Mathematics 265 Introduction to Calculus I

Study Guide

Introduction

Thinking skills and problem-solving activities are indispensable to every area of our lives. To some extent, we are all problem solvers. The problem solver’s work is mostly a tangle of guesswork, analogy, wishful thinking, observing patterns, and frustration. To become a master problem solver may be as inaccessible as acquiring the skills of a virtuoso, but everyone can become a better, more confident problem solver.^[1]

In this introductory calculus course, you will learn to use functions to solve problems. This course covers approximately 34 sections of the textbook Single Variable Calculus, 5th ed, by James Stewart [Scarborough, ON : Nelson, 2002]. These sections are reproduced in a textbook customized for Athabasca University.

We do not cover all of the material presented in the original textbook because, as the author indicates, the textbook “contains more material than can be covered in any one course” (2006, p. 8). Furthermore, we do not follow the author’s suggested order; instead, we ask you to use your intuition before we get into formal definitions. For instance, we do not require that you understand the formal definition of limit in order to evaluate limits, we first consider what a limit is in a visual, intuitive way, and once you have learned to apply theorems to evaluate limits, we then study the formal definition. We are very selective about the material we ask you to read and the exercises we recommend that you do from the textbook.

Recommendations for Learning Mathematics

We have some recommendations for you to become a better, more confident problem solver.

• Be prepared to practise. To master mathematical concepts, all we need to do is practise, practise and practise again. You must do at least all of the exercises presented in this guide, and we also encourage you to try non-assigned exercises from your textbook. A steady pace, practice and patience will take you farther than any short cuts. You may have to change your study habits, because those that were sufficient for success in high school are not good enough for university.

  We provide you with answers to all of the assigned problems in the Appendix of this Study Guide, and with hints to the “Learning from Mistakes” exercises of each chapter. Do not look at our answers and hints before making at least one honest try to find the answer (if there is one). In other words, do not give up too easily. The process of learning from an exercise starts with asking the right questions: What do I want to know? What do I have to explain? What do I know about this? What is this exercise saying? What is this exercise testing?

• Read your Study Guide and textbook. The textbook is not a collection of solved problems; you are expected to understand the course material, and put it to use to solve the problems yourself. Hunting through a section to find a worked-out exercise that is similar to the assigned problem is totally inappropriate. Solving a mathematical problem requires an understanding of the concepts discussed.

  Never read mathematics without pencil and paper in hand. Read the corresponding assigned section before trying the exercises. You should treat the examples as exercises, and try to solve them without the author’s help. Be prepared to check all the details as you read, and above all, read critically. Take nothing for granted.

  If you do not understand a statement, go back in the section, or to a previous section, to see if you missed or misunderstood something. If you still fail to understand a concept after puzzling over it for a while, mark the place, continue reading, and ask your tutor for help on that point.

  You will be asked to do exercises as concepts arise, do them at that precise moment, not later: you must make sure you understand what is presented before going on to the next concept.

  You must learn to make reference notes that are useful to you. Some reference tables are provided as examples to show you how information may be displayed for ease of use.

  Once again, do not limit yourself to the assigned exercises. Instead, try to do as many exercises as you have time for, do not avoid those that appear to be challenging—the fun in learning is the satisfaction of being able to respond positively to a challenge. The time you spend in reading the Study Guide and the textbook will save you time in the long run. You should note any difficulty you have with the homework, and ask your tutor for help.

• Raise your expectations. In high school, students are accustomed to finding that the first few weeks of a new course are nothing more than a review of the last course, or even the last few courses. When a new idea is finally presented, the class spends several days working on it before moving ahead. Students quickly learn that there is no major crisis if they do not follow what is said the first time, or if they miss a class. The concept is repeated soon (and sometimes ad nauseam). In this course, however, the review of previous material is minimal; new concepts are introduced in the first few pages of Unit 2. Moreover, every subsequent concept assumes mastery of the previous ones, and involves the presentation of new ideas. This is why you must study steadily and constantly throughout each unit. Mathematics cannot be learned on a hurry.

  With the preliminaries presented in Unit 1, we intend to review or complement what you already know of algebra and trigonometry. After this first unit, we assume that you will be capable of working out the basics.

  For example, to find the solutions of the equation ${\displaystyle 5x^2+6=7}$, you would expect your high school teacher to write

  ${\displaystyle 5x^2+6=7\text{,}}$

  hence, ${\displaystyle 5x^2=7-6=1\text{,}}$

  which implies ${\displaystyle x^2=\frac{1}{5}\text{,}}$

  and we conclude that ${\displaystyle x=\pm \sqrt{\frac{1}{5}}\text{.}}$

  In this course, we would probably say, ${\displaystyle 5x^2+6=7}$, so ${\displaystyle x^2=1∕5}$, and ${\displaystyle x=\pm \sqrt{1∕5}}$. We would give no further details. If you cannot work out the details, review your basics and make sure you understand the algebraic process. Ask your tutor for help if necessary. Do not leave it for another day. Lack of proficiency on the basics is the major cause of students’ frustration in mathematics courses.

  Questions are posed throughout the Study Guide. These questions are not rhetorical; rather, they are designed to encourage you to think through and reflect on the concepts and processes to arrive at an answer. We expect you to answer the questions in order to get a deeper understanding of the mathematics involved in calculus.

• Learn to evaluate your own work. Solutions for all odd-numbered exercises are provided in the Student Solution Manual. Solutions of even-numbered exercises are not given, because you are expected to start relying more on your own knowledge and less on external verification to determine whether an answer is correct. The intention is that, eventually, you will be able to correct other people’s work, but you must start with your own. To help you to achieve this goal, each unit ends with a section titled “Learning from Mistakes”—a series of questions with common errors that you are asked to identify and correct. Hints are found at the end of this guide. Complete solutions are provided in the Student Manual.

• Ask for help. You may have the opportunity of consulting other students or knowledgeable professionals, or you may even want to hire a private tutor. Discussing problems and course material with someone can help you to gain insight into difficult concepts. But take warning: others cannot understand the concept for you. It may be tempting to let others do your homework for you, but you will not learn if you do so. You must take the time to learn! It is unwise to pretend that you understand when you do not.

• Use the evaluation mechanisms. A test is a learning experience, not just a bureaucratic way to quantify your knowledge. Study your self-graded exercises and graded TME s carefully; make sure you understand why you made mistakes, and how you can avoid making them again. The best time to understand missing concepts is right away, when the ideas are still fresh in your mind. In a test or assignment, you must show that you have mastered the concepts and can solve problems. Rote memorization and regurgitation are not what we are looking for. Your grade reflects what you know and how well you know it, as demonstrated by your personal performance, not on the basis of a comparison of your performance with that of others. Do not rely on the good will of the marker to get credit for your work. The marker will grade what is on the paper, not what you might have intended to do.

• Prepare for the examinations. Sample examinations are provided to allow you to practise. But we do not want you to have false expectations because you did well in a practice examination. The best preparation for an examination is to try different exercises at random. At the end of each unit, we give you a list of exercises from the “Review” section of each chapter. Use this list to prepare for the examinations. Do the exercises in a random order.

  Making summaries of each unit is another excellent way to prepare yourself for an examination. And never cram for a mathematics examination.

  Note: All questions in the assignments and sample examinations are worth several points, because several steps are required to solve them. Please show all your work and provide proper justification for all your answers.

  IMPORTANT: On the graded exams in the Möbius platform, each question is worth 1 point. Partial scores are possible when a question comprises multiple parts. In these cases, each part of the question is equally weighted for a total of 1 point for the whole question.

In mathematics, it pays to be persistent: people who succeed in mathematics are those who do not give up after the first attempt at solving a problem, but keep asking themselves, “What else can I try?”

References in This Study Guide

References that provide page numbers may refer to the textbook, Readings from Stewart: Single Variable Calculus, Fifth Edition [Scarborough, ON : Thomson Nelson, 2006], or to this Study Guide. We have endeavoured to ensure that there is no ambiguity.

Within the Study Guide, examples, theorems, definitions, etc., are numbered; for example, Theorem 3.23. Note that the first number points to the unit where the item is found.

Definitions marked with an asterisk (*) are not formal definitions, but they do indicate how you should think about the concept presented. Some formal definitions are provided in the last section of Unit 3.

Chapter Review

Your textbook provides a “Review” section at the end of each chapter. We recommend that you study the “Concept Check” and the “True-False Quiz” in each of these reviews.

When you come to the review, you should feel that you have learned enough to do the exercises presented. If you need assistance, consult your tutor.

Proofs

A proof is a logical explanation of why something is true. We present some proofs in this course. At a minimum, you should be able to understand a proof’s arguments, and to prove simple results independently. The section titled “Proofs of Theorems,” pages 373-378 of the textbook, provides proofs of several foundational theorems.

Other formal definitions and proofs can be found in advanced calculus textbooks. We recommend:

Apostol, Tom A. Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd ed. New York: Addison Wesley, 1975.

You may also want to look into the websites below.

Wikipedia: The Free Encyclopedia. Mathematics. Retrieved November 7, 2007, from: http://en.wikipedia.org/wiki/Mathematics

Wikipedia: The Free Encyclopedia. Mathematical Proof. Retrieved November 7, 2007, from: http://en.wikipedia.org/wiki/Mathematical_proof

It is fair to say that students are not penalized if they do not master formal definitions and proofs, but are rewarded if they do.

Technology

In this course, we do not use technological aids in our work. You are not required to have a particular calculator, or to use a computer algebra system or software. The textbook does have exercises that require a graphing calculator or computer—they are indicated with special icons. You are welcome to try them, if you have the appropriate technology and know how to use it. Read the introduction titled “To the Student” on page 7 of the textbook to understand the meaning of the various icons used.

In the examinations, you are allowed to use a simple scientific calculator; however, graphing calculators, programmable calculators, and hand-held computers are not allowed.

You are not allowed to use tapes, cell phones, iPods, a BlackBerry, or other electronic or digital devices, or to consult with other people during the examinations.

Footnote

^[1] Blitzer, Robert. Introductory Algebra for College Students, 2nd ed., p. 123. Upper Saddle River, NJ: Prentice Hall, 1998.