Calculus Series and Differential Equations Mathematics 1b Spring 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 1b you will become more well versed in the language of modern mathematics and learn more about its applications to other disciplines Math 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 methods 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 environmental 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 We will start the semester with integration You should already be familiar with the definite integral its definition as the limit of Riemann sums and its calculation using antiderivatives and u substitution The definite integral enables us to tackle a multitude of problems in a wide array of fields we will use the notion of Riemann sums slicing approximating and summing to apply integration in various contexts More important than any one particular application is the ability to apply the integration as appropriate in problem solving we will devote time to developing your skill in doing this We ll spend a short time at the very beginning of the course looking at methods of integration including the integration analogues of both the Chain Rule and Product Rule for differentiation We ll look at some transformations of integrals that enable us to tackle them more efficiently and then move on to applications of the definite integral 2 In the second unit of the course we will approximate familiar functions like e x e x sin x and cos x2 by polynomials The functions listed are challenging to evaluate and some are challenging to integrate By using polynomials of increasingly large degree we can get increasingly good approximations to the functions In fact we will find that each of these functions has a representation as an infinite polynomial study infinite sums 3 3 You already are aware that a rational number such as 31 can be represented by an infinite sum 10 100 3 3 for the case at hand Actually irrational numbers such as e and have representations as 1000 10000 infinite sums as well We will end with differential equations equations modeling rates of change Differential equations permeate quantitative analysis throughout the sciences in physics chemistry biology environmental 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 Text Single Variable Calculus Concepts and Contexts by James Stewart Second edition Brooks Cole 2001 This text is available at the Harvard Coop There will be supplementary material available as well Class and Problem Sessions Math 1b is taught in sections that meet three hours per week Each section of Math 1b has a Course Assistant who will be in class collect and correct homework assignments and hold weekly problem sessions You are strongly encouraged to attend these problem sessions as they are an integral part of the course and will be generally be devoted to working problems and amplifying the lecture material The pace of the course is rather fast so these sessions should be particularly valuable to you in learning the material A schedule of all problem sessions will be posted on the course web site feel free to go to any Math 1b Course Assistant s Problem Session 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 1 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 assignment 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 assistants 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 two homework scores will be dropped if you are in a TTh section and your lowest three homework scores will be dropped if you are in a MWF section 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 However write ups you hand in must be your own work you must be comfortable explaining