Courses tagged with "Taking derivatives" (37)

This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable.  Put differently, complex analysis is the theory of the differentiation and integration of functions that depend on one complex variable.  Such functions, beautiful on their own, are immediately useful in Physics, Engineering, and Signal Processing.  Because of the algebraic properties of the complex numbers and the inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear.  Simply put, you will be working with lines and sets and very specific functions on the complex planedrawing pictures of them and teasing out all of their idiosyncrasies.  You will again find yourself calculating line integrals, just as in multivariable calculus.  However, the techniques you learn in this course will help you get past many of the seeming dead-ends you ran up against in…

This course will introduce you to the fundamentals of probability theory and random processes. The theory of probability was originally developed in the 17th century by two great French mathematicians, Blaise Pascal and Pierre de Fermat, to understand gambling. Today, the theory of probability has found many applications in science and engineering. Engineers use data from manufacturing processes to sample characteristics of product quality in order to improve the products being produced. Pharmaceutical companies perform experiments to determine the effect of a drug on humans and use the results to make decisions about treatment of illnesses, while economists observe the state of the economy over periods of time and use the information to forecast the economic future. In this course, you will learn the basic terminology and concepts of probability theory, including random experiments, sample spaces, discrete distribution, probability density function, expected values, and conditional probability. You will al…

Mathematics is about structure, about reasoning, and about modeling.  This course braids these three threads together.  Mathematical logic began as the study of the reasoning used in mathematics, but it turns out to be useful in describing the mathematical concept of structure and in modeling automated reasoningthat is, modeling computation. The logical approach to structure gives an alternate perspective on such other mathematical subjects as combinatorics and abstract algebra.  This, for the most part, is described by the area of model theory, which is the focus of Unit 1. In Unit 2, we will look at modeling computation.  The central fact of these models, from a logical standpoint, is that once we can handle a computation as a definable mathematical object, we can prove that certain computations are impossible.  The most famous such proof is Gödel’s Incompleteness Theorem, showing that it is impossible to compute truth in a system sufficiently strong to describe natural number arithmetic.

This course will introduce you to a number of statistical tools and techniques that are routinely used by modern statisticians for a wide variety of applications. First, we will review basic knowledge and skills that you learned in MA121: Introduction to Statistics [1]. Units 2-5 will introduce you to new ways to design experiments and to test hypotheses, including multiple and nonlinear regression and nonparametric statistics. You will learn to apply these methods to building models to analyze complex, multivariate problems. You will also learn to write scripts to carry out these analyses in R, a powerful statistical programming language. The last unit is designed to give you a grand tour of several advanced topics in applied statistics. [1] http://www.saylor.org/courses/ma121/…

This course is designed to provide you with a simple and straightforward introduction to econometrics.  Econometrics is an application of statistical procedures to the testing of hypotheses about economic relationships and to the estimation of parameters.  Regression analysis is the primary procedure commonly used by researchers and managers whether their employments are within the goods or the resources market and/or within the agriculture, the manufacturing, the services, or the information sectors of an economy. Completion of this course in econometrics will help you progress from a student of economics to a practitioner of economics.  By completing this course, you will gain an overview of econometrics, develop your ability to think like an economist, hone your skills building and testing models of consumer and producer behavior, and synthesize the results you find through analyses of data pertaining to market-based economic systems.  In essence, professional economists conduct studies that combine…

This course is a continuation of the first-semester course titled Introduction to Computer Science I (CS101 [1]).  It will introduce you to a number of more advanced Computer Science topics, laying a strong foundation for future academic study in the discipline.  We will begin with a comparison between Javathe programming language utilized last semesterand C++, another popular, industry-standard programming language.  We will then discuss the fundamental building blocks of Object-Oriented Programming, reviewing what we learned last semester and familiarizing ourselves with some more advanced programming concepts.  The remaining course units will be devoted to various advanced Computer Science topics, including the Standard Template Library, Exceptions, Recursion, Searching and Sorting, and Template Classes.  By the end of the class, you will have a solid understanding of Java and C++ programming, as well as a familiarity with the major issues that programmers routinely address in a professional setting.

This course is the second installment of Single-Variable Calculus.  In Part I (MA101 [1]), we studied limits, derivatives, and basic integrals as a means to understand the behavior of functions.  While this end goal remains the same, we will now focus on adapting what we have learned to applications.  By the end of this course, you should have a solid understanding of functions and how they behave.  You should also be able to apply the concepts we have learned in both Parts I and II of Single-Variable Calculus to a variety of situations. We will begin by revisiting and building upon what we know about the integral.  We will then explore the mathematical applications of integration before delving into the second major topic of this course: series.  The course will conclude with an introduction to differential equations. [1] http:///courses/ma101/…

Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions.  You may find that these courses share many of the same basic concepts, and that Multivariable Calculus will simply extend your knowledge of functions to functions of several variables.  The transition from single variable relationships to many variable relationships is not as simple as it may seem; you will find that multi-variable functions, in some cases, will yield counter-intuitive results. The structure of this course very much resembles the structure of Single-Variable Calculus I and II.  We will begin by taking a fresh look at limits and continuity.  With functions of many variables, you can approach a limit from many different directions.  We will then move on to derivatives and the process by which we generalize them to higher dimensions.  Finally, we will look at multiple integrals, or integration over regions of space as opposed to intervals. The goal of Mu…

“Why is math important?  Why do I have to learn math?”  These are typical questions that you have most likely asked at one time or another in your education.  While you may learn things in math class that you will not use again, the study of mathematics is still an important one for human development.  Math is widely-used in daily activities (e.g. shopping, cooking, etc.) and in most careers (e.g. medicine, teaching, engineering, construction, business, statistics in psychology, etc.).  Math is also considered a “universal language.”  One of the fundamental reasons why you learn math is to help you tackle problems, both mathematical and non-mathematical, with clear, concise, and logical steps.  In this course, you will study important fundamental math concepts. This course begins your journey into the “Real World Math” series.  These courses are intended not just to help you learn basic algebra and geometry topics, but also to show you how these topics are used in everyday life.  In thi…

This introductory mathematics course is for you if you have a solid foundation in arithmetic (that is, you know how to perform operations with real numbers, including negative numbers, fractions, and decimals).  Numbers and basic arithmetic are used often in everyday life in both simple situations, like estimating how much change you will get when making a purchase in a store, as well as in more complicated ones, like figuring out how much time it would take to pay off a loan under interest. The subject of algebra focuses on generalizing these procedures.  For example, algebra will enable you to describe how to calculate change without specifying how much money is to be spent on a purchase-it will teach you the basic formulas and steps you need to take no matter what the specific details of the situation are.  Likewise, accountants use algebraic formulas to calculate the monthly loan payments for a loan of any size under any interest rate.  In this course, you will learn how to work with formulas that a…

“Everything is numbers.”  This phrase was uttered by the lead character, Dr. Charlie Epps, on the hit television show “NUMB3RS.”  If everything has a mathematical underpinning, then it follows that everything is somehow mathematically connected, even if it is only in some odd, “six degrees of separation (or Kevin Bacon)” kind of way. Geometry is the study of space (for now, mainly two-dimensional, with some three-dimensional thrown in) and the relationships of objects contained inside.  It is one of the more relatable math courses, because it often answers that age-old question, “When am I ever going to use this in real life?”  Look around you right now.  Do you see any triangles?  Can you spot any circles?  Do you see any books that look like they are twice the size of other books?  Does your wall have paint on it? In geometry, you will explore the objects that make up our universe.  Most people never give a second thought to how things are constructed, but there are geometric ru…

In this course, you will study basic algebraic operations and concepts, as well as the structure and use of algebra. This includes solving algebraic equations, factoring algebraic expressions, working with rational expressions, and graphing linear equations. You will apply these skills to solve real-world problems (word problems). Each unit will have its own application problems, depending on the concepts you have been exposed to. This course is also intended to provide you with a strong foundation for intermediate algebra and beyond. It will begin with a review of some math concepts formed in pre-algebra, such as ordering operations and simplifying simple algebraic expressions, to get your feet wet. You will then build on these concepts by learning more about functions, graphing of functions, evaluation of functions, and factorization. You will spend time on the rules of exponents and their applications in distribution of multiplication over addition/subtraction. This course provides students the opportuni…

Partial differential equations (PDEs) describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position. For example, the heat equation can be used to describe the change in heat distribution along a metal rod over time. PDEs arise as part of the mathematical modeling of problems connected to different branches of science, such as physics, biology, and chemistry. In these fields, experiment and observation provide information about the connections between rates of change of an important quantity, such as heat, with respect to different variables. These connections must be exploited to find an explicit way of calculating the unknown quantity, given the values of the independent variables  that is, to derive certain laws of nature. While we do not know why partial differential equations provide what has been termed the “unreasonable effectiveness of mathematics in the natural sciences” (the title of a 1960 paper by physicist…

This course is designed to introduce you to the study of calculus. You will learn concrete applications of how calculus is used and, more importantly, why it works. Calculus is not a new discipline; it has been around since the days of Archimedes. However, Isaac Newton and Gottfried Leibniz, two seventeenth-century European mathematicians concurrently working on the same intellectual discovery hundreds of miles apart, were responsible for developing the field as we know it today. This brings us to our first question, what is today’s calculus? In its simplest terms, calculus is the study of functions, rates of change, and continuity. While you may have cultivated a basic understanding of functions in previous math courses, in this course you will come to a more advanced understanding of their complexity, learning to take a closer look at their behaviors and nuances. In this course, we will address three major topics: limits, derivatives, and integrals, as well as study their respective foundations and appl…

Real Analysis II is the sequel to Saylor’s Real Analysis I, and together these two courses constitute the foundations of real analysis in mathematics. In this course, you will build on key concepts presented in Real Analysis I, particularly the study of the real number system and real-valued functions defined on all or part (usually intervals) of the real number line. The main objective of MA241 [1] was to introduce you to the concept and theory of differential and integral calculus as well as the mathematical analysis techniques that allow us to understand and solve various problems at the heart of sciencenamely, questions in the fields of physics, economics, chemistry, biology, and engineering. In this course, you will build on these techniques with the goal of applying them to the solution of more complex mathematical problems. As long as a problem can be modeled as a functional relation between two quantities, each of which can be expressed as a set of real numbers, the techniques used for real-value…

Precalculus I is designed to prepare you for Precalculus II, Calculus, Physics, and higher math and science courses. In this course, the main focus is on five types of functions: linear, polynomial, rational, exponential, and logarithmic. In accompaniment with these functions, you will learn how to solve equations and inequalities, graph, find domains and ranges, combine functions, and solve a multitude of real-world applications. In this course, you will not only be learning new algebraic techniques that are necessary for other math and science courses, but you will be learning to become a critical thinker. You will be able to determine what is the best approach to take such as numerical, graphical, or algebraic to solve a problem given particular information. Then you will investigate and solve the problem, interpret the answer, and determine if it is reasonable. A few examples of applications in this course are determining compound interest, growth of bacteria, decay of a radioactive substance, and the…

This course is a continuation of MA001: Beginning Algebra [1]. Algebra allows us to formulate real-world problems in an abstract mathematical term or equation. These equations can then be solved by using techniques you will learn in this course. For example, if I can ride my bicycle at 5 miles per hour and I live 12 miles from work, how long will it take me to get to work? Or, suppose I am a pitcher for the St. Louis Cardinals and my fast ball is 95 miles per hour, how much time does the hitter have to react to the baseball? And, can you explain why an object thrown up into the air will come back down? If so, can you tell how long it will take for the object to hit the ground? These are all examples of problems that can be stated as an algebraic equation and then solved. In this course you will study compound inequalities and solve systems of linear equations. You will then study radicals and rational exponents, followed by quadratic equations and techniques used to solve these equations. Finally, you will…