Course Syllabus

Weekly homeworks

First Test: Thursday October 12,

 

1) Axioms of fields:
   Examples: R,Q, {0,1}
   Axioms of  ordered fields:
   Examples: R,Q 
  
2) Completeness Axiom
    (two versions)
3) Definition of Countable sets,
   Examples: N,Z,NxN, Q
   Example of uncountable set: R
4) Definition of the limit of a sequence ,
5) Algebraic rules for limits (sum, product etc)
6) Defintion of Cauchy sequences
7) Definition of sup and inf
8) Definition of a limit point of a sequence
9) Theorem: a point is the limit point of a sequence
            iff it is the limit of a subsequence.         
10) Bolzano Weierstrass
11) Definition of "increasing" and "decreasing" sequences
12) Theorem: If a sequence is  monoton and bounded 
             then it is convergent.

   

Second Test: Tuesday November 21,

 

1)  Definition of continuity 
    a) using sequences
    b) "eps-delta" definition 
2)  Definition of uniform continuity
3)  Properties of continuous functions  f on 
    closed and bounded intervals I (and
    the proofs):
    a) f is bounded on I
    b) f assumes its inf/sup on I
    c) f is uniform continuous on I.
    d) f is Riemann integrable on [a,b]
4)  Intermediate Value Theorem
5)  Definition of "f is Riemann integrable on [a,b]".
    Definition of the Riemann integral of f over [a,b].   
6)  Examples of functions defined on [a,b] which
    are not Riemann integrable
7)  Improper Riemann integrals (two kinds).
8)  Definition of derivative of a function.
9)  Rules of differentiation and their proofs
    a) sums differences
    b) product rule
    c) chain rule  
    

Final Examination: Monday December 11, 1-3

    See  First and Second Test, and beyond that:

 1) Mean Value Theorem/Theorem of Rolle  and their proof.
 2) Fundamental Theorem of Calculus (I/II) and its proof.
 3) Taylor polynomial, and Taylor's theorem and its proof.
 4) Def. of strictly monoton. Relation between
    "f is strictly monoton increasing/decreasing"
    "f is  monoton increasing/decreasing"  and
    properties of derivative
 5) Inverse functions and their derivatives