance analyses should, to the extent possible, be re-cast as a sequence of sub-problems: . Vertical-Plane Formulation (two-dimensional (2D) problem illustrated in top right panel on cover) — Considers the vertical plane containing two problem-specific locations and the center of a spherical earth, and utilizes plane trigonometry as the primary analysis method. This formulation provides closed-form solutions. Spherical-Surface Formulation (2D problem illustrated in bottom left panel on cover) — Considers two or three problem-specific locations on the surface of a spherical earth; utilizes spherical trigonometry as the primary analysis method.
    [Show full text]