Calculus Series and Differential Equations Mathematics S 1b Summer 2003 Course Information and Syllabus Course Content and Goals About four hundred years ago Galileo wrote The book of the universe is written in the language of mathematics Although the language of mathematics has evolved over time the statement has as much validity today as it did when it was written In Mathematics S 1b you will become more vell versed in the language of modern mathematics and learn about its applications to other disciplines Math S 1b is a second semester calculus course for students who have previously been introduced to the basic ideas of differential and integral calculus Over the semester we will study three related topics topics that form a central part of the language of modern science applications and techniques of integration infinite series and the representation of functions by infinite polynomials known as power series differential equations The material we take up in this course has applications in physics chemistry biology enviromental science astronomy economics and statistics We want you to leave the course not only with computational ability but with the ability to use these notions in their natural scientific contexts and with an appreciation of their mathematical beauty and power In your previous math courses you studied differential calculus and were introduced to integral calculus You studied the Fundamental Theorem of Calculus which illuminates the connection between differentiation and integration We begin this course by looking at various applications of the definite integral The definite integral enables us to tackle many problems including calculuating the net change in amount given a varying density determining volume and arclength and computing physical quantities In order to compute integrals we will study some techniques of integration such as the integration analogues of both the Product Rule and Chain Rule for differentiation We will briefly look at some alternative transformations of integrals that enable us to tackle them more efficiently The goal is not to transform you into an integration automaton we live in the computer age but to have you acquire familiarity with the techniques and the ability to apply them to some standard situations More important is the ability to apply the integration as appropriate in problem solving we will devote time to developing your skill in doing this In the second unit of the course we will study infinite sums You already are aware that a rational number 3 3 3 3 such as 31 can be represented by an infinite sum 10 100 1000 10000 for the case at hand Actually irrational numbers such as e and have representations as infinite sums as well In fact we will find that many functions such as f x ex and f x sin x can be represented by infinite polynomials known as power series We will learn to compute understand and manipulate these representations Polynomial approximations based on these power series representations are widely used by engineers physicists and many other scientists We will end with differential equations equations modeling rates of change Differential equations permeate quantitative analysis throughout the sciences in physics chemistry biology enviromental science astronomy and social sciences In a beautiful and succinct way they provide a wealth of information By the end of the course you will appreciate the power and usefulness differential equations and you will see how the work we have done with both series and integration comes into play in analyzing their solutions Class Meeting Times Tuesdays and Thursdays 1 00 3 30 pm in Science Center 110 Instructor Robin Gottlieb gottlieb math harvard edu Office Science Center 429 phone 617 495 7882 Office Hours Tuesdays and Thursdays 3 30 4 30 pm Morning Sessions Every day 11 10 am 12 00 noon in SC 102b The daily morning sessions conducted by Andrew Lobb are an integral part of the course All exams except for the final exam will take place in morning sessions Text Stewart s Calculus Concepts and Contexts 2nd edition This text is published by Brooks Cole and is available at the Harvard Coop or via the internet There will be supplementary material available on the web 1 The assumption is that students come into the course having seen most of the material in Chapters 1 4 and 5 1 5 5 We ll cover most of Chapters 5 8 plus topics covered in supplementary materials Homework Problems are an integral part of the course it is virtually impossible to learn the material and to do well in the course without working through the homework problems in a thoughtful manner Don t just crank through computations and write down answers think about the problems posed the strategy you employ the meaning of the computations you perform and the answers you get It is often in this reflection that the greatest learning takes place An assignment will be given at each class meeting Unless otherwise specified the problem set is due at the following class meeting and will be returned graded at the subsequent class If you miss a class then you are responsible for obtaining the assignment and handing it in on time Solutions put together by the course assistant will be available on the course website When your homework assignments are returned to you you can consult the solutions for help with any mistakes you might have made Problem sets must be turned in on time When computing your final homework grade your lowest homework score will be dropped Note that homework problems will sometimes look a bit different from problems specifically explicitly discussed in class To do mathematics you need to think about the material not simply follow recipes Following preset recipes is something computers are great at We want you to be able to do more than this Giving you problems different from those done in class is consistent with our goal of teaching you the art of applying ideas of integration and differentiation to different contexts Feel free to use a calculator or computer to check or investigate problems for homework However an answer with the explanation because my calculator says so will not receive credit Use the calculator as a learning tool not as a crutch Calculators will not be allowed on examinations due in part to equity issues You are welcome to collaborate with other students on solving homework problems in fact you are encouraged to do so and we will provided you with contact information for your