Math 10860, Honors Calculus 2 Topics covered May 4, 2018 The final will be on Wednesday May 9, 4.15pm-6.15pm in 107 Pasquerilla Center (the usual classroom). The exam will be a mixture of definitions & theorem statements, things to prove, and problems. Here is a little more detail on that; I will endeavour not to throw curveballs, and to base the exam predominantly on what is discussed in points 1, 2 and 3 below. I have, however, also included a list of all the topics we have covered this semester, and for at least a small portion of the exam, everything from that list is fair game. 1. Definitions and Theorem statements: Every concept we discussed this semester had a formal definition, and every theorem had a precise statement, with necessary hypotheses. I might ask a question that, in part, asks for a complete and correct statement of: • Def. of Darboux integral • The fundamental theorems of calculus • Continuity and differentiability of the inverse • The inverse of a continuous function on an interval • Def. of exponential, log and trigonometric functions (really? Def. of trigonometric functions again? Wasn't that on the midterm? OK, so let me revise this to just def. of exponential, log) • Def. of the Taylor polynomial, and Taylor's theorem with intergral and/or Lagrange remainder • Def. of convergence of a sequence 1 • Bolzano-Weierstrass theorem • Def. of Cauchy sequence • Def. of summability, • The Cauchy criterion for summability, the vanishing condition, the bound- edness criterion, the comparison test, the limit comparison test, the ratio test, the integral test, Leibniz' theorem on alternating series • Def. of absolute convergence and conditional convergence • Def. of pointwise and uniform convergence of sequences of functions • Def. of summability and uniform summability of sequences of functions • The properties of uniform limits with regards continuity, integrability and differentiability • The Weierstrass M-test • Def. of power series, Taylor series. Maclarin series • Def. of uniform continuity 2. Things to prove: There will be some questions that ask you to provide proofs of facts that we have proven in class, or were straightforward enough to be left as exercises in class. I promise that those questions will come from this list: • If Q is a partition that includes all the points of P , and some more, then L(f; P ) ≤ L(f; Q) ≤ U(f; Q) ≤ U(f; P ) • If f : R ! R is differentiable and f 0(x) = f(x) for all x, then there is a constant c such that f(x) = cex for all x • If two polynomials of degree n agree at a point a to order n, they are the same polynomials • If a sequence converges to a limit, that limit is unique P P P • If an converges to limit `a and bn converges to limit `b then (an +bn) converges to limit `a + `b • Every sequence has a monotone subsequence • The ratio test for convergence of series • The comparison and limit comparison tests for convergence of series • Absolute convergence implies convergence • Uniformly continuous functions on a closed interval are integrable 2 3. Problems: The exam will also have a few problems: computations, such as evaluating an integral, or calculating a Taylor polynomial, determining whether an improper integral, sequence or series converges or not, proofs of small propositions that follow from theorems that you know. I will try very hard to resist the urge to litter the exam with what I might consider \nice" problems that could take half-a-day to figure out; for determining what problems to pose, I will draw heavily for inspiration on the homeworks and quizzes. Here is a complete list of the topics that we have covered this semester. • Partitions, Darboux sums, definition of (Darboux) integral • U(f; P ) − L(f; P ) < " criterion for integrability • Example of integrable function with infinitely many discontinuities • Splitting the interval of integration, linearity of integral, upper and lower bounds on integral in terms of upper and lower bounds on function • Integrability closed under sums, scalar multiplication, multiplication, changing at finitely many values, taking absolute value, taking max, min, positive and negative parts, Cauchy-Schwarz inequality • Integrability of continuous functions • The fundamental theorems of calculus, parts 1 and 2 R x • Defining functions via F (x) = a f, and manipulating the results (especially differentiating) when simple limits replaced by functions • Improper integrals over infinite ranges, improper integrals of unbounded func- R 1 α R 1 α tions, comparison test for improper integrals, status of 0 1=t dt and 1 1=t dt • One-to-one and invertible functions, graphs of inverse functions • Continuous one-to-one functions on an interval are monotone, determining range (so domain of inverse) • Continuity and differentiability of inverse functions • Defining and differentiating f(x) = xa for rational a • Defining the logarithm function via an integral, and the exponential function as its inverse, basic properties of exp and log (algebraic identities, derivatives), definition of e, ex, ax • Characterizing exp as solution to a differential equation 3 • Exponential functions grow faster than polynomial functions • Informal (geometric) definition of sin; cos • Definition of π, formal definitions of cos, sin via integration, basic properties of the trigonometric functions (identities, derivatives) • Characterizing sin; cos as solutions to a differential equation, angle addition formulae • The hyperbolic functions | definition, explicit formulae, basic properties All of the above was the material covered up to the first midterm. • Primitives, or antiderivatives, definition and notation, unique up to additive constants, linearity • Common primitives (log; exp, trigonometric, hyperbolic and related) • Integration by parts, definite and indefinite versions, reduction formulae • Integration by substitution, general principle, definite and indefinite versions, using trigonometric substitutions • Integrating products of powers of sin and cos, half-angle formulae • Substitution t = tan(x=2) • Method of partial fractions • Taylor polynomial of a function about a point, agreement to order n at a point, uniqueness of the Taylor polynomial • Finding Taylor polynomial via expanding a series for derivative and then inte- grating, e.g., tan−1, log • Taylors theorem with integral remainder formula, proof via integration by parts, derivation of Lagrange remainder form assuming continuity of (n+1)st derivative • Analyzing remainder term of Taylor series for exp, sin, cos, log, tan−1, estimating functions at points • Taylor's theorem with Lagrange remainder term via Cauchy Mean Value Theorem • Sequences, convergence, convergence to 1 and −∞ • Limits of sums, products, reciprocals scalar multiples of sequences, limits of rational functions 4 • Interaction between limits of sequences and limits of functions, limit of (an), limit of a function of a sequence • Bounded and monotone sequences, every sequence has a monotone subsequence, Bolzano-Weierstrass theorem, Cauchy sequences All of the above was the material covered up to the second midterm. • Series, summability, divergence of series, the harmonic series, the geometric series • Closure of summable sequences under linear combinations, Cauchy criterion for summability, the vanishing condition, the boundedness criterion, the comparison test, the limit comparison test, the ratio test, the integral test, Leibniz' theorem on alternating series • Absolute and conditional convergence, absolute convergence implies conver- gence, rearrangements of absolutely convergent sequences, rearrangements of conditionally convergent sequences • Pointwise convergence of functions to limit, nuisance examples showing pointwise limit not well-behaved • Uniform converge of functions on an interval to limit, preservation of continuity, limit of integral is integral of limit, conditions under which derivative of limit is limit of derivatives • Uniform convergence/summability of sequences of functions, Weierstrass M-test for uniform summability • Power series, Taylor series, Maclaurin series, uniform convergence of power series below any point of convergence, radius of convergence, integrating and differentiation term-by-term inside radius of convergence, Binomial series • Uniform continuity, uniformly continuous functions of closed interval are inte- grable, continuous functions on closed interval are uniformly continuous 5.