Math 10850, fall 2017, University of Notre Dame Notes on final December 12, 2017 When and where The final exam will be on Wednesday, December 13, 4.15-6.15pm in 129 Hayes-Healy. What you need to know The final is cumulative, although there will be a slant towards later material. Here is a summary of the topics that we have covered this semester, roughly in chronological order, followed by a list of theorems whose proofs we have seen, and definitions that you should know correct statements of. There is some overlap between these lists. Note that some items appear in square parentheses (like [this]); these are things that we have seen but that I won't examine on the final. I'm including them on the list for completeness. You should be able to answer questions like: • Logic: { What is a proposition? { What are conjunction, disjunction, negation, implication, bidirectional implica- tion? { What are the contrapositive and converse of an implication? { What is a truth table? { What are the truth tables of conjunction, disjunction, etc.? { What are tautologies and contradictions? { How do you use a truth table to check if a proposition is a tautology or a contradiction? 1 { What are logical implication and logical equivalence? { How do you use logical implication and logical equivalence to check if a propo- sition is a tautology or a contradiction? { What are the basic logical equivalences such as associativity, commutativity, distributivity, De Morgan's laws, the law of implication? { What is a predicate? { What are universal, existential and unique existential quantification? { How do you negate a quantified predicate? • Axioms of real numbers { What are the thirteen axioms that we take as the axioms of the real number system? { How are the basic symbols of inequality defined? { How do you derive other basic properties of numbers from the axioms? { What is the absolute value function? { What is the triangle inequality? { How do you solve equalities and inequalities involving the absolute value function? { What is an inductive set? { How are the natural numbers defined? { What is the principle of mathematical induction? { What is the principle of complete induction? { How do you prove that addition of many terms is associative? { What is the well-ordering principle? { How do you structure a proof by induction? { What is summation notation? { What is a definition of a sequence of terms by recursion? n { What is the binomial coefficient k ? { What is the binomial theorem (concerning (x + y)n)? { What is the Bernoulli inequality? { What are the integers, and the rationals? p { How do you show that 2 is irrational? • Functions { What is a function (of real numbers)? 2 { What are domain, range and codomain? { What are the sum, difference, product and ratio/quotient (division) of functions? { How is the domain of a function built from additional, multiplication, etc., obtained from the domain of the constituent parts? { What are constant, linear, power, polynomial and rational functions? { What is the composition of functions? { How is the domain of a function built from composition obtained from the domain of the constituent parts? { What is the graph of a function? { What do the graphs of constant, linear, power and polynomial functions look like? { What are circles and ellipses? { What are the sin and cosine functions? • Limits { What is the definition of a function tending to a limit? { How do you compute basic limits using the formal "-δ definition? { How do you establish that a limit does not exist? { What is the limit of sin x=x as x approaches 0? { What are one-sided limits (from the left/below and right/above)? { What does it mean to say that a function tends to +1 or −∞ approaching a point? { What are limits at infinity? { What are the basic rules for adding, multiplying, dividing limits? • Continuity { What is the definition of a function being continuous at a point? { What are the different ways in which a function might be not continuous? { What conditions ensure that a composition of functions is continuous at a point? { What is an example of a function that is defined everywhere but continuous nowhere? { What is an example of a function that is defined everywhere, is continuous at all irrationals, but not continuous at any rational? { How is the continuity of a function at a point related to the one-sided limits of the function at that point? 3 { What does the Intermediate Value Theorem say? { Why are the Intermediate Value Theorem and the Extreme Value Theorem false if we work exclusively in the rational numbers? { Why are the Intermediate Value Theorem and the Extreme Value Theorem false if we do not assume continuity of functions, on closed intervals? { How is the function x1=n defined for natural numbers n? { What can you say about odd-degree polynomials, with regards to real roots? { What can you say about even-degree polynomials, with regards to boundedness? • Completeness { What does it mean to say that a set is bounded above, or below? { What is an upper bound, or lower bound, for a set? { What is a least upper bound for a set, and a greatest lower bound? { What is the supremum sup A of a set A, and the infimum inf A? { Which subsets of the reals have suprema and infima? { What is the completeness axiom for the reals? { How does the completeness axiom allow a proof of the Intermediate Value Theorem? { What is the Archimedean property? { If a function is continuous at a point, what can you conclude about boundedness around that point? { How does the completeness axiom allow a proof that continuous functions on closed intervals are bounded? { How does the completeness axiom allow a proof that continuous functions on closed intervals achieve their maxima and minima? { What does it mean to say that a set is dense in the reals? • Differentiation { What does it mean to say that a function is differentiable at a point? { What is the linearization of a function at a point? { What is the equation of the tangent line to the graph of a function at a point? { What is the physical interpretation of the derivative? { What can be said about the derivatives of sums and constant multiples of a function/functions? { What is the product rule for differentiation? { What is the product rule for the product of more than two functions? 4 { What is the reciprocal rule for differentiation? { What is the quotient rule for differentiation? { What is the chain rule for differentiation? { What is the derivative of a function that is expressed as a composition of more than two functions? { What are higher derivatives of a function? { What is the product rule for higher derivatives of the product of two functions? { What are the derivatives of basic functions such as xn, x1=n, 1=xn, and the trigonometric functions? { What is the relationship between differentiability and continuity? { What is a maximum point, and a minimum point, of a function, on a subset of the domain? { What is the maximum value, and a the minimum value, of a function, on a subset of the domain? { What is a local maximum point, and a local minimum point, of a function, on a subset of the domain? { What is a critical point of a function? { What is the connection between local extrema and critical points? (Fermat's principle) { What is Rolle's theorem? { What is the Mean Value Theorem? { What can be said about a function whose derivative is 0 on an interval? { What can be said about two functions whose derivatives agree on an interval? { What does it mean for a function to be increasing or decreasing on an interval? { What is the connection between intervals of increase/decrease and the first derivative? { What steps does one take to sketch the graph of a function? { What are the first and second derivative tests for local maxima/minima? { What is L'H^opital'srule (in all its various forms)? { What is the Cauchy Mean Value Theorem? { [What does it mean for a function to be convex, or concave, on an interval?] { [What is the connection between intervals of convexity/concavity and the first and second derivatives?] { [What is Darboux's theorem? (the intermediate value property of derivatives)] 5 You should be able to prove things like: • (−a)(−b) = ab • ab = 0 implies one of a; b is 0 • If x; y ≥ 0, then x2 ≥ y2 is equivalent to x ≥ y • the triangle inequality • the general associative property • the binomial theorem • Bernoulli's inequality p • the irrationality of 2 • the limit, if it exists, is unique • the limit of a sum, product or quotient is the sum, product or quotient of the limits (in all its incarnations: limits at infinity, infinite limits, finite limits) • the squeeze theorem for limits • continuity of sums, products and quotients of continuous functions • if g is continuous at a and f is continuous at g(a) then f ◦ g is continuous at a • the Dirichlet function is continuous nowhere • the Stars over Babylon function (Thomae's function, the popcorn function) is continuous at all irrationals, but not continuous at any rational • a function that is continuous and positive at a point is positive on some interval around that point • a function is continuous at a point if and only if it is both left- and right-continuous at that point • every non-negative number has a unique non-negative nth root, for every even n, and every number has a unique nth root, for every odd n • the nth root functions are continuous • the Intermediate Value Theorem • N is not bounded above • N satisfies the Archimedean property 6 • 1=n can be made arbitrarily small • a function that is continuous at a point is bounded in some