Courses tagged with "Taking derivatives" (37)

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 17th 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 calculus today?  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 ap…

If you invest in financial markets, you may want to predict the price of a stock in six months from now on the basis of company performance measures and other economic factors. As a college student, you may be interested in knowing the dependence of the mean starting salary of a college graduate, based on your GPA. These are just some examples that highlight how statistics are used in our modern society. To figure out the desired information for each example, you need data to analyze. The purpose of this course is to introduce you to the subject of statistics as a science of data. There is data abound in this information age; how to extract useful knowledge and gain a sound understanding in complex data sets has been more of a challenge. In this course, we will focus on the fundamentals of statistics, which may be broadly described as the techniques to collect, clarify, summarize, organize, analyze, and interpret numerical information. This course will begin with a brief overview of the discipline of stat…

This course is an introduction to linear algebra.  It has been argued that linear algebra constitutes half of all mathematics.  Whether or not everyone would agree with that, it is certainly true that practically every modern technology relies on linear algebra to simplify the computations required for Internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between. Linear algebra can be viewed either as the study of linear equations or as the study of vectors.  It is tied to analytic geometry; practically speaking, this means that almost every fact you will learn in this course has a picture associated with it.  Learning to connect the facts with their geometric interpretation will be very useful for you. The book which is used in the course focuses both on the theoretical aspects as well as the applied aspects of linear algebra.  As a result, you will be able to learn the geometric interpretations of many of the algebraic concepts…

This course has been designed to provide you with a clear, accessible introduction to discrete mathematics. Discrete mathematics describes processes that consist of a sequence of individual steps (as compared to calculus, which describes processes that change in a continuous manner). The principal topics presented in this course are logic and proof, induction and recursion, discrete probability, and finite state machines. As you progress through the units of this course, you will develop the mathematical foundations necessary for more specialized subjects in computer science, including data structures, algorithms, and compiler design. Upon completion of this course, you will have the mathematical know-how required for an in-depth study of the science and technology of the computer age.

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 17th-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 a…

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.  In this course (Part II), we will extend our differentiation and integration abilities and apply the techniques we have learned. Additional integration techniques, in particular, are a major part of the course.  In Part I, we learned how to integrate by various formulas and by reversing the chain rule through the technique of substitution.  In Part II, we will learn some clever uses of substitution, how to reverse the product rule for differentiation through a technique called integration by parts, and how to rewrite trigonometric and rational integrands that look impossible into simpler forms.  Series, while a major topic in their own right, also serve to extend our integration reach: they culminate in an application that lets you integrate almost any function you’d like. Integration allows us to calculat…

This course is an introduction to linear algebra.  It has been argued that linear algebra constitutes half of all mathematics.  Whether or not everyone would agree with that, it is certainly true that practically every modern technology relies on linear algebra to simplify the computations required for Internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between. Linear algebra can be viewed either as the study of linear equations or as the study of vectors.  It is tied to analytic geometry; practically speaking, this means that almost every fact you will learn in this course has a picture associated with it.  Learning to connect the facts with their geometric interpretation will be very useful for you. The book which is used in the course focuses both on the theoretical aspects as well as the applied aspects of linear algebra.  As a result, you will be able to learn the geometric interpretations of many of the algebraic concepts…

In this course, you will look at the properties behind the basic concepts of probability and statistics and focus on applications of statistical knowledge.  You will learn about how statistics and probability work together.  The subject of statistics involves the study of methods for collecting, summarizing, and interpreting data.  Statistics formalizes the process of making decisions, and this course is designed to help you use statistical literacy to make better decisions.  Note that this course has applications for the natural sciences, economics, computer science, finance, psychology, sociology, criminology, and many other fields. We read data in articles and reports every day.  After finishing this course, you should be comfortable evaluating an author's use of data.  You will be able to extract information from articles and display that information effectively.  You will also be able to understand the basics of how to draw statistical conclusions. This course will begin with descriptive statistic…

Precalculus II continues the in-depth study of functions addressed in Precalculus I by adding the trigonometric functions to your function toolkit. In this course, you will cover families of trigonometric functions, as well as their inverses, properties, graphs, and applications. Additionally, you will study trigonometric equations and identities, the laws of sines and cosines, polar coordinates and graphs, parametric equations and elementary vector operations. You might be curious how the study of trigonometry, or “trig,” as it is more often referred to, came about and why it is important to your studies still. Trigonometry, from the Greek for “triangle measure,” studies the relationships between the angles of a triangle and its sides and defines the trigonometric functions used to describe those relationships. Trigonometric functions are particularly useful when describing cyclical phenomena and have applications in numerous fields, including astronomy, navigation, music theory, physics, chemistry…

Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently CONSTANT. Solving an algebra problem, like y = 2x + 5, merely produces a pairing of two predetermined numbers, although an infinite set of pairs. Algebra is even useful in rate problems, such as calculating how the money in your savings account increases because of the interest rate R, such as Y = X0+Rt, where t is elapsed time and X0 is the initial deposit. With compound interest, things get complicated for algebra, as the rate R is itself a function of time with Y = X0 + R(t)t. Now we have a rate of change which itself is changing. Calculus came to the rescue, as Isaac Newton introduced the world to mathematics specifically designed to handle those things that change. Calculus is among the most important and useful developments of human thought. Even though it is over…

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.  In this course (Part II), we will extend our differentiation and integration abilities and apply the techniques we have learned. Additional integration techniques, in particular, are a major part of the course.  In Part I, we learned how to integrate by various formulas and by reversing the chain rule through the technique of substitution.  In Part II, we will learn some clever uses of substitution, how to reverse the product rule for differentiation through a technique called integration by parts, and how to rewrite trigonometric and rational integrands that look impossible into simpler forms.  Series, while a major topic in their own right, also serve to extend our integration reach: they culminate in an application that lets you integrate almost any function you’d like. Integration allows us to calculat…

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…

The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems.  Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations.  The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting.  The most important aspect of this course is that you will learn what it means to prove a mathematical proposition.  We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions.  The environments we use are propositions and predicates, finite sets and…

Differential equations are, in addition to a topic of study in mathematics, the main language in which the laws and phenomena of science are expressed.  In basic terms, a differential equation is an expression that describes how a system changes from one moment of time to another, or from one point in space to another.  When working with differential equations, the ultimate goal is to move from a microscopic view of relevant physics to a macroscopic view of the behavior of a system as a whole. Let’s look at a simple differential equation.  Based on previous math and physics courses, you know that a car that is constantly accelerating in the x-direction obeys the equation d2x/dt2 = a, where a is the applied acceleration.  This equation has two derivations with respect to time, so it is a second-order differential equation; because it has derivations with respect to only one variable (in this example, time), it is known as an  ordinary differential equation, or an ODE. Let’s say that we want to sol…

The study of “abstract algebra” grew out of an interest in knowing how attributes of sets of mathematical objects behave when one or more properties we associate with real numbers are restricted.  For example, we are familiar with the notion that real numbers are closed under multiplication and division (that is, if we add or multiply a real number, we get a real number).  But if we divide one integer by another integer, we may not get an integer as a resultmeaning that integers are not closed under division.  We also know that if we take any two integers and multiply them in either order, we get the same resulta principle known as the commutative principle of multiplication for integers.  By contrast, matrix multiplication is not generally commutative.  Students of abstract algebra are interested in these sorts of properties, as they want to determine which properties hold true for any set of mathematical objects under certain operations and which types of structures result when we perform certain o…

This course is designed to introduce you to the rigorous examination of the real number system and the foundations of calculus of functions of a single real variable. Analysis lies at the heart of the trinity of higher mathematics algebra, analysis, and topology because it is where the other two fields meet. In calculus, you learned to find limits, and you used these limits to give a rigorous justification for ideas of rate of change and areas under curves. Many of the results that you learned or derived were intuitive in many cases you could draw a picture of the situation and immediately “see” whether or not the result was true. This intuition, however, can sometimes be misleading. In the first place, your ability to find limits of real-valued functions on the real line was based on certain properties of the underlying field on which undergraduate calculus is founded: the real numbers. Things may have become slightly more complicated when you began to work in other spaces. For instance, you may r…

This course will introduce students to the field of computer science and the fundamentals of computer programming.  It has been specifically designed for students with no prior programming experience, and does not require a background in Computer Science.  This course will touch upon a variety of fundamental topics within the field of Computer Science and will use Java, a high-level, portable, and well-constructed computer programming language developed by Sun Microsystems, to demonstrate those principles.  We will begin with an overview of the topics we will cover this semester and a brief history of software development.  We will then learn about Object-Oriented programming, the paradigm in which Java was constructed, before discussing Java, its fundamentals, relational operators, control statements, and Java I/0.  The course will conclude with an introduction to algorithmic design.  By the end of the course, you should have a strong understanding of the fundamentals of Computer Science and the Java p…

Linear algebra is the study of vector spaces and linear mappings between them.  In this course, we will begin by reviewing topics you learned in Linear Algebra I, starting with linear equations, followed by a review of vectors and matrices in the context of linear equations.  The review will refresh your knowledge of the fundamentals of vectors and of matrix theory, how to perform operations on matrices, and how to solve systems of equations.  After the review, you should be able to understand complex numbers from algebraic and geometric viewpoints to the fundamental theorem of algebra.  Next, we will focus on eigenvalues and eigenvectors.  Today, these have applications in such diverse fields as computer science (Google's PageRank algorithm), physics (quantum mechanics, vibration analysis, etc.), economics (equilibrium states of Markov models), and more.  We will end with the spectral theorem, which provides a decomposition of the vector space on which operators act, and singular-value decomposition, w…

Numerical analysis is the study of the methods used to solve problems involving continuous variables.  It is a highly applied branch of mathematics and computer science, wherein abstract ideas and theories become the quantities describing things we can actually touch and see.  The real number line is an abstraction where many interesting and useful ideas live, but to actually realize these ideas, we are forced to employ approximations of the real numbers.  For example, consider marking a ruler at sqrt{2}.  We know that sqrt{2} approx 1.4142, but if we put the mark there, we know we are in error for there is an infinite sequence of nonzero digits following the 2.  Even more: a number doesn’t have any width, yet any mark we make would have a width, and in that width lives an infinite number of real numbers.  You may ask yourself: isn’t it sufficient to represent sqrt{2} with 1.414?  This is the kind of question that this course will explore.  We have been trying to answer such questions for over 2,0…

This course is a continuation of Abstract Algebra I: we will revisit structures like groups, rings, and fields as well as mappings like homomorphisms and isomorphisms.  We will also take a look at ring factorization, which will lead us to a discussion of the solutions of polynomials over abstracted structures instead of numbers sets.  We will end the section on rings with a discussion of general lattices, which have both set and logical properties, and a special type of lattice known as Boolean algebra, which plays an important role in probability.  We will also visit an important topic in mathematics that you have likely encountered already: vector spaces.  Vector spaces are central to the study of linear algebra, but because they are extended groups, group theory and geometric methods can be used to study them. Later in this course, we will take a look at more advanced topics and consider several useful theorems and counting methods.  We will end the course by studying Galois theoryone of the most im…