Calculus
Lutzer/Goodwill – Preliminary Edition
August 2010
To the Student
If you’re reading this, you’re already ahead of the game. The most important thing you can do with this book is read it. The trick is that many students––perhaps even you––are unaccustomed to reading technical writing, which is a skill unto itself. So with the few sentences that we have to spend here, we want to offer some simple advice. First and foremost, be patient. Learning takes time and requires significant mental effort.
Effective reading strategies
Many people find that a scan-then-dig strategy works well when reading technical writing. Your first pass through a section is meant only to familiarize you with the general organization of topics and terms in the section so that you have a mental “road map” of the learning that you’ll do. The second pass is where you dig into the main ideas and develop the relevant skills.
Note our use of the word “dig.” Learning technical material, as with the physical act of digging, is an active endeavor that often requires significant effort, even for professionals. Recently when reading a book about applications of physics to the medical sciences, one of us found himself saying, “That doesn’t seem right. That would mean the relationship is something like …” after which there was a lot of writing on scrap paper, a few occurrences of, “Oh …” and finally, “Now I get it.”
The point is that in order to really learn calculus you have to do calculus. We hope that you will check calculations as you go, so that you get used to the ways that various terms interact and develop a basic understanding of what they mean; but we’ve also provided you with many opportunities in a section to do what you’re reading about, and so solidify the idea or technique. These opportunities are labeled as “Try It Yourself” examples. We provide the answer here, so that you can check yourself. If your answer disagrees with ours, and you can’t figure out why, we’ve posted a full solution online.
Effective practice
When taking a quiz or exam, and more importantly, later, when you’re actually using calculus to solve real world problems, you’ll have to think creatively about how to combine the ideas and techniques that you learn here. You won’t develop that creative mathematical thinking by watching someone else do it. You have to practice it yourself. We’ve already mentioned this kind of thing as part of effective reading strategies, but here we mean to offer advice about making homework exercises effective learning tools.
Specifically, many students find that having a solutions manual is helpful, and we have worked with Dr. Justin Young to provide a good one for this text. Our advice is to use it as a stepping stone, rather than a crutch. Each time you complete an exercise by reading part or all of its solution in the manual, try to do a similar exercise on your own. Otherwise, other people are doing the creative “heavy lifting” for you, and you may end up in the situation of saying, “I don’t understand why I got all of the homework exercises right but did poorly on the exam.”
In closing
Our last piece of advice is to talk about what you’re learning. The best way to solidify and synthesize ideas is often to articulate them to other people. And if “other people” includes your teacher, he or she can help you by hearing and responding to the way that you describe ideas and techniques.
Preface
This project started when a former student spoke to us about calculus texts. He said, “The texts that are out there are good reference books, but they don’t actually teach you anything. If I couldn’t get it from the teacher, there’s no way I could learn it from the text.” After a review of the most popular texts on the market today, we understood what he meant. While correct and concise, these texts are written in a style that’s difficult to parse for readers who are unaccustomed to technical writing, and they often neglect to provide motivation of their content at a practical level that speaks to modern students. The result is that both students and teachers tend to skim these texts rather than actually read them. Although they make great reference books in later years, during the actual learning process calculus texts function as little more than very heavy exercise sets. That’s a far cry from the ideal.
Ideally, a calculus text should spark curiosity in its readers. It should hold students’ attention, motivate the material in relevant and interesting ways, and answer the spectrum of students’ questions from “How do we do this?” to “Why do we do this?” to “How can I make sense of this?” It should engage students in the discussion and discovery of ideas, develop their intuition, and help them learn about the power, beauty, and versatility of calculus.
So that’s what we set out to do. Our goal is nothing less than to provide students with an accessible presentation that enhances their learning, complements their classroom experience, and strengthens their understanding of this beautiful subject. This book revolves around getting students’ attention –– and keeping it –– with an engaging presentation. By referring to their own experiences and interests, we help students to develop their intuition, so that the ideas and techniques of calculus make sense to them.
Engaging Students
Our textbook is written primarily for students who are interested in physical and medical sciences, mathematics, engineering, or computer science, but also provides inlets to the subject and exercises for students whose interests lie in business or social science. Most of these students are interested in applications, so we often use applications in our examples and in the preliminary discussion of a new topic. For example, the idea of a limit is introduced using a thought experiment in which we track the position of a planet that passes between us and the sun; the concept of the derivative is introduced using clinical data about the radius of an artery; derivatives of trigonometric functions are motivated by a vibrating guitar string; Riemann sums are introduced to calculate the net displacement of an accelerating vehicle based on data from a radar gun; the idea of a sequence is motivated by concentrations of medication in a patient’s bloodstream; and the list goes on.
We recognize that the applications must be presented at an elementary level, with broad generalizations, so that people who are new to a subject can quickly get a rough understanding without being drawn away from the mathematical topic at hand. At the same time, we have found that students feel a connection to the subject when it is presented in context, and that they appreciate a glimpse of the rich tapestry that lies beyond what they are learning at the moment.
We also understand that introducing skills and concepts in simple settings can benefit students by allowing them to focus on the relevant mathematics without distraction, so we’ve worked hard to minimize technical details from other disciplines.
Calling on students’ experience
Part of engaging students is helping them learn by connecting the ideas of calculus to their experience and observations. So doing provides them with a way to understand the material in terms of familiar facets of life, so calculus makes sense to them rather than being a random collection of formulas. Here are some examples that we use to accomplish this:
Newton’s method: We begin our discussion of Newton’s method by asking the question, “Suppose a model rocket is coming back down, after its flight. It’s descending at 6 feet per second, and is 30 feet above the ground. How much longer before it lands?” Students always respond with the expected answer of five seconds based on their experience with motion. Pedagogically, what’s important about their answer is that they’ve assumed a constant rate of change. That’s Newton’s method in a nutshell.
Mean Value Theorem: We begin by asking you to think about a race against a friend. The friend promises to run at a steady, constant rate from start to finish, and though you may speed up and slow down at will, by some fluke you tie. Could you have been running faster than your friend at each and every moment, or slower the whole time? No, because one of you would have won the race uncontested. If you start and finish together, there must have been some time at which you were running at the same speed. This discussion opens the door to the Mean Value Theorem.
Substitution: When we apply the substitution technique to definite integrals, we begin by determining a change in altitude in two different ways––one calculation based on time, and the other based on position. This allows us to help students understand the substitution technique by characterizing it as a change in our point of view, from initial and terminal times to starting and stopping locations.
Making the presentation accessible
Because we intend for this book to be read, not just carried around, we’ve worked hard to make it as accessible to students as possible. In part, we accomplish this by writing in a colloquial style during the development of ideas, so that by the time students read the formal language of a theorem, they already know what it says.
We use language that today’s students in quantitative disciplines take for granted. This does not mean using modern slang, but that terms such as zoom, compress, resolution, and bundle, can make the narrative flow more easily for today’s readers. These words are part of the language in a technically savvy society. For example, students often have trouble writing a formula for f(t+h) when working with the difference quotient, but they readily understand the idea of find/replace based on their experience with word processors. So we have them find all occurrences of t and replace them with (t+h).
We also use color to help students follow longer or more complicated calculations, and to reinforce similarities across examples. We make common use of underbraces to provide guidance as students read equations, and often include margin notes to answer common questions without interrupting the flow of the presentation.
Promoting active reading
Some years ago, we attended a seminar in which Dr. P.K. Imbrie spoke about ways of engaging students, even in large classes. He suggested breaking up lectures with small "try it yourself" moments that keep students actively involved in the learning process. There are other ways, of course, but Dr. Imbrie’s idea made a lot of sense to us as authors: students who are inexperienced at learning mathematics from a book often read in a passive way, as if it’s a novel, but learning mathematics is an active endeavor. To promote active learning while students read, we often include a “Try It Yourself” example immediately after one whose solution was fully laid out. The answers to these Try It Yourself examples are provided in the text, and if students need to see a full solution they can access it online. These examples act as pedagogical "stepping stones," half-way between examples and exercises, allowing students to try something for themselves but with a virtual safety net.
Reading v. Spelling in mathematics
For many students, mathematical equations and theorems are just a collection of symbols that they are expected to manipulate in order to “magically” arrive at an answer to a question. The equation itself is completely devoid of meaning for them. So starting from the first section of the book, we teach students how to interpret the equations they see, and to assess the ones they write themselves so that mathematics becomes a communication of ideas. We consider this skill to be a kind of quantitative literacy.
This is closely related to English literacy: when reading, we don’t view each symbol (letter) separately, but look at groups of symbols to derive meaning. For example, the following sentence would be confusing if read letter by letter (try it, out loud):
“T-h-e-n-e-x-t-t-i-m-e-I-s-l-e-e-p-,-i-t-w-o-n-’-t-b-e-o-n-m-y-c-a-l-c-u-l-u-s-b-o-o-k-.-It-d-i-d-n-’-t-h-e-l-p-l-a-s-t-t-i-m-e-a-n-d-n-o-w-I-h-a-v-e-a-c-r-i-c-k-i-n-m-y-n-e-c-k-.”
Instead, we group the letters and read,
“The next time I sleep, it won’t be on my calculus book. It didn’t help last time and now I have a crick in my neck.”
Each grouping (word) conveys a concept, and together they communicate an idea. We help students move beyond spelling to reading mathematics by demonstrating this skill in the presentation and by including exercises that ask students to make the transition from English to mathematical notation and back.
Additionally, we often use dimensional analysis to check that calculations of physical objects and phenomena make sense. When done at an elementary level, this helps students internalize the meaning of the difference quotient, which has units such as meters per second, and understand why Riemann sums approximate the net change in, for example, position (when the summands have the form rate X time). Dimensional analysis also helps students to remember formulas such as those for cylindrical shells and washers, which are used when calculating volumes of solids of revolution.
Helping students avoid common errors
We have found that students often benefit from learning about common mistakes, so we pause to discuss them at the end of each section. For example, students sometimes forget whether the recursion formula in Newton’s method requires the derivative in the numerator or the denominator; and they sometimes make use the same letter for the substitution as the original variable of an integral, which invariably leads to errors. So in the section on substitution we discuss ways to remember formulas and avoid common errors.
Exercises and projects
There’s a wide gap between understanding calculus and being able to do calculus. And like any activity, practice is how we bridge that gap. Since different schools require different levels of rigor, we've designed the exercise sets to include various levels of difficulty. We've also sorted them according to the particular skills that are required, and we've separated skill exercises from those that develop concepts or demonstrate relevance by way of application. At the end of each chapter you'll find a Chapter Review, and projects that have been distilled from those we’ve assigned to teams of students in our own classes through the years. Some focus on purely mathematical ideas, while others ask students to use their skill and understanding in context.
Helping students learn the “art” of calculus –– decision making
When working with integration or infinite series, students have to decide which technique is appropriate to each problem. Knowing when to use the ratio test, for example, is just as important as knowing how to use it, and making that initial decision is itself a skill. In order to help students develop that skill, we include designated groups of exercises called Mixed Practice sets that require students to use techniques from previous sections on integration techniques and convergence tests, respectively. This helps students learn to analyze problems and decide on appropriate methods at the same time that they are learning the mechanics that each choice entails.
Proofs
Proofs lie at the heart of deeper mathematical understanding, but the very word “poof” often triggers first-year students to stop listening. First-year students tend to find formal proofs unsatisfactory as answers to the question, “Why?” Indeed, from their point of view, formal proofs obfuscate facts more often than they illuminate them. With this in mind, we frequently precede proofs with informal discussions that focus on the salient ideas students are about to see, and put them in a context that allows students to intuit the relationship between them. This often makes the formal proofs easier for students to follow, and leaves students more satisfied with the proof and its role it plays.
The Role of Technology
Generally speaking, today’s students are technology savvy. They are familiar with chat rooms, have pages on Facebook, maintain blogs, send emails and text messages, and are generally comfortable interacting online. It’s important to understand that their mathematical education, like the rest of their lives, has been affected by the presence of technology. Many of today’s students begin a math problem by reaching for their graphing calculator because they’ve been trained to think in terms of graphs, which have always been readily accessible. Using computational technology is as normal to them as driving a car.
In principle, we believe that computational technology is a powerful tool that can help expedite the problem solving process, but only if the user knows what to look for––much in the same way that your car’s maximum speed is irrelevant if you’re lost. So we tend to focus on ideas and concepts, and bring in technology where we believe it’s appropriate.
We also believe that numerical techniques should not be discussed and then discarded. They are important tools for engineers, scientists, and applied mathematicians, so we regularly ask students to use them to approximate quantities that are otherwise impossible for them to calculate. To facilitate students’ use of these algorithms, we have provided instructions for implementing them on a generic spreadsheet program (a tool that is often used by engineers and scientists, but which has received little attention from teachers of mathematics) in several places throughout the book.