Notes on Calculus. Allen Olsen Lexington HS, Lexington MA September 8, 2015

Transcription

1 Notes on Calculus Allen Olsen Lexington HS, Lexington MA 024 September 8, 205

2 Contents Foreword 4 2 Limits 9 3 Properties of the Absolute Value Function 4 4 Limits (2) 7 5 Limit Theorems 9 6 More About Limits 29 7 Continuity 32 8 The Boundary Axiom 35 9 The Least-Upper-Bound Principle 36 0 A Theorem about Continuous Functions 38 Bolzano s Theorem 39 2 The Intermediate Value Theorem 42 3 Rules for Derivatives 42 4 Chain Rule 47 5 Exponential Functions 49 6 The Extreme Value Theorem 55 7 The Critical Point Theorem 60 8 The Mean Value Theorem 64 9 Riemann Sums and Integrals The Fundamental Theorem of Calculus 77 2

3 2 Improper Integrals Partial Fractions L Hôpital s Rule Taylor s Theorem Taylor Series Infinite Series and Convergence Tests Calculus of a Complex Variable - Differentiation 4 28 Calculus of a Complex Variable - Integration 6 29 Appendix: Logic, Set Theory, Intervals, Neighborhoods 25 3

4 Foreword It is customary these days to frame curriculum discussions in terms of Big Ideas, Essential Questions, and the like. In mathematics, surely the biggest of big ideas is the idea of Proof: i.e., the idea that the validity of mathematical results stems not from the authority of the person asserting them, butfromtheabilityofanyonewhatsoevertodemonstratetheirtruth by starting from a small set of initial assumptions and definitions and using logical reasoning. The methodology and rigor of proofs is indeed what sets mathematics apart from other disciplines. In recent years, however, there has been a shift in the high-school curriculum away from this big idea. The recent Common Core de-emphasizes geometry, where most of the instruction in methods of proof once resided, in favor of a beefed-up algebra and trigonometry curriculum. But the missing discussion of geometric proofs has not been replaced by any instruction on doing algebra-based proofs. This shift therefore doesn t just reduce students understanding of geometry; it reduces as well their understanding of algebra and trigonometry. In trig, for example, exact values for trigonometric functions of certain angles, for example, sin 8 = + 5,stemfromproofs 2 based on the geometry of regular polygons. These identities provide valuable practice of the angle-addition formulas for students, in a closed form expression, and this has considerable pedagogical value. We see from this example that proof is a way to get results that cannot be obtained otherwise, or can be obtained only with more difficulty. The curriculum of the AP Calculus Exam contains all of the important theorems and techniques and examples of their use, but is devoid of proofs that would expose to our students the important properties of the real numbers that lead to these theorems. There are so many subtleties in the structure of the real numbers, and it seems important to me that we take time to use the tools of calculus to share with our students the beauty of this subject and of the underlying mathematics. In other words, I think our students deserve to know why Calculus works and not just how Calculus works. There is a cost, of course; to take one example, learning how to prove the product rule takes considerably longer than learning how to use it. Writing a proof is not something that you can learn by reading; you have to actually do it, and get it wrong many times before you get it right. In the view of many AP teachers, the primary goal of an AP Calculus class is to cover the 4

5 AP curriculum and give students sufficient exposure to, and practice solving, the kinds of calculus questions that usually appear on the AP Exams. Not only do proofs not appear on the exam, but learning to write them takes, potentially, a lot of valuable class time away from covering a very challenging curriculum. These concerns are understandable. When I broach the subject of adding new units into the curriculum, I frequently hear the response, tell me what IshouldtakeoutsothatIhavethetimetoteachtheseextratopics. But this argument, of course, is fallacious. Teaching and learning are not just zero-sum games, and not all topics are created equal. In my mind, talking about where these results come from provides not merely additional quantity, but additional quality to the course. It is easy when teaching Calculus to forget that the AP Exam does not measure understanding; it measures instead proficiency of technique. Why then, are increasing numbers of colleges not awarding college credit for successfully passing the AP Exam? I would like to suggest that we teachers should go farther to ensure our students understand the concepts. We want them to be able to build on their understanding after they are in college, in science, math or engineering courses. We should be teaching them the Calculus they will need in college, instead of just preparing them for one exam. My Calculus students ask questions like the following. They really aren t on the AP exam, but they represent the next level of conceptual understanding, the why questions that come after the how questions. I think they are good questions, and they deserve good answers. How can you figure out whether a particular limit even exists? And what its value is? Is infinity a number? What happens when you try to add something to it or divide by it? What is the difference between the Intermediate Value Theorem and the Mean Value Theorem? f Why doesn t (f g) = f g? And = f g g? What is the difference between real numbers and rational numbers? Can a function defined only on the rational numbers be continuous? 5

6 How come lim + x isn t just? How do you know it even exists? x x This set of notes is an attempt to enrich the classroom experience of the High School Calculus student by providing a resource to them and their teachers for answering questions like these. They grew out of my experience teaching AP Calculus AB and AP Calculus BC over the last ten years to High School students at Lexington High School in Lexington, MA, and Lawrence High School in Lawrence, MA. They are a supplement to the material found in the standard AP Calculus textbooks, a set of appendices, if you will. My goal is to make available additional mathematical rigor to the calculus curriculum, primarily by covering the specific properties of real numbers that make it possible to prove the theorems in the AP Calculus curriculum. The proofs are self-contained in that I supply every ingredient necessary to do or understand them. But I did not set out to write a Calculus textbook; I offer this as a supplement to the many fine textbooks that now exist. Once I set out on this project, there were two main obstacles I encountered, one mathematical, and the other pedagogical. Proofs of theorems like the Mean Value Theorem or the Intermediate Value Theorem rely on very esoteric results in real analysis, of the kind taught to math majors in college. In my initial researches, I was not able, for example, to come up with a proof of the Extreme Value Theorem that did not rely on the concept of compact sets. Since the pedagogic goal of this project was accessibility of the proofs, implying simplicity, brevity, and understandability, many of the extant proofs taught in college were non-starters. So my first mathematical goal was to see if I could find the subset of real analysis that was sufficient to be able to prove these results, but small enough to be taught in a High School setting. There followed months of effort to go back into the history of Calculus and understand the development of the subject and how its practitioners viewed it conceptually. What properties of real numbers distinguished them from other numbers? Which of these properties had the leverage to be used for these proofs? After considerable effort, I settled on the Least Upper Bound Principle. Starting with this, I was able to go back and write proofs of all the theorems using only on this principle. The result, I think, shows the conceptual unity of Calculus and its intimate dependence on the properties of the real number system. This reformulation was a big win pedagogically, because it provided a simple, unified conceptual basis for understanding where 6

7 Calculus comes from. Using the Least Upper Bound Principle allows us to write a nifty, accessible proof that the limit lim n (+/n) n exists and why its value is where it is. Ihavetriedtowritetheproofsinaclearform,sinceagoodwaytolearn how to do proofs is to copy proofs that others have written. The process of writing them out slows down the brain and gets it to appreciate the details the reasons for the steps. After doing this for a while, one s brain starts to really pay attention to the definitions and theorems that underlie the reasoning, and to generate a conceptual framework on which to build one s understanding of calculus. My hope is that this work will contribute to increased understanding of where this stuff comes from on the part of our students. There are three areas of note in the material ahead. First, the creation of the present epsilon-delta definition of limit is a milestone in human thought, the culmination of 200 years of work since the invention of Calculus by Newton. The math required to master it is just a few properties of the absolute value function. An important proof is the proof that limits are unique, even though the assignment of a delta function to a given epsilon is not unique, because any smaller positive function could work as well. I try to encourage students to see the definition of limit as modular, with one module constituting the y-values and epsilon, and the other consisting of the x-values and delta. Then when it comes time to talk about different types of limits, new modules can simply be switched in to the constant proof framework and all the limit theorems still hold. This is where limits at infinity and infinite limits show up for the first time. A big idea of the course is that infinity is not a real number, but rather, that the machinery of limits is what gives it its definition and its properties. Students should have just enough practice proving limits by using the definition to see how ungainly it is, and to motivate the importance of the limit theorems that follow. Second, the property of continuity, though not as powerful as differentiability, provides sufficient power to prove not only the Intermediate Value Theorem and the Mean Value Theorem, but also Bolzano s Theorem, the Extreme Value Theorem, and the Critical Point Theorem. The advantages of this approach are many: first, students can finally see the true relation of the IVT to the MVT and can appreciate the power of continuity. The way we usually teach continuity is as kind of a throwaway definition. It turns out to be very important. In these notes you can begin to see the hierarchy of 7

8 properties of real functions: continuity is a stronger property than merely having a limit, and being differentiable is a stronger property yet. Furthermore, one can see that the concept of continuity would be much less useful without the Least Upper Bound Principle. The third area is the proof of the theorem that continuous functions are Riemann integrable. The idea of a Riemann sum is central to the Integral Calculus, but to the uninitiated, it seems rather rickety. Not only are we allowed to vary the subintervals in the partition of the closed interval as we construct the sum, but we can even vary the way of choosing the point in each subinterval at which we calculate the representative value of the integrand. Then we are to apply an ill-defined limiting process, and somehow the continuity of the integrand forces it all to come together to produce a unique value. In our proof of this theorem, I isolate these different aspects and demonstrate how the Least Upper Bound Principle allows us to control the variation of the sum as it presses to convergence. One pedagogical benefit of establishing this theorem on a solid basis is that when Improper Integrals are introduced, one can immediately understand why we have to be concerned that discontinuities of the integrand might affect convergence of the integral. The text of these notes was taken directly from lesson plans that I taught in the school year in my the two BC Calculus sections. These notes are dedicated to the students in these sections, who suffered through early versions and who supplied many helpful corrections and suggestions. The responsibility for any remaining errors and omissions are, of course, mine. I also dedicate these notes to my colleagues. It is my hope that these notes will be useful those who read them and will deepen their love for this fascinating subject. Allen Olsen August, 205 8

9 2 Limits Achilles and the Tortoise is a famous paradox of motion by Zeno of Elea (490 BC 430 BC), that proves it is impossible for the fastest runner in the world (Achilles) to win a race against the slowest runner (the Tortoise), if the latter is given a head start. Here s how it goes. The Tortoise starts one hundred paces in front of Achilles. The race starts! Achilles blazes through the first one hundred paces in a flash. By the time he has reached the Tortoise s starting line, the Tortoise has laboriously moved twenty paces forward. Achilles presses ahead and covers the additional twenty paces in an instant. But while he is doing this, the Tortoise has moved another four paces forward. During the time it takes Achilles to reach the Tortoise s location at the beginning of each one of these intervals, the Tortoise moves some distance forward, so that when Achilles reaches where the Tortoise was, the Tortoise is always still ahead. Since the Tortoise is always ahead, the Tortoise can never be overtaken by Achilles!!! Now, we know that this result is absurd. But the great thing about this paradox the reason people are still talking about it 2, 500 years later is that it s hard to figure out what, specifically, is wrong with Zeno s reasoning. Can you do it? Could you explain your reasoning to Zeno in terms he would understand? Try to work out the following exercises before consulting the answers at the end of the section. Exercise 2. In the above paradox, Achilles is running five times faster than the Tortoise. Using algebra, calculate how many paces he has to run before he reaches the Tortoise. Exercise 2.2 Zeno s paradox plays out during successive intervals of time, which get shorter and shorter. His logic is solid he proves convincingly that at the end of any finite number of intervals, the Tortoise is still ahead of Achilles. Yet you showed in exercise 2. that Achilles catches the Tortoise after 25 steps. This is a contradiction. How do you resolve it? 9

10 Since Zeno proves without a doubt that the Tortoise is always ahead after afinitenumberofintervals,wehavetoconsiderthepossibilitythatittakes an infinite number of the increasingly smaller intervals for Achilles to catch the Tortoise. Is that possible in a finite amount of time? We can prove that it is by writing an expression for the total distance Achilles runs before he meets the Tortoise. You could calculate the total time, but since Achilles is running at a constant speed, the distance and the time are proportional. Thus the possibility of Achilles running through an infinite number of decreasing time intervals in a finite time is the same as the possibility of him running through an infinite number of decreasing distance intervals in a finite amount of space. Exercise 2.3 Set up an infinite sum that describes how far Achilles runs before he catches up to the Tortoise. Do you recognize what kind of series this is? Use Euclid s method for summing it up. Is the sum of the series finite or infinite? Describe what s happening in terms of the limit lim n 5 n =0. So at last we see the resolution of the paradox. There s a hidden limit buried within the problem in this case lim n 5 n. By proving this limit converges to 0, we can show that the infinite number of intervals extend only over a finite distance, or, equivalently, that Achilles can cover them in a finite amount of time. Thus he can catch the Tortoise and pass him. But wait! Did I say prove? Wedidnothingofthekind!Ourresolutionto the mystery of Achilles and the Tortoise is a new mystery what is a limit, and why do we say that lim n 5 n =0? Let stacklethesecondmystery first. We want to consider the behavior of (/5) n as n gets larger and larger. We start with the inequality 0 < /5 < andmultiplyrepeatedlyby/5: 0 < (/5) 2 < (/5) < 0 < (/5) 3 < (/5) 2 < (/5) < 0 < (/5) n < (/5) n < < (/5) 2 < (/5) < This shows that (/5) n is always positive, but always less than (/5) n,so as n increases, (/5) n gets smaller and smaller, and closer to 0, but never equal to 0. Thus lim n 5 n =0. Exercise 2.4 What do you think? Is this a good proof? 0

11 It is not enough to show that (/5) n gets closer and closer to 0 for its limit to be 0; for this we have to show that there is no number E>0such that (/5) n E for all values of n. In other words, (/5) n has to be able to get infinitely close to 0. We will prove in the next exercise that no such number E can exist. Exercise 2.5 (a) Assume that an E>0existssuchthat(/5) n E regardless of what n is. Show that E<. (b) Given this E, canyoufindapositiveintegern such that (/5) N <E? (c) Show that this means that no E can exist with the properties we hypothesized. This means that (/5) n can get infinitely close to 0. Zeno s paradox is an interesting and challenging paradox that, even today, pushes us to think really hard about the nature of infinity, infinite subdivisions, and the behavior of functions as their arguments approach certain values. In analyzing this paradox, we discovered that. There is a limiting process (the number of intervals, n, hastogoto ). 2. There is a sum of an infinite number of terms (the distance that Achilles travels before reaching the Tortoise) that is finite. 3. Achilles can, therefore, in a finite amount of time, and a finite distance, catch up with the Tortoise, and pass him. But our conclusion relies on our ability to prove that lim n 5 n =0. We have made progress on that front. We have shown that. (/5) n gets closer and closer to 0 as n increases. 2. It gets, in fact, infinitely close to 0 (there is no number L>0 that is less than (/5) n for all n). Now can we say that lim n 5 n =0? Alas, no, not yet.

12 The problem is that we can t prove that the limit exists without a definition of what a limit is. So we must turn our attention to that. We will do that in the next section of these Notes. Solution to Exercise 2.: Achilles is running five times as fast as the Tortoise, so when they meet, Achilles will have run five times as far. Let x be the distance the Tortoise moves in that time; then 5x is the distance Achilles runs, which is 00 + x paces. Setting these equal, we see they meet when the Tortoise has moved x = 25 paces, and Achilles has run 25 paces. Solution to Exercise 2.2: If Zeno s argument is correct, then it can t take a finite number of intervals for Achilles to catch the Tortoise. The only other possibility is that it takes an infinite number of intervals. Solution to exercise 2.3: The distance Achilles runs is / = 00(+(/5) + (/5) 2 +(/5) ). This geometric series has an infinite number of terms in it, so we need to be careful. Consider the sequence of partial sums: S = 00 S 2 = 00(+(/5)) S 3 = 00 +(/5) + (/5) 2 S n = 00 +(/5) + (/5) (/5) n Now we can solve for S n using Euclid s method as follows: S n = 00 +(/5) + (/5) (/5) n (/5) S n = 00 (/5) + (/5) 2 +(/5) (/5) n +(/5) n Subtracting, most of the terms cancel and we get ( (/5)) S n =00( (/5) n ), or (/5) n S n =00. (/5) 2

13 The only part of this expression that depends on n is (/5) n. We are given that lim n ( 5 )n =0. Thenwecanwrite 0 S = lim S n =00 =25. n 4/5 This agrees with our answer to exercise 2.. Solution to exercise 2.4: No. By the same argument, + (/5) n is always positive, but always less than + (/5) n, so as n increases, + (/5) n gets smaller and smaller, and closer to 0, but never equal to 0. But lim n + n 5 =,not0. Theproblemhereisthatthefunctiondoesnot get close enough to 0 for 0 to be its limit. Solution to Exercise 2.5: (a) From what was said above, E (/5) n < soe<. (b) Since 0 <E<, ln E<0and ln E>0. Pick an integer N> ln E ln 5 > 0. Then we have ln 5 < N ln E 0 < ln 5 + N ln E =ln 5 E /N < 5 E /N (/5) N < E. (c) This is a contradiction, since we assumed that for all n, (/5) n E. Thus, no such E exists and (/5) n gets arbitrarily close to 0 as n increases. 3