MTWRF 9:10-10:05 at BP (Bolz Hall) 124
| Instructor: Sasha Leibman | office: MW406 office hours: Monday 12-1pm, Thursday 10:10-11am, and by appointment |
| e-mail: leibman.1@osu.edu |
Textbook: M. Spivak Calculus, 4th edition
Mentoring:
Two mentors (two experienced students) Adarsh and Levi
will be helping you with this course and grading your homework.
Meetings with them will be held
Thursdays from 4 to 6 p.m. and Sundays from 2 to 4 p.m.
in Cockins Hall 240.
Lecture notes, version of 10/29
Exams:
Midterm 1 –
solutions
[topics,
review problems –
solutions]
Midterm 2 –
solutions
[topics,
review problems –
solutions]
Midterm 3 is going to be on Monday, November 3.
It will be devoted to differentiation and mean value theorems:
topics,
review problems
(Midterm 4 is planned to be around November 17 and devoted to integration and Taylor polynomials.)
(Final is scheduled for December 12 or 15,
will be comprehensive, but focused mainly on the last, "more advanced" topics of the course.)
Homework:
| Homework 1 | due by Tuesday, September 2 – solutions |
| Homework 2 | due by Tuesday, September 9 – solutions |
| Homework 3 | due by Tuesday, September 16 – solutions |
| Homework 4 | due by Tuesday, September 30 – solutions |
| Homework 5 | due by Tuesday, October 7 – solutions |
| Homework 6 | due by Tuesday, October 14 – solutions |
| Homework 7 | due by Tuesday, October 28 – solutions |
Studied topics:
| August 26: |
Logical and set-theretical notations, mappings, types of proofs (Lecture notes 0.2-0.5)
Axioms of real numbers (Lecture notes 1.1, Spivak pp.1-11 and 135) |
| August 27: |
Axioms of real numbers (Lecture notes 1.1, Spivak pp.1-11 and 135)
Some elementary properties of real numbers (Lecture notes 1.2, Spivak pp.1-11) |
| August 28: |
Some elementary properties of real numbers (Lecture notes 1.2, Spivak pp.1-11)
The order relation ">" on R (Lecture notes 1.3, Spivak pp.10-11) The absolute value (Lecture notes 1.4, Spivak pp.11-12) |
| August 29: |
The absolute value (Lecture notes 1.4, Spivak pp.11-12)
Squares and square roots of real numbers (Lecture notes 1.5, Spivak p.12) |
| September 2: |
Existence of square roots (Lecture notes 1.5)
The arithmetic–geometric mean inequality and the Cauchy-Schwarz inequality (Lecture notes 1.6, Spivak ex.7 on p.15, ex.18,19 on pp.17-18) |
| September 3: |
The triangle inequality in the plane (Lecture notes 1.6)
Natural numbers and the principle of induction (Lecture notes 1.7, Spivak pp.21-26 and ex.25 on p.24) The Archimedian property of N (Lecture notes 1.7, Spivak pp.138-139) |
| September 4: |
Bernoulli's inequality and its corollaries (Lecture notes 1.7, Spivak ex.19 on p.32)
Natural numbers are closed under addition and multiplication (Lecture notes 1.7) Inductive definitions, powers, finite sums and products (Lecture notes 1.7, Spivak pp.21-26) "Modified" induction principle (Lecture notes 1.7) |
| September 5: |
Binomial coefficients and the binomial formula (Lecture notes 1.7, Spivak ex.3 on p.27)
Integers (Lecture notes 1.8, Spivak p.25) |
| September 8: |
Divisibility of integers, coprime and prime integers
(Lecture notes 1.8, Spivak ex.17 on p.31)
The principle of complete induction (Lecture notes 1.7, Spivak p.23) |
| September 9: |
The fundamental theorem of arithmetic (Lecture notes 1.8, Spivak ex.17 on p.31)
The well ordering of N (Lecture notes 1.7, Spivak p.23) |
| September 10: |
Rational numbers (Lecture notes 1.10, Spivak pp.25-26)
Irrationality of √2 (Lecture notes 1.10, Spivak pp.25-26) Intervals in R, infinite points, limit points, dense sets (Lecture notes 1.9) Denseness of Q in R (Lecture notes 1.10, Spivak ex. 5 on p.140) |
| September 11: |
Another proof of irrationality of √2 (Lecture notes 1.10)
The extended real line (Lecture notes 1.9) Supremum and infimum (Lecture notes 1.11, Spivak p.134) |
| September 12: |
Supremum and infimum (Lecture notes 1.11, Spivak p.134)
The nested intervals principle (Lecture notes 1.12, Spivak ex.14 on p.142) Binary expansion of real numbers (Lecture notes 1.12, cf. Spivak ex.7 on p.492) |
| September 15: |
Base d numerical systems (Lecture notes 1.12, cf. Spivak ex.7 on p.492)
Existence of R (Lecture notes 1.13, Spivak Ch.29) |
| September 16: |
Uniqueness of R (Lecture notes 1.13, Spivak Ch.29)
Cardinality of infinite sets, countable and uncountable sets (Lecture notes 1.14, Spivak pp.449-450) |
| September 17: |
Cantor's theorem (Lecture notes 1.14, Spivak pp.449-450)
Algebraic and transcendental numbers (Lecture notes 1.14, Spivak pp.449-450) |
| September 18: |
Cantor's set (Lecture notes 1.14)
Review problems. |
| September 19: | Review problems. |
| September 22: | Midterm 1. |
| September 23: |
Limits of sequences (Lecture notes 2.1, Spivak Ch.22)
Properties of converging sequences and their limits (Lecture notes 2.2, Spivak Ch.22) The squeeze theorem (Lecture notes 2.2, Spivak Ch.22) |
| September 24: |
Arithmetic properties of limits (Lecture notes 2.2, Spivak Ch.22)
Infinite limits (Lecture notes 2.2, Spivak Ch.22) |
| September 25: |
Some standard limits (Lecture notes 2.3, Spivak Ch.22)
Monotone sequences (Lecture notes 2.4, Spivak Ch.22) |
| September 26: |
Number e (Lecture notes 2.4, Spivak Ch.22)
Newton's algorithm for computing √ (Lecture notes 2.4, Spivak Ch.22) Cauchy criterion of convergence (Lecture notes 2.5, Spivak Ch.22) |
| September 29: |
Subsequences and limit points (Lecture notes 2.6, Spivak pp.458,467)
The Bolzano-Weierstrass theorem (Lecture notes 2.6, Spivak p.458) |
| September 30: | limsup and liminf (Lecture notes 2.7, Spivak p.467) |
| October 1: |
Properties and application of limsup and liminf
(Lecture notes 2.7, Spivak p.467 and problems 16,18 on p.465)
The limit of a function at a limit point of its domain (Lecture notes 3.1, Spivak pp.90-99) The sequential definition of the limit of a function (Lecture notes 3.1, Spivak Theorem 1 on pp.456-457) |
| October 2: | Properties of limits: uniqueness, inequalities, arithmetical properties (Lecture notes 3.2, Spivak pp.102-104) |
| October 3: |
Limit of the composition (Lecture notes 3.2)
The squeeze theorem (Lecture notes 3.2, Spivak pp.102-104) Variants of limits: infinite, at infinity, one-sided (Lecture notes 3.3-3.4, Spivak p.106) |
| October 6: |
Functional limits of all kinds (Lecture notes 3.3-3.4, Spivak p.106)
Limits of a monotone function (Lecture notes 3.5) |
| October 7: |
The Cauchy criterion for functional limits (Lecture notes 3.6)
Functions, continuous at a point (Lecture notes 4.1, Spivak Ch.6) Properties of functions continuous at a point (Lecture notes 4.1, Spivak Ch.6) |
| October 8: |
Kinds of discontinuity (Lecture notes 4.2)
Discontinuities of a monotone function (Lecture notes 4.2, Spivak Ch.6) |
| October 9: | The intermediate value theorem and its corollaries (Lecture notes 4.3, Spivak Ch.7) |
| October 10: |
Max/min values of a function continuous on a closed bounded interval
(Lecture notes 4.4, Spivak pp.122-124 and 135-138)
Uniformly continuous functions (Lecture notes 4.5, Spivak pp.144-146) |
| October 13: |
Uniformly continuous functions (Lecture notes 4.5, Spivak pp.144-146)
Extension of a uniformly continuous function by continuity (Lecture notes 4.5, Spivak pp.144-146) |
| October 14: | The exponential, logarithmic, and power functions (Lecture notes 4.6) |
| October 15: |
Functions translating addition or multiplication to addition or multiplication
(Lecture notes 4.6)
Review problems. |
| October 20: | Midterm 2. |
| October 21: |
The derivative of a function at a point (Lecture notes 5.1, Spivak Ch.9)
Continuity and algebraic properties of derivatives (Lecture notes 5.1, Spivak Ch.9) The derivative of the composition (Lecture notes 5.1, Spivak Ch.9) |
| October 22: |
The derivative of the inverse function (Lecture notes 5.1, Spivak Ch.9)
The derivatives of basic functions (Lecture notes 5.1, Spivak Ch.9) |
| October 23: | Convex functions (Lecture notes 5.2, Spivak Appendix after Ch.11) |
| October 24: |
Natural exponential and logarithmic functions (Lecture notes 5.3, Spivak Ch.15)
Local properties of differentiable functions (Lecture notes 5.4, Spivak ex.69 on p.217) Extremal points (Lecture notes 5.4, Spivak pp.188-192) The mean value theorem (Lecture notes 5.5, Spivak pp.193-194) |
| October 27: |
The mean value theorems (Lecture notes 5.5, Spivak pp.193-194)
Analysing the behavior of a function given its derivative (Lecture notes 5.6, Spivak Ch.11) Limits of the derivative (Lecture notes 5.6, Spivak p.203) |
| October 28: |
Darboux's theorem (Lecture notes 5.6, Spivak ex.60 on p.214)
Trigonometric functions (Lecture notes 5.7, Spivak Ch.15) |
| October 29: |
Trigonometric functions (Lecture notes 5.7, Spivak Ch.15)
Functions with discontinuous or non-differentiable derivatives (Lecture notes 5.7, Spivak Ch.15) |
| October 29: |
Higher order derivatives (Lecture notes 5.8, Spivak pp.161-163)
L'Hospital's rules (Lecture notes 5.9, Spivak pp.204-205 and ex.54-56 on pp.213-214) |
Plans for the nearest future:
Review problems.
Midterm on Monday, November 3.