Course Logistics for Math 195. Instructor: Alexander Hahn, Professor of Mathematics, 238 Hayes-Healy or Honors Program Office.

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1 Course Logistics for Math 195 Instructor: Alexander Hahn, Professor of Mathematics, 238 Hayes-Healy or Honors Program Office. Class Times and Location: MWF, 9:35 am to 10:25 am, 214 DeBartolo Hall, and T, 11:00 am to 11:50 am, 129 Hayes-Healy Building. University Honor Code: In effect throughout the semester Text: Alexander J. Hahn, Basic Calculus: From Archimedes to Newton to its Role in Science, Springer-Verlag, Course Website: Contains: Course logistics; current homework assignments; listing of the course teams; links to the web site for the text and to complete solutions of the homework problems. Daily Homework: Study and problem solving assignments will be due for every class period. You won t be handing anything in, but you will need to be prepared. Your approach should be a mix between rolling up your sleeves with paper and pencil by your side and collaboration with fellow students. Structure of Classes: At the beginning of each class, I ll ask if you have any questions about the assigned material and problems. Your questions should grow out of your studies and I ll expect them to be specific. Make a serious effort to understand the solutions of the problems on your own before you turn to the solutions on the course website. After your questions are addressed in class there will be a quiz. After the quiz, the class will continue with a combination of explanations and questions/answers - often with you at the board. Quizzes: These will have two questions each, will about 5 minutes long, and will count 10 points. Both questions will be routine: one based on the study material and the other on a problem. There will be around 35 of these in the course of the semester. Teams: Early in the semester I will ask you to group yourselves into teams of about 3 to 5 students each. In general, a team should consist of students who are likely to study together (at least occasionally). Team Assignments: There will be one problem set per Chapter of the text for a total of seven for the semester (or one about every two weeks). These assignments - taken from the Additional Exercises Sections on the course

2 website - are to be done as follows: Do the problems on your own first. At a later time, discuss your solutions at a meeting of your team and exchange ideas about your approaches. If no one has a complete solution to a problem, your team should brainstorm and try to produce one. Drawing on the best work of its members, each team compiles one solution set that is to be handed in. Each team member is to grade - carefully and honestly - his/her own performance on each team assignment from 0 to 10 in two categories: First, the effort expended in solving the problems independently, and secondly, the understanding of the solutions at the time the assignment is handed in by the team. These scores should be placed on the cover sheet next to the name of each team member. These team assignments will be corrected/graded and receive an additional 30 points maximum, so that the highest score that an individual student can achieve is 50 points Examinations: Wednesday, October 8 and Wednesday, November 19 both in our DeBartolo classroom and at the usual time. Final Exam: Monday, December 15 from 8 am to 10 am. (Location to be announced). Absences: From a quiz: if excused, an average grade will be recorded; if not, a zero will be recorded. Absence from an exam: makeup only if circumstances are serious. Grading Scheme: The total possible points for the quizzes will be around 350. The seven team assignments will count a total of 350 points. The two examinations will be 100 points each, and the final will be 200. An additional maximum of 200 points will be awarded for class participation. Grand total: 1300 points. Office Hours: By Appointment with Wendy Wolfe. Either in 238 Hayes- Healy or the Honors Program Office. Course Consultants: Four Honors Program students who took the course last year (and performed very well) will be available to you for assistance, each for a total of 2 hours per week. They will contact you by and introduce themselves to you. They are Alicia Lachiondo, Alicia.J.Lachiondo.1@nd.edu Joey Minta, Joseph.P.Minta.1@nd.edu Paul, Switaj, Paul.J.Switaj.1@nd.edu Noel Teske, Noelle.M.Teske.5@nd.edu Calculators: To be used only in elementary mathematics mode, rather than calculus or graphing calculator mode.

3 The Course Goals: Mathematics courses often present elements of the subject with an exclusive emphasis on a mastery of routines without making a serious effort to demonstrate the larger impact of this discipline. Interesting historical, scientific, and cultural connections are often left to footnotes and exercises (if they are engaged at all). The purpose of this course is: to review essential aspects of geometry, trigonometry, and analytic geometry, and then to develop the calculus, in modern form, but from within the scientific agenda of the times of creation of these subjects. In terms of key phrases, the course proceeds as follows. Eratosthenes's estimate of the size of the Earth; the geometry with which Aristarchus estimated the sizes of the Earth, Moon and Sun, and the distances between them (stellar parallax is included here); a study of the mathematics of the Ptolemaic system; conic sections and Archimedes's computation of the area of a parabolic section; analytic geometry in the context of the astronomy of Galileo and Kepler; a modern course in essential calculus based on the work of Leibniz and Newton; and, finally, a serious look into the heart of Newton's Principia Mathematica, the most famous and influential book of mathematics and science ever written. In the process, you will engage the practice of a lot of modern mathematics (including geometry, algebra, trigonometry, analytic geometry, and differential and integral calculus) often in novel ways Outline of the Material (taken from the text): 1. The Greeks Measure the Universe The Pythagoreans Measure Length The Measure of Angles Eratosthenes Measures the Earth Right Triangles Aristarchus Sizes up the Universe The Sandreckoner Postscript 21 Exercises Ptolemy and the Dynamics of the Universe A Geometry of Shadows and the Motion of the Sun Geometry in the Almagest The Solar Model The Modern Perspective Another Look at the Solar Model Epicycles 43

4 2.7. Postscript 45 Exercises Archimedes Measures Area The Conic Sections The Question of Area Playing with Squares The Area of the Parabolic Section The Method Postscript 63 Exercises A New Astronomy and a New Geometry A New Astronomy The Studies of Galileo The Geometry of Descartes Circles and Trigonometry The Ellipse Cavalieri's Principle Kepler's Analysis of the Orbits The Method of Successive Approximations Computing Orbital Information Postscript 100 Exercises The Calculus of Leibniz Straight Lines Tangent Lines to Curves Areas and Differentials The Fundamental Theorem of Calculus Functions 126 A. The Derivative 129 B. Antiderivatives Some Applications 133 A. Finding Maximum and Minimum Values 133 B. Volumes 134 C. Lengths of Curves Postscript 138 Exercises The Calculus of Newton 145

5 6.1. Areas under Simple Curves The Fundamental Theorem (again) Computing Definite Integrals Moving Points The Trajectory of a Projectile Application to Ballistics? Postscript 171 Exercises The Principia Equal Areas in Equal Times Analyzing Centripetal Force The Inverse Square Law Test Case: The Orbit of the Moon The Law of Universal Gravitation Incredible Consequences Postscript 195 Exercises Analysis of Functions Putting a Limit to the Test Continuous Functions Differentiability Derivatives as Rates of Change About Derivatives 220 A. Computing Derivatives 220 B. Some Theoretical Concerns Derivatives of Trigonometric Functions Increase and Decrease of Functions Maximum and Minimum Values Postscript 238 Exercises 239

Basic Calculus. Alexander J. Hahn. From /vrcnimeaes to 1 Newton to its Kole in Jcience. Шк) Springer

Basic Calculus. Alexander J. Hahn. From /vrcnimeaes to 1 Newton to its Kole in Jcience. Шк) Springer Basic Calculus From /vrcnimeaes to 1 Newton to its Kole in Jcience Alexander J. Hahn Deportment of Mathematics University of Notre Dame Шк) Springer Contents iiiiiimmiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii