MA 341 Fall 2011
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Analytic Geometry
Guide Study Georgia End-Of-Course Tests Georgia ANALYTIC GEOMETRY TABLE OF CONTENTS INTRODUCTION ...........................................................................................................5 HOW TO USE THE STUDY GUIDE ................................................................................6 OVERVIEW OF THE EOCT .........................................................................................8 PREPARING FOR THE EOCT ......................................................................................9 Study Skills ........................................................................................................9 Time Management .....................................................................................10 Organization ...............................................................................................10 Active Participation ...................................................................................11 Test-Taking Strategies .....................................................................................11 Suggested Strategies to Prepare for the EOCT ..........................................12 Suggested Strategies the Day before the EOCT ........................................13 Suggested Strategies the Morning of the EOCT ........................................13 Top 10 Suggested Strategies during the EOCT .........................................14 TEST CONTENT ........................................................................................................15 -
Hydraulic Cylinder Replacement for Cylinders with 3/8” Hydraulic Hose Instructions
HYDRAULIC CYLINDER REPLACEMENT FOR CYLINDERS WITH 3/8” HYDRAULIC HOSE INSTRUCTIONS INSTALLATION TOOLS: INCLUDED HARDWARE: • Knife or Scizzors A. (1) Hydraulic Cylinder • 5/8” Wrench B. (1) Zip-Tie • 1/2” Wrench • 1/2” Socket & Ratchet IMPORTANT: Pressure MUST be relieved from hydraulic system before proceeding. PRESSURE RELIEF STEP 1: Lower the Power-Pole Anchor® down so the Everflex™ spike is touching the ground. WARNING: Do not touch spike with bare hands. STEP 2: Manually push the anchor into closed position, relieving pressure. Manually lower anchor back to the ground. REMOVAL STEP 1: Remove the Cylinder Bottom from the Upper U-Channel using a 1/2” socket & wrench. FIG 1 or 2 NOTE: For Blade models, push bottom of cylinder up into U-Channel and slide it forward past the Ram Spacers to remove it. Upper U-Channel Upper U-Channel Cylinder Bottom 1/2” Tools 1/2” Tools Ram 1/2” Tools Spacers Cylinder Bottom If Ram-Spacers fall out Older models have (1) long & (1) during removal, push short Ram Spacer. Ram Spacers 1/2” Tools bushings in to hold MUST be installed on the same side Figure 1 Figure 2 them in place. they were removed. Blade Models Pro/Spn Models Need help? Contact our Customer Service Team at 1 + 813.689.9932 Option 2 HYDRAULIC CYLINDER REPLACEMENT FOR CYLINDERS WITH 3/8” HYDRAULIC HOSE INSTRUCTIONS STEP 2: Remove the Cylinder Top from the Lower U-Channel using a 1/2” socket & wrench. FIG 3 or 4 Lower U-Channel Lower U-Channel 1/2” Tools 1/2” Tools Cylinder Top Cylinder Top Figure 3 Figure 4 Blade Models Pro/SPN Models STEP 3: Disconnect the UP Hose from the Hydraulic Cylinder Fitting by holding the 1/2” Wrench 5/8” Wrench Cylinder Fitting Base with a 1/2” wrench and turning the Hydraulic Hose Fitting Cylinder Fitting counter-clockwise with a 5/8” wrench. -
An Introduction to Topology the Classification Theorem for Surfaces by E
An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5. -
Canada Archives Canada Published Heritage Direction Du Branch Patrimoine De I'edition
Rhetoric more geometrico in Proclus' Elements of Theology and Boethius' De Hebdomadibus A Thesis submitted in Candidacy for the Degree of Master of Arts in Philosophy Institute for Christian Studies Toronto, Ontario By Carlos R. Bovell November 2007 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-43117-7 Our file Notre reference ISBN: 978-0-494-43117-7 NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. -
Lesson 3: Rectangles Inscribed in Circles
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection . -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
Mathematicians
MATHEMATICIANS [MATHEMATICIANS] Authors: Oliver Knill: 2000 Literature: Started from a list of names with birthdates grabbed from mactutor in 2000. Abbe [Abbe] Abbe Ernst (1840-1909) Abel [Abel] Abel Niels Henrik (1802-1829) Norwegian mathematician. Significant contributions to algebra and anal- ysis, in particular the study of groups and series. Famous for proving the insolubility of the quintic equation at the age of 19. AbrahamMax [AbrahamMax] Abraham Max (1875-1922) Ackermann [Ackermann] Ackermann Wilhelm (1896-1962) AdamsFrank [AdamsFrank] Adams J Frank (1930-1989) Adams [Adams] Adams John Couch (1819-1892) Adelard [Adelard] Adelard of Bath (1075-1160) Adler [Adler] Adler August (1863-1923) Adrain [Adrain] Adrain Robert (1775-1843) Aepinus [Aepinus] Aepinus Franz (1724-1802) Agnesi [Agnesi] Agnesi Maria (1718-1799) Ahlfors [Ahlfors] Ahlfors Lars (1907-1996) Finnish mathematician working in complex analysis, was also professor at Harvard from 1946, retiring in 1977. Ahlfors won both the Fields medal in 1936 and the Wolf prize in 1981. Ahmes [Ahmes] Ahmes (1680BC-1620BC) Aida [Aida] Aida Yasuaki (1747-1817) Aiken [Aiken] Aiken Howard (1900-1973) Airy [Airy] Airy George (1801-1892) Aitken [Aitken] Aitken Alec (1895-1967) Ajima [Ajima] Ajima Naonobu (1732-1798) Akhiezer [Akhiezer] Akhiezer Naum Ilich (1901-1980) Albanese [Albanese] Albanese Giacomo (1890-1948) Albert [Albert] Albert of Saxony (1316-1390) AlbertAbraham [AlbertAbraham] Albert A Adrian (1905-1972) Alberti [Alberti] Alberti Leone (1404-1472) Albertus [Albertus] Albertus Magnus -
Circumference and Area of Circles 353
Circumference and 7-1 Area of Circles MAIN IDEA Find the circumference Measure and record the distance d across the circular and area of circles. part of an object, such as a battery or a can, through its center. New Vocabulary circle Place the object on a piece of paper. Mark the point center where the object touches the paper on both the object radius and on the paper. chord diameter Carefully roll the object so that it makes one complete circumference rotation. Then mark the paper again. pi Finally, measure the distance C between the marks. Math Online glencoe.com • Extra Examples • Personal Tutor • Self-Check Quiz in. 1234 56 1. What distance does C represent? C 2. Find the ratio _ for this object. d 3. Repeat the steps above for at least two other circular objects and compare the ratios of C to d. What do you observe? 4. Graph the data you collected as ordered pairs, (d, C). Then describe the graph. A circle is a set of points in a plane center radius circumference that are the same distance from a (r) (C) given point in the plane, called the center. The segment from the center diameter to any point on the circle is called (d) the radius. A chord is any segment with both endpoints on the circle. The diameter of a circle is twice its radius or d 2r. A diameter is a chord that passes = through the center. It is the longest chord. The distance around the circle is called the circumference. -
Thermodynamics of Spacetime: the Einstein Equation of State
gr-qc/9504004 UMDGR-95-114 Thermodynamics of Spacetime: The Einstein Equation of State Ted Jacobson Department of Physics, University of Maryland College Park, MD 20742-4111, USA [email protected] Abstract The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation δQ = T dS connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with δQ and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air. arXiv:gr-qc/9504004v2 6 Jun 1995 The four laws of black hole mechanics, which are analogous to those of thermodynamics, were originally derived from the classical Einstein equation[1]. With the discovery of the quantum Hawking radiation[2], it became clear that the analogy is in fact an identity. How did classical General Relativity know that horizon area would turn out to be a form of entropy, and that surface gravity is a temperature? In this letter I will answer that question by turning the logic around and deriving the Einstein equation from the propor- tionality of entropy and horizon area together with the fundamental relation δQ = T dS connecting heat Q, entropy S, and temperature T . -
History of Mathematics
Georgia Department of Education History of Mathematics K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by using manipulatives and a variety of representations, working independently and cooperatively to solve problems, estimating and computing efficiently, and conducting investigations and recording findings. There is a shift towards applying mathematical concepts and skills in the context of authentic problems and for the student to understand concepts rather than merely follow a sequence of procedures. In mathematics classrooms, students will learn to think critically in a mathematical way with an understanding that there are many different ways to a solution and sometimes more than one right answer in applied mathematics. Mathematics is the economy of information. The central idea of all mathematics is to discover how knowing some things well, via reasoning, permit students to know much else—without having to commit the information to memory as a separate fact. It is the reasoned, logical connections that make mathematics coherent. The implementation of the Georgia Standards of Excellence in Mathematics places a greater emphasis on sense making, problem solving, reasoning, representation, connections, and communication. History of Mathematics History of Mathematics is a one-semester elective course option for students who have completed AP Calculus or are taking AP Calculus concurrently. It traces the development of major branches of mathematics throughout history, specifically algebra, geometry, number theory, and methods of proofs, how that development was influenced by the needs of various cultures, and how the mathematics in turn influenced culture. The course extends the numbers and counting, algebra, geometry, and data analysis and probability strands from previous courses, and includes a new history strand. -
A Comparison of Differential Calculus and Differential Geometry in Two
1 A Two-Dimensional Comparison of Differential Calculus and Differential Geometry Andrew Grossfield, Ph.D Vaughn College of Aeronautics and Technology Abstract and Introduction: Plane geometry is mainly the study of the properties of polygons and circles. Differential geometry is the study of curves that can be locally approximated by straight line segments. Differential calculus is the study of functions. These functions of calculus can be viewed as single-valued branches of curves in a coordinate system where the horizontal variable controls the vertical variable. In both studies the derivative multiplies incremental changes in the horizontal variable to yield incremental changes in the vertical variable and both studies possess the same rules of differentiation and integration. It seems that the two studies should be identical, that is, isomorphic. And, yet, students should be aware of important differences. In differential geometry, the horizontal and vertical units have the same dimensional units. In differential calculus the horizontal and vertical units are usually different, e.g., height vs. time. There are differences in the two studies with respect to the distance between points. In differential geometry, the Pythagorean slant distance formula prevails, while in the 2- dimensional plane of differential calculus there is no concept of slant distance. The derivative has a different meaning in each of the two subjects. In differential geometry, the slope of the tangent line determines the direction of the tangent line; that is, the angle with the horizontal axis. In differential calculus, there is no concept of direction; instead, the derivative describes a rate of change. In differential geometry the line described by the equation y = x subtends an angle, α, of 45° with the horizontal, but in calculus the linear relation, h = t, bears no concept of direction. -
Circle Theorems
Circle theorems A LEVEL LINKS Scheme of work: 2b. Circles – equation of a circle, geometric problems on a grid Key points • A chord is a straight line joining two points on the circumference of a circle. So AB is a chord. • A tangent is a straight line that touches the circumference of a circle at only one point. The angle between a tangent and the radius is 90°. • Two tangents on a circle that meet at a point outside the circle are equal in length. So AC = BC. • The angle in a semicircle is a right angle. So angle ABC = 90°. • When two angles are subtended by the same arc, the angle at the centre of a circle is twice the angle at the circumference. So angle AOB = 2 × angle ACB. • Angles subtended by the same arc at the circumference are equal. This means that angles in the same segment are equal. So angle ACB = angle ADB and angle CAD = angle CBD. • A cyclic quadrilateral is a quadrilateral with all four vertices on the circumference of a circle. Opposite angles in a cyclic quadrilateral total 180°. So x + y = 180° and p + q = 180°. • The angle between a tangent and chord is equal to the angle in the alternate segment, this is known as the alternate segment theorem. So angle BAT = angle ACB. Examples Example 1 Work out the size of each angle marked with a letter. Give reasons for your answers. Angle a = 360° − 92° 1 The angles in a full turn total 360°. = 268° as the angles in a full turn total 360°.