Three-Cornered Things | Weekly wanderings through higher math—from not so high a vantage point. Differential Equations Differential Equations (Math 3301) Here are my online notes for my differential equations course that I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations. I’ve tried to make these notes as self contained as possible and so all the information needed to read through them is either from a Calculus or Algebra class or contained in other sections of the notes. A couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. Here is a listing and brief description of the material in this set of notes. Basic Concepts
Intuitive Understanding Of Euler’s Formula Euler's identity seems baffling: It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! Not according to 1800s mathematician Benjamin Peirce: It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth. Argh, this attitude makes my blood boil! Euler's formula describes two equivalent ways to move in a circle. That's it? Starting at the number 1, see multiplication as a transformation that changes the number (1 * e^(i*pi))Regular exponential growth continuously increases 1 by some rate; imaginary exponential growth continuously rotates a numberGrowing for "pi" units of time means going pi radians around a circleTherefore, e^(i*pi) means starting at 1 and rotating pi (halfway around a circle) to get to -1 That's the high-level view -- let's dive into the details. It follows the post -- watch together, or at your leisure. The equals sign is overloaded.
The AP Calculus BC Exam Exam Content In 1956, 386 students took what was then known as the AP Mathematics Exam. By 1969, still under the heading of AP Mathematics, it had become Calculus AB and Calculus BC. The Calculus BC exam covers the same differential and integral calculus topics that are included in the Calculus AB exam, plus additional topics in differential and integral calculus, and polynomial approximations and series. This is material that would be included in a two-semester calculus sequence at the college level. Because graphing calculator use is an integral part of the course, the exam contains questions that require students to use a graphing calculator. If students take the BC exam, they cannot take the AB exam in the same year because the exams share some questions. Multiple-Choice Questions For sample multiple-choice questions, refer to the Course Description. AP Calculus Course Description, Effective Fall 2012 (.pdf/2.28MB) Free-Response Questions AP Calculus Free-Response Question Collections
The Thirty Greatest Mathematicians Click for a discussion of certain omissions. Please send me e-mail if you believe there's a major flaw in my rankings (or an error in any of the biographies). Obviously the relative ranks of, say Fibonacci and Ramanujan, will never satisfy everyone since the reasons for their "greatness" are different. I'm sure I've overlooked great mathematicians who obviously belong on this list. Please e-mail and tell me! Following are the top mathematicians in chronological (birth-year) order. Earliest mathematicians Little is known of the earliest mathematics, but the famous Ishango Bone from Early Stone-Age Africa has tally marks suggesting arithmetic. Early Vedic mathematicians The greatest mathematics before the Golden Age of Greece was in India's early Vedic (Hindu) civilization. Top Thales of Miletus (ca 624 - 546 BC) Greek domain Apastambha (ca 630-560 BC) India Pythagoras of Samos (ca 578-505 BC) Greek domain Panini (of Shalatula) (ca 520-460 BC) Gandhara (India) Tiberius(?) Geocentrism vs.
CALCULUS.ORG Geometry The word geometry is Greek for geos - meaning earth and metron - meaning measure. Geometry was extremely important to ancient societies and was used for surveying, astronomy, navigation, and building. Geometry, as we know it is actually known as Euclidean geometry which was written well over 2000 years ago in Ancient Greece by Euclid, Pythagoras, Thales, Plato and Aristotle just to mention a few. Geometry is the study of angles and triangles, perimeter, area and volume. Terms (Undefined) Point Points show position. Terms (Defined) Line Segment A line segment is a straight line segment which is part of the straight line between two points.
THE CALCULUS PAGE PROBLEMS LIST Problems and Solutions Developed by : D. A. And brought to you by : Beginning Differential Calculus : Problems on the limit of a function as x approaches a fixed constant limit of a function as x approaches plus or minus infinity limit of a function using the precise epsilon/delta definition of limit limit of a function using l'Hopital's rule ... Beginning Integral Calculus : Problems using summation notation Problems on the limit definition of a definite integral Problems on u-substitution Problems on integrating exponential functions Problems on integrating trigonometric functions Problems on integration by parts Problems on integrating certain rational functions, resulting in logarithmic or inverse tangent functions Problems on integrating certain rational functions by partial fractions Problems on power substitution Problems on integration by trigonometric substitution ... Sequences and Infinite Series :
GEM1518K - Mathematics in Art & Architecture - Project Submission “For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.” - M.C. Escher Contents Page 1 Introduction The Art of Alhambra Our Area of Focus Our Aim Page 2 The Principals behind Tessellations - Translation - Rotation - Reflection - Glide Reflection Page 3 Mathematics in Escher's Art - Translation - Rotation - Glide Reflection - Combination Page 4 Our Original Tessellations - The Catch (Translation) - Under the Sea (Rotation) - The Herd (Glide Reflection) - Dumbo & Butterfly (Combination) Page 5 Possible links with Architecture Conclusion References Mathematics in Escher's Art In this page we attempt to try to use simple mathematical terms to explain a few of Escher's pieces. One of the basic principals behind his tessellations is the use of what we call an "addition and subtraction" method within the grid. Translation One of Escher's first explorations into the tessllations is that of the use of translation. Rotation Glide Reflection Combination
Calculus, Contemporary Calculus, Hoffman Contemporary Calculus Dale HoffmanBellevue Collegedhoffman@bellevuecollege.edu A free on-line calculus text Many of these materials were developed for the Open Course Library Project of the Washington State Colleges as part of a Gates Foundation grant. The goal of this project was to create materials that would be FREE (on the web) to anyone who wanted to use or modify them (and not have to pay $200 for a calculus book). They have been used by several thousand students. The textbook sections, in color, are available free in pdf format at the bottom of this page. The links below are to pdf files. Chapter 0 -- Review and Preview Chapter 1 -- Functions, Graphs, Limits and Continuity Chapter 2 -- The Derivative Chapter 3 -- Derivatives and Graphs Chapter 4 -- The Integral Chapter 5 -- Applications of Definite Integrals Chapter 6 -- Introduction to Differential Equations Chapter 7 -- Inverse Trigonometric Functions Chapter 8 -- Improper Integrals and Integration Techniques
Learning Calculus: Overcoming Our Artificial Need for Precision Accepting that numbers can do strange, new things is one of the toughest parts of math: There’s numbers between the numbers we count with? (Yes — decimals)There’s a number for nothing at all? (Sure — zero)The number line is two dimensional? Calculus is a beautiful subject, but challenges some long-held assumptions: Numbers don’t have to be perfectly accurate? Today’s post introduces a new way to think about accuracy and infinitely small numbers. Counting Numbers vs. Not every number is the same. Our first math problems involve counting: we have 5 apples and remove 3, or buy 3 books at $10 each. We later learn about fractions and decimals, and things get weird: What’s the smallest fraction? It gets worse. We’re hit with a realization: we have limited accuracy for quantities that are measured, not counted. What do I mean? That’s cute, but you didn’t answer my question — what number is it? You may pout, open your calculator and say it’s “18.8495…”. We don’t know! Why? But I need exact numbers!
Web-Based Study Guides The Mth 253-256 sequence forms the core mathematics sequence for engineering, mathematics, and some science majors at Oregon State University. These courses cover sequences and series, multivariable calculus, vector calculus, and differential equations and have a total enrollment each year of approximately 1000. The purpose of this project is to develop Web-based study guides for these courses that can be used by students currently enrolled in these courses and serve as a resource for the OSU community. Web-based study guides take advantage of the power of hypertext links. In a book topics are ordered linearly. Accessing the Study Guides The study guides can be accessed by clicking on the buttons above or on the links below. There are also calculator tutorials for the TI-85 and HP 38G. Questions and Comments Please see the copyright page for information on contacting the authors of these pages. Personnel The principal investigators for this project are Dennis Garity and Satish Reddy. Funding