The midterm will be take place during scheduled class time on Friday, July 24. The final exam will take place during scheduled class time on Friday, August 7, 2020. Homework will be collected on the due dates in our schedule: you will submit them on Canvas. Quizzes will be administered during class time on the scheduled quiz days, and will be submitted on Canvas.
No makeup exams and no makeup quizzes will be allowed, without proper documentation. Let's make this class run smoothly, because there is a lot to cover and timeliness is of the essence! The official policy is the following: If you know that you are going to miss an exam or cannot take the final exam at the scheduled time, please notify your instructor at least two classes in advance. If you miss a midterm exam for any acceptable reason (e.g. religious obligation, documented illness), that midterm exam score will be replaced by an estimated score based on your performance on the other midterm. If you miss both exams for acceptable reasons, your midterm scores will be replaced by estimated scores based on your performance on the final. If you miss the final exam and have not rescheduled it in advance, you will score zero on the final or receive an incomplete in the course, depending on the circumstances. You may not reschedule a final exam after the exam has started. In order to be excused from an exam for medical reasons, you must either produce a note from a doctor, or you must obtain prior permission from the instructor to miss the exam. Self diagnosis and self medication are not acceptable for this purpose.
Homework: Red homework problems will be graded (see Homework tab). The other homework problems need not be turned in, but all assigned problems are considered fair game for quizzes and exams. This course will be based 100% on our course textbook, which is mostly problems. Much of the theory is left as problems. I myself will work many problems, both in lecture and in 'problem sessions' held on ZOOM during class time. There are way too many problems in there for our short five week timeframe. Therefore, let us look at these homeworks as the foundation of this course. The graded problems are but the tip of the iceberg. The full extent of the subject will need much more work and devotion than that. The ideal for us is to work as many problems as possible, for with each worked problem we advance a mile into the conceptual territory we should wish to map out.
From my experience last spring, I think the best solution is to use a scanning app on your phone, rather than convert a jpeg. Scanning apps automatically generate a pdf and the way it generates it is much better for printing purposes: much smaller size, simple black-and-white coloring, doesn't eat up toner and take 5 minutes for the printer to spool. Please use a scanning app (or equivalent) to scan all submitted assigments.
Homework problems should be TeXed up! If you know LaTeX you should typset your homeowrks. In any case, I want a pdf, even if that means taking a picture and converting it to pdf. It is important for grading purposes, because I want to make comments and remarks on pdfs, which I can return to you easily.
You are allowed to use the internet and other resources (e.g. the recommended supplementary books). However! First of all, you must cite your sources when you use them. If you outright copy a proof, you are required to confess to the fact, and it will cost you a point (it remains to be determined exactly how I'll score the homeworks, but suffice it to say I'll give you roughly a 20% discount). I encourage you all to work together on homeworks, maybe even divide up the problems into smaller groups and then get together to share solutions. We'll discuss strategies in class.
No late homework will be accepted, to help develop regularity of habits, as well as to make everybody's lives more bearable--we are on a tight schedule of five weeks! I will drop your lowest homework score to counterbalance, and I will also do the same with quizzes.
Students With Disabilities: If you qualify for accommodations because of a disability, please submit your accommodation letter from Disability Services to your faculty member in a timely manner (at least one week before the exam) so that your needs can be addressed. Disability Services determines accommodations based on documented disabilities in the academic environment. Information on requesting accommodations is located on the Disability Services website www.colorado.edu/disabilityservices/students. Contact Disability Services at 303-492-8671 or email@example.com for further assistance. If you have a temporary medical condition or injury, see Temporary Medical Conditions under the Students tab on the Disability Services website and discuss your needs with your professor.
Student Classroom and Course-Related Behavior:
Students and faculty each have responsibility for maintaining an appropriate learning environment. Those who fail to adhere to such behavioral standards may be subject to discipline. Professional courtesy and sensitivity are especially important with respect to individuals and topics dealing with race, color, national origin, sex, pregnancy, age, disability, creed, religion, sexual orientation, gender identity, gender expression, veteran status, political affiliation or political philosophy. Class rosters are provided to the instructor with the student's legal name. I will gladly honor your request to address you by an alternate name or gender pronoun. Please advise me of this preference early in the semester so that I may make appropriate changes to my records. For more information, see the policies on classroom behavior and the Student Code of Conduct.
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Pascal, in the opening lines of his Of the Geometrical Spirit (c 1657) says, `We may have three principal objects in the study of truth: one to discover it when it is sought; another to demonstrate when it is possessed; and a third, to discriminate it from the false when it is examined. I do not speak of the first [this is speculation, or insight, and involves ingenuity and creativity--these are not uniformly distributed among people and so do not constitute a method]; I treat particularly of the second, and it includes the third. For if we know the method of proving the truth, we shall have, at the same time, that of discriminating it, since, in examining whether the proof that is given of it is in conformity with the rules that are understood, we shall know whether it is exactly demonstrated.`
He then proceeds to explain the method of proof. Step one is establishing clear definitions, step two is proofs, `never advancing any proposition which could not be demonstrated by truths already known,` whether those be previously proven propositions, our established definitions or the assumed axioms of the subject.
Pascal elaborates more fully on the proper methods of proof in another essay, The Art of Persuasion: 'This art, which I call the art of persuading, and which, properly speaking, is simply the process of perfect methodical proofs, consists of three essential parts: of defining the terms of which we should avail ourselves by clear definitions; of proposing principles or evident axioms to prove the thing in question; and of always mentally substituting in the demonstrations the definition in the place of the thing defined.'
'The reason of this method is evident,' says Pascal, 'since it would be useless to propose what it is sought to prove, and to undertake the demonstration of it, if all the terms which are not intelligible had not first been clearly defined; and since it is necessary in the same manner that the demonstration should be preceded by the demand for the evident principles that are necessary to it, for if we do not secure the foundation we cannot secure the edifice; and since, in fine, it is necessary in demonstrating mentally, to substitute the definitions in the place of the things defined, as otherwise there might be an abuse of the different meanings that are encountered in the terms. It is easy to see that, by observing this method, we are sure of convincing, since the terms all being understood, and perfectly exempt from ambiguity by the definitions, and the principles being granted, if in the demonstration we always mentally substitute the definitions for the things defined, the invincible force of the conclusions cannot fail of having its whole effect.'
Significant portions of these two essays are devoted to discussing key nuances of the method, which require skill, even art, to master:
Concerning definitions, he makes the all-important distinction between primitive definitions and complex definitions, which reflects the distinction between obvious and non-obviousterms.
Primitive definitions are those of things we directly perceive (but do not yet understand)--his examples are motion (a primitive term of the subject of mechanics), number, equality, greater than, less than (primitive terms of the subject of arithmetic), space (a primitive term of geometry), and time (a primitive of both mechanics and geometry today). We do not define these terms because we all understand what we mean by them, even though we do not know their hidden nature and deeper meaning (which is, after all, the avowed purpose of the subsequent theoretical development of each subject).
Complex definitions are constructed out of primitive terms and previously defined terms. For example, in arithmetic a composite integer is defined in terms of divisibility and prime number, both of which are also composite terms (defined as they are in terms of the primitives product and 1).
Concerning the proper construction of definitions in general, the basic principle is this: We have complete freedom in defining complex terms, subject only to the restriction that the result is unambiguous in meaning ('You can't argue with a definition.'). As Pascal putis it, 'The only definitions recognized in geometry are what the logicians call definitions of name, that is, the arbitrary application of names to things which are clearly designated by terms perfectly known...Their utility and use is to elucidate and abbreviate discourse...Hence it appears that definitions are very arbitrary, and that they are never subject to contradiction; for nothing is more permissible than to give to a thing which has been clearly designated, whatever name we choose...For geometricians, and all those who proceed methodically, only impose names on things to abbreviate discourse, and not to diminish or change the idea of the things of which they are discoursing.'
Concerning the proper usage of definitions in proofs, the guiding principle is the equality of the defined term and the thing it defines (for example 'even number' and 'an integer divisible by 2'), because it is by this equality that a term succeeds in abbreviating a concept. As with equations involving variables, their usage comes via substitution: the term may be substituted for the thing defined when writing a proof, and the thing defined may be substituted for the term when reading a proof.
Concerning axioms there is this to say: It is an art, requiring insight and ingenuity, to find, or, may better, distill a subject's foundational principles to a short list of axioms. These are, first of all, propositions, whose truth is granted, usually on empirical grounds (as is the case with the meaning of primitive terms). There are no other reasons than self-evidence in experiential terms or pragmatic usefulness (subject of course to topicality) for the axioms. Axioms serve as the foundation to a subject (as the five postulates of Euclid serve as the foundation to Euclidean geometry). All other truths of the subject derive, logically, from them (e.g. the Pythagorean Theorem derives logically from the five postulates of Euclid).
Two special topics round out his discussion, both in Of the Geometrical Spirit.
First, the topic of infinity. Actually, there are two sides to infinity, the infinitely large and the infinitely small or infinitessimal. Pascal, of course, lived roughly a generation before Newton's and Leibniz' invention of calculus, so it is not surprising to see one of the fundamental issues of analysis discussed minutely here. Numbers and measurement were united into one single framework by Descartes, Pascal's immediate predecessor, and all things to which measurement applies are subject to the issues involved here: time, space, motion.
'That is, in a word, whatever movement, whatever number, whatever space, whatever time there may be, there is always a greater and a less than these: so that they all stand betwixt nothingness and the infinite, being always infinitely distant from these extremes. All these truths cannot be demonstrated; and yet they are the foundations and principles of geometry. But as the cause that renders them incapable of demonstration is not their obscurity, but on the contrary their extreme obviousness, this lack of proof is not a defect, but rather a perfection. From which we see that geometry can neither define objects nor prove principles; but for this single and advantageous reason that both are in an extreme natural clearness, which convinces reason more powerfully than discourse. For what is more evident than this truth, that a number whatever it may be, can be increased, can be doubled? Again, may not the speed of a movement be doubled, and may not a space be doubled in the same manner?'
Secondly, the topic of dimension. A point, or equivalently the real number representing that point on a line, is 0-dimensional. An interval is 1-dimensional. Even though points are the elements of lines, they are infinitessimals in terms of dimension 1, and they do not add up to an interval (this is why it is necessary to develop the integral, to resolve this paradox--it will require a new concept, that of the limit). Pascal drives his point home with a simple example: a soldier versus an army.
'It is annoying to dwell upon such trifles; but there are times for trifling. It suffices to say to minds clear on this matter that two negations of extension cannot make an extension. But as there are some who pretend to elude this light by this marvellous answer, that two negations of extension can as well make an extension as two units, neither of which is a number, can make a number by their combination; it is necessary to reply to them that they might in the same manner deny that twenty thousand men make an army, although no single one of them is an army; that a thousand houses make a town, although no single one is a town; or that the parts make the whole, although no single one is the whole; or, to remain in the comparison of numbers, that two binaries make a quaternary, and ten tens a hundred, although no single one is such.'
The stage has been set for the development of calculus, whose full logical articulation is analysis.
Analysis I provides the conceptual architecture behind calculus in one variable, namely its logical structure in terms of the basic conceptual categories:
topology of the real numbers (limits, continuity, compactness, connectedness, sequences and series)
differentiability, local linearization, and higher order polynomial approximation (Taylor polynomials)
integration and anti-differentiation
function series (especially power series and Taylor series)
The most interesting theorems concern the interrelations between these concepts:
The Heine-Borel Theorem characterizes compact subsets of the real line (those for which any open cover can be reduced to a finite subcover): they are precisely the closed and bounded subsets (e.g. closed intervals), and simultaneously they are the sequentially compact subsets (those which contain the limits of their convergent sequences). Thus, `closedness'+'boundedness`=`compactness`, and we have just gained a richer undersanding: whether we look at boundary points, limits of sequences, or finite reducibility of open covers, we are discussing the same concept, compactness.
Cantor's Theorem says the real numbers are uncountable. Moreover, any open interval has the same cardinality as the whole real line. This gives you some notion of the `size` of real numbers--`size,` now, understood in crisp functional terms--the real numbers (and therefore all open intervals) are so huge, in terms of the number of points they contain, that they cannot even be enumerated ad infinitum.
Baire's Theorem says that the real numbers cannot be written as the countable union of nowhere-dense sets. Forget about joining up a finite set of points to make a continuous line. You can't do it even with countably many--that was Cantor's theorem. But in fact, you can't even do it with countably many countable meager sets (a 'meager' subset here means having no interior, such as an interval may have, e.g. a single point has no interior). That is how big the real numbers are--a fascinating conceptual revelation.
The Extreme Value Theorem and the Intermediate Value Theorem together say that continuous functions take closed intervals to closed intervals. The EVT guarantees the endpoints are in the image of the function, and the IVT guarantees the interior points are in the image. This has practical value for numerical approximations, which we will discuss. There is a unifying concept joining these theorems: connectedness. Continuous functions take connected subsets to connected subsets.
The Mean Value Theorem is the workhorse of calculus. The derivative dictates, or determines, the y-values of a differentiable function on an open interval. This theorem is invaluable, for it gives you the meaning of calculus: the derivative, which measures the local, instantaneous growth-rate of a function, drives the overall growth rate of the function. This is the 'why' of calculus: Why do we need the derivative? Because it allows us to understand the qualitative behavior of differentiable functions. Several immediate results follow, including optimization and L'Hospital's Rule, but also, later, the Fundamental Theorem of Calculus.
Having motivated differentiability, it's time to understand which functions possess this desirable property. In breaking down this class of functions, we will make rigorous comparisons between it and several other classes: continuous functions, k-times continuously differentiable functions, smooth functions, analytic functions (those representable by their Taylor series), and polynomials. The climax is the Weierstrass Approximation Theorem, which says that on closed intervals (i.e. compact sets) even continuous functions may be approximated by polynomials. Taylor's Theorem is thereby surpassed in generality, if not in computational efficacy. Nevertheless, there are many numerical applications of the result, which we will discuss.
Concerning the above list of function classes, we must next ask how each class behaves with respect to convergence (sequences and series of functions!). And supposing convergence, can we differentiate or integrate term-by-term? In other words, what is the topology of these function classes? Do limits of sequences of functions of each class remain in the class (are the classes closed)? What else may be required to ensure this?