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Calculus I, II, & III a Problem-Based Approach with Early Transcendentals

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Calculus I, II, & III a Problem-Based Approach with Early Transcendentals Edited: 12:12pm, January 17, 2017 JOURNAL OF INQUIRY-BASED LEARNING IN MATHEMATICS Calculus I, II, & III A Problem-Based Approach with Early Transcendentals Mahavier < Allen, Browning, Daniel, & So Lamar University Contents To the Instructor v About these Notes . v Grading . vi Some Guidance . vii History of the Course . x Student Feedback . xi To the Student xv How this Class Works . xv A Common Pitfall . xv Boardwork . xvi How to Study each Day . xvi What is Calculus? . xvii 1 Functions, Limits, Velocity, and Tangents 1 1.1 Average Speed and Velocity . 1 1.2 Limits and Continuity . 3 1.3 Instantaneous Velocity and Growth Rate . 7 1.4 Tangent Lines . 10 2 Derivatives 11 2.1 Derivatives of Polynomial, Root, and Rational Functions . 12 2.2 Derivatives of the Trigonometric Functions . 16 2.3 The Chain Rule . 17 2.4 Derivatives of the Inverse Trigonometric Functions . 19 2.5 Derivatives of the Exponential and Logarithmic Functions . 21 2.6 Logarithmic and Implicit Differentiation . 23 3 Applications of Derivatives 27 3.1 Linear Approximations . 27 3.2 Limits involving Infinity and Asymptotes . 28 3.3 Graphing . 30 3.4 The Theorems of Calculus . 34 ii CONTENTS iii 3.5 Maxima and Minima . 37 3.6 Related Rates . 39 4 Integrals 41 4.1 Riemann Sums and Definite Integrals . 41 4.2 Anti-differentiation and the Indefinite Integral . 45 4.3 The Fundamental Theorem of Calculus . 47 4.4 Applications . 50 5 Techniques of Integration 58 5.1 Change of Variable Method . 58 5.2 Partial Fractions . 59 5.3 Integration by Parts . 60 5.4 Integrals of Trigonometric Functions . 62 5.5 Integration by Trigonometric Substitution . 63 5.6 The Hyperbolic Functions . 64 5.7 Improper Integrals . 64 6 Sequences & Series 67 6.1 Sequences . 67 6.2 Series with Positive Constant Terms . 70 6.3 Alternating Series . 77 6.4 Power Series . 81 7 Conic Sections, Parametric Equations, and Polar Coordinates 86 7.1 Conic Sections . 86 7.2 Parametric Equations . 88 7.3 Polar Coordinates . 92 8 Vectors and Lines 95 9 Cross Product and Planes 101 10 Limits and Derivatives 104 11 Optimization and Lagrange Multipliers 115 12 Integration 120 13 Line Integrals, Flux, Divergence, Gauss’ and Green’s Theorem 129 14 Practice Problems 138 14.1 Chapter 1 Practice . 138 14.2 Chapter 2 Practice . 142 Mahavier < Allen, Browning, Daniel, & So math.lamar.edu/faculty/mahavier/ CONTENTS iv 14.3 Chapter 3 Practice . 146 14.4 Chapter 4 Practice . 151 14.5 Chapter 5 Practice . 160 14.6 Chapter 6 Practice . 167 14.7 Chapter 7 Practice . 176 14.8 Chapter 8 Practice . 180 14.9 Chapter 9 Practice . 182 14.10 Chapter 10 Practice . 184 14.11 Chapter 11 Practice . 189 14.12 Chapter 12 Practice . 191 14.13 Chapter 13 Practice . 196 15 Appendices 199 15.1 Additional Materials on Limits . 199 15.2 The Extended Reals . 200 15.3 Exponential and Logarithmic Functions . 201 15.4 Trigonometry . 205 15.5 Summation Formulas . 206 Endnotes to Instructor 207 Mahavier < Allen, Browning, Daniel, & So math.lamar.edu/faculty/mahavier/ To the Instructor “A good lecture is usually systematic, complete, precise – and dull; it is a bad teaching instrument.” - Paul Halmos About these Notes These notes constitute a self-contained sequence for teaching, using an inquiry-based pedagogy, most of the traditional topics of Calculus I, II, & III in classes of up to 40 students. The goal of the course is to have students constantly working on the problems and presenting their attempts in class every day, supplemented by appropriate guidance and input from the instructor. For lack of a better word, I’ll call such input mini-lectures. When the presentations in class and the questions that arise become the focus, a team-effort to understand and then explain becomes the soul of the course. The instructor will likely need to provide input that fills in gaps in pre-calculus knowledge, ties central concepts together, foreshadows skills that will be needed in forthcoming material, and discusses some real-life applications of the techniques that are learned. The key to success is that mini-lectures are given when the need arises and after students have worked hard. At these times, students’ minds are primed to take in the new knowledge just when they need it. Sometimes, when the problems are hard and the students are struggling, the instructor may need to serve as a motivational speaker, reminding students that solving and explaining problems constitute the two skills most requested by industry. Businesses want graduates who can understand a problem, solve it, and communicate the solution to their peers. That is what we do, in a friendly environment, every day in class. At the same time, we get to discover the amazing mathematics that is calculus in much the same way as you discover new mathematics – by trying new things and seeing if they work! Even though the vast majority of students come to like the method over time, I spend a non- trivial amount of time in the beginning explaining my motivation in teaching this way and encour- aging students both in my office and in the classroom. I explain that teaching this way is much harder for me, but that the benefit I see in students’ future course performances justifies my extra workload. And I draw parallels, asking a student what she is good at. Perhaps she says basketball. Since I don’t play basketball, I’ll ask her, “If I come to your dorm or house this evening and you lecture to me about basketball and I watch you play basketball, how many hours do you think will be required before I can play well?” She understands, but I still reiterate, that I can’t learn to play basketball simply by watching it. I can learn a few rules and I can talk about it, but I can’t do it. To learn to play basketball, I’ll need to pick up the ball and shoot and miss and shoot again. And I’ll need to do a lot of that to improve. Mathematics is no different. Sometimes a student will say, “But I learn better when I see examples.” I’ll ask if that is what happened in his pre-calculus class and he’ll say “Yes.” Then I’ll point out that the first problems in this course, the very ones he is struggling with, are material from pre-calculus and the fact that he is struggling shows clearly that he did not learn. His pre-calculus teacher did not do him any favors. As soon as a problem is just v To the Instructor vi barely outside of the rote training he saw in class and regurgitated on tests, he is unable to work the problems. This brings home the fact that rote training has not been kind to him and that, as hard as it is on me, I am going to do what I think is best for him. This blunt, but kind honesty with my students earns their trust and sets the stage for a successful course. I’ve taught calculus almost every long semester since 1988. When I first taught calculus, I lectured. A very few years later when I taught calculus, I sent my students to the board one day each week. Using each of these methods, I saw students who demonstrated a real talent for mathematics by asking good questions and striving to understand concepts deeply. Some of these students did poorly on rote tests. Others did well. But I was only able to entice a small number of either group to consider a major in mathematics. Today when I teach Calculus, students present every day and mathematical inquiry is a major part of what occurs in the classroom. And the minors and majors flow forward from the courses because they see the excitement that we as mathematicians know so well from actually doing mathematics. Many are engineers who choose to earn a double major or a second degree. Others simply change their major to mathematics. The key to this sea change is that the course I now teach is taught just as I teach sailing. I simply let them sail and offer words of encouragement along with tidbits of wisdom, tricks of the trade and the skills that come from a half-life of doing mathematics. And “just sailing” is fun, but a lecture on how to sail is not, at least not until you have done enough of it to absorb information from the lecture. When teaching calculus, no two teachers will share exactly the same goals. More than anything else, I want to excite them about mathematics by enabling them to succeed at doing mathematics and explaining to others what they have done. In addition to this primary goal, I want to train the students to read mathematics carefully and critically. I want to teach them to work on problems on their own rather than seeking out other materials and I emphasize this, even as I allow them to go to other sources because of the size of the classes. I want them to see that there are really only a few basic concepts in calculus, the limiting processes that recur in continuity, differentia- bility, integration, arc-length, and Gauss’ theorem. I want them to note that calculus in multiple dimensions really is a parallel to calculus in one dimension. For these reasons, I try to write tersely without trying to give lengthy “intuitive” introductions, and I write without graphics. I believe it is better to define annulus carefully and make the students interpret the English than to draw a picture and say “This is an annulus.” The latter is more efficient but does not train critical reading skills.
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