| lectures: CC 246, MWF 10:30 | |
| recitation: BE 184, TR 10:30 or 11:30 |
| Lecturer: Sasha Leibman | office: MW406 office hours: Wednesday 2-3:30, Thursday 3-4:30 |
| e-mail: leibman@math.ohio-state.edu | |
| phone: 292-0663 |
| Recitation: Dr. David Ralston |
Textbook: A. Browder, Mathematical Analysis, an introduction.
Exercises: Set 1, Set 2, Set 3, Set 4, Set 5, Set 6, Set 7, Set 8, Set 9, Set 10
Homework:
| HW1: | Due on Thursday, October 1 -- | Exercises 2, 4, 7, and 11 from Set 2 |
| HW2: | Due on Friday, October 9 -- | Exercises 10, 13, and 15(abc) from Set 3 |
| HW3: | Due on Friday, October 16 -- | Exercises 5, 8, 9, and 12 from Set 4 |
| HW4: | Due on Friday, October 23 -- | Exercises 1 and 3 from Set 5 and exercises 14 and 21 from Browder, p.52 |
| HW5: | Due on Friday, November 6 -- | Exercises 1 and 8 from Set 7 |
| HW6: | Due on Friday, November 13 -- | Exercises 4, 5 and 10 from Set 8 |
| HW7: | Due on Friday, November 20 -- | Exercises 4, 10 and 14 from Set 9 |
| HW8: | Due on Wednesday, November 25 -- | Exercises 6, 8 and 15 from Set 10 |
Solutions to some problems from: Set 2, Set 3, Set 4, Set 5, Set 6, Set 7, Set 8, Set 9
Studied topics:
| November 23: |
Sin and cos functions.
Darboux' theorem (intermediate value theorem for derivatives). (Browder, p.84) |
| November 20: |
Rolle's and Lagrange's mean value theorems.
Relation between the increase/decrease of a function on an interval
and the sign of its derivative ont his interval.
(Browder, pp.83-84)
Sin and cos functions. |
| November 18: |
"Calculus" - the derivative of the sum, product, composition of functions.
The derivative of the inverse function.
(Browder, pp.81-82)
Relation between the increas/decrease/extremality of a function at a point and the sign of its derivative at this point. (Browder, p.83) |
| November 16: | The derivative of a function at a point - several equivalent definitions (Browder, 4.1) |
| November 13: |
Uniformly continuous functions. (Browder, p.61)
Continuous function on a bounded closed interval is uniformly continuous. (Browder, p.61) |
| November 9: |
Increasing functions. Cantor's ladder. Inverse of an increasing function.
Intermediate value theorem. (Browder, p.59) Continuous function on a bounded closed interval attains its maximum and minimum. (Browder, p.59) |
| November 6: |
Properties of functions continuous at a point and on a set. (Browder, pp.57-58)
Limits and discontinuities of monotone functions. (Browder, p.58) |
| November 4: |
Limit of a compositin of two functions.
Cauchy criterion for existence of limits. Functions, continuous at a point. (Browder, p.57) Classification of discontinuities. |
| November 2: |
Limit of a function at a point
(epsilon-delta and sequential definitions) (Browder, p.55)
Elementary properties of limits. |
| October 28: |
Rearrangement of series (Browder, 2.4)
Cauchy product of series (Browder, pp.49-50) |
| October 26: | Abel's summation by parts formula. Dirichlet's and Abel's tests. (Browder, p.44) |
| October 23: |
Cauchy condensation test. (Browder, pp.42-43)
Series with both positive and negative terms. Conditional convergence. Alternating series test. (Browder, pp.43,45) Abel's summation by parts formula. (Browder, p.44) |
| October 21: |
Comparison test and its corollaries. (Browder, p.40)
Root and ratio tests. (Browder, pp.41-42) Cauchy condensation test. (Browder, pp.42-43) |
| October 19: |
Infinite series. (Browder, 2.3)
Cauchy criterion for series. Series with nonnegative terms. (Browder, p.40) Absolute convergence. (Browder, p.40) Comparison test. (Browder, p.40) |
| October 16: |
limsup and liminf as the maximal and the minimal limit points. (Browder, pp.34-35)
Cauchy criterion of convergence. (Browder, p.36) |
| October 14: |
Subsequences. (Browder, p.34)
Limit points and subsequential limits of a sequence. Bolzano-Weierstrass theorem. (Browder, p.35) limsup and liminf. (Browder, pp.34-35) |
| October 12: |
Elementary properties of limits, squeeze theorem. (Browder, pp.28-30)
Standard limits. (Browder, pp.30-31) Monotone sequences. Number e. (Browder, pp.33-34) |
| October 9: |
x^(1/n) (Browder, p. 18)
Sequences, limits and their elementary properties. (Browder, pp.28-30) |
| October 7: |
Expansion of real numbers in "base b" numerical system. (See Browder, example 2.7)
The "continued fractions" representation of real numbers. (See Browder, 2.2) Functions on R: absolute value, linear functions, x^n, x^(1/n). (Browder, 1.6) |
| October 5: |
Intervals in R.
Intersection of a nested sequence of bounded closed intervals
is nonempty. (Browder, 1.7)
R is uncountable. Irrational, algebraic and transcendental numbers. (Browder, 1.7) Expansion of real numbers in "base b" numerical system. |
| October 2: |
The set of rational numbers Q is not complete. (Browder, p.14)
Existence of R. (Browder, 1.9) Extended set of real numbers, sup and inf. (Browder, p.15) Archimedian property of R. Q is dense in R. (Browder, pp.15-16) |
| September 30: |
The set of infinite 0,1-sequences is uncountable.
Upper and lower bounds, supremum and infimum. Complete ordered sets. Definition of the set of real numbers. (Browder, 1.5) |
| September 28: | Countable sets. The set of rational numbers is countable. (Browder, 1.3) |
| September 25: |
Equivalence relations.
Integers and rational numbers as equivalence classes. (Browder, 1.2)
Ordered fields. (Browder, 1.5) Root of 2 is not rational. (Browder, 1.4) Finite sets. Countable sets. (Browder, 1.3) |
| September 23: |
Natural numbers. Induction. (See Browder, beginning of section 1.2)
Relations. Equivalence relations. (Browder, section 1.1) |