Calculus PDF

© 2022 Table of Contents Preface vi Outline...

© 2022 Table of Contents Preface vi Outline ix 1 Review 1 1.1 Functions........................................ 2 1.2 Inverse Functions.................................... 14 1.3 Trig Functions...................................... 21 1.4 Solving Trig Equations................................. 27 1.5 Solving Trig Equations with Calculators, Part I.................... 37 1.6 Solving Trig Equations with Calculators, Part II................... 50 1.7 Exponential Functions................................. 55 1.8 Logarithm Functions.................................. 58 1.9 Exponential and Logarithm Equations........................ 65 1.10 Common Graphs.................................... 73 2 Limits 86 2.1 Tangent Lines and Rates of Change......................... 87 2.2 The Limit........................................ 97 2.3 One-Sided Limits.................................... 108 2.4 Limit Properties..................................... 115 2.5 Computing Limits.................................... 121 2.6 Infinite Limits...................................... 130 2.7 Limits at Infinity, Part I................................. 141 2.8 Limits at Infinity, Part II................................. 150 2.9 Continuity........................................ 160 2.10 The Definition of the Limit............................... 168 3 Derivatives 183 3.1 The Definition of the Derivative............................ 185 3.2 Interpretation of the Derivative............................ 192 3.3 Differentiation Formulas................................ 202 3.4 Product and Quotient Rule............................... 211 3.5 Derivatives of Trig Functions............................. 217 3.6 Derivatives of Exponentials & Logarithms...................... 228 i  Table of Contents 3.7 Derivatives of Inverse Trig Functions......................... 233 3.8 Derivatives of Hyperbolic Functions.......................... 239 3.9 Chain Rule....................................... 241 3.10 Implicit Differentiation................................. 253 3.11 Related Rates...................................... 262 3.12 Higher Order Derivatives................................ 279 3.13 Logarithmic Differentiation............................... 284 4 Derivative Applications 287 4.1 Rates of Change.................................... 289 4.2 Critical Points...................................... 293 4.3 Minimum and Maximum Values............................ 301 4.4 Finding Absolute Extrema............................... 311 4.5 The Shape of a Graph, Part I............................. 318 4.6 The Shape of a Graph, Part II............................. 329 4.7 The Mean Value Theorem............................... 340 4.8 Optimization....................................... 347 4.9 More Optimization................................... 365 4.10 L’Hospital’s Rule.................................... 384 4.11 Linear Approximations................................. 390 4.12 Differentials....................................... 393 4.13 Newton’s Method.................................... 396 4.14 Business Applications................................. 400 5 Integrals 407 5.1 Indefinite Integrals................................... 408 5.2 Computing Indefinite Integrals............................. 414 5.3 Substitution Rule for Indefinite Integrals....................... 424 5.4 More Substitution Rule................................. 437 5.5 Area Problem...................................... 451 5.6 Definition of the Definite Integral........................... 461 5.7 Computing Definite Integrals............................. 473 5.8 Substitution Rule for Definite Integrals........................ 487 6 Applications of Integrals 497 6.1 Average Function Value................................ 498 6.2 Area Between Curves................................. 501 6.3 Volume with Rings................................... 514 6.4 Volume with Cylinders................................. 525 6.5 More Volume Problems................................ 534 6.6 Work........................................... 546 7 Integration Techniques 553 7.1 Integration by Parts................................... 556 © Paul Dawkins Calculus – ii –  Table of Contents 7.2 Integrals Involving Trig Functions........................... 569 7.3 Trig Substitutions.................................... 582 7.4 Partial Fractions.................................... 598 7.5 Integrals Involving Roots................................ 607 7.6 Integrals Involving Quadratics............................. 610 7.7 Integration Strategy.................................. 619 7.8 Improper Integrals................................... 627 7.9 Comparison Test for Improper Integrals....................... 636 7.10 Approximating Definite Integrals........................... 646 8 More Applications of Integrals 653 8.1 Arc Length....................................... 654 8.2 Surface Area...................................... 661 8.3 Center Of Mass..................................... 667 8.4 Hydrostatic Pressure and Force............................ 671 8.5 Probability........................................ 676 9 Parametric and Polar 680 9.1 Parametric Equations................................. 681 9.2 Tangents with Parametric Equations......................... 704 9.3 Area with Parametric Equations............................ 712 9.4 Arc Length with Parametric Equations........................ 715 9.5 Surface Area with Parametric Equations....................... 719 9.6 Polar Coordinates................................... 721 9.7 Tangents with Polar Coordinates........................... 732 9.8 Area with Polar Coordinates.............................. 734 9.9 Arc Length with Polar Coordinates.......................... 741 9.10 Surface Area with Polar Coordinates......................... 744 9.11 Arc Length and Surface Area Revisited........................ 745 10 Series and Sequences 747 10.1 Sequences....................................... 748 10.2 More on Sequences.................................. 759 10.3 Series - Basics..................................... 765 10.4 Convergence & Divergence of Series......................... 772 10.5 Special Series..................................... 782 10.6 Integral Test....................................... 792 10.7 Comparison & Limit Comparison Test......................... 803 10.8 Alternating Series Test................................. 815 10.9 Absolute Convergence................................. 822 10.10 Ratio Test........................................ 826 10.11 Root Test........................................ 834 10.12 Strategy for Series................................... 838 10.13 Estimating the Value of a Series............................ 841 © Paul Dawkins Calculus – iii –  Table of Contents 10.14 Power Series...................................... 854 10.15 Power Series and Functions.............................. 863 10.16 Taylor Series...................................... 871 10.17 Applications of Series................................. 883 10.18 Binomial Series..................................... 888 11 Vectors 890 11.1 Basic Concepts..................................... 891 11.2 Vector Arithmetic.................................... 896 11.3 Dot Product....................................... 903 11.4 Cross Product...................................... 912 12 3 Dimensional Space 918 12.1 The 3-D Coordinate System.............................. 919 12.2 Equations of Lines................................... 926 12.3 Equations of Planes.................................. 933 12.4 Quadric Surfaces.................................... 936 12.5 Functions of Several Variables............................ 943 12.6 Vector Functions.................................... 951 12.7 Calculus with Vector Functions............................ 962 12.8 Tangent and Normal Vectors............................. 966 12.9 Arc Length with Vector Functions........................... 971 12.10 Curvature........................................ 974 12.11 Velocity and Acceleration............................... 977 12.12 Cylindrical Coordinates................................ 980 12.13 Spherical Coordinates................................. 983 13 Partial Derivatives 990 13.1 Limits.......................................... 991 13.2 Partial Derivatives................................... 997 13.3 Interpretations of Partial Derivatives......................... 1007 13.4 Higher Order Partial Derivatives............................ 1011 13.5 Differentials....................................... 1015 13.6 Chain Rule....................................... 1016 13.7 Directional Derivatives................................. 1026 14 Applications of Partial Derivatives 1035 14.1 Tangent Planes..................................... 1036 14.2 Gradient Vector..................................... 1040 14.3 Relative Extrema.................................... 1043 14.4 Absolute Extrema.................................... 1052 14.5 Lagrange Multipliers.................................. 1061 15 Multiple Integrals 1076 15.1 Double Integrals.................................... 1077 © Paul Dawkins Calculus – iv –  Table of Contents 15.2 Iterated Integrals.................................... 1081 15.3 Double Integrals over General Regions....................... 1089 15.4 Double Integrals in Polar Coordinates........................ 1102 15.5 Triple Integrals..................................... 1114 15.6 Triple Integrals in Cylindrical Coordinates...................... 1123 15.7 Triple Integrals in Spherical Coordinates....................... 1126 15.8 Change of Variables.................................. 1131 15.9 Surface Area...................................... 1145 15.10 Area and Volume Revisited.............................. 1148 16 Line Integrals 1149 16.1 Vector Fields...................................... 1150 16.2 Line Integrals - Part I.................................. 1155 16.3 Line Integrals - Part II................................. 1167 16.4 Line Integrals of Vector Fields............................. 1171 16.5 Fundamental Theorem for Line Integrals....................... 1175 16.6 Conservative Vector Fields.............................. 1179 16.7 Green’s Theorem.................................... 1187 17 Surface Integrals 1196 17.1 Curl and Divergence.................................. 1197 17.2 Parametric Surfaces.................................. 1202 17.3 Surface Integrals.................................... 1210 17.4 Surface Integrals of Vector Fields........................... 1219 17.5 Stokes’ Theorem.................................... 1230 17.6 Divergence Theorem.................................. 1236 A Calculus I Extras 1238 A.1 Proof of Various Limit Properties........................... 1239 A.2 Proof of Various Derivative Properties........................ 1255 A.3 Proof of Trig Limits................................... 1268 A.4 Proofs of Derivative Applications Facts........................ 1273 A.5 Proof of Various Integral Properties.......................... 1283 A.6 Area and Volume Formulas.............................. 1295 A.7 Types of Infinity..................................... 1299 A.8 Summation Notation.................................. 1303 A.9 Constant of Integration................................. 1306 Index 1311 © Paul Dawkins Calculus –v– Preface First, here’s a little bit of history on how this material was created (there’s a reason for this, I promise). A long time ago (2002 or so) when I decided I wanted to put some mathematics stuff on the web I wanted a format for the source documents that could produce both a pdf version as well as a web version of the material. After some investigation I decided to use MS Word and MathType as the easiest/quickest method for doing that. The result was a pretty ugly HTML (i.e web page code) and had the drawback of the mathematics were images which made editing the mathematics painful. However, it was the quickest way or dealing with this stuff. Fast forward a few years (don’t recall how many at this point) and the web had matured enough that it was now much easier to write mathematics in LATEX (https://en.wikipedia.org/wiki/LaTeX) and have it display on the web (LATEX was my first choice for writing the source documents). So, I found a tool that could convert the MS Word mathematics in the source documents to LATEX. It wasn’t perfect and I had to write some custom rules to help with the conversion but it was able to do it without “messing” with the mathematics and so I didn’t need to worry about any math errors being introduced in the conversion process. The only problem with the tool is that all it could do was convert the mathematics and not the rest of the source document into LATEX. That meant I just converted the math into LATEX for the website but didn’t convert the source documents. Now, here’s the reason for this history lesson. Fast forward even more years and I decided that I really needed to convert the source documents into LATEX as that would just make my life easier and I’d be able to enable working links in the pdf as well as a simple way of producing an index for the material. The only issue is that the original tool I’d use to convert the MS Word mathematics had become, shall we say, unreliable and so that was no longer an option and it still has the problem on not converting anything else into proper LATEX code. So, the best option that I had available to me is to take the web pages, which already had the mathematics in proper LATEX format, and convert the rest of the HTML into LATEX code. I wrote a set of tools to do his and, for the most part, did a pretty decent job. The only problem is that the tools weren’t perfect. So, if you run into some “odd” stuff here (things like , , , , etc.) please let me know the section with the code that I missed. I did my best to find all the “orphaned” HTML code but I’m certain I missed some on occasion as I did find my eyes glazing over every once in a while as I went over the converted document. Now, with that out of the way, let’s get into the actual preface of the material. Here are the notes for my Calculus courses that I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn Calculus vi  Preface or needing a refresher in Calculus. This document contains the majority, if not all, of the topics that are typically taught in full set of Calculus courses (i.e. Calculus I, Calculus II and Calculus III). Note that there are also topics that for a variety of reasons (mostly time issues) I am not able to cover in my classes. These topics are included for those that wish to learn those topics and/or for instructors that are using this material for their course and wish to cover the topics. Also, even in sections that I do cover in my classes I may not actually cover all the examples listed here (again for time reasons) and they are provided for those that wish to see another example or two. I’ve tried to make these material as self-contained as possible and so all the information needed to get started reading through them is either from an Algebra or Trig class. Note that, outside of the Review chapter, I am assuming that you do recall the Algebra and/or Trig needed at various points and won’t, for the most part, be reviewing or covering that as those topics arise. For the most part I will simply assume that you recall those topics or can go back and refresh them as needed. This is not meant to be “difficult” with the reader but simply an acknowledgment that I have to assume you have the prerequisite knowledge at some point so we can focus on the topics we’re trying to learn rather than spending all our time refreshing knowledge that you really are supposed to know (or at least be somewhat familiar with) prior to getting into a Calculus course. There is also the reality that if I included discussion/refresher of all the prerequisite material at every step these pages would eventually be so long that it would be hard to focus in on the material that we’re actually trying to learn. Also note that not spending time refreshing prerequisite material will extend into later topics in the material as well. For example, when discussing the integration techniques typically covered in a Calculus II course it is assumed that you know basic integration and don’t need more that a cursory, at best, refresher/reminder of basic integration and basic substitutions. Another example of this would be moving into multi-variable Calculus, i.e. the material typically taught in a Calculus III course. Once we move into multi-variable Calculus it is assumed that you understand single variable Calculus and can do basic differentiation and integration. So, if you are jumping into the middle of the material to learn a particular topic and run across something that you don’t know there is a good chance that you are missing some knowledge of a prerequisite material and will need to find it in this set of material to cover that prior to learning the topic you wish to learn. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. 1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn calculus I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasn’t covered in class. 2. Because I want these notes to provide some more examples for you to read through, I don’t always work the same problems in class as those given in the notes. Likewise, even if I do work some of the problems in here I may work fewer problems in class than are presented here. © Paul Dawkins Calculus – vii –  Preface 3. Sometimes questions in class will lead down paths that are not covered here. I tried to anticipate as many of the questions as possible when writing these up, but the reality is that I can’t anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that I’ve not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are. 4. This is somewhat related to the previous three items, but is important enough to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class. © Paul Dawkins Calculus – viii – Outline Here is a listing (and brief description) of the material in this set of notes. 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, Exponential/Logarithm Functions and Solving Exponential/Logarithm Equations. Functions - In this section we will cover function notation/evaluation, determining the domain and range of a function and function composition. Inverse Functions - In this section we will define an inverse function and the notation used for inverse functions. We will also discuss the process for finding an inverse function. Trig Functions - In this section we will give a quick review of trig functions. We will cover the basic notation, relationship between the trig functions, the right triangle definition of the trig functions. We will also cover evaluation of trig functions as well as the unit circle (one of the most important ideas from a trig class!) and how it can be used to evaluate trig func- tions. Solving Trig Equations - In this section we will discuss how to solve trig equations. The answers to the equations in this section will all be one of the “standard” angles that most students have memorized after a trig class. However, the process used here can be used for any answer regardless of it being one of the standard angles or not. Solving Trig Equations with Calculators, Part I - In this section we will discuss solving trig equations when the answer will (generally) require the use of a calculator (i.e. they aren’t one of the standard angles). Note however, the process used here is identical to that for when the answer is one of the standard angles. The only difference is that the answers in here can be a little messy due to the need of a calculator. Included is a brief discussion of inverse trig functions. Solving Trig Equations with Calculators, Part II - In this section we will continue our discussion of solving trig equations when a calculator is needed to get the answer. The equations in this section tend to be a little trickier than the ”normal” trig equation and are not always covered in a trig class. Exponential Functions - In this section we will discuss exponential functions. We will cover the basic definition of an exponential function, the natural exponential function, i.e. ex , as well as the properties and graphs of exponential functions ix  Outline Logarithm Functions - In this section we will discuss logarithm functions, evaluation of log- arithms and their properties. We will discuss many of the basic manipulations of logarithms that commonly occur in Calculus (and higher) classes. Included is a discussion of the natural (ln(x)) and common logarithm (log(x)) as well as the change of base formula. Exponential and Logarithm Equations - In this section we will discuss various methods for solving equations that involve exponential functions or logarithm functions. Common Graphs - In this section we will do a very quick review of many of the most common functions and their graphs that typically show up in a Calculus class. 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. Tangent Lines and Rates of Change - In this section we will introduce two problems that we will see time and again in this course : Rate of Change of a function and Tangent Lines to functions. Both of these problems will be used to introduce the concept of limits, although we won’t formally give the definition or notation until the next section. The Limit - In this section we will introduce the notation of the limit. We will also take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us. We will be estimating the value of limits in this section to help us understand what they tell us. We will actually start computing limits in a couple of sections. One-Sided Limits - In this section we will introduce the concept of one-sided limits. We will discuss the differences between one-sided limits and limits as well as how they are related to each other. Limit Properties - In this section we will discuss the properties of limits that we’ll need to use in computing limits (as opposed to estimating them as we’ve done to this point). We will also compute a couple of basic limits in this section. Computing Limits - In this section we will looks at several types of limits that require some work before we can use the limit properties to compute them. We will also look at computing limits of piecewise functions and use of the Squeeze Theorem to compute some limits. Infinite Limits - In this section we will look at limits that have a value of infinity or negative infinity. We’ll also take a brief look at vertical asymptotes. Limits At Infinity, Part I - In this section we will start looking at limits at infinity, i.e. limits in which the variable gets very large in either the positive or negative sense. We will concentrate on polynomials and rational expressions in this section. We’ll also take a brief look at horizontal asymptotes. Limits At Infinity, Part II - In this section we will continue covering limits at infinity. We’ll be looking at exponentials, logarithms and inverse tangents in this section. © Paul Dawkins Calculus –x–  Outline Continuity - In this section we will introduce the concept of continuity and how it relates to limits. We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval. The Definition of the Limit - In this section we will give a precise definition of several of the limits covered in this section. We will work several basic examples illustrating how to use this precise definition to compute a limit. We’ll also give a precise definition of continuity. Derivatives - In this chapter we will start looking at the next major topic in a calculus class, deriva- tives. This chapter is devoted almost exclusively to finding derivatives. We will be looking at one application of them in this chapter. We will be leaving most of the applications of derivatives to the next chapter. The Definition of the Derivative - In this section we define the derivative, give various nota- tions 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 inter- pretations 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! © Paul Dawkins Calculus – xi –  Outline Implicit Differentiation - In this section we will discuss implicit differentiation. Not every func- tion 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 0 (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 sec- tion, 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 differenti- ation works for higher order derivatives. Logarithmic Differentiation - In this section we will discuss logarithmic differentiation. Loga- rithmic 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. Derivative Applications - In the previous chapter we focused almost exclusively on the computation of derivatives. In this chapter will focus on applications of derivatives. It is important to always remember that we didn’t spend a whole chapter talking about computing derivatives just to be talking about them. There are many very important applications to derivatives. The two main applications that we’ll be looking at in this chapter are using derivatives to determine information about graphs of functions and optimization problems. These will not be the only appli- cations however. We will be revisiting limits and taking a look at an application of derivatives that will allow us to compute limits that we haven’t been able to compute previously. We will also see how derivatives can be used to estimate solutions to equations. Rates of Change - In this section we review the main application/interpretation of deriva- tives from the previous chapter (i.e. rates of change) that we will be using in many of the applications in this chapter. Critical Points - In this section we give the definition of critical points. Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. We will work a number of examples illustrating how to find them for a wide variety of functions. Minimum and Maximum Values - In this section we define absolute (or global) minimum and maximum values of a function and relative (or local) minimum and maximum values of a function. It is important to understand the difference between the two types of minimum/- maximum (collectively called extrema) values for many of the applications in this chapter © Paul Dawkins Calculus – xii –  Outline and so we use a variety of examples to help with this. We also give the Extreme Value Theo- rem and Fermat’s Theorem, both of which are very important in the many of the applications we’ll see in this chapter. Finding Absolute Extrema - In this section we discuss how to find the absolute (or global) minimum and maximum values of a function. In other words, we will be finding the largest and smallest values that a function will have. The Shape of a Graph, Part I - In this section we will discuss what the first derivative of a function can tell us about the graph of a function. The first derivative will allow us to identify the relative (or local) minimum and maximum values of a function and where a function will be increasing and decreasing. We will also give the First Derivative test which will allow us to classify critical points as relative minimums, relative maximums or neither a minimum or a maximum. The Shape of a Graph, Part II - In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down. The second derivative will also allow us to identify any inflection points (i.e. where concavity changes) that a function may have. We will also give the Second Derivative Test that will give an alternative method for identifying some critical points (but not all) as relative minimums or relative maximums. The Mean Value Theorem - In this section we will give Rolle’s Theorem and the Mean Value Theorem. With the Mean Value Theorem we will prove a couple of very nice facts, one of which will be very useful in the next chapter. Optimization Problems - In this section we will be determining the absolute minimum and/or maximum of a function that depends on two variables given some constraint, or relationship, that the two variables must always satisfy. We will discuss several methods for determining the absolute minimum or maximum of the function. Examples in this section tend to center around geometric objects such as squares, boxes, cylinders, etc. More Optimization Problems - In this section we will continue working optimization problems. The examples in this section tend to be a little more involved and will often involve situations that will be more easily described with a sketch as opposed to the ’simple’ geometric objects we looked at in the previous section. L’Hospital’s Rule and Indeterminate Forms - In this section we will revisit indeterminate forms and limits and take a look at L’Hospital’s Rule. L’Hospital’s Rule will allow us to evaluate some limits we were not able to previously. Linear Approximations - In this section we discuss using the derivative to compute a linear approximation to a function. We can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do with when we have the function there really are reasons that one might want to do this. We give two ways this can be useful in the examples. © Paul Dawkins Calculus – xiii –  Outline Differentials - In this section we will compute the differential for a function. We will give an application of differentials in this section. However, one of the more important uses of differentials will come in the next chapter and unfortunately we will not be able to discuss it until then. Newton’s Method - In this section we will discuss Newton’s Method. Newton’s Method is an application of derivatives that will allow us to approximate solutions to an equation. There are many equations that cannot be solved directly and with this method we can get approxi- mations to the solutions to many of those equations. Business Applications - In this section we will give a cursory discussion of some basic ap- plications of derivatives to the business field. We will revisit finding the maximum and/or minimum function value and we will define the marginal cost function, the average cost, the revenue function, the marginal revenue function and the marginal profit function. Note that this section is only intended to introduce these concepts and not teach you everything about them. Integrals In this chapter we will be looking at integrals. Integrals are the third and final major topic that will be covered in this class. As with derivatives this chapter will be devoted almost exclusively to finding and computing integrals. Applications will be given in the following chapter. There are really two types of integrals that we’ll be looking at in this chapter : Indefinite Integrals and Definite Integrals. The first half of this chapter is devoted to indefinite integrals and the last half is devoted to definite integrals. As we will see in the last half of the chapter if we don’t know indefinite integrals we will not be able to do definite integrals. Indefinite Integrals - In this section we will start off the chapter with the definition and proper- ties of indefinite integrals. We will not be computing many indefinite integrals in this section. This section is devoted to simply defining what an indefinite integral is and to give many of the properties of the indefinite integral. Actually computing indefinite integrals will start in the next section. Computing Indefinite Integrals - In this section we will compute some indefinite integrals. The integrals in this section will tend to be those that do not require a lot of manipulation of the function we are integrating in order to actually compute the integral. As we will see starting in the next section many integrals do require some manipulation of the function before we can actually do the integral. We will also take a quick look at an application of indefinite integrals. Substitution Rule for Indefinite Integrals - In this section we will start using one of the more common and useful integration techniques - The Substitution Rule. With the substitution rule we will be able integrate a wider variety of functions. The integrals in this section will all require some manipulation of the function prior to integrating unlike most of the integrals from the previous section where all we really needed were the basic integration formulas. More Substitution Rule - In this section we will continue to look at the substitution rule. The problems in this section will tend to be a little more involved than those in the previous sec- tion. © Paul Dawkins Calculus – xiv –  Outline Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals. We will be approximating the amount of area that lies between a function and the x-axis. As we will see in the next section this problem will lead us to the definition of the definite integral and will be one of the main interpretations of the definite integral that we’ll be looking at in this material. Definition of the Definite Integral - In this section we will formally define the definite integral, give many of its properties and discuss a couple of interpretations of the definite integral. We will also look at the first part of the Fundamental Theorem of Calculus which shows the very close relationship between derivatives and integrals Computing Definite Integrals - In this section we will take a look at the second part of the Fundamental Theorem of Calculus. This will show us how we compute definite integrals without using (the often very unpleasant) definition. The examples in this section can all be done with a basic knowledge of indefinite integrals and will not require the use of the substitution rule. Included in the examples in this section are computing definite integrals of piecewise and absolute value functions. Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals. The only real requirements to being able to do the examples in this section are being able to do the substitution rule for indefinite integrals and understanding how to compute definite integrals in general. Applications of Integrals In this last chapter of this course we will be taking a look at a couple of Applications of Integrals. There are many other applications, however many of them require integration techniques that are typically taught in Calculus II. We will therefore be focusing on applications that can be done only with knowledge taught in this course. Because this chapter is focused on the applications of integrals it is assumed in all the examples that you are capable of doing the integrals. There will not be as much detail in the integration process in the examples in this chapter as there was in the examples in the previous chapter. 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 © Paul Dawkins Calculus – xv –  Outline 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. Integration Techniques In this chapter we are going to be looking at various integration techniques. There are a fair number of them and some will be easier than others. The point of the chapter is to teach you these new techniques and so this chapter assumes that you’ve got a fairly good working knowledge of basic integration as well as substitutions with integrals. In fact, most integrals involving “simple” substitutions will not have any of the substitution work shown. It is going to be assumed that you can verify the substitution portion of the integration yourself. Also, most of the integrals done in this chapter will be indefinite integrals. It is also assumed that once you can do the indefinite integrals you can also do the definite integrals and so to conserve space we concentrate mostly on indefinite integrals. There is one exception to this and that is the Trig Substitution section and in this case there are some subtleties involved with definite integrals that we’re going to have to watch out for. Outside of that however, most sections will have at most one definite integral example and some sections will not have any definite integral examples. 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 func- tions. In particular we concentrate integrating products of sines and cosines as well as prod- ucts 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, © Paul Dawkins Calculus – xvi –  Outline 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 Cal- culus 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 integra- tion 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. More Applications of Integrals In this section we’re going to take a look at some of the Applications of Integrals. It should be noted as well that these applications are presented here, as opposed to Calculus I, simply because many of the integrals that arise from these applications tend to require techniques that we discussed in the previous chapter. 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 exam- ple, 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 © Paul Dawkins Calculus – xvii –  Outline 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 section we will be looking at parametric equa- tions and polar coordinates. While the two subjects don’t appear to have that much in common on the surface we will see that several of the topics in polar coordinates can be done in terms of parametric equations and so in that sense they make a good match in this chapter We will also be looking at how to do many of the standard calculus topics such as tangents and area in terms of parametric equations and polar coordinates. 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 dy d2 y dx and dx2 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 elimi- nating 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 equa- tion). 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 coordi- nate system to the ‘normal’ Cartesian/Rectangular coordinate system. We will derive formu- las 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. dy Tangents with Polar Coordinates - In this section we will discuss how to find the derivative 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 © Paul Dawkins Calculus – xviii –  Outline 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. Series and Sequences In this chapter we’ll be taking a look at sequences and (infinite) series. In fact, this chapter will deal almost exclusively with series. However, we also need to understand some of the basics of sequences in order to properly deal with series. We will therefore, spend a little time on sequences as well. Series is one of those topics that many students don’t find all that useful. To be honest, many students will never see series outside of their calculus class. However, series do play an important role in the field of ordinary differential equations and without series large portions of the field of partial differential equations would not be possible. In other words, series is an important topic even if you won’t ever see any of the applications. Most of the applications are beyond the scope of most Calculus courses and tend to occur in classes that many students don’t take. So, as you go through this material keep in mind that these do have applications even if we won’t really be covering many of them in this class. Sequences - In this section we define just what we mean by sequence in a math class and give the basic notation we will use with them. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. We will also give many of the basic facts and properties we’ll need as we work with sequences. More on Sequences - In this section we will continue examining sequences. We will deter- mine if a sequence is an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. We will also determine a sequence is bounded below, bounded above and/or bounded. Series - The Basics - In this section we will formally define an infinite series. We will also give many of the basic facts, properties and ways we can use to manipulate a series. We will also briefly discuss how to determine if an infinite series will converge or diverge (a more in depth discussion of this topic will occur in the next section). Convergence/Divergence of Series - In this section we will discuss in greater detail the con- vergence and divergence of infinite series. We will illustrate how partial sums are used to © Paul Dawkins Calculus – xix –  Outline determine if an infinite series converges or diverges. We will also give the Divergence Test for series in this section. Special Series - In this section we will look at three series that either show up regularly or have some nice properties that we wish to discuss. We will examine Geometric Series, Telescoping Series, and Harmonic Series. Integral Test - In this section we will discuss using the Integral Test to determine if an infi- nite series converges or diverges. The Integral Test can be used on an infinite series pro- vided the terms of the series are positive and decreasing. A proof of the Integral Test is also given. Comparison Test/Limit Comparison Test - In this section we will discuss using the Comparison Test and Limit Comparison Tests to determine if an infinite series converges or diverges. In order to use either test the terms of the infinite series must be positive. Proofs for both tests are also given. Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. The Alternating Series Test can be used only if the terms of the series alternate in sign. A proof of the Alternating Series Test is also given. Absolute Convergence - In this section we will have a brief discussion of absolute con- vergence and conditionally convergent and how they relate to convergence of infinite se- ries. Ratio Test - In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. The Ratio Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. A proof of the Ratio Test is also given. Root Test - In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. The Root Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. A proof of the Root Test is also given. Strategy for Series - In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or diverge. Note as well 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. A summary of all the various tests, as well as conditions that must be met to use them, we discussed in this chapter are also given in this section. Estimating the Value of a Series - In this section we will discuss how the Integral Test, Com- parison Test, Alternating Series Test and the Ratio Test can, on occasion, be used to estimate the value of an infinite series. Power Series - In this section we will give the definition of the power series as well as the definition of the radius of convergence and interval of convergence for a power series. We © Paul Dawkins Calculus – xx –  Outline will also illustrate how the Ratio Test and Root Test can be used to determine the radius and interval of convergence for a power series. Power Series and Functions - In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. However, use of this formula does quickly illustrate how functions can be represented as a power series. We also discuss differentiation and integration of power series. Taylor Series - In this section we will discuss how to find the Taylor/Maclaurin Series for a function. This will work for a much wider variety of function than the method discussed in the previous section at the expense of some often unpleasant work. We also derive some well known formulas for Taylor series of ex , cos(x) and sin(x) around x = 0. Applications of Series - In this section we will take a quick look at a couple of applications of series. We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. We will also see how we can use the first few terms of a power series to approximate a function. Binomial Series - In this section we will give the Binomial Theorem and illustrate how it can be used to quickly expand terms in the form (a + b)n when n is an integer. In addition, when n is not an integer an extension to the Binomial Theorem can be used to give a power series representation of the term. Vectors This is a fairly short chapter. We will be taking a brief look at vectors and some of their properties. We will need some of this material in the next chapter and those of you heading on towards Calculus III will use a fair amount of this there as well. Basic Concepts - 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 interpreta- tion of the sum and difference of two vectors. We also define and give a geometric interpre- tation 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. Three Dimensional Space In this chapter we will start taking a more detailed look at three dimen- sional space (3-D space or R3 ). This is a very important topic for Calculus III since a good portion of Calculus III is done in three (or higher) dimensional space. © Paul Dawkins Calculus – xxi –  Outline We will be looking at the equations of graphs in 3-D space as well as vector valued functions and how we do calculus with them. We will also be taking a look at a couple of new coordinate systems for 3-D space. The 3-D Coordinate System - In this section we will introduce the standard three dimensional coordinate system as well as some common notation and concepts needed to work in three dimensions. Equations of Lines - In this section we will derive the vector form and parametric form for the equation of lines in three dimensional space. We will also give the symmetric equations of lines in three dimensional space. Note as well that while these forms can also be useful for lines in two dimensional space. Equations of Planes - In this section we will derive the vector and scalar equation of a plane. We also show how to write the equation of a plane from three points that lie in the plane. Quadric Surfaces - In this section we will be looking at some examples of quadric sur- faces. Some examples of quadric surfaces are cones, cylinders, ellipsoids, and elliptic paraboloids. Functions of Several Variables - In this section we will give a quick review of some important topics about functions of several variables. In particular we will discuss finding the domain of a function of several variables as well as level curves, level surfaces and traces. Vector Functions - In this section we introduce the concept of vector functions concentrating primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well. We will illustrate how to find the domain of a vector function and how to graph a vector function. We will also show a simple relationship between vector functions and parametric equations that will be very useful at times. Calculus with Vector Functions - In this section here we discuss how to do basic calculus, i.e. limits, derivatives and integrals, with vector functions. Tangent, Normal and Binormal Vectors - In this section we will define the tangent, normal and binormal vectors. Arc Length with Vector Functions - In this section we will extend the arc length formula we used early in the material to include finding the arc length of a vector function. As we will see the new formula really is just an almost natural extension of one we’ve already seen. Curvature - In this section we give two formulas for computing the curvature (i.e. how fast the function is changing at a given point) of a vector function. Velocity and Acceleration - In this section we will revisit a standard application of derivatives, the velocity and acceleration of an object whose position function is given by a vector function. For the acceleration we give formulas for both the normal acceleration and the tangential acceleration. Cylindrical Coordinates - In this section we will define the cylindrical coordinate system, an alternate coordinate system for the three dimensional coordinate system. As we will see © Paul Dawkins Calculus – xxii –  Outline cylindrical coordinates are really nothing more than a very natural extension of polar coordi- nates into a three dimensional setting. Spherical Coordinates - In this section we will define the spherical coordinate system, yet another alternate coordinate system for the three dimensional coordinate system. This co- ordinates system is very useful for dealing with spherical objects. We will derive formulas to convert between cylindrical coordinates and spherical coordinates as well as between Cartesian and spherical coordinates (the more useful of the two). Partial Derivatives In Calculus I and in most of Calculus II we concentrated on functions of one variable. In Calculus III we will extend our knowledge of calculus into functions of two or more variables. Despite the fact that this chapter is about derivatives we will start out the chapter with a section on limits of functions of more than one variable. In the remainder of this chapter we will be looking at differentiating functions of more than one variable. As we will see, while there are differences with derivatives of functions of one variable, if you can do derivatives of functions of one variable you shouldn’t have any problems differentiating functions of more than one variable. You’ll just need to keep one subtlety in mind as we do the work. Limits - In the section we’ll take a quick look at evaluating limits of functions of several vari- ables. 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 impor- tant 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 par- tial 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 vari- ables. 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 vari- ables can, in turn, be written in terms of different variables. We will also give a nice method © Paul Dawkins Calculus – xxiii –  Outline for writing down the chain rule for pretty much any situation you might run into when deal- ing 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 a several applications of partial derivatives. Most of the applications will be extensions to applications to ordinary derivatives that we saw back in Calculus I. For instance, we will be looking at finding the absolute and relative extrema of a function and we will also be looking at optimization. Both (all three?) of these subjects were major applications back in Calculus I. They will, however, be a little more work here because we now have more than one variable. 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 previ- ous 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 Calculus I we moved on to the subject of integrals once we had finished the discussion of derivatives. The same is true in this course. Now that we have finished our discussion of derivatives of functions of more than one variable we need to move on to integrals of functions of two or three variables. © Paul Dawkins Calculus – xxiv –  Outline Most of the derivatives topics extended somewhat naturally from their Calculus I counterparts and that will be the same here. However, because we are now involving functions of two or three variables there will be some differences as well. There will be new notation and some new issues that simply don’t arise when dealing with functions of a single variable. Double Integrals - In this section we will formally define the double integral as well as giving a quick interpretation of the double integral. Iterated Integrals - In this section we will show how Fubini’s Theorem can be used to evaluate double integrals where the region of integration is a rectangle. Double Integrals over General Regions - In this section we will start evaluating double inte- grals over general regions, i.e. regions that aren’t rectangles. We will illustrate how a double integral of a function can be interpreted as the net volume of the solid between the surface given by the function and the xy-plane. Double Integrals in Polar Coordinates - In this section we will look at converting integrals (including dA) in Cartesian coordinates into Polar coordinates. The regions of integration in these cases will be all or portions of disks or rings and so we will also need to convert the original Cartesian limits for these regions into Polar coordinates. Triple Integrals - In this section we will define the triple integral. We will also illustrate quite a few examples of setting up the limits of integration from the three dimensional region of integration. Getting the limits of integration is often the difficult part of these problems. Triple Integrals in Cylindrical Coordinates - In this section we will look at converting inte- grals (including dV ) in Cartesian coordinates into Cylindrical coordinates. We will also be converting the original Cartesian limits for these regions into Cylindrical coordinates. Triple Integrals in Spherical Coordinates - In this section we will look at converting integrals (including dV ) in Cartesian coordinates into Spherical coordinates. We will also be converting the original Cartesian limits for these regions into Spherical coordinates. Change of Variables - In previous sections we’ve converted Cartesian coordinates in Polar, Cylindrical and Spherical coordinates. In this section we will generalize this idea and dis- cuss how we convert integrals in Cartesian coordinates into alternate coordinate systems. Included will be a derivation of the dV conversion formula when converting to Spherical co- ordinates. Surface Area - In this section we will show how a double integral can be used to determine the surface area of the portion of a surface that is over a region in two dimensional space. Area and Volume Revisited - In this section we summarize the various area and volume formulas from this chapter. Line Integrals In this section we are going to start looking at Calculus with vector fields (which we’ll define in the first section). In particular we will be looking at a new type of integral, the line integral and some of the interpretations of the line integral. We will also take a look at one of the more important theorems involving line integrals, Green’s Theorem. © Paul Dawkins Calculus – xxv –  Outline 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 theo- rem 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 interest- ing application of Green’s Theorem that we can use to find the area of a two dimensional region. Surface Integrals In the previous chapter we looked at evaluating integrals of functions or vector fields where the points came from a curve in two- or three-dimensional space. We now want to extend this idea and integrate functions and vector fields where the points come from a surface in three-dimensional space. These integrals are called surface integrals. Curl and Divergence - In this section we will introduce the concepts of the curl and the di- vergence 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, © Paul Dawkins Calculus – xxvi –  Outline 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 In this appendix is material that didn’t fit into other sections for a variety of reasons. Also, in order to not obscure the mechanics of actually working problems, most of the proofs of various facts and formulas are in this chapter as opposed to being in the section with the fact/formula. 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 Facts/Formulas/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/Formulas - In this section we prove many of the facts that we saw in the Applications of Derivatives chapter. Proof of Various Integral Facts/Formulas/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. Summa- tion 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 in- volving 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 stu- dents get different answers to the same integral. We include two examples of this kind of situation. © Paul Dawkins Calculus – xxvii – 1 Review Technically a student coming into a Calculus class is supposed to know both Algebra and Trigonom- etry. Unfortunately, the reality is often much different. Most students enter a Calculus class woefully unprepared for both the algebra and the trig that is in a Calculus class. This is very unfortunate since good algebra skills are absolutely vital to successfully completing any Calculus course and if your Calculus course includes trig (as this one does) good trig skills are also important in many sections. The above statement is not meant to denigrate your favorite Algebra or Trig instructor. It is simply an acknowledgment of the fact that many of these courses, especially Algebra courses, are aimed at a more general audience and so do not always put the time into topics that are vital to a Calculus course and/or the level of difficulty is kept lower than might be best for students heading on towards Calculus. Far too often the biggest impediment to students being successful in a Calculus course is they do not have sufficient skills in the underlying algebra and trig that will be in many of the calculus problems we’ll be looking at. These students end up struggling with the algebra and trig in the problems rather than working to understand the calculus topics which in turn negatively impacts their grade in a Calculus course. The intent of this chapter, therefore, is to do a very cursory review of some algebra and trig skills that are vital to a calculus course that many students just didn’t learn as well as they should have from their Algebra and Trig courses. This chapter does not include all the algebra and trig skills that are needed to be successful in a Calculus course. It only includes those topics that most students are particularly deficient in. For instance, factoring is also vital to completing a standard calculus class but is not included here as it is assumed that if you are taking a Calculus course then you do know how to factor. Likewise, it is assumed that if you are taking a Calculus course then you know how to solve linear and quadratic equations so those topics are not covered here either. For a more in depth review of Algebra topics you should check out the full set of Algebra notes at http://tutorial.math.lamar.edu. Note that even though these topics are very important to a Calculus class we rarely cover all of them in the actual class itself. We simply don’t have the time to do that. We will cover certain portions of this chapter in class, but for the most part we leave it to the students to read this chapter on their own to make sure they are ready for these topics as they arise in class. 1 Chapter 1 : Review Section 1.1 : Functions 1.1 Functions In this section we’re going to make sure that you’re familiar with functions and function notation. Both will appear in almost every section in a Calculus class so you will need to be able to deal with them. First, what exactly is a function? The simplest definition is an equation will be a function if, for any x in the domain of the equation (the domain is all the x’s that can be plugged into the equation), the equation will yield exactly one value of y when we evaluate the equation at a specific x. This is usually easier to understand with an example. Example 1 Determine if each of the following are functions. (a) y = x2 + 1 (b) y 2 = x + 1 Solution (a) y = x2 + 1 This first one is a function. Given an x, there is only one way to square it and then add 1 to the result. So, no matter what value of x you put into the equation, there is only one possible value of y when we evaluate the equation at that value of x. (b) y 2 = x + 1 The only difference between this equation and the first is that we moved the exponent off the x and onto the y. This small change is all that is required, in this case, to change the equation from a function to something that isn’t a function. To see that this isn’t a function is fairly simple. Choose a value of x, say x = 3 and plug this into the equation. y2 = 3 + 1 = 4 Now, there are two possible values of y that we could use here. We could use y = 2 or y = −2. Since there are two possible values of y that we get from a single x this equation isn’t a function. Note that this only needs to be the case for a single value of x to make an equation not be a function. For instance, we could have used x = −1 and in this case, we would get a single y (y = 0). However, because of what happens at x = 3 this equation will not be a function. © Paul Dawkins Calculus –2– Chapter 1 : Review Section 1.1 : Functions Next, we need to take a quick look at function notation. Function notation is nothing more than a fancy way of writing the y in a function that will allow us to simplify notation and some of our work a little. Let’s take a look at the following function. y = 2x2 − 5x + 3 Using function notation, we can write this as any of the following. f (x) = 2x2 − 5x + 3 g (x) = 2x2 − 5x + 3 h (x) = 2x2 − 5x + 3 R (x) = 2x2 − 5x + 3 w (x) = 2x2 − 5x + 3 y (x) = 2x2 − 5x + 3... Recall that this is NOT a letter times x, this is just a fancy way of writing y. So, why is this useful? Well let’s take the function above and let’s get the value of the function at x = −3. Using function notation we represent the value of the function at x = −3 as f (−3). Function notation gives us a nice compact way of representing function values. Now, how do we actually evaluate the function? That’s really simple. Everywhere we see an x on the right side we will substitute whatever is in the parenthesis on the left side. For our function this gives, f (−3) = 2(−3)2 − 5 (−3) + 3 = 2 (9) + 15 + 3 = 36 Let’s take a look at some more function evaluation. Example 2 Given f (x) = −x2 + 6x − 11 find each of the following. (a) f (2) (b) f (−10) (c) f (t) (d) f (t − 3) (e) f (x − 3) (f) f (4x − 1) © Paul Dawkins Calculus –3– Chapter 1 : Review Section 1.1 : Functions Solution (a) f (2) f (2) = −(2)2 + 6(2) − 11 = −3 (b) f (−10) f (−10) = −(−10)2 + 6 (−10) − 11 = −100 − 60 − 11 = −171 Be careful when squaring negative numbers! (c) f (t) f (t) = −t2 + 6t − 11 Remember that we substitute for the x’s WHATEVER is in the parenthesis on the left. Often this will be something other than a number. So, in this case we put t’s in for all the x’s on the left. (d) f (t − 3) f (t − 3) = −(t − 3)2 + 6 (t − 3) − 11 = −t2 + 12t − 38 Often instead of evaluating functions at numbers or single letters we will have some fairly complex evaluations so make sure that you can do these kinds of evaluations. (e) f (x − 3) f (x − 3) = −(x − 3)2 + 6 (x − 3) − 11 = −x2 + 12x − 38 The only difference between this one and the previous one is that we changed the t to an x. Other than that, there is absolutely no difference between the two! Don’t get excited if an x appears inside the parenthesis on the left. (f) f (4x − 1) f (4x − 1) = −(4x − 1)2 + 6 (4x − 1) − 11 = −16x2 + 32x − 18 This one is not much different from the previous part. All we did was change the equation that we were plugging into the function. All throughout a calculus course we will be finding roots of functions. A root of a function is nothing more than a number for which the function is zero. In other words, finding the roots of a function, g (x), is equivalent to solving g (x) = 0 © Paul Dawkins Calculus –4– Chapter 1 : Review Section 1.1 : Functions Example 3 Determine all the roots of f (t) = 9t3 − 18t2 + 6t Solution So, we will need to solve, 9t3 − 18t2 + 6t = 0 First, we should factor the equation as much as possible. Doing this gives,