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18.01 Single Variable CalculusAs taught in: Fall 2006 You can use calculus to find the slope of any function at a given point. (Image by MIT OpenCourseWare.) This introductory calculus course covers differentiation and integration of functions of one variable, with applications.
Course DescriptionThis course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space. MIT OpenCourseWare offers another version of 18.02, from the Spring 2006 term. Both versions cover the same material, although they are taught by different faculty and rely on different textbooks. Multivariable Calculus (18.02) is taught during the Fall and Spring terms at MIT, and is a required subject for all MIT undergraduates. Course DescriptionDifferential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams. This unit is the second in the MSXR209 series of five units on mathematical modelling. In this unit you are asked to relate the stages of the mathematical modelling process to a previously formulated mathematical model. This example, that of skid mark produced by vehicle tyres, is typical of accounts of modelling that you may see in books, or produced in the workplace. The aim of this unit is to help you to draw out and to clarify mathematical modelling ideas by considering the example. It assumes that you have studied Modelling pollution in the Great Lakes (MSXR209_1). Once market research has been conducted the data needs to be analysed and presented. This unit provides examples of how to achieve this using tally charts and graphs. It also provides suggestions of how best to draw conclusions from your research and present your findings. This unit looks at complex numbers. You will learn how they are defined, examine their geometric representation and then move on to looking at the methods for finding the nth roots of complex numbers and the solutions to simple polynominal equations. You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The unit includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry. This unit is the third in the MSXR209 series of five units on mathematical modellng. It provides an overview of the processes involved in developing models, starting by explaining how to specify the purpose of the model. It then moves on to look at aspects involved in creating models, such as simplifying problems, choosing variables and parameters, formulating relationships and finding solutions. You will also look at interpreting results and evaluating models. This unit assumes that you have previously studied Modelling pollution in the Great Lakes (MSXR209_1) and Analysing skid marks (MSXR209_2). Diagrams, charts and graphs are used by all sorts of people to express information in a visual way, whether it's in a report by a colleague or a plan from your interior designer. This unit will teach you how to interpret these tools and how to use them yourself to convey information more effectively. This unit extends the ideas introduced in the unit on first-order differential equations to a particular type of second-order differential equations which has a variety of applications. The unit assumes that you have previously had a basic grounding in calculus, know something about first-order differential equations and some familiarity with complex numbers. This unit is aimed at teachers who wish to review how they go about the practice of teaching maths, those who are considering becoming maths teachers, or those who are studying maths courses and would like to understand more about the teaching process. This Unit will introduce you to a number of ways of representing data graphically and of summarising data numerically. You will learn the uses for pie charts, bar charts, histograms and scatterplots. You will also be introduced to various ways of summarising data and methods for assessing location and dispersion. Graphs are a common way of presenting information. However, like any other type of representation, graphs rely on shared understandings of symbols and styles to convey meaning. Also, graphs are normally drawn specifically with the intention of presenting information in a particularly favourable or unfavourable light, to convince you of an argument or to influence your decisions. This unit will help you to identify and use information in maths and statistics, whether for your work, study or personal purposes. Experiment with some of the key resources in this subject area, and learn about the skills which will enable you to plan searches for information, so you can find what you are looking for more easily. Discover the meaning of information quality, and learn how to evaluate the information you come across. You will also be introduced to the many different ways of organising your own information, and learn how to reference it properly in your work. Finally, discover how to keep up to date with the latest developments in your area of interest by using tools such as RSS and mailing lists. This unit introduces the topic of differential equations. The subject is developed without assuming that you have come across it before, but it is taken for granted that you have a basic grounding in calculus. In particular, you will need to have a good grasp of the basic rules for differentiation and integration. This unit is concerned with the technique of expressing a periodic function as a sum of terms, where each term is a constant, a sine function or a cosine function. There is a strong analogy with the technique of expressing a (non-periodic) function as a Taylor series, which is a sum of terms that are powers of the independent variable(s); in both cases, working with just the first few terms generally gives a useful approximation. This unit assumes the following background knowledge: the definition of the period; forced oscillations and resonance; integration by parts.  This unit is concerned with two main topics. In Section 1, you will learn about another kind of graphical display, the boxplot. A boxplot is a fairly simple graphic, which displays certain summary statistics of a set of data. Boxplots are particularly useful for assessing quickly the location, dispersion, and symmetry or skewness of a set of data, and for making comparisons of these features in two or more data sets. Boxplots can also be useful for drawing attention to possible outliers in a data set. The other topic, which is covered in Sections 2 and 3, is that of dealing with data presented in tabular form. You are, no doubt, familiar with such tables: they are common in the media and in reports and other documents. Yet it is not always straightforward to see at first glance just what information a table of data is providing, and it often helps to carry out certain calculations and/or to draw appropriate graphs to make this clearer. In this unit, some other kinds of data tables and some different approaches are covered. In our everyday lives we use we use language to develop ideas and to communicate them to other people. In this unit we examine ways in which language is adapted to express mathematical ideas. This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognise mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the Calculator Book, Tapping into Mathematics With the TI-83 Graphics Calculator. The unit ends by asking you to reflect on the process of studying mathematics. In order to complete this unit you will need to have obtained a Texas Instruments TI-83 calculator and the book Tapping into Mathematics With the TI-83 Graphics Calculator (ISBN 0201175479). In this unit you will see first how to convert vectors from geometric form, in terms of a magnitude and direction, to component form, and then how conversion in the opposite sense is accomplished. The ability to convert between these different forms of a vector is useful in certain problems involving displacement and velocity, as shown in Section 2, in which you will also work with bearings. This unit is the fourth in the MSXR209 series of five units on mathematical modelling. In this unit you will be taken through the whole modelling process in detail, from creating a first simple model, through evaluating it, to the subsequent revision of the model by changing one of the assumptions. The problem that will be examined is one based on heat transfer. This unit assumes you have studied Modelling pollution in the Great Lakes (MSXR209_1), Analysing skid marks (MSXR209_2) and Developing modelling skills (MSXR209_3). This unit is the first in the MSXR209 series of five units that introduce the idea of modelling with mathematics. This unit centres on a mathematical model of how pollution levels in the Great Lakes of North America vary over a period of time. It demonstrates that, by keeping the model as simple as possible extremely complex systems can be understood and predicted. This is the fifth and final unit in the MSXR209 series on mathematical modelling. In this unit we revisit the model developed in the first unit of this series on pollution in the Great Lakes of North America. Here we evaluate and revise the original model by comparing its predictions against data from the lakes before finally reflecting on the techniques used. This unit assumes you have studied Modelling pollution in the Great Lakes (MSXR209_1), Analysing skid marks (MSXR209_2), Developing modelling skills (MSXR209_3) and Modelling heat transfer (MSXR209_4). This unit lays the foundation of the subject of mechanics. Mechanics is concerned with how and why objects stay put, and how and why they move. In particular, this unit - Modelling Static Problems - considers why objects stay put. And it assumes that you have a good working knowledge of vectors. This unit is intended to develop your understanding of Newtonian mechanics in relation to oscillating systems. In addition to a basic grounding in calculus, this unit assumes that you have some understanding of how to solve second-order linear constant-coefficient differential equations; how to take the dot product of two vectors; of solving statics problems; and of applying Newton's second law to mechanical problems. This unit lays the foundation of Newtonian mechanics and in particular the procedure for solving dynamics problems. The preresquisite skills needed for this unit are the ability to solve first and second-order differential equations, a knowledge of vectors, and an understanding of the concept of a force This unit shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this unit are an understanding of ordinary differential equations and basic familiarity with partial differential equations. This unit is intended to further develop your understanding of Newtonian mechanics in relation to oscillating systems. In addition to a basic grounding in solving systems of differential equations, this unit assumes that you have some understanding of eigenvalues and eigenvectors. Do fractions and decimals make you apprehensive about maths? Do you lack confidence in dealing with numbers? If so, then this unit is for you. The unit will explain the basics of working with positive and negative numbers and how to multiply and divide with fractions and decimals. Do fractions and decimals make you apprehensive about maths? Do you lack confidence in dealing with numbers? If so, then this unit is for you. The unit will explain the basics of working with positive and negative numbers and how to multiply and divide with fractions and decimals. Number systems and the rules for combining numbers can be daunting. This unit will help you to understand the detail of rational and real numbers, complex numbers and integers. You will also be introduced to modular arithmetic and the concept of a relation between elements of a set. This unit looks at a wide variety of ways of comparing prices and the construction of a price index. You will also look at the Retail Price Index (RPI) and the Consumer Price Index (CPI), indices used by the UK Government to calculate the percentage by which prices in general have risen over any given period. You wil also look at the important statistical and mathematical ideas that contribute to the construction of a price index. From politics to cookery, ratios, proportions and percentages are part of everyday life. This unit is designed to help you become more familiar with how figures can be manipulated, then you can check whether that discount really is as big as they claim! Sometimes the best way to understand a set of data is to sketch a simple graph. This exercise can reveal hidden trends and meanings not clear from just looking at the numbers. In this unit you will review the various approaches to sketching graphs and learn some more advanced techniques. Scientific calculators are a wonderful invention, but they're only as good as the people who use them. If you often get an unexpected – or ridiculous – result when you press the ‘enter’ button, this unit is for you. Learn how to do a calculation correctly and get the right answer every time. From paving your patio to measuring the ingredients for your latest recipe, squares, roots and powers really are part of everyday life. This unit reviews the basics of all three and also describes scientific notation, which is a convenient way of writing or displaying large numbers. Many of us struggle to understand the fractions, formulas and calculations needed for everyday tasks. This unit provides an overview of Open University course Y162 Starting with maths, which is designed to help develop the skills needed for higher level study while also developing knowledge and understanding about maths. Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum. This unit shows how various situations can be modelled by a system of linear differential equations. The prerequisite requirements to gain full advantage from this unit are a basic understanding of differential equations, a familiarity with the properties of matrices and determinants and some understanding of eigenvalues and eigenvectors. This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane. Attempts to answer problems in areas as diverse as science, technology and economics involve solving simultaneous linear equations. In this unit we look at some of the equations that represent points, lines and planes in mathematics. We explore concepts such as Euclidean space, vectors, dot products and conics. This unit focuses on your initial encounters with research. It invites you to think about how perceptions of mathematics have influenced you in your prior learning, your teaching and the attitudes of learners.  Complete College Algebra/Pre-calculus online text Courtesy of Blinn College and Texas A&M University
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