This course will be oranized in an experimental fashion, which clearly separates different forms of work from each other, as well as passing requirements from letter grade criteria. More precisely, there will be three components of this course: (a) theory, (b) technical skills, and (c) more conceptual problem solving.

For instance, by completing the theory part, and passing all 3 quizzes, you'd earn at leat a B- for the course even before the final, but failing all 3 quizzes would limit your course grade by a B+ at best.

In addition to the theory of one-variable calculus, you are encouraged to study more advanced topics from our textbook (such as e.g. the topology of metric spaces) in the regime of a reading course; we don't plan however to factor this work into passing or letter grade criteria.

** Week **

Jan 27 29: Open and closed sets / Heine-Borel's thm. Compactness. Weirestrass' theorem

Feb 3 5: Limits of sequences / Subseqences, lower and upper limits

Feb 10 12: Cauchy sequences, completeness. Connectedness / Special sequences

Feb 17 19: Limits of functions / Continuity

Feb 24 26: Uniform continuity, maxima and minima / Derivatives

Mar 3 5: Ferma's, Rolle's, Lagrange's, and Taylor's thms / l'Hospital's rule. Computation of limits using Taylor's formula

Mar 10 12: Riemann integral / Properties of integrals

Mar 17 19: The fundamental thm of calculus / Techniques of integration

[Mar 24 26: Spring break]

Apr 31 2: Infinite series; Cauchy, root, ratio, integral, etc. tests

Apr 7 9: Power series

Apr 14 16: Sequences and series of functions. Uniform convergence

Apr 21 23: Equicontinuity, Arzela-Ascoli's thm / Weierstrass' approximation thm

Apr 28 30: Review / Exercises

May 5 7: (RRR week) Exercises

The introductory section of your notes should contain

This roughly corresponds to the material on pp. 24-30 of the textbook

In principle each portion of a mathematical text can be nothing but: Definition, Theorem (i.e the formulation of it), or Proof. For expositional purposes one might also add: Remark, Example (which in fact is also a definition, theorem, and proof together), Exercise (i.e. the formulation of it). So, structure your text accordingly. If a sentense you are writing doesn't fit any of these types, it better be removed.

and definition of upper / lower limit (lim sup / lim inf) of a bounded sequence.

Definition of a complete set (in R). Theorem: A subset in \R is complete if and only if it is closed. Corollary: Every Cauchy sequence in \R converges.

Definition of a connected set (in R). Theorem: A subset in \R is connected if and only if together with every two points x,y it contains the entire interval [x,y] (see section "Connectedness" in Rudin's book).

For a function f: X --> |R, x_0 a limit point of X \subset |R, and a number A, define what it means that lim_{x --> x_0} f(x) =A.

Proposition: This is equivalent to saying that for every sequence (x_n) in X\{x_0} which tends to x_0, the sequence (f(x_n)) tends to A.

Corollary about limits of sums, products, and ratios of two functions which are known to have limits as x tends to x_0.

Definition of continuity of a function f: X --> R at x_0 \in X.

Proposition': f is continuous at x_0 if and only if lim f(x_n) = f(lim x_n) for every sequence (x_n) in X converging to x_0.

Equivalence of four definitions of continuous functions X --> |R (in terms of \epsilon-\delta, limits of sequences, closed sets, open sets).

Corollary: Continuity of sums, products, and ratios of continuous fnctions.

Cotinuity of the composition of continuous functions.

Examples: Continuity of constant, linear, polynomial, rational functions; continuity of arithmetic operations.

Theorem. A continuous image of a compact set is compact.

Corollary. A continuous real-valued function on a compact set achieve its maximum and minimum values.

Corollary. A continuous fucntion f: [a,b] --> |R achieves its maximum and minimum values.

Theorem. A continuous function on a compact set is uniformly continuous.

Theorem. A continuous image of a connected set is connected.

Corollary. A continuous real-valued function on a connected set between any two values assumes all intermediate ones.

Corollary. The range of a continuous function f: [a,b] --> |R is the closed interval [m,M] between the minimum and maximum values.

Corollary. A continuous f: [a,b] --> |R has a zero on (a,b) if f(a) and f(b) have opposite signs.

Definition of the derivative at a point.

Differentiability at a point implies continuity at this point.

Formulas for the derivatives of the sum, product, ratio, and composition of two differentialble functions.

Example of a function |R --> |R differentiable everywhere, whose derivative is discontinuous at the origin.

Definition of global and local maximum and minimum.

Fermat's, Rolle's, Lagrange's and Taylor's theorems.

Generalized Mean Value Theorem, l'Hospital's Rule, Taylor's formula with remainder in the form of Cauchy, an example of application to computation of limits.

Here is the practice set on limits of functions , and the answers are:

a/b, ln 2, m/n, mn(n-m)/2, n(n+1)/2, 1/2, 1, 1/4, 1, 17/5

Quiz on this matrial is scheduled for Th, April 2.

Notions of partition of [a,b] , lower and upper Riemann sum L(P,f), U(P,f), lower and upper integral as sup_P L(P,f) and inf_P U(P,f) respectively. Riemann integrability (as equality between lower and upper integrals).

Proposition about increase of L(P,f) and decrease of U(P,f) under refinement of partitions, the inequality L(P_1, f)\leq U(P_2,f) for arbitrary partitions P_1, P_2, and (in cosequence) the respective inequlity between the lower and upper integrals.

The Riemann integrability criterion: Existence for every \epsilon>0 of a patition P such that U(P,f)-L(P,f) < \epsilon.

Theorems about Riemann intgrability of functions: (a) continuous on [a,b], (b) bounded functions having at most finitely many discontinuity points, (c) monotone on [a,b].

Let f, f_1,f_2 be Riemann-integrable on [a,b], then

f_1+f_2 and cf are (and the repective equalities for the integrals hold true);

if f_1\leq f_2, then \int f_1 dx \leq \int f_2 dx

if a < c < b, then f is Riemann-integrable on [a,c], [c,b], and the integral is additive with respect to the interval of integration.

Theorem about Riemann-integrability of the composition of continuous function with a Riemann-integrable one, and the corollaries about integrability of the product of two Rimann-integrable functions, and of the absolute value of a Riemann-integrable function.

Change of variables in Riemann integrables.

The Fundamental theorem of calculus (if F'=f \in R[a,b], then \int_a^b fdx = F(b)-F(a)).

Another form: if f \in R[a,b], then F(x):=\int_a^x fdx is continuous, and d F/dx (x) = f(x) at every continuity point x of f.

Corollary: integration by parts.

Here is the practice set on integration (and here are the answers )

Definition of convergent series. The Cauchy criterion of convergence.

\lim a_n =0 as a ncessary condition for convergence for \sum a_n, but not sufficient

The integral criterion (of convergence of series of decreasing positive terms).

Definition of absolutely convergent series.

Thm: An absolutely convergent series converges, and its sum does not change under reordering.

Theorem about (sum and) product of absolutely convergent series, and its application to power series.

The multiplicative property of the exponential function.

Here is the practice set on series. (

Counter-examples (from the opening section of Chapter 7 in Rudin's book) to independence of double limits on the order, to conservation of continuity, differentiability, integrability properties under passing to the limit, and to the possibility of termwise differentiation and integration of the sum of a series of functions.

Existence of continuous nowhere differeniable functions |R --> |R.

For every continuous function on [a,b], there is a sequence of polynomials which tends to this function uniformly on [a,b].

THIS MATERIAL CONCLUDES THE THEORETICAL COMPONENT OF OUR COURSE

Here is the set of problems for our semminar discussions on April 30, May 5, and May 7.