Algebra
Calculus
Differential Equations
Extras
Misc

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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.

Preliminaries In this chapter we will do a quick review of some topics that are absolutely essential to being successful in an Algebra class. We review exponents (integer and rational), radicals, polynomials, factoring polynomials, rational expressions and complex numbers. Integer Exponents In this section we will start looking at exponents. We will give the basic properties of exponents and illustrate some of the common mistakes students make in working with exponents. Examples in this section we will be restricted to integer exponents. Rational exponents will be discussed in the next section. Rational Exponents In this section we will define what we mean by a rational exponent and extend the properties from the previous section to rational exponents. We will also discuss how to evaluate numbers raised to a rational exponent. Radicals In this section we will define radical notation and relate radicals to rational exponents. We will also give the properties of radicals and some of the common mistakes students often make with radicals. We will also define simplified radical form and show how to rationalize the denominator. Polynomials In this section we will introduce the basics of polynomials a topic that will appear throughout this course. We will define the degree of a polynomial and discuss how to add, subtract and multiply polynomials. Factoring Polynomials In this section we look at factoring polynomials a topic that will appear in pretty much every chapter in this course and so is vital that you understand it. We will discuss factoring out the greatest common factor, factoring by grouping, factoring quadratics and factoring polynomials with degree greater than 2. Rational Expressions In this section we will define rational expressions. We will discuss how to reduce a rational expression lowest terms and how to add, subtract, multiply and divide rational expressions. Complex Numbers In this section we give a very quick primer on complex numbers including standard form, adding, subtracting, multiplying and dividing them.

Graphing and Functions In this chapter we’ll look at two very important topics in an Algebra class. First, we will start discussing graphing equations by introducing the Cartesian (or Rectangular) coordinates system and illustrating use of the coordinate system to graph lines and circles. We will also formally define a function and discuss graph functions and combining functions. We will also discuss inverse functions. Graphing In this section we will introduce the Cartesian (or Rectangular) coordinate system. We will define/introduce ordered pairs, coordinates, quadrants, and x and y-intercepts. We will illustrate these concepts with a couple of quick examples Lines In this section we will discuss graphing lines. We will introduce the concept of slope and discuss how to find it from two points on the line. In addition, we will introduce the standard form of the line as well as the point-slope form and slope-intercept form of the line. We will finish off the section with a discussion on parallel and perpendicular lines. Circles In this section we discuss graphing circles. We introduce the standard form of the circle and show how to use completing the square to put an equation of a circle into standard form. The Definition of a Function In this section we will formally define relations and functions. We also give a “working definition” of a function to help understand just what a function is. We introduce function notation and work several examples illustrating how it works. We also define the domain and range of a function. In addition, we introduce piecewise functions in this section. Graphing Functions In this section we discuss graphing functions including several examples of graphing piecewise functions. Combining Functions In this section we will discuss how to add, subtract, multiply and divide functions. In addition, we introduce the concept of function composition. Inverse Functions In this section we define one-to-one and inverse functions. We also discuss a process we can use to find an inverse function and verify that the function we get from this process is, in fact, an inverse function.

Common Graphs In this chapter we will look at graphing some of the more common functions you might be asked to graph. We graph parabolas, ellipses, hyperbolas and rational functions in this chapter. We will also look at transformations of functions and introduce the concept of symmetry. Lines, Circles and Piecewise Functions This section is here only to acknowledge that we’ve already talked about graphing these in a previous chapter. Parabolas In this section we will be graphing parabolas. We introduce the vertex and axis of symmetry for a parabola and give a process for graphing parabolas. We also illustrate how to use completing the square to put the parabola into the form \(f(x)=a(x-h)^{2}+k\). Ellipses In this section we will graph ellipses. We introduce the standard form of an ellipse and how to use it to quickly graph an ellipse. Hyperbolas In this section we will graph hyperbolas. We introduce the standard form of a hyperbola and how to use it to quickly graph a hyperbola. Miscellaneous Functions In this section we will graph a couple of common functions that don’t really take all that much work to do but will be needed in later sections. We’ll be looking at the constant function, square root, absolute value and a simple cubic function. Transformations In this section we will be looking at vertical and horizontal shifts of graphs as well as reflections of graphs about the \(x\) and \(y\)-axis. Collectively these are often called transformations and if we understand them they can often be used to allow us to quickly graph some fairly complicated functions. Symmetry In this section we introduce the idea of symmetry. We discuss symmetry about the x-axis, y-axis and the origin and we give methods for determining what, if any symmetry, a graph will have without having to actually graph the function. Rational Functions In this section we will discuss a process for graphing rational functions. We will also introduce the ideas of vertical and horizontal asymptotes as well as how to determine if the graph of a rational function will have them.

Polynomial Functions In this chapter we will take a more detailed look at polynomial functions. We will discuss dividing polynomials, finding zeroes of polynomials and sketching the graph of polynomials. We will also look at partial fractions (even though this doesn’t really involve polynomial functions). Dividing Polynomials In this section we’ll review some of the basics of dividing polynomials. We will define the remainder and divisor used in the division process and introduce the idea of synthetic division. We will also give the Division Algorithm. Zeroes/Roots of Polynomials In this section we’ll define the zero or root of a polynomial and whether or not it is a simple root or has multiplicity \(k\). We will also give the Fundamental Theorem of Algebra and The Factor Theorem as well as a couple of other useful Facts. Graphing Polynomials In this section we will give a process that will allow us to get a rough sketch of the graph of some polynomials. We discuss how to determine the behavior of the graph at \(x\)-intercepts and the leading coefficient test to determine the behavior of the graph as we allow x to increase and decrease without bound. Finding Zeroes of Polynomials As we saw in the previous section in order to sketch the graph of a polynomial we need to know what it’s zeroes are. However, if we are not able to factor the polynomial we are unable to do that process. So, in this section we’ll look at a process using the Rational Root Theorem that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes. Partial Fractions In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression. What we will be asking here is what “smaller” rational expressions did we add and/or subtract to get the given rational expression. This is a process that has a lot of uses in some later math classes. It can show up in Calculus and Differential Equations for example.

Exponential and Logarithm Functions In this chapter we will introduce two very important functions in many areas : the exponential and logarithm functions. We will look at their basic properties, applications and solving equations involving the two functions. If you are in a field that takes you into the sciences or engineering then you will be running into both of these functions. Exponential Functions In this section we will introduce exponential functions. We will give some of the basic properties and graphs of exponential functions. We will also discuss what many people consider to be the exponential function, \(f(x) = {\bf e}^{x}\). Logarithm Functions In this section we will introduce logarithm functions. We give the basic properties and graphs of logarithm functions. In addition, we discuss how to evaluate some basic logarithms including the use of the change of base formula. We will also discuss the common logarithm, \(\log(x)\), and the natural logarithm, \(\ln(x)\). Solving Exponential Equations In this section we will discuss a couple of methods for solving equations that contain exponentials. Solving Logarithm Equations In this section we will discuss a couple of methods for solving equations that contain logarithms. Also, as we’ll see, with one of the methods we will need to be careful of the results of the method as it is always possible that the method gives values that are, in fact, not solutions to the equation. Applications In this section we will look at a couple of applications of exponential functions and an application of logarithms. We look at compound interest, exponential growth and decay and earthquake intensity.

Systems of Equations In this chapter we will take a look at solving systems of equations. We will solve linear as well as nonlinear systems of equations. We will also take a quick look at using augmented matrices to solve linear systems of equations. Linear Systems with Two Variables In this section we will solve systems of two equations and two variables. We will use the method of substitution and method of elimination to solve the systems in this section. We will also introduce the concepts of inconsistent systems of equations and dependent systems of equations. Linear Systems with Three Variables In this section we will work a couple of quick examples illustrating how to use the method of substitution and method of elimination introduced in the previous section as they apply to systems of three equations. Augmented Matrices In this section we will look at another method for solving systems. We will introduce the concept of an augmented matrix. This will allow us to use the method of Gauss-Jordan elimination to solve systems of equations. We will use the method with systems of two equations and systems of three equations. More on the Augmented Matrix In this section we will revisit the cases of inconsistent and dependent solutions to systems and how to identify them using the augmented matrix method. Nonlinear Systems In this section we will take a quick look at solving nonlinear systems of equations. A nonlinear system of equations is a system in which at least one of the equations is not linear, i.e. has degree of two or more. Note as well that the discussion here does not cover all the possible solution methods for nonlinear systems. Solving nonlinear systems is often a much more involved process than solving linear systems.

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.

Basic Concepts In this chapter we introduce many of the basic concepts and definitions that are encountered in a typical differential equations course. We will also take a look at direction fields and how they can be used to determine some of the behavior of solutions to differential equations. Definitions In this section some of the common definitions and concepts in a differential equations course are introduced including order, linear vs. nonlinear, initial conditions, initial value problem and interval of validity. Direction Fields In this section we discuss direction fields and how to sketch them. We also investigate how direction fields can be used to determine some information about the solution to a differential equation without actually having the solution. Final Thoughts In this section we give a couple of final thoughts on what we will be looking at throughout this course.

Laplace Transforms In this chapter we introduce Laplace Transforms and how they are used to solve Initial Value Problems. With the introduction of Laplace Transforms we will now be able to solve some Initial Value Problems that we wouldn’t be able to solve otherwise. We will solve differential equations that involve Heaviside and Dirac Delta functions. We will also give brief overview on using Laplace transforms to solve nonconstant coefficient differential equations. In addition, we will define the convolution integral and show how it can be used to take inverse transforms. The Definition In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the definition. Laplace Transforms In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms. Inverse Laplace Transforms In this section we ask the opposite question from the previous section. In other words, given a Laplace transform, what function did we originally have? We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table. Step Functions In this section we introduce the step or Heaviside function. We illustrate how to write a piecewise function in terms of Heaviside functions. We also work a variety of examples showing how to take Laplace transforms and inverse Laplace transforms that involve Heaviside functions. We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions. Solving IVP's with Laplace Transforms In this section we will examine how to use Laplace transforms to solve IVP’s. The examples in this section are restricted to differential equations that could be solved without using Laplace transform. The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to. Nonconstant Coefficient IVP's In this section we will give a brief overview of using Laplace transforms to solve some nonconstant coefficient IVP’s. We do not work a great many examples in this section. We only work a couple to illustrate how the process works with Laplace transforms. IVP's With Step Functions This is the section where the reason for using Laplace transforms really becomes apparent. We will use Laplace transforms to solve IVP’s that contain Heaviside (or step) functions. Without Laplace transforms solving these would involve quite a bit of work. While we do work one of these examples without Laplace transforms, we do it only to show what would be involved if we did try to solve one of the examples without using Laplace transforms. Dirac Delta Function In this section we introduce the Dirac Delta function and derive the Laplace transform of the Dirac Delta function. We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. We also give a nice relationship between Heaviside and Dirac Delta functions. Convolution Integrals In this section we give a brief introduction to the convolution integral and how it can be used to take inverse Laplace transforms. We also illustrate its use in solving a differential equation in which the forcing function (i.e. the term without any y’s in it) is not known. Table Of Laplace Transforms This section is the table of Laplace Transforms that we’ll be using in the material. We give as wide a variety of Laplace transforms as possible including some that aren’t often given in tables of Laplace transforms.

Series Solutions to DE's In this chapter we are going to take a quick look at how to represent the solution to a differential equation with a power series. We will also look at how to solve Euler’s differential equation. In addition, we will do a quick review of power series and Taylor series to help with work in the chapter. Review : Power Series In this section we give a brief review of some of the basics of power series. Included are discussions of using the Ratio Test to determine if a power series will converge, adding/subtracting power series, differentiating power series and index shifts for power series. Review : Taylor Series In this section we give a quick reminder on how to construct the Taylor series for a function. Included are derivations for the Taylor series of \({\bf e}^{x}\) and \(\cos(x)\) about \(x = 0\) as well as showing how to write down the Taylor series for a polynomial. Series Solutions In this section we define ordinary and singular points for a differential equation. We also show who to construct a series solution for a differential equation about an ordinary point. The method illustrated in this section is useful in solving, or at least getting an approximation of the solution, differential equations with coefficients that are not constant. Euler Equations In this section we will discuss how to solve Euler’s differential equation, \(ax^{2}y'' + b x y' +c y = 0\). Note that while this does not involve a series solution it is included in the series solution chapter because it illustrates how to get a solution to at least one type of differential equation at a singular point.

Boundary Value Problems & Fourier Series In this chapter we will introduce two topics that are integral to basic partial differential equations solution methods. The first topic, boundary value problems, occur in pretty much every partial differential equation. The second topic, Fourier series, is what makes one of the basic solution techniques work. Boundary Value Problems In this section we’ll define boundary conditions (as opposed to initial conditions which we should already be familiar with at this point) and the boundary value problem. We will also work a few examples illustrating some of the interesting differences in using boundary values instead of initial conditions in solving differential equations. Eigenvalues and Eigenfunctions In this section we will define eigenvalues and eigenfunctions for boundary value problems. We will work quite a few examples illustrating how to find eigenvalues and eigenfunctions. In one example the best we will be able to do is estimate the eigenvalues as that is something that will happen on a fairly regular basis with these kinds of problems. Periodic Functions & Orthogonal Functions In this section we will define periodic functions, orthogonal functions and mutually orthogonal functions. We will also work a couple of examples showing intervals on which \(\cos\left(\frac{n \pi x}{L}\right)\) and \(\sin\left(\frac{n \pi x}{L}\right)\) are mutually orthogonal. The results of these examples will be very useful for the rest of this chapter and most of the next chapter. Fourier Sine Series In this section we define the Fourier Sine Series, i.e. representing a function with a series in the form \(\sum\limits_{n = 1}^\infty {{B_n}\sin \left( {\frac{{n\pi x}}{L}} \right)} \). We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function. Fourier Cosine Series In this section we define the Fourier Cosine Series, i.e. representing a function with a series in the form \(\sum\limits_{n = 0}^\infty {{A_n}\cos \left( {\frac{{n\pi x}}{L}} \right)} \). We will also define the even extension for a function and work several examples finding the Fourier Cosine Series for a function. Fourier Series In this section we define the Fourier Series, i.e. representing a function with a series in the form \(\sum\limits_{n = 0}^\infty {{A_n}\cos \left( {\frac{{n\pi x}}{L}} \right)} + \sum\limits_{n = 1}^\infty {{B_n}\sin \left( {\frac{{n\pi x}}{L}} \right)} \). We will also work several examples finding the Fourier Series for a function. Convergence of Fourier Series In this section we will define piecewise smooth functions and the periodic extension of a function. In addition, we will give a variety of facts about just what a Fourier series will converge to and when we can expect the derivative or integral of a Fourier series to converge to the derivative or integral of the function it represents.

Calculus

This is a full set of material for all the topics that are traditionally covered in the Calculus I, Calculus II, and Calculus III courses. Limits, derivatives, and integrals for single variable functions and multi-variable functions are all covered and examples are worked and the solution process examined in detail. Applications of limits, derivatives and integrals are all examined and examples worked. Major theorems such as Intermediate Value Theorem, Mean Value Theorem, Fundamental Theorem of Calculus, Green’s Theorem, Stoke’s Theorem, and Divergence Theorem are covered and discussed. Practice problems with full solutions are included to reinforce a students understanding of the concepts and topics covered.

Calculus I covers limits, derivatives and basic integration. Calculus II covers advanced integration techniques, parametric equations, polar coordinates, series, vectors and 3D space. Calculus III covers 3D space, partial derivatives, double and triple integrals, line integrals, and surface integrals. Of course, applications are also presented where applicable.

Select the book to see a full listing of topics covered.

Review In this chapter we give a brief review of selected topics from Algebra and Trig that are vital to surviving a Calculus course. Included are Functions, Trig Functions, Solving Trig Equations and Equations, Exponential/Logarithm Functions and Solving Exponential/Logarithm Equations.

Limits In this chapter we introduce the concept of limits. We will discuss the interpretation/meaning of a limit, how to evaluate limits, the definition and evaluation of one-sided limits, evaluation of infinite limits, evaluation of limits at infinity, continuity and the Intermediate Value Theorem. We will also give a brief introduction to a precise definition of the limit and how to use it to evaluate limits.

Derivatives In this chapter we introduce Derivatives. We cover the standard derivatives formulas including the product rule, quotient rule and chain rule as well as derivatives of polynomials, roots, trig functions, inverse trig functions, hyperbolic functions, exponential functions and logarithm functions. We also cover implicit differentiation, related rates, higher order derivatives and logarithmic differentiation.

Chapter overview The Definition of the Derivative In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function. Interpretation of the Derivative In this section we give several of the more important interpretations of the derivative. We discuss the rate of change of a function, the velocity of a moving object and the slope of the tangent line to a graph of a function. Differentiation Formulas In this section we give most of the general derivative formulas and properties used when taking the derivative of a function. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers. Product and Quotient Rule In this section we will give two of the more important formulas for differentiating functions. We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate. Derivatives of Trig Functions In this section we will discuss differentiating trig functions. Derivatives of all six trig functions are given and we show the derivation of the derivative of \(\sin(x)\) and \(\tan(x)\). Derivatives of Exponential and Logarithm Functions In this section we derive the formulas for the derivatives of the exponential and logarithm functions. Derivatives of Inverse Trig Functions In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent. Derivatives of Hyperbolic Functions In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine. Chain Rule In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule! Implicit Differentiation In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e.g. y = f(x) and yet we will still need to know what f'(x) is. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates (the next section). Related Rates In this section we will discuss the only application of derivatives in this section, Related Rates. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one (or more) quantities in the problem. This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work. Higher Order Derivatives In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives. Logarithmic Differentiation In this section we will discuss logarithmic differentiation. Logarithmic differentiation gives an alternative method for differentiating products and quotients (sometimes easier than using product and quotient rule). More importantly, however, is the fact that logarithm differentiation allows us to differentiate functions that are in the form of one function raised to another function, i.e. there are variables in both the base and exponent of the function.

Applications of Derivatives In this chapter we will cover many of the major applications of derivatives. Applications included are determining absolute and relative minimum and maximum function values (both with and without constraints), sketching the graph of a function without using a computational aid, determining the Linear Approximation of a function, L’Hospital’s Rule (allowing us to compute some limits we could not prior to this), Newton's Method (allowing us to approximate solutions to equations) as well as a few basic Business applications.

Integrals In this chapter we will give an introduction to definite and indefinite integrals. We will discuss the definition and properties of each type of integral as well as how to compute them including the Substitution Rule. We will give the Fundamental Theorem of Calculus showing the relationship between derivatives and integrals. We will also discuss the Area Problem, an important interpretation of the definite integral.

Applications of Integrals In this chapter we will take a look at some applications of integrals. We will look at Average Function Value, Area Between Curves, Volume (both solids of revolution and other solids) and Work.

Chapter overview Average Function Value In this section we will look at using definite integrals to determine the average value of a function on an interval. We will also give the Mean Value Theorem for Integrals. Area Between Curves In this section we’ll take a look at one of the main applications of definite integrals in this chapter. We will determine the area of the region bounded by two curves. Volumes of Solids of Revolution / Method of Rings In this section, the first of two sections devoted to finding the volume of a solid of revolution, we will look at the method of rings/disks to find the volume of the object we get by rotating a region bounded by two curves (one of which may be the \(x\) or \(y\)-axis) around a vertical or horizontal axis of rotation. Volumes of Solids of Revolution/Method of Cylinders In this section, the second of two sections devoted to finding the volume of a solid of revolution, we will look at the method of cylinders/shells to find the volume of the object we get by rotating a region bounded by two curves (one of which may be the \(x\) or \(y\)-axis) around a vertical or horizontal axis of rotation. More Volume Problems In the previous two sections we looked at solids that could be found by treating them as a solid of revolution. Not all solids can be thought of as solids of revolution and, in fact, not all solids of revolution can be easily dealt with using the methods from the previous two sections. So, in this section we’ll take a look at finding the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution. Work In this section we will look at is determining the amount of work required to move an object subject to a force over a given distance.

Extras In this chapter proofs of many of the facts/properties/theorems given through out the material are given. Also included are a brief review of summation notation, a discussion on the different 'types' of infinity and a discussion about a subtlety involved with the constant of integration from indefinite integrals.

Chapter overview Proof of Various Limit Properties In this section we prove several of the limit properties and facts that were given in various sections of the Limits chapter. Proof of Various Derivative Properties In this section we prove several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.

Proof of Trig Limits In this section we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions using the definition of the derivative. Proofs of Derivative Applications Facts In this section we prove many of the facts that we saw in the Applications of Derivatives chapter. Proof of Various Integral Properties In this section we prove some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter. Area and Volume Formulas In this section we derive the formulas for finding area between two curves and finding the volume of a solid of revolution. Types of Infinity In this section we have a discussion on the types of infinity and how these affect certain limits. Note that there is a lot of theory going on 'behind the scenes' so to speak that we are not going to cover in this section. This section is intended only to give you a feel for what is going on here. To get a fuller understanding of some of the ideas in this section you will need to take some upper level mathematics courses. Summation Notation In this section we give a quick review of summation notation. Summation notation is heavily used when defining the definite integral and when we first talk about determining the area between a curve and the \(x\)-axis. Constant of Integration In this section we have a discussion on a couple of subtleties involving constants of integration that many students don’t think about when doing indefinite integrals. Not understanding these subtleties can lead to confusion on occasion when students get different answers to the same integral. We include two examples of this kind of situation.

Integration Techniques In this chapter we will look at several integration techniques including Integration by Parts, Integrals Involving Trig Functions, Trig Substitutions and Partial Fractions. We will also look at Improper Integrals including using the Comparison Test for convergence/divergence of improper integrals.

Chapter overview Integration by Parts In this section we will be looking at Integration by Parts. Of all the techniques we’ll be looking at in this class this is the technique that students are most likely to run into down the road in other classes. We also give a derivation of the integration by parts formula. Integrals Involving Trig Functions In this section we look at integrals that involve trig functions. In particular we concentrate integrating products of sines and cosines as well as products of secants and tangents. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions. Trig Substitutions In this section we will look at integrals (both indefinite and definite) that require the use of a substitutions involving trig functions and how they can be used to simplify certain integrals. Partial Fractions In this section we will use partial fractions to rewrite integrands into a form that will allow us to do integrals involving some rational functions. Integrals Involving Roots In this section we will take a look at a substitution that can, on occasion, be used with integrals involving roots. Integrals Involving Quadratics In this section we are going to look at some integrals that involve quadratics for which the previous techniques won’t work right away. In some cases, manipulation of the quadratic needs to be done before we can do the integral. We will see several cases where this is needed in this section. Integration Strategy In this section we give a general set of guidelines for determining how to evaluate an integral. The guidelines give here involve a mix of both Calculus I and Calculus II techniques to be as general as possible. Also note that there really isn’t one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. Improper Integrals In this section we will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. Collectively, they are called improper integrals and as we will see they may or may not have a finite (i.e. not infinite) value. Determining if they have finite values will, in fact, be one of the major topics of this section. Comparison Test for Improper Integrals It will not always be possible to evaluate improper integrals and yet we still need to determine if they converge or diverge (i.e. if they have a finite value or not). So, in this section we will use the Comparison Test to determine if improper integrals converge or diverge. Approximating Definite Integrals In this section we will look at several fairly simple methods of approximating the value of a definite integral. It is not possible to evaluate every definite integral (i.e. because it is not possible to do the indefinite integral) and yet we may need to know the value of the definite integral anyway. These methods allow us to at least get an approximate value which may be enough in a lot of cases.

Applications of Integrals In this chapter we’ll take a look at a few applications of integrals. We will look at determining the arc length of a curve, the surface area of a solid of revolution, the center of mass of a region bounded by two curves, the hydrostatic force/pressure on a plate submerged in water and a quick look at computing the mean of a probability density function. The applications given here tend to result in integrals that are typically covered in a Calculus II course.

Chapter overview Arc Length In this section we’ll determine the length of a curve over a given interval. Surface Area In this section we’ll determine the surface area of a solid of revolution, i.e. a solid obtained by rotating a region bounded by two curves about a vertical or horizontal axis. Center of Mass In this section we will determine the center of mass or centroid of a thin plate where the plate can be described as a region bounded by two curves (one of which may the \(x\) or \(y\)-axis). Hydrostatic Pressure and Force In this section we’ll determine the hydrostatic pressure and force on a vertical plate submerged in water. The plates used in the examples can all be described as regions bounded by one or more curves/lines. Probability Many quantities can be described with probability density functions. For example, the length of time a person waits in line at a checkout counter or the life span of a light bulb. None of these quantities are fixed values and will depend on a variety of factors. In this section we will look at probability density functions and computing the mean (think average wait in line or average life span of a light blub) of a probability density function.

Parametric Equations and Polar Coordinates In this chapter we will introduce the ideas of parametric equations and polar coordinates. We will also look at many of the basic Calculus ideas (tangent lines, area, arc length and surface area) in terms of these two ideas.

Series & Sequences In this chapter we introduce sequences and series. We discuss whether a sequence converges or diverges, is increasing or decreasing, or if the sequence is bounded. We will then define just what an infinite series is and discuss many of the basic concepts involved with series. We will discuss if a series will converge or diverge, including many of the tests that can be used to determine if a series converges or diverges. We will also discuss using either a power series or a Taylor series to represent a function and how to find the radius and interval of convergence for this series.

Vectors In this (very brief) chapter we will take a look at the basics of vectors. Included are common notation for vectors, arithmetic of vectors, dot product of vectors (and applications) and cross product of vectors (and applications).

Chapter overview Vectors - The Basics In this section we will introduce some common notation for vectors as well as some of the basic concepts about vectors such as the magnitude of a vector and unit vectors. We also illustrate how to find a vector from its starting and end points. Vector Arithmetic In this section we will discuss the mathematical and geometric interpretation of the sum and difference of two vectors. We also define and give a geometric interpretation for scalar multiplication. We also give some of the basic properties of vector arithmetic and introduce the common \(i\), \(j\), \(k\) notation for vectors. Dot Product In this section we will define the dot product of two vectors. We give some of the basic properties of dot products and define orthogonal vectors and show how to use the dot product to determine if two vectors are orthogonal. We also discuss finding vector projections and direction cosines in this section. Cross Product In this section we define the cross product of two vectors and give some of the basic facts and properties of cross products.

3-Dimensional Space In this chapter we will start looking at three dimensional space. This chapter is generally prep work for Calculus III and so we will cover the standard 3D coordinate system as well as a couple of alternative coordinate systems. We will also discuss how to find the equations of lines and planes in three dimensional space. We will look at some standard 3D surfaces and their equations. In addition we will introduce vector functions and some of their applications (tangent and normal vectors, arc length, curvature and velocity and acceleration).

3-Dimensional Space In this chapter we will start looking at three dimensional space. This chapter is generally prep work for Calculus III and so we will cover the standard 3D coordinate system as well as a couple of alternative coordinate systems. We will also discuss how to find the equations of lines and planes in three dimensional space. We will look at some standard 3D surfaces and their equations. In addition we will introduce vector functions and some of their applications (tangent and normal vectors, arc length, curvature and velocity and acceleration).

Partial Derivatives In this chapter we’ll take a brief look at limits of functions of more than one variable and then move into derivatives of functions of more than one variable. As we’ll see if we can do derivatives of functions with one variable it isn’t much more difficult to do derivatives of functions of more than one variable (with a very important subtlety). We will also discuss interpretations of partial derivatives, higher order partial derivatives and the chain rule as applied to functions of more than one variable. We will also define and discuss directional derivatives.

Chapter overview Limits In the section we’ll take a quick look at evaluating limits of functions of several variables. We will also see a fairly quick method that can be used, on occasion, for showing that some limits do not exist. Partial Derivatives In this section we will look at the idea of partial derivatives. We will give the formal definition of the partial derivative as well as the standard notations and how to compute them in practice (i.e. without the use of the definition). As you will see if you can do derivatives of functions of one variable you won’t have much of an issue with partial derivatives. There is only one (very important) subtlety that you need to always keep in mind while computing partial derivatives. Interpretations of Partial Derivatives In the section we will take a look at a couple of important interpretations of partial derivatives. First, the always important, rate of change of the function. Although we now have multiple ‘directions’ in which the function can change (unlike in Calculus I). We will also see that partial derivatives give the slope of tangent lines to the traces of the function. Higher Order Partial Derivatives In the section we will take a look at higher order partial derivatives. Unlike Calculus I however, we will have multiple second order derivatives, multiple third order derivatives, etc. because we are now working with functions of multiple variables. We will also discuss Clairaut’s Theorem to help with some of the work in finding higher order derivatives. Differentials In this section we extend the idea of differentials we first saw in Calculus I to functions of several variables. Chain Rule In the section we extend the idea of the chain rule to functions of several variables. In particular, we will see that there are multiple variants to the chain rule here all depending on how many variables our function is dependent on and how each of those variables can, in turn, be written in terms of different variables. We will also give a nice method for writing down the chain rule for pretty much any situation you might run into when dealing with functions of multiple variables. In addition, we will derive a very quick way of doing implicit differentiation so we no longer need to go through the process we first did back in Calculus I. Directional Derivatives In the section we introduce the concept of directional derivatives. With directional derivatives we can now ask how a function is changing if we allow all the independent variables to change rather than holding all but one constant as we had to do with partial derivatives. In addition, we will define the gradient vector to help with some of the notation and work here. The gradient vector will be very useful in some later sections as well. We will also give a nice fact that will allow us to determine the direction in which a given function is changing the fastest.

Applications of Partial Derivatives In this chapter we will take a look at several applications of partial derivatives. We will find the equation of tangent planes to surfaces and we will revisit on of the more important applications of derivatives from earlier Calculus classes. We will spend a significant amount of time finding relative and absolute extrema of functions of multiple variables. We will also introduce Lagrange multipliers to find the absolute extrema of a function subject to one or more constraints.

Chapter overview Tangent Planes and Linear Approximations In this section formally define just what a tangent plane to a surface is and how we use partial derivatives to find the equations of tangent planes to surfaces that can be written as \(z=f(x,y)\). We will also see how tangent planes can be thought of as a linear approximation to the surface at a given point. Gradient Vector, Tangent Planes and Normal Lines In this section discuss how the gradient vector can be used to find tangent planes to a much more general function than in the previous section. We will also define the normal line and discuss how the gradient vector can be used to find the equation of the normal line. Relative Minimums and Maximums In this section we will define critical points for functions of two variables and discuss a method for determining if they are relative minimums, relative maximums or saddle points (i.e. neither a relative minimum or relative maximum). Absolute Minimums and Maximums In this section we will how to find the absolute extrema of a function of two variables when the independent variables are only allowed to come from a region that is bounded (i.e. no part of the region goes out to infinity) and closed (i.e. all of the points on the boundary are valid points that can be used in the process). Lagrange Multipliers In this section we’ll see discuss how to use the method of Lagrange Multipliers to find the absolute minimums and maximums of functions of two or three variables in which the independent variables are subject to one or more constraints. We also give a brief justification for how/why the method works.

Multiple Integrals In this chapter will be looking at double integrals, i.e. integrating functions of two variables in which the independent variables are from two dimensional regions, and triple integrals, i.e. integrating functions of three variables in which the independent variables are from three dimensional regions. Included will be double integrals in polar coordinates and triple integrals in cylindrical and spherical coordinates and more generally change in variables in double and triple integrals.

Line Integrals In this chapter we will introduce a new kind of integral : Line Integrals. With Line Integrals we will be integrating functions of two or more variables where the independent variables now are defined by curves rather than regions as with double and triple integrals. We will also investigate conservative vector fields and discuss Green’s Theorem in this chapter.

Chapter overview Vector Fields In this section we introduce the concept of a vector field and give several examples of graphing them. We also revisit the gradient that we first saw a few chapters ago. Line Integrals - Part I In this section we will start off with a quick review of parameterizing curves. This is a skill that will be required in a great many of the line integrals we evaluate and so needs to be understood. We will then formally define the first kind of line integral we will be looking at : line integrals with respect to arc length. Line Integrals - Part II In this section we will continue looking at line integrals and define the second kind of line integral we’ll be looking at : line integrals with respect to \(x\), \(y\), and/or \(z\). We also introduce an alternate form of notation for this kind of line integral that will be useful on occasion. Line Integrals of Vector Fields In this section we will define the third type of line integrals we’ll be looking at : line integrals of vector fields. We will also see that this particular kind of line integral is related to special cases of the line integrals with respect to x, y and z. Fundamental Theorem for Line Integrals In this section we will give the fundamental theorem of calculus for line integrals of vector fields. This will illustrate that certain kinds of line integrals can be very quickly computed. We will also give quite a few definitions and facts that will be useful. Conservative Vector Fields In this section we will take a more detailed look at conservative vector fields than we’ve done in previous sections. We will also discuss how to find potential functions for conservative vector fields. Green's Theorem In this section we will discuss Green’s Theorem as well as an interesting application of Green’s Theorem that we can use to find the area of a two dimensional region.

Surface Integrals In this chapter we look at yet another kind on integral : Surface Integrals. With Surface Integrals we will be integrating functions of two or more variables where the independent variables are now on the surface of three dimensional solids. We will also look at Stokes’ Theorem and the Divergence Theorem.

Chapter overview Curl and Divergence In this section we will introduce the concepts of the curl and the divergence of a vector field. We will also give two vector forms of Green’s Theorem and show how the curl can be used to identify if a three dimensional vector field is conservative field or not. Parametric Surfaces In this section we will take a look at the basics of representing a surface with parametric equations. We will also see how the parameterization of a surface can be used to find a normal vector for the surface (which will be very useful in a couple of sections) and how the parameterization can be used to find the surface area of a surface. Surface Integrals In this section we introduce the idea of a surface integral. With surface integrals we will be integrating over the surface of a solid. In other words, the variables will always be on the surface of the solid and will never come from inside the solid itself. Also, in this section we will be working with the first kind of surface integrals we’ll be looking at in this chapter : surface integrals of functions. Surface Integrals of Vector Fields In this section we will introduce the concept of an oriented surface and look at the second kind of surface integral we’ll be looking at : surface integrals of vector fields. Stokes' Theorem In this section we will discuss Stokes’ Theorem. Divergence Theorem In this section we will discuss the Divergence Theorem.

Extras

Cheat Sheets and Tables

Below are summaries of the options available. Select an option on the left to access it.

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.

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.

Algebra

Chapter overview Exponents Absolute Value Radicals Rationalizing Functions Multiplying Polynomials Factoring Simplifying Rational Expressions Graphing and Common Graphs Solving Equations, Part I Solving Equations, Part II Solving Systems of Equations Solving Inequalities Absolute Value Equations and Inequalities

Trigonometry

Chapter overview Trig Function Evaluation Graphs of Trig Functions Trig Formulas Solving Trig Equations Inverse Trig Functions

Exponentials & Logarithms

Chapter overview Basic Exponential Functions Basic Logarithm Functions Logarithm Properties Simplifying Logarithms Solving Exponential Equations Solving Logarithm Equations

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. General Errors Algebra Errors Trig Errors Common Errors Calculus Errors

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. The Definition Arithmetic Conjugate and Modulus Polar and Exponential Forms Powers and Roots

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. General Tips Taking Notes Getting Help Doing Homework Problem Solving Studying For an Exam Taking an Exam Learn From Your Errors

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Preliminaries In this chapter we will do a quick review of some topics that are absolutely essential to being successful in an Algebra class. We review exponents (integer and rational), radicals, polynomials, factoring polynomials, rational expressions and complex numbers. Integer Exponents In this section we will start looking at exponents. We will give the basic properties of exponents and illustrate some of the common mistakes students make in working with exponents. Examples in this section we will be restricted to integer exponents. Rational exponents will be discussed in the next section. Rational Exponents In this section we will define what we mean by a rational exponent and extend the properties from the previous section to rational exponents. We will also discuss how to evaluate numbers raised to a rational exponent. Radicals In this section we will define radical notation and relate radicals to rational exponents. We will also give the properties of radicals and some of the common mistakes students often make with radicals. We will also define simplified radical form and show how to rationalize the denominator. Polynomials In this section we will introduce the basics of polynomials a topic that will appear throughout this course. We will define the degree of a polynomial and discuss how to add, subtract and multiply polynomials. Factoring Polynomials In this section we look at factoring polynomials a topic that will appear in pretty much every chapter in this course and so is vital that you understand it. We will discuss factoring out the greatest common factor, factoring by grouping, factoring quadratics and factoring polynomials with degree greater than 2. Rational Expressions In this section we will define rational expressions. We will discuss how to reduce a rational expression lowest terms and how to add, subtract, multiply and divide rational expressions. Complex Numbers In this section we give a very quick primer on complex numbers including standard form, adding, subtracting, multiplying and dividing them.

Graphing and Functions In this chapter we’ll look at two very important topics in an Algebra class. First, we will start discussing graphing equations by introducing the Cartesian (or Rectangular) coordinates system and illustrating use of the coordinate system to graph lines and circles. We will also formally define a function and discuss graph functions and combining functions. We will also discuss inverse functions. Graphing In this section we will introduce the Cartesian (or Rectangular) coordinate system. We will define/introduce ordered pairs, coordinates, quadrants, and x and y-intercepts. We will illustrate these concepts with a couple of quick examples Lines In this section we will discuss graphing lines. We will introduce the concept of slope and discuss how to find it from two points on the line. In addition, we will introduce the standard form of the line as well as the point-slope form and slope-intercept form of the line. We will finish off the section with a discussion on parallel and perpendicular lines. Circles In this section we discuss graphing circles. We introduce the standard form of the circle and show how to use completing the square to put an equation of a circle into standard form. The Definition of a Function In this section we will formally define relations and functions. We also give a “working definition” of a function to help understand just what a function is. We introduce function notation and work several examples illustrating how it works. We also define the domain and range of a function. In addition, we introduce piecewise functions in this section. Graphing Functions In this section we discuss graphing functions including several examples of graphing piecewise functions. Combining Functions In this section we will discuss how to add, subtract, multiply and divide functions. In addition, we introduce the concept of function composition. Inverse Functions In this section we define one-to-one and inverse functions. We also discuss a process we can use to find an inverse function and verify that the function we get from this process is, in fact, an inverse function.

Common Graphs In this chapter we will look at graphing some of the more common functions you might be asked to graph. We graph parabolas, ellipses, hyperbolas and rational functions in this chapter. We will also look at transformations of functions and introduce the concept of symmetry. Lines, Circles and Piecewise Functions This section is here only to acknowledge that we’ve already talked about graphing these in a previous chapter. Parabolas In this section we will be graphing parabolas. We introduce the vertex and axis of symmetry for a parabola and give a process for graphing parabolas. We also illustrate how to use completing the square to put the parabola into the form \(f(x)=a(x-h)^{2}+k\). Ellipses In this section we will graph ellipses. We introduce the standard form of an ellipse and how to use it to quickly graph an ellipse. Hyperbolas In this section we will graph hyperbolas. We introduce the standard form of a hyperbola and how to use it to quickly graph a hyperbola. Miscellaneous Functions In this section we will graph a couple of common functions that don’t really take all that much work to do but will be needed in later sections. We’ll be looking at the constant function, square root, absolute value and a simple cubic function. Transformations In this section we will be looking at vertical and horizontal shifts of graphs as well as reflections of graphs about the \(x\) and \(y\)-axis. Collectively these are often called transformations and if we understand them they can often be used to allow us to quickly graph some fairly complicated functions. Symmetry In this section we introduce the idea of symmetry. We discuss symmetry about the x-axis, y-axis and the origin and we give methods for determining what, if any symmetry, a graph will have without having to actually graph the function. Rational Functions In this section we will discuss a process for graphing rational functions. We will also introduce the ideas of vertical and horizontal asymptotes as well as how to determine if the graph of a rational function will have them.

Polynomial Functions In this chapter we will take a more detailed look at polynomial functions. We will discuss dividing polynomials, finding zeroes of polynomials and sketching the graph of polynomials. We will also look at partial fractions (even though this doesn’t really involve polynomial functions). Dividing Polynomials In this section we’ll review some of the basics of dividing polynomials. We will define the remainder and divisor used in the division process and introduce the idea of synthetic division. We will also give the Division Algorithm. Zeroes/Roots of Polynomials In this section we’ll define the zero or root of a polynomial and whether or not it is a simple root or has multiplicity \(k\). We will also give the Fundamental Theorem of Algebra and The Factor Theorem as well as a couple of other useful Facts. Graphing Polynomials In this section we will give a process that will allow us to get a rough sketch of the graph of some polynomials. We discuss how to determine the behavior of the graph at \(x\)-intercepts and the leading coefficient test to determine the behavior of the graph as we allow x to increase and decrease without bound. Finding Zeroes of Polynomials As we saw in the previous section in order to sketch the graph of a polynomial we need to know what it’s zeroes are. However, if we are not able to factor the polynomial we are unable to do that process. So, in this section we’ll look at a process using the Rational Root Theorem that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes. Partial Fractions In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression. What we will be asking here is what “smaller” rational expressions did we add and/or subtract to get the given rational expression. This is a process that has a lot of uses in some later math classes. It can show up in Calculus and Differential Equations for example.

Exponential and Logarithm Functions In this chapter we will introduce two very important functions in many areas : the exponential and logarithm functions. We will look at their basic properties, applications and solving equations involving the two functions. If you are in a field that takes you into the sciences or engineering then you will be running into both of these functions. Exponential Functions In this section we will introduce exponential functions. We will give some of the basic properties and graphs of exponential functions. We will also discuss what many people consider to be the exponential function, \(f(x) = {\bf e}^{x}\). Logarithm Functions In this section we will introduce logarithm functions. We give the basic properties and graphs of logarithm functions. In addition, we discuss how to evaluate some basic logarithms including the use of the change of base formula. We will also discuss the common logarithm, \(\log(x)\), and the natural logarithm, \(\ln(x)\). Solving Exponential Equations In this section we will discuss a couple of methods for solving equations that contain exponentials. Solving Logarithm Equations In this section we will discuss a couple of methods for solving equations that contain logarithms. Also, as we’ll see, with one of the methods we will need to be careful of the results of the method as it is always possible that the method gives values that are, in fact, not solutions to the equation. Applications In this section we will look at a couple of applications of exponential functions and an application of logarithms. We look at compound interest, exponential growth and decay and earthquake intensity.

Systems of Equations In this chapter we will take a look at solving systems of equations. We will solve linear as well as nonlinear systems of equations. We will also take a quick look at using augmented matrices to solve linear systems of equations. Linear Systems with Two Variables In this section we will solve systems of two equations and two variables. We will use the method of substitution and method of elimination to solve the systems in this section. We will also introduce the concepts of inconsistent systems of equations and dependent systems of equations. Linear Systems with Three Variables In this section we will work a couple of quick examples illustrating how to use the method of substitution and method of elimination introduced in the previous section as they apply to systems of three equations. Augmented Matrices In this section we will look at another method for solving systems. We will introduce the concept of an augmented matrix. This will allow us to use the method of Gauss-Jordan elimination to solve systems of equations. We will use the method with systems of two equations and systems of three equations. More on the Augmented Matrix In this section we will revisit the cases of inconsistent and dependent solutions to systems and how to identify them using the augmented matrix method. Nonlinear Systems In this section we will take a quick look at solving nonlinear systems of equations. A nonlinear system of equations is a system in which at least one of the equations is not linear, i.e. has degree of two or more. Note as well that the discussion here does not cover all the possible solution methods for nonlinear systems. Solving nonlinear systems is often a much more involved process than solving linear systems.

Derivatives In this chapter we introduce Derivatives. We cover the standard derivatives formulas including the product rule, quotient rule and chain rule as well as derivatives of polynomials, roots, trig functions, inverse trig functions, hyperbolic functions, exponential functions and logarithm functions. We also cover implicit differentiation, related rates, higher order derivatives and logarithmic differentiation. The Definition of the Derivative In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function. Interpretation of the Derivative In this section we give several of the more important interpretations of the derivative. We discuss the rate of change of a function, the velocity of a moving object and the slope of the tangent line to a graph of a function. Differentiation Formulas In this section we give most of the general derivative formulas and properties used when taking the derivative of a function. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers. Product and Quotient Rule In this section we will give two of the more important formulas for differentiating functions. We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate. Derivatives of Trig Functions In this section we will discuss differentiating trig functions. Derivatives of all six trig functions are given and we show the derivation of the derivative of \(\sin(x)\) and \(\tan(x)\). Derivatives of Exponential and Logarithm Functions In this section we derive the formulas for the derivatives of the exponential and logarithm functions. Derivatives of Inverse Trig Functions In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent. Derivatives of Hyperbolic Functions In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine. Chain Rule In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule! Implicit Differentiation In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e.g. y = f(x) and yet we will still need to know what f'(x) is. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates (the next section). Related Rates In this section we will discuss the only application of derivatives in this section, Related Rates. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one (or more) quantities in the problem. This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work. Higher Order Derivatives In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives. Logarithmic Differentiation In this section we will discuss logarithmic differentiation. Logarithmic differentiation gives an alternative method for differentiating products and quotients (sometimes easier than using product and quotient rule). More importantly, however, is the fact that logarithm differentiation allows us to differentiate functions that are in the form of one function raised to another function, i.e. there are variables in both the base and exponent of the function.

Applications of Integrals In this chapter we will take a look at some applications of integrals. We will look at Average Function Value, Area Between Curves, Volume (both solids of revolution and other solids) and Work. Average Function Value In this section we will look at using definite integrals to determine the average value of a function on an interval. We will also give the Mean Value Theorem for Integrals. Area Between Curves In this section we’ll take a look at one of the main applications of definite integrals in this chapter. We will determine the area of the region bounded by two curves. Volumes of Solids of Revolution / Method of Rings In this section, the first of two sections devoted to finding the volume of a solid of revolution, we will look at the method of rings/disks to find the volume of the object we get by rotating a region bounded by two curves (one of which may be the \(x\) or \(y\)-axis) around a vertical or horizontal axis of rotation. Volumes of Solids of Revolution/Method of Cylinders In this section, the second of two sections devoted to finding the volume of a solid of revolution, we will look at the method of cylinders/shells to find the volume of the object we get by rotating a region bounded by two curves (one of which may be the \(x\) or \(y\)-axis) around a vertical or horizontal axis of rotation. More Volume Problems In the previous two sections we looked at solids that could be found by treating them as a solid of revolution. Not all solids can be thought of as solids of revolution and, in fact, not all solids of revolution can be easily dealt with using the methods from the previous two sections. So, in this section we’ll take a look at finding the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution. Work In this section we will look at is determining the amount of work required to move an object subject to a force over a given distance.

Extras In this chapter proofs of many of the facts/properties/theorems given through out the material are given. Also included are a brief review of summation notation, a discussion on the different 'types' of infinity and a discussion about a subtlety involved with the constant of integration from indefinite integrals. Proof of Various Limit Properties In this section we prove several of the limit properties and facts that were given in various sections of the Limits chapter. Proof of Various Derivative Properties In this section we prove several of the rules/formulas/properties of derivatives that we saw in Derivatives Chapter.

Proof of Trig Limits In this section we give proofs for the two limits that are needed to find the derivative of the sine and cosine functions using the definition of the derivative. Proofs of Derivative Applications Facts In this section we prove many of the facts that we saw in the Applications of Derivatives chapter. Proof of Various Integral Properties In this section we prove some of the facts and formulas from the Integral Chapter as well as a couple from the Applications of Integrals chapter. Area and Volume Formulas In this section we derive the formulas for finding area between two curves and finding the volume of a solid of revolution. Types of Infinity In this section we have a discussion on the types of infinity and how these affect certain limits. Note that there is a lot of theory going on 'behind the scenes' so to speak that we are not going to cover in this section. This section is intended only to give you a feel for what is going on here. To get a fuller understanding of some of the ideas in this section you will need to take some upper level mathematics courses. Summation Notation In this section we give a quick review of summation notation. Summation notation is heavily used when defining the definite integral and when we first talk about determining the area between a curve and the \(x\)-axis. Constant of Integration In this section we have a discussion on a couple of subtleties involving constants of integration that many students don’t think about when doing indefinite integrals. Not understanding these subtleties can lead to confusion on occasion when students get different answers to the same integral. We include two examples of this kind of situation.

Integration Techniques In this chapter we will look at several integration techniques including Integration by Parts, Integrals Involving Trig Functions, Trig Substitutions and Partial Fractions. We will also look at Improper Integrals including using the Comparison Test for convergence/divergence of improper integrals. Integration by Parts In this section we will be looking at Integration by Parts. Of all the techniques we’ll be looking at in this class this is the technique that students are most likely to run into down the road in other classes. We also give a derivation of the integration by parts formula. Integrals Involving Trig Functions In this section we look at integrals that involve trig functions. In particular we concentrate integrating products of sines and cosines as well as products of secants and tangents. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions. Trig Substitutions In this section we will look at integrals (both indefinite and definite) that require the use of a substitutions involving trig functions and how they can be used to simplify certain integrals. Partial Fractions In this section we will use partial fractions to rewrite integrands into a form that will allow us to do integrals involving some rational functions. Integrals Involving Roots In this section we will take a look at a substitution that can, on occasion, be used with integrals involving roots. Integrals Involving Quadratics In this section we are going to look at some integrals that involve quadratics for which the previous techniques won’t work right away. In some cases, manipulation of the quadratic needs to be done before we can do the integral. We will see several cases where this is needed in this section. Integration Strategy In this section we give a general set of guidelines for determining how to evaluate an integral. The guidelines give here involve a mix of both Calculus I and Calculus II techniques to be as general as possible. Also note that there really isn’t one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. Improper Integrals In this section we will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. Collectively, they are called improper integrals and as we will see they may or may not have a finite (i.e. not infinite) value. Determining if they have finite values will, in fact, be one of the major topics of this section. Comparison Test for Improper Integrals It will not always be possible to evaluate improper integrals and yet we still need to determine if they converge or diverge (i.e. if they have a finite value or not). So, in this section we will use the Comparison Test to determine if improper integrals converge or diverge. Approximating Definite Integrals In this section we will look at several fairly simple methods of approximating the value of a definite integral. It is not possible to evaluate every definite integral (i.e. because it is not possible to do the indefinite integral) and yet we may need to know the value of the definite integral anyway. These methods allow us to at least get an approximate value which may be enough in a lot of cases.

Applications of Integrals In this chapter we’ll take a look at a few applications of integrals. We will look at determining the arc length of a curve, the surface area of a solid of revolution, the center of mass of a region bounded by two curves, the hydrostatic force/pressure on a plate submerged in water and a quick look at computing the mean of a probability density function. The applications given here tend to result in integrals that are typically covered in a Calculus II course. Arc Length In this section we’ll determine the length of a curve over a given interval. Surface Area In this section we’ll determine the surface area of a solid of revolution, i.e. a solid obtained by rotating a region bounded by two curves about a vertical or horizontal axis. Center of Mass In this section we will determine the center of mass or centroid of a thin plate where the plate can be described as a region bounded by two curves (one of which may the \(x\) or \(y\)-axis). Hydrostatic Pressure and Force In this section we’ll determine the hydrostatic pressure and force on a vertical plate submerged in water. The plates used in the examples can all be described as regions bounded by one or more curves/lines. Probability Many quantities can be described with probability density functions. For example, the length of time a person waits in line at a checkout counter or the life span of a light bulb. None of these quantities are fixed values and will depend on a variety of factors. In this section we will look at probability density functions and computing the mean (think average wait in line or average life span of a light blub) of a probability density function.

Parametric Equations and Polar Coordinates In this chapter we will introduce the ideas of parametric equations and polar coordinates. We will also look at many of the basic Calculus ideas (tangent lines, area, arc length and surface area) in terms of these two ideas. Parametric Equations and Curves In this section we will introduce parametric equations and parametric curves (i.e. graphs of parametric equations). We will graph several sets of parametric equations and discuss how to eliminate the parameter to get an algebraic equation which will often help with the graphing process. Tangents with Parametric Equations In this section we will discuss how to find the derivatives \(\frac{dy}{dx}\) and \(\frac{d^{2}y}{dx^{2}}\) for parametric curves. We will also discuss using these derivative formulas to find the tangent line for parametric curves as well as determining where a parametric curve in increasing/decreasing and concave up/concave down. Area with Parametric Equations In this section we will discuss how to find the area between a parametric curve and the \(x\)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus I techniques on the resulting algebraic equation). Arc Length with Parametric Equations In this section we will discuss how to find the arc length of a parametric curve using only the parametric equations (rather than eliminating the parameter and using standard Calculus techniques on the resulting algebraic equation). Surface Area with Parametric Equations In this section we will discuss how to find the surface area of a solid obtained by rotating a parametric curve about the \(x\) or \(y\)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus techniques on the resulting algebraic equation). Polar Coordinates In this section we will introduce polar coordinates an alternative coordinate system to the ‘normal’ Cartesian/Rectangular coordinate system. We will derive formulas to convert between polar and Cartesian coordinate systems. We will also look at many of the standard polar graphs as well as circles and some equations of lines in terms of polar coordinates. Tangents with Polar Coordinates In this section we will discuss how to find the derivative \(\frac{dy}{dx}\) for polar curves. We will also discuss using this derivative formula to find the tangent line for polar curves using only polar coordinates (rather than converting to Cartesian coordinates and using standard Calculus techniques). Area with Polar Coordinates In this section we will discuss how to the area enclosed by a polar curve. The regions we look at in this section tend (although not always) to be shaped vaguely like a piece of pie or pizza and we are looking for the area of the region from the outer boundary (defined by the polar equation) and the origin/pole. We will also discuss finding the area between two polar curves. Arc Length with Polar Coordinates In this section we will discuss how to find the arc length of a polar curve using only polar coordinates (rather than converting to Cartesian coordinates and using standard Calculus techniques). Surface Area with Polar Coordinates In this section we will discuss how to find the surface area of a solid obtained by rotating a polar curve about the \(x\) or \(y\)-axis using only polar coordinates (rather than converting to Cartesian coordinates and using standard Calculus techniques). Arc Length and Surface Area Revisited In this section we will summarize all the arc length and surface area formulas we developed over the course of the last two chapters.

Vectors In this (very brief) chapter we will take a look at the basics of vectors. Included are common notation for vectors, arithmetic of vectors, dot product of vectors (and applications) and cross product of vectors (and applications). Vectors - The Basics In this section we will introduce some common notation for vectors as well as some of the basic concepts about vectors such as the magnitude of a vector and unit vectors. We also illustrate how to find a vector from its starting and end points. Vector Arithmetic In this section we will discuss the mathematical and geometric interpretation of the sum and difference of two vectors. We also define and give a geometric interpretation for scalar multiplication. We also give some of the basic properties of vector arithmetic and introduce the common \(i\), \(j\), \(k\) notation for vectors. Dot Product In this section we will define the dot product of two vectors. We give some of the basic properties of dot products and define orthogonal vectors and show how to use the dot product to determine if two vectors are orthogonal. We also discuss finding vector projections and direction cosines in this section. Cross Product In this section we define the cross product of two vectors and give some of the basic facts and properties of cross products.

Partial Derivatives In this chapter we’ll take a brief look at limits of functions of more than one variable and then move into derivatives of functions of more than one variable. As we’ll see if we can do derivatives of functions with one variable it isn’t much more difficult to do derivatives of functions of more than one variable (with a very important subtlety). We will also discuss interpretations of partial derivatives, higher order partial derivatives and the chain rule as applied to functions of more than one variable. We will also define and discuss directional derivatives. Limits In the section we’ll take a quick look at evaluating limits of functions of several variables. We will also see a fairly quick method that can be used, on occasion, for showing that some limits do not exist. Partial Derivatives In this section we will look at the idea of partial derivatives. We will give the formal definition of the partial derivative as well as the standard notations and how to compute them in practice (i.e. without the use of the definition). As you will see if you can do derivatives of functions of one variable you won’t have much of an issue with partial derivatives. There is only one (very important) subtlety that you need to always keep in mind while computing partial derivatives. Interpretations of Partial Derivatives In the section we will take a look at a couple of important interpretations of partial derivatives. First, the always important, rate of change of the function. Although we now have multiple ‘directions’ in which the function can change (unlike in Calculus I). We will also see that partial derivatives give the slope of tangent lines to the traces of the function. Higher Order Partial Derivatives In the section we will take a look at higher order partial derivatives. Unlike Calculus I however, we will have multiple second order derivatives, multiple third order derivatives, etc. because we are now working with functions of multiple variables. We will also discuss Clairaut’s Theorem to help with some of the work in finding higher order derivatives. Differentials In this section we extend the idea of differentials we first saw in Calculus I to functions of several variables. Chain Rule In the section we extend the idea of the chain rule to functions of several variables. In particular, we will see that there are multiple variants to the chain rule here all depending on how many variables our function is dependent on and how each of those variables can, in turn, be written in terms of different variables. We will also give a nice method for writing down the chain rule for pretty much any situation you might run into when dealing with functions of multiple variables. In addition, we will derive a very quick way of doing implicit differentiation so we no longer need to go through the process we first did back in Calculus I. Directional Derivatives In the section we introduce the concept of directional derivatives. With directional derivatives we can now ask how a function is changing if we allow all the independent variables to change rather than holding all but one constant as we had to do with partial derivatives. In addition, we will define the gradient vector to help with some of the notation and work here. The gradient vector will be very useful in some later sections as well. We will also give a nice fact that will allow us to determine the direction in which a given function is changing the fastest.

Applications of Partial Derivatives In this chapter we will take a look at several applications of partial derivatives. We will find the equation of tangent planes to surfaces and we will revisit on of the more important applications of derivatives from earlier Calculus classes. We will spend a significant amount of time finding relative and absolute extrema of functions of multiple variables. We will also introduce Lagrange multipliers to find the absolute extrema of a function subject to one or more constraints. Tangent Planes and Linear Approximations In this section formally define just what a tangent plane to a surface is and how we use partial derivatives to find the equations of tangent planes to surfaces that can be written as \(z=f(x,y)\). We will also see how tangent planes can be thought of as a linear approximation to the surface at a given point. Gradient Vector, Tangent Planes and Normal Lines In this section discuss how the gradient vector can be used to find tangent planes to a much more general function than in the previous section. We will also define the normal line and discuss how the gradient vector can be used to find the equation of the normal line. Relative Minimums and Maximums In this section we will define critical points for functions of two variables and discuss a method for determining if they are relative minimums, relative maximums or saddle points (i.e. neither a relative minimum or relative maximum). Absolute Minimums and Maximums In this section we will how to find the absolute extrema of a function of two variables when the independent variables are only allowed to come from a region that is bounded (i.e. no part of the region goes out to infinity) and closed (i.e. all of the points on the boundary are valid points that can be used in the process). Lagrange Multipliers In this section we’ll see discuss how to use the method of Lagrange Multipliers to find the absolute minimums and maximums of functions of two or three variables in which the independent variables are subject to one or more constraints. We also give a brief justification for how/why the method works.

Line Integrals In this chapter we will introduce a new kind of integral : Line Integrals. With Line Integrals we will be integrating functions of two or more variables where the independent variables now are defined by curves rather than regions as with double and triple integrals. We will also investigate conservative vector fields and discuss Green’s Theorem in this chapter. Vector Fields In this section we introduce the concept of a vector field and give several examples of graphing them. We also revisit the gradient that we first saw a few chapters ago. Line Integrals - Part I In this section we will start off with a quick review of parameterizing curves. This is a skill that will be required in a great many of the line integrals we evaluate and so needs to be understood. We will then formally define the first kind of line integral we will be looking at : line integrals with respect to arc length. Line Integrals - Part II In this section we will continue looking at line integrals and define the second kind of line integral we’ll be looking at : line integrals with respect to \(x\), \(y\), and/or \(z\). We also introduce an alternate form of notation for this kind of line integral that will be useful on occasion. Line Integrals of Vector Fields In this section we will define the third type of line integrals we’ll be looking at : line integrals of vector fields. We will also see that this particular kind of line integral is related to special cases of the line integrals with respect to x, y and z. Fundamental Theorem for Line Integrals In this section we will give the fundamental theorem of calculus for line integrals of vector fields. This will illustrate that certain kinds of line integrals can be very quickly computed. We will also give quite a few definitions and facts that will be useful. Conservative Vector Fields In this section we will take a more detailed look at conservative vector fields than we’ve done in previous sections. We will also discuss how to find potential functions for conservative vector fields. Green's Theorem In this section we will discuss Green’s Theorem as well as an interesting application of Green’s Theorem that we can use to find the area of a two dimensional region.

Surface Integrals In this chapter we look at yet another kind on integral : Surface Integrals. With Surface Integrals we will be integrating functions of two or more variables where the independent variables are now on the surface of three dimensional solids. We will also look at Stokes’ Theorem and the Divergence Theorem. Curl and Divergence In this section we will introduce the concepts of the curl and the divergence of a vector field. We will also give two vector forms of Green’s Theorem and show how the curl can be used to identify if a three dimensional vector field is conservative field or not. Parametric Surfaces In this section we will take a look at the basics of representing a surface with parametric equations. We will also see how the parameterization of a surface can be used to find a normal vector for the surface (which will be very useful in a couple of sections) and how the parameterization can be used to find the surface area of a surface. Surface Integrals In this section we introduce the idea of a surface integral. With surface integrals we will be integrating over the surface of a solid. In other words, the variables will always be on the surface of the solid and will never come from inside the solid itself. Also, in this section we will be working with the first kind of surface integrals we’ll be looking at in this chapter : surface integrals of functions. Surface Integrals of Vector Fields In this section we will introduce the concept of an oriented surface and look at the second kind of surface integral we’ll be looking at : surface integrals of vector fields. Stokes' Theorem In this section we will discuss Stokes’ Theorem. Divergence Theorem In this section we will discuss the Divergence Theorem.

Basic Concepts In this chapter we introduce many of the basic concepts and definitions that are encountered in a typical differential equations course. We will also take a look at direction fields and how they can be used to determine some of the behavior of solutions to differential equations. Definitions In this section some of the common definitions and concepts in a differential equations course are introduced including order, linear vs. nonlinear, initial conditions, initial value problem and interval of validity. Direction Fields In this section we discuss direction fields and how to sketch them. We also investigate how direction fields can be used to determine some information about the solution to a differential equation without actually having the solution. Final Thoughts In this section we give a couple of final thoughts on what we will be looking at throughout this course.

Laplace Transforms In this chapter we introduce Laplace Transforms and how they are used to solve Initial Value Problems. With the introduction of Laplace Transforms we will now be able to solve some Initial Value Problems that we wouldn’t be able to solve otherwise. We will solve differential equations that involve Heaviside and Dirac Delta functions. We will also give brief overview on using Laplace transforms to solve nonconstant coefficient differential equations. In addition, we will define the convolution integral and show how it can be used to take inverse transforms. The Definition In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the definition. Laplace Transforms In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms. Inverse Laplace Transforms In this section we ask the opposite question from the previous section. In other words, given a Laplace transform, what function did we originally have? We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table. Step Functions In this section we introduce the step or Heaviside function. We illustrate how to write a piecewise function in terms of Heaviside functions. We also work a variety of examples showing how to take Laplace transforms and inverse Laplace transforms that involve Heaviside functions. We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions. Solving IVP's with Laplace Transforms In this section we will examine how to use Laplace transforms to solve IVP’s. The examples in this section are restricted to differential equations that could be solved without using Laplace transform. The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to. Nonconstant Coefficient IVP's In this section we will give a brief overview of using Laplace transforms to solve some nonconstant coefficient IVP’s. We do not work a great many examples in this section. We only work a couple to illustrate how the process works with Laplace transforms. IVP's With Step Functions This is the section where the reason for using Laplace transforms really becomes apparent. We will use Laplace transforms to solve IVP’s that contain Heaviside (or step) functions. Without Laplace transforms solving these would involve quite a bit of work. While we do work one of these examples without Laplace transforms, we do it only to show what would be involved if we did try to solve one of the examples without using Laplace transforms. Dirac Delta Function In this section we introduce the Dirac Delta function and derive the Laplace transform of the Dirac Delta function. We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. We also give a nice relationship between Heaviside and Dirac Delta functions. Convolution Integrals In this section we give a brief introduction to the convolution integral and how it can be used to take inverse Laplace transforms. We also illustrate its use in solving a differential equation in which the forcing function (i.e. the term without any y’s in it) is not known. Table Of Laplace Transforms This section is the table of Laplace Transforms that we’ll be using in the material. We give as wide a variety of Laplace transforms as possible including some that aren’t often given in tables of Laplace transforms.

Series Solutions to DE's In this chapter we are going to take a quick look at how to represent the solution to a differential equation with a power series. We will also look at how to solve Euler’s differential equation. In addition, we will do a quick review of power series and Taylor series to help with work in the chapter. Review : Power Series In this section we give a brief review of some of the basics of power series. Included are discussions of using the Ratio Test to determine if a power series will converge, adding/subtracting power series, differentiating power series and index shifts for power series. Review : Taylor Series In this section we give a quick reminder on how to construct the Taylor series for a function. Included are derivations for the Taylor series of \({\bf e}^{x}\) and \(\cos(x)\) about \(x = 0\) as well as showing how to write down the Taylor series for a polynomial. Series Solutions In this section we define ordinary and singular points for a differential equation. We also show who to construct a series solution for a differential equation about an ordinary point. The method illustrated in this section is useful in solving, or at least getting an approximation of the solution, differential equations with coefficients that are not constant. Euler Equations In this section we will discuss how to solve Euler’s differential equation, \(ax^{2}y'' + b x y' +c y = 0\). Note that while this does not involve a series solution it is included in the series solution chapter because it illustrates how to get a solution to at least one type of differential equation at a singular point.

Boundary Value Problems & Fourier Series In this chapter we will introduce two topics that are integral to basic partial differential equations solution methods. The first topic, boundary value problems, occur in pretty much every partial differential equation. The second topic, Fourier series, is what makes one of the basic solution techniques work. Boundary Value Problems In this section we’ll define boundary conditions (as opposed to initial conditions which we should already be familiar with at this point) and the boundary value problem. We will also work a few examples illustrating some of the interesting differences in using boundary values instead of initial conditions in solving differential equations. Eigenvalues and Eigenfunctions In this section we will define eigenvalues and eigenfunctions for boundary value problems. We will work quite a few examples illustrating how to find eigenvalues and eigenfunctions. In one example the best we will be able to do is estimate the eigenvalues as that is something that will happen on a fairly regular basis with these kinds of problems. Periodic Functions & Orthogonal Functions In this section we will define periodic functions, orthogonal functions and mutually orthogonal functions. We will also work a couple of examples showing intervals on which \(\cos\left(\frac{n \pi x}{L}\right)\) and \(\sin\left(\frac{n \pi x}{L}\right)\) are mutually orthogonal. The results of these examples will be very useful for the rest of this chapter and most of the next chapter. Fourier Sine Series In this section we define the Fourier Sine Series, i.e. representing a function with a series in the form \(\sum\limits_{n = 1}^\infty {{B_n}\sin \left( {\frac{{n\pi x}}{L}} \right)} \). We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function. Fourier Cosine Series In this section we define the Fourier Cosine Series, i.e. representing a function with a series in the form \(\sum\limits_{n = 0}^\infty {{A_n}\cos \left( {\frac{{n\pi x}}{L}} \right)} \). We will also define the even extension for a function and work several examples finding the Fourier Cosine Series for a function. Fourier Series In this section we define the Fourier Series, i.e. representing a function with a series in the form \(\sum\limits_{n = 0}^\infty {{A_n}\cos \left( {\frac{{n\pi x}}{L}} \right)} + \sum\limits_{n = 1}^\infty {{B_n}\sin \left( {\frac{{n\pi x}}{L}} \right)} \). We will also work several examples finding the Fourier Series for a function. Convergence of Fourier Series In this section we will define piecewise smooth functions and the periodic extension of a function. In addition, we will give a variety of facts about just what a Fourier series will converge to and when we can expect the derivative or integral of a Fourier series to converge to the derivative or integral of the function it represents.

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