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4434Contemporary Calculus I
http://www.merlot.org/merlot/viewMaterial.htm?id=913152
This is a textbook for differential calculus with explanations, examples, worked solutions, problem sets and answers. It has been reviewed by calculus instructors and class-tested by them and the author.Topics are typically introduced by way of applications, and the text contains the usual theorems and techniques of a first course in calculus. Besides technique practice and applications of the techniques, the examples and problem sets are also designed to help students develop a visual and conceptual understanding of the main ideas of differential calculus.This is the first of a three-part calculus series. The others are available on the author's website. Statistics Using Technology
http://www.merlot.org/merlot/viewMaterial.htm?id=908731
This is a statistics textbook to be used in an introductory statistics class. This book uses technology to calculate probabilities. The approach to this textbook is to ask people to interpret statistics and think statistically. Chapter 1: Statistical Basics Section 1.1: What is Statistics? Section 1.2: Sampling Methods Section 1.3: Experimental Design Section 1.4: How Not to Do Statistics Chapter 2: Graphical Descriptions of Data Section 2.1: Qualitative Data Section 2.2: Quantitative Data Section 2.3: Other Graphical Representations of Data Chapter 3: Numerical Descriptions of Data Section 3.1: Measures of Center Section 3.2: Measures of Spread Section 3.3: Ranking Chapter 4: Probability Section 4.1: Empirical Probability Section 4.2: Theoretical Probability Section 4.3: Conditional Probability Section 4.4: Counting Techniques Chapter 5: Discrete Probability Distributions Section 5.1: Basics of Probability Distributions Section 5.2: Binomial Probability Distribution Section 5.3: Mean and Standard Deviation of Binomial Distribution Chapter 6: Continuous Probability Distributions Section 6.1: Uniform Distribution Section 6.2: Graphs of the Normal Distribution Section 6.3: Finding Probabilities for the Normal Distribution Section 6.4: Assessing Normality Section 6.5: Sampling Distribution and the Central Limit Theorem Chapter 7: One-Sample Inference Section 7.1: Basics of Hypothesis Testing Section 7.2: One-Sample Proportion Test Section 7.3: One-Sample Test for the Mean Chapter 8: Estimation Section 8.1: Basics of Confidence Intervals Section 8.2: One-Sample Interval for the Proportion Section 8.3: One-Sample Interval for the Mean Chapter 9: Two-Sample Inference Section 9.1: Paired Samples for Two Means Section 9.2: Independent Samples for Two Means Section 9.3: Two Proportions Chapter 10: Regression and Correlation Section 10.1: Regression Section 10.2: Correlation Section 10.3: Inference for Regression and Correlation Chapter 11: Chi-Square and ANOVA Tests Section 11.1: Chi-Square Test for Independence Section 11.2: Chi-Square Goodness of Fit Section 11.3: Analysis of Variance (ANOVA)APEX Calculus I
http://www.merlot.org/merlot/viewMaterial.htm?id=908754
A college level treatment of standard Calculus topics beginning with limits and ending with iterated (multivariable) integration. The material is presented in a "traditional format," designed to make the transition from popular Calculus books (such as Stewart or Thomas/Finney) straightforward. This text is suitable for most "Calc I," "Calc II" and "Calc III" courses. This text is currently in use in MA 123, 124 and 215, Calculus and Analytic Geometry I, II & III at Virginia Military Institute.The links above are for the first portion of the book. The second part of the book can be read here or purchased here. The third part of the book is not yet completely finished.License: Creative Commons Attribution Sharealike Noncommercial. This license is very open. It allows reuse, remixing, and distribution, but prohibits commercial use and requires any remixes use the same license as the original. This limits where the content can be remixed into, but on the other hand ensures that no-one can remix the content then put the remix under a more restrictive license. The non-commercial clause can make getting printed copies of remixes challenging depending upon how strictly the authors interpret the clause.Logic and Proofs
http://www.merlot.org/merlot/viewMaterial.htm?id=907321
Logic is a remarkable discipline. It is deeply tied to mathematics and philosophy, as correctness of argumentation is particularly crucial for these abstract disciplines. Logic systematizes and analyzes steps in reasoning: correct steps guarantee the truth of their conclusion given the truth of their premise(s); incorrect steps allow the formulation of counterexamples, i.e., of situations in which the premises are true, but the conclusion is false.Recognizing (and having conceptual tools for recognizing) the correctness or incorrectness of steps is crucial in order to critically evaluate arguments, not just in philosophy and mathematics, but also in ordinary life. This skill is honed by working in two virtual labs. In the ProofLab you learn to construct complex arguments in a strategically guided way, whereas in the TruthLab the emphasis is on finding counterexamples systematically.Who should take this course?This is an introductory course designed for students from a broad range of disciplines, from mathematics and computer science to drama and creative writing. The highly interactive presentation makes it possible for any student to master the material. Concise multimedia lectures introduce each chapter; they discuss, in detail, the central notions and techniques presented in the text, but also articulate and motivate the learning objectives for each chapter.Open & Free VersionThe Open & Free, Logic & Proofs course includes the first five chapters of Logic & Proofs, providing a basic introduction to sentential logic. A full version of Logic & Proofs, including both sentential and predicate logic, is also available without technical or instructor support to independent users, for a small fee. No credit is awarded for completing either the Open & Free, Logic & Proofs course or the full, unsupported Logic & Proofs course.Probability and Statistics
http://www.merlot.org/merlot/viewMaterial.htm?id=906143
'This module consists of three units:Unit 1: Descriptive Statistics and Probability DistributionsDescriptive statistics in unit one is developed either as an extension of secondary mathematics or as an introduction to first time learners of statistics. It introduces the measures of dispersion in statistics. The unit also introduces the concept of probability and the theoretical treatment of probability.Unit 2: Random variables and Test DistributionsThis unit requires Unit 1 as a prerequisite. It develops from the moment and moment generating functions, Markov and Chebychev inequalities, special univariate distributions, bivariate probability distributions and analyses conditional probabilities. The unit gives insights into the analysis of correlation coefficients and distribution functions of random variables such as the Chi-square, t and F.Unit 3: Probability TheoryThis unit builds up from unit 2. It analyses probability using indicator functions. It introduces Bonferoni inequality random vectors,, generating functions, characteristic functions and statistical independence random samples. It develops further the concepts of functions of several random variables and independence of X and S2 in normal samples order statistics. The unit summarises with the treatment of convergence and limit theorems.'Analysis 1
http://www.merlot.org/merlot/viewMaterial.htm?id=906080
'Part 1 consists of three units:Unit 1 - Analysis on the real lineIn this unit we start by decomposing the set of real numbers into its subsets. We then define the so called standard metric on to be able to study its structure which consists of concepts like open and closed intervals, neighbourhoods, interior and limit points leading to examples of open and closed subsets of . Countability of such subsets and numerical sequences are also essential in the structure of as a metric space. We also study functions defined on sets of real numbers with respect to concepts of continuity differentiability and integrability.Unit 2 – Vector AnalysisThis unit deals with mainly vector calculus and its applications. Thus we specifically consider concepts like gradient, divergence and curl. This leads to well known theorems like Green’s stokes divergence theorems and other related results. The curvilinear coordinates system is also given a treat in this unit.Unit 3 – Complex AnalysisThe main objective in the study of this unit is to define a function of a complex variable and then look at its degree of smoothness such as existence of limit, continuity and differentiability along the lines of calculus of functions of a real variable. Also a look at integration and power series involving a function of a complex variable will complete this unit.In addition to downloading the files below, you can also view them directly:Module part 1 (Scribd)Intro to module (YouTube)Overview of module (YouTube)'Basic Mathematics
http://www.merlot.org/merlot/viewMaterial.htm?id=906086
'This module consists of three units which are as follows:Unit 1: (i) Sets and Functions (ii) Composite FunctionsThis unit starts with the concept of a set. It then intoroduces logic which gives the learner techniques for distinguishing between correct and incorrect arguments using propositions and their connectives. A grasp of sets of real numbers on which we define elementary functions is essential. The need to have pictorial representations of a function necessitates the study of its graph. Note that the concept of a function can also be viewed as an instruction to be carried out on a set of objects. This necessitates the study of arrangements of objects in a certain order, called permutations and combinations.Unit 2: Binary OperationsIn this unit we look at the concept of binary operations. This leads to the study of elementary properties of integers such as congruence. The introduction to algebraic structures is simply what we require to pave the way for unit 3.Unit 3: Groups, Subgroups and HomomorphismThis unit is devoted to the study of groups and rings. These are essentially sets of numbers or objects which satisfy some given axioms. The concepts of subgroup and subring are also important to study here. For the sake of looking at cases of fewer axiomatic demands we will also study the concepts of homomorphisms and isomorphisms. Here we will be reflecting on the concept of a mapping or a function from either one group to the other or from one ring to the other in order to find out what properties such a function has.'Brief Calculus
http://www.merlot.org/merlot/viewMaterial.htm?id=906090
'This short text is designed more for self-study or review than for classroom use; full solutions are given for nearly all the end-of-chapter problems.''The focus is mainly on integration and differentiation of functions of a single variable, although iterated integrals are discussed. Infinitesimals are used when appropriate, and are treated more rigorously than in old books like Thompson's Calculus Made Easy, but in less detail than in Keisler's Elementary Calculus: An Approach Using Infinitesimals.Numerical examples are given using the open-source computer algebra system Yacas, and Yacas is also used sometimes to cut down on the drudgery of symbolic techniques such as partial fractions. Proofs are given for all important results, but are often relegated to the back of the book, and the emphasis is on teaching the techniques of calculus rather than on abstract results.'Calculus
http://www.merlot.org/merlot/viewMaterial.htm?id=906096
'This is a four unit module. The first two units cover the basic concepts of the differential and integral calcualus of functions of a single variable. The third unit is devoted to sequences of real numbers and infinite series of both real numbers and of some special functions. The fourth unit is on the differential and integral calculus of functions of several variables.Starting with the definitions of the basic concepts of limit and continuity of functions of a single variable the module proceeds to introduce the notions of differentiation and integration, covering both methods and applications.Definitions of convergence for sequences and infinite series are given. Tests for convergence of infinite series are presented, including the concepts of interval and radius of convergence of a power series.Partial derivatives of functions of several variables are introduced and used in formulating Taylor’s theorem and finding relative extreme values.'Differential Equations
http://www.merlot.org/merlot/viewMaterial.htm?id=906101
'This module consisst of two units, namely Introduction to Ordinary differential equations and higher order differential equations respectively. In unit one both homogeneous and non-homogeneous ordinary differential equations are discussed and their solutions obtained with a variety of techniques.Some of these techniques include the variation of parameters, the method of undetermined coefficients and the inverse operators. In unit two series solutions of differential equations are discussed .Also discussed are partial differential equations and their solution by separation of variables. Other topics discussed are Laplace transforms, Fourier series, Fourier transforms and their applications.Outline: SyllabusUnit 1: Introduction to Ordinary Differential EquationsLevel 2. Priority A. Calculus 3 is prerequisite.First order differential equations and applications. Second order differential equations. Homogeneous equations with constant coefficients. Equations with variable coefficients. Non-homogeneous equations. Undetermined coefficients. Variations of parameters. Inverse differential operators.Unit 2: Higher Order Differential Equations and ApplicationsLevel 2. Priority B. Differential Equations 1 is prerequisite.Series solution of second order linear ordinary differential equations. Special functions. Methods of separation of variables applied to second order partial differential equations. Spherical harmonics. Laplace transform and applications. Fourier series, Fourier transform and applications'