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Honors Mathematics II Functions of a Single Variable Honors Mathematics II Functions of a Single Variable Dr. Horst Hohberger University of Michigan - Shanghai Jiaotong University Joint Institute Fall Term 2015 Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 1 / 624 Introduction Office Hours, Email, TAs I Please read the Course profile, which has been uploaded tothe Resources section on the SAKAI course site. I My office is Room 218 in in the Law School Building. I My email is [email protected] and I’ll try to answer email queries within 24 hours. I Office hours will be announced on SAKAI. I Please also make use of the chat room on SAKAI for asking questions, making comments or giving feedback on the course. I The recitation class schedule and the TA office hours and contact details will be announced on SAKAI. I Of course, you can also contact the TA with questions outside of office hours, but please observe basic rules of politeness. Itisnot appropriate to phone your TA at 10 pm in the evening and request help for the exercise set due the next day. Your TA is there to help you, but is not a 24/7 help line. Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 2 / 624 Introduction Coursework Policy The following applies mostly to the paper-based homework at the beginning of this term: I Hand in your coursework on time, by the date given on each set of course work. Late work will not be accepted unless you come to me personally and I find your explanation for the lateness acceptable. I You can be deducted up to 10% of the awarded marks for an assignment if you fail to write neatly and legibly. Messiness will be penalized! Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 3 / 624 Introduction Use of Wikipedia and Other Sources; Honor Code Policy When faced with a particularly difficult problem, you may want to refer to other textbooks or online sources such as Wikipedia. Here are a few guidelines: I Outside sources may treat a similar sounding subject matter at a much more advanced or a much simpler level than this course. This means that explanations you find are much more complicated or far too simple to help you. For example, when looking up the “induction axiom” you may find many high-school level explanations that are not sufficient for our problems; on the other hand, wikipedia containsa lot of information relating to formal logic that is far beyond what we are discussing here. I If you do use any outside sources to help you solve a homework problem, you are not allowed to just copy the solution; this is considered a violation of the Honor Code. Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 4 / 624 Introduction Use of Wikipedia and Other Sources; Honor Code Policy I The correct way of using outside sources is to understand the contents of your source and then to write in your own words and without referring back to the source the solution of the problem. Your solution should differ in style significantly from the published solution. If you are not sure whether you are incorporating too much material from your source in your solutions, then you must cite the source that you used. I You may cooperate with other students in finding solutions to assignments, but you must write your own answers. Do not simply copy answers from other students. It is acceptable to discuss the problems orally, but you may not look at each others’ written notes. Do not show your written solutions to any other student. This would be considered a violation of the Honor Code. Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 5 / 624 Introduction Use of Wikipedia and Other Sources; Honor Code Policy In this course, the following actions are examples of violations of the Honor Code: I Showing another student your written solution to a problem. I Sending a screenshot of your solution via QQ, email or other means to another student. I Showing another student the written solution of a third student; distributing some student’s solution to other students. I Viewing another student’s written solution. I Copying your solution in electronic form (LATEX source, PDF, JPG image etc.) to the computer hardware (flash drive, hard disk etc.) of another student. Having another student’s solution in electronic form on your computer hardware. If you have any questions regarding the application of the Honor Code, please contact me or any of the TAs. Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 6 / 624 Introduction Grading Policy I It is JI policy to curve the grades for standard undergraduate courses so that the median grade is a “B.” For this Honors course, the grade will be curved to achieve a median grade of “B+.” I The grade will be composed of the course work and the exams as follows: I First midterm exam: 20% I Second midterm exam: 25% I Final exam: 30% I Course work: 25% Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 7 / 624 Introduction What is Mathematics? Mathematics is about certainty. Mathematics is about modeling. Mathematics is about thinking. Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 8 / 624 Introduction Course Focus Please forget everything you have learned in school; you haven’t learned it. Edmund Landau, Foundations of Analysis, 1929 The course will focus on functions of one variable, in particular I Foundations: Set Theory, Logic, Numbers, Algebraic Structures. I Functions, Limits and Continuity, I Differentiation I Integration The main textbook for this course is Calculus, M. Spivak, 3rd Edition. However, additional material from other sources will be added to the lecture. Further textbook references will then be given. Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 9 / 624 Introduction Notation The notation used in these notes should be fairly well-known from school; some symbols will be defined explicitly to make sure that theyP are understood unambiguously (for example, the summation symbol ). A particular convention needs some explaining: when an equality is a definition (and not a statement that can be proven), the equal symbol will sometimes have a colon added on the side of the defined quantity. For example, A [ B := fx : x 2 A or x 2 Bg indicates that that the union A [ B is being defined through this equality. Another example might involve n(n + 1) 1 + 2 + ··· + n = =: a , 2 n indicating that the number an is defined to be n(n + 1)=2. Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 10 / 624 Introduction Notation On the other hand, it would be wrong to write n(n + 1) 1 + 2 + ··· + n := 2 because the objects on the left and the right are already defined and the equality is a statement that can be proven; it does not define anything. Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 11 / 624 Basic Tools and Definitions Part I Basic Tools and Definitions Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 12 / 624 Basic Tools and Definitions Basic Concepts in Logic Basic Concepts in Set Theory The Natural Numbers and Mathematical Induction The Rational Numbers Roots, Bounds & Real Numbers Complex Numbers Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 13 / 624 Basic Tools and Definitions Basic Concepts in Logic Basic Concepts in Logic Basic Concepts in Set Theory The Natural Numbers and Mathematical Induction The Rational Numbers Roots, Bounds & Real Numbers Complex Numbers Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 14 / 624 Basic Tools and Definitions Basic Concepts in Logic Propositional Logic, Statements Reference L. Loomis and S. Sternberg, Advanced Calculus, Sections 0.1-0.4. Freely available here (clickable link): http://www.math.harvard.edu/∼shlomo/docs/Advanced_Calculus.pdf. A statement (also called a proposition) is anything we can regard as being either true or false. We do not define here what the words “statement”, “true” or “false” mean. This is beyond the purview of mathematics and falls into the realm of philosophy. Instead, we apply the principle that “we know it when we see it.” We will generally not use examples from the “real world” as statements. The reason is that in general objects in the real world are much to loosely define for the application of strict logic to make anysense. For example, the statement “It is raining.” may be considered true by some people (“Yes, raindrops are falling out of the sky.”) while at the same time false by others (“No, it is merely drizzling.”) Furthermore, important information is missing (Where is it raining? When is it raining?). Dr. Hohberger (UM-SJTU JI) Functions of a Single Variable Fall 2015 15 / 624 Basic Tools and Definitions Basic Concepts in Logic Statements 1.1.1. Examples. I “3 > 2” is a true statement. I “x3 > 0” is not a statement, because we can not decide whether it is true or not. I “the cube of any number is strictly positive (> 0)” is a false statement. The last example can be written using a statement variable x: I “For any number x, x3 > 0” The first part of the statement isa quantifier (“for any number x”), while the second part is called a statement frame or predicate (“x3 > 0”). A statement frame becomes a statement (which can then be either true or false) when the variable takes on a specific value; for example, 13 > 0 is a true statement and 03 > 0 is a false statement.
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