Site Map
This page lists every top-level destination on Paul's Online Notes. If your browser has JavaScript disabled, use this page as a starting point.
Courses
- Algebra — This course covers the standard topics from an Algebra course. We start by covering exponents, radicals and factoring so we have the basics down. We will learn how to solve linear, quadratic equations and inequalities as well as some applications of these techniques. In addition, we will introduce function notation, domains and ranges of functions, and finding roots of functions. We also examine graphs of lines, circles, parabolas, ellipses, hyperbolas, and transformations. Finally, we will look at exponential functions, logarithmic functions, solutions to exponential and logarithmic functions, and solving linear systems of equations with two or three variables.
- Calculus I — This course covers the standard topics from a typical Calculus I course. We start off with a quick review of some prerequisite material. We will then introduce limits including how to compute limits and we will define and discuss continuity. Derivatives are then introduced and we will learn how to differentiate various functions, the product and quotient rules, and the chain rule. The final topic of the course will be integrals and we will explore some basic methods of computing integrals. Throughout the course we will discuss various applications of limits, derivatives and integrals.
- Calculus II — In this course we pick up where Calculus I left off and learn some advanced integration techniques such as integration by parts and partial fractions as well as some applications of integrals. We then introduce the concept of parametric equations and we learn how to perform various Calculus operations with them. Next, we introduce sequences and series, including Taylor series, and investigate how to determine if they will converge or diverge. In the course we will also explore vectors, vector operations, 3D space, and various coordinate systems.
- Calculus III — In Calculus III we formally move into 3D space so we start off with some topics in 3D space to make sure that we all have the same background. We will spend most of the course learning how to differentiate and integrate functions of two, or more, variables. This will include double, triple, line and surface integrals. We will learn how to apply Lagrange multipliers to determine the optimal values of multi-variable functions subject to a constraint. We will also discuss the very important theorems: Green’s Theorem, Stokes’ Theorem and the Divergence Theorem.
- Differential Equations — In this course we will look at various techniques for finding solutions to differential equations. We will spend most of our time solving homogeneous and nonhomogeneous first and second order differential equations and learn how to extend these solution methods to higher order differential equations. We will learn how to apply Laplace transforms to solve differential equations and we will learn how to solve systems of linear differential equations. We finish off the course with some topics that are typically not covered in a differential equations course but are included for your knowledge. These are series solutions, boundary value problems, Fourier series and some introductory topics in solving partial differential equations.
Extras
- Algebra Trig Review — This is a series of problems, with solutions, in selected topics in Algebra and Trigonometry that are vital for being successful in a Calculus course. The material presented here is intended to be a refresher for these topics and so does not go into great detail in the solutions outside of what is needed to remind you of the topics.
- Common Math Errors — This is a series of errors and misconceptions that students in a math course will often run into. In most of the sections, the errors/misconceptions are presented in the incorrect form and the correct from. There will also be discussion about the error or misconception in most cases and what to watch out for so you don’t make the error/misconception. Most of the sections are accessible to anyone with some basic Algebra background. There are, however, some errors involving Trigonometry and Calculus.
- Complex Number Primer — Many students never really see complex numbers and then are, all of a sudden, expected to know them. This quick primer is intended to introduce you to the basics of complex numbers as well as arithmetic involving complex numbers and some of the more common operations involving complex numbers. Hopefully, if you are in the situation of all of a sudden needing to know about complex numbers, this primer will give you all the information you need.
- How To Study Math — This is a list of general tips on how to study and be successful in a math course. They have been broken down into distinct topics (taking notes, homework, exams, etc.). Every person is different so you may not find all of them useful but hopefully you can use this as a guide to determine what works for you in being successful in your math course.
- Cheat Sheets and Tables — A series of cheat sheets and tables for quick reference. Topics include Algebra, Trigonometry, and Calculus. There are also tables for common derivatives and integrals as well as a table of Laplace transforms.