Math 10860, Honors 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 • The fundamental theorems of calculus • Continuity and differentiability of the inverse • The inverse of a continuous on an • 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

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 comparison test, the , the integral test, Leibniz’ theorem on alternating • Def. of absolute convergence and conditional convergence • Def. of pointwise and of 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 , . 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 , 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 ) < ε 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 over infinite ranges, improper integrals of unbounded func- R 1 α R ∞ 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 as its inverse, basic properties of exp and log (algebraic identities, ), 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 , definition and notation, unique up to additive constants, linearity

• Common primitives (log, exp, trigonometric, hyperbolic and related)

• , 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 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

• Sequences, convergence, convergence to ∞ 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), 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, of series, the harmonic series, the

• Closure of summable sequences under linear combinations, Cauchy criterion for summability, the vanishing condition, the boundedness criterion, the comparison test, the , the ratio test, the integral test, Leibniz’ theorem on

• 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,

• Uniform continuity, uniformly continuous functions of closed interval are inte- grable, continuous functions on closed interval are uniformly continuous

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