MTWRF 9:10-10:05 at BO (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 12/9
Handouts:
Examples of integration
e and π are transcendental
The sum of ∑1/n2
by Yile Huang.
(This is a very nice solutions of the so-called Basel's problem.
It however uses improper two-dimensional integration
and is beyond the scope of our course.)
Exams:
Midterm 1 –
solutions
[topics,
review problems –
solutions]
Midterm 2 –
solutions
[topics,
review problems –
solutions]
Midterm 3 –
solutions
[topics,
review problems –
solutions]
Midterm 4 –
solutions
[topics,
review problems –
solutions]
Final exam
is scheduled for December 12, 8am-10am, and December 15, 10am-12pm, –
you may choose any of these two exams.
It will be comprehensive,
but focused mainly on the last, "more advanced" topics of the course.
Here is a list of
review problems –
solutions
covering the last 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 |
| Homework 8 | due by Wednesday, November 12 – solutions |
| Homework 9 | due by Tuesday, November 25 – solutions |
| Homework 10 | due by Tuesday, December 9 – 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 30: |
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) |
| October 31: | Review problems. |
| November 3: | Midterm 3. |
| November 4: | Riemann integrable functions (Lecture notes 6.1, Spivak Ch.13) |
| November 5: |
Riemann integrable functions (Lecture notes 6.1, Spivak Ch.13)
Properties of integral (Lecture notes 6.1, Spivak Ch.13, pp.267-269 and ex.38 on p.281) |
| November 6: |
Mean value theorems for integrals (Lecture notes 6.3, Spivak ex.23 on p.277)
Integral inequalities (Lecture notes 6.4, Spivak p.270 and ex.39 on p.281) |
| November 7: |
The fundamental theorem of calculus (Lecture notes 6.5, Spivak Ch.14 pp.285-289)
Technique of integration: integration by parts and substitutions (Lecture notes 6.6, Spivak Ch.19) |
| November 10: |
Technique of integration (Lecture notes 6.6, Spivak Ch.19)
Improper integrals (Lecture notes 6.7, Spivak ex.39 and 43 on pp.394, 397) |
| November 12: |
Improper integrals (Lecture notes 6.7, Spivak ex.39 and 43 on pp.394, 397)
The Gamma function (Lecture notes 6.7, Spivak ex.40 on pp.394) Absolutely convergent improper integrals (Lecture notes 6.7, Spivak ex.39 and 43 on pp.394, 397) |
| November 13: |
Rectifiable curves, the arc-length of a curve (Lecture notes 6.8, Spivak ex.25 on p.277)
Taylor polynomials, the Taylor-Maclaurin formula (Lecture notes 7.1, Spivak Chapter 20) |
| November 14: |
Operations on functions and their Taylor polynomials
(Lecture notes 7.2, Spivak ex.9 on pp.433-434)
Taylor polynomials of basic functions (Lecture notes 7.3, Spivak pp.424-428) |
| November 17: |
Application of Taylor polynomials:
calculation of indeterminate limits, finding derivatives,
determining the local behavior of functions (Lecture notes 7.4, Spivak pp.417-418) The remainder in Taylor's formula (Lecture notes 7.5, Spivak pp.421-424 and exercise 19 on p.436) |
| November 18: | Review. |
| November 19: | Midterm 4. |
| November 20: |
The Taylor remainders for exp, sin, cos, log (Lecture notes 7.5, Spivak pp.424-428)
e is irrational (Lecture notes 7.5, Spivak pp.424, 429) Series: partial sums, convergence (Lecture notes 8.1, Spivak Ch.22) The Cauchy criterion for series (Lecture notes 8.1, Spivak p.473) Series with nonnegative terms (Lecture notes 8.2, Spivak p.474) |
| November 21: |
The comparison, limit comparison, root, ratio, integral, and condensation tests
(Lecture notes 8.2, Spivak pp.474-479 and ex.9,23 on p.493,496)
Absolute and conditional convergence of series (Lecture notes 8.1, Spivak pp.480-481) |
| November 24: |
Leibniz's alternating series test (Lecture notes 8.3, Spivak pp.480-482)
Abel's summation by parts formula (Lecture notes 8.3, Spivak ex.36 on p.392) Dirichlet's and Abel's tests (Lecture notes 8.3, Spivak ex.22 on p.495) Groupings of series (Lecture notes 8.4) |
| November 25: |
Rearrangements of series, Riemann's theorem
(Lecture notes 8.4, Spivak pp.483-486)
Double series. Unordered sums (Lecture notes 8.5) |
| December 1: |
Cauchy's product of series (Lecture notes 8.5, Spivak pp.486-488)
Infinite products (Lecture notes 8.6, Spivak ex.28 on p.497) |
| December 2: |
Pointwise and uniform convergence of functional sequences
(Lecture notes 9.1, Spivak pp.499-504)
"The good properties" of the uniform limits of functional sequences (Lecture notes 9.1, Spivak pp.505-506) |
| December 3: |
Uniformly convergent functional series (Lecture notes 9.2, Spivak p.506)
Absolute uniform convergence of functional series. The Weierstrass M-test (Lecture notes 9.2, Spivak pp.507-508) |
| December 4: |
Examples of converging functional series (Lecture notes 9.2, Spivak p.508)
Power series; radius and interval of convergence (Lecture notes 10.1, Spivak pp.510-511) |
| December 5: | Operations on power series (Lecture notes 10.1, Spivak pp.511-513) |
| December 8: |
Taylor series (Lecture notes 10.2, Spivak pp.515-516)
Analytic functions (Lecture notes 10.3) |
| December 9: |
Rigidity of analytic functions (Lecture notes 10.3, Spivak ex.13 on p.520)
Abel's theorem (Lecture notes 10.4, Spivak ex.19 on p.522) |
| December 10: | Review problems. |
Plans for the nearest future:
Final exam on Monday December 15, 10am-12pm (at BO 124)