Dror Bar-Natan: Classes: 2003-04: | FEEDBACK |

**Agenda:** Understand calculus and its rigorous foundations.

**Instructor:** Dror Bar-Natan, drorbn@math.toronto.edu,
Sidney Smith 5016G, 416-946-5438. Office hours: Thursdays 12:30-1:30.

**Classes:** Tuesdays 10-11 and Thursdays 9-11 at Mining
Building 128.

**Tutorials:** Mondays at 2-4, in three groups divided by the last
non-zero digit of your student number:

digit | tutor | place | |

1-3 | Vicentiu Tipu, vtipu@math.toronto.edu, SS 4055, 8-4794 | RW 142 | |

4-6 | Ching-Nam Hung, cnhung@math.toronto.edu, SS 4052, 8-3484 | LM 157 | |

7-9 | Cristian Ivanescu, cristian@math.toronto.edu, SS 6027, 8-2095 | UC 328 |

# | Week of ... | |

Fall Semester: | ||

1 | September 8 | First week of classes - no tutorials!
Handout: About This Class Homework Assignment 1 Class notes for Tuesday September 9, 2003 (degrees and radians, the basic trigonometric functions, basic identities) Class notes for Thursday September 11, 2003 (rotating coordinate systems, sin and cos of sums of angles, double and half angles, products of sin's and cos's, a word about AM modulation) |

2 | September 15 |
Handout:
The 13 Postulates
Homework Assignment 2 Class notes for Tuesday September 16, 2003 (some key points this course is about, the first 9 postulates and some consequences thereof) Class notes for Thursday September 18, 2003 (more consequences of P1-P9, 1+1=0 on Vulcan, P10-P12, consequences of P10-P12) |

3 | September 22 |
We took our
Class Photo
this week!
Homework Assignment 3 Class notes for Tuesday September 23, 2003 (the triangle inequality, the definition of the natural nnumbers and the axiom of induction) Class notes for Thursday September 25, 2003 (more on induction, irrationality of sqrt(2), every set of natural numbers has a minimal member, recursive definitions) |

4 | September 29 |
Dror: This week's office hours are on Friday 9:30 - 10:30.
Homework Assignment 4 Class notes for Tuesday September 30, 2003 (Equivalent formulations of the induction principle and the notion of a function) Handout: Visualization Handout: JHU's Zucker on Math Courses Class notes for Thursday October 2, 2003 (the set-theoretic definition of a function, function composition and properties of compositions, graphs and visualization) |

5 | October 6 |
Homework Assignment 5
Class notes for Tuesday October 7, 2003 (23x27=27x23, (-3)x(-5)=3x5, the pincushion function, distances in the plane and their basic properties, circles) Class notes for Thursday October 9, 2003 (first things about limits) |

6 | October 13 |
Monday is Thanksgiving.
Homework Assignment 6 Class notes for Tuesday October 14, 2003 (limits of sums and products) Class notes for Thursday October 16, 2003 (last words about limits and first words about continuity; continuity of compositions) Friday October 17 is the last date to drop down to Math 133 or Math 135. |

7 | October 20 |
Monday: Term Exam 1
took place in the tutorials. See the
Solution of Term Exam 1.
Homework Assignment 7 Class notes for Tuesday October 21, 2003 (the three fun theorems: statements, some philosophy, corollaries) Handout: Math 137's "How to Solve Problems" Class notes for Thursday October 23, 2003 (polynomials have roots, upper bounds, least upper bounds, P13) Friday October 24 is the last date to drop down to Math 137. |

8 | October 27 |
Handout: Monsters
Homework Assignment 8 Class notes for Tuesday October 28, 2003 (comments on P13: alternative formulation, N is unbounded) Class notes for Thursday October 30, 2003 (proofs of the three fun theorems, differentiability, diffrentiable is continuous) |

9 | November 3 |
Homework Assignment 9
Class notes for Tuesday November 4, 2003 (continuous but non-differentiable functions, derivatives and approximations, notations of derivatives, finding derivatives is fun and easy) Class notes for Thursday November 6, 2003 (differentiation is fun and easy: proofs; the chain rule) |

10 | November 10 |
Homework Assignment 10
Math 137's "Guidelines to submitting problem sets" Class notes for Tuesday November 11, 2003 (proof of the chain rule, local min/max) Class notes for Thursday November 13, 2003 (necessary and sufficient conditions of extrema using the mean value theorem) |

11 | November 17 |
Dror was away on Tuesday, in Banff. Class was taught by
our tutor Ching-Nam (Julian) Hung.
Homework Assignment 11 Class notes for Tuesday November 18, 2003 (Julian on mean value theorems, L'Hopitals's rule) A Little on Convexity Class notes for Thursday November 20, 2003 (proof of L'Hopitals's rule, convexity, studying functions, confused start of inverse functions) |

12 | November 24 |
Homework Assignment 12
Class notes for Tuesday November 25, 2003 (inverse functions: definition, existence, continuity, differentiability; rational powers) Some problems with Areas Class notes for Thursday November 27, 2003 (proofs of the continuity and differentiability theorems for inverse functions, a word about areas) |

13 | December 1 |
Monday: Term Exam 2
took place in the tutorials. See the
Solution of Term Exam 2.
Class notes for Tuesday December 2, 2003 (the definition of the integral) Handout: Integration Homework Assignment 12a Class notes for Thursday December 4, 2003 (the basics about partitions and upper and lower sums) |

Winter Semester: | ||

14 | January 5 |
No tutorials this week!
Handout: Integration Homework Assignment 14 Class notes for Tuesday January 6, 2004 (uniform continuity and integrability) Class notes for Thursday January 8, 2004 (continuity on a closed interval implies uniform continuity, minor integration theorems) |

15 | January 12 |
Monday: TA evaluation forms will be filled during tutorials.
Homework Assignment 15 Class notes for Tuesday January 13, 2004 (the first fundamental theorem of calculus) Handout: A Continuous but Non-Differentiable Derivative Class notes for Thursday January 15, 2004 (differentiating integrals and integrating derivatives, defining angles and defining pi) |

16 | January 19 |
Homework Assignment 16
Class notes for Tuesday January 20, 2004 (the definitions of cos and sin, their derivatives, the equation f''=-f and some consequences) Class notes for Thursday January 22, 2004 (end of trigonometry chapter) |

17 | January 26 |
Homework Assignment 17
Class notes for Tuesday January 27, 2004 (the definition of log and of exp) Class notes for Thursday January 29, 2004 (end of log/exp chapter) |

18 | February 2 |
Homework Assignment 18
Class notes for Tuesday February 3, 2004 (formula day - more log formulas, integration formulas) Class notes for Thursday February 5, 2004 (integration by parts, π is irrational) |

19 | February 9 |
Monday: Term Exam 3
took place in the tutorials. See the
Solution of Term Exam 3.
Around this week will be your last chance to drop this course. Homework Assignment 19 Class notes for Tuesday February 10, 2004 (more on π is irrational, simple substitutions) Class notes for Thursday February 12, 2004 (complicated substitutions, the ultimate trigonometric substitution, rational functions, the ultimate integration technique) |

Reading | February 16 | |

20 | February 23 |
Homework Assignment 20
The Cosmopolitan Integral Class notes for Tuesday February 24, 2004 (the cosmopolitan integral: arc length) Letter regarding the Undergraduate Mathematics Contest Class notes for Thursday February 26, 2004 (volume and surface area via integration, Taylor polynomials) |

21 | March 1 |
Homework Assignment 21
Brook Taylor Class notes for Tuesday March 2, 2004 (Taylor polynomials, a high-derivative criterion for min/max) Class notes for Thursday March 4, 2004 (the remainder formula and applications, irrationality of e) |

22 | March 8 |
Homework Assignment 22
See James Cook's solution of the HW22 Just For Fun problem Class notes for Tuesday March 9, 2004 (sequences - early material) Class notes for Thursday March 11, 2004 (the Bolzano-Weierstrass theorem, Cauchy sequences) |

23 | March 15 |
Homework Assignment 23
Handout: Series Class notes for Tuesday March 16, 2004 (series - early material) Handout: The strange resummation theorem Class notes for Thursday March 18, 2004 (convergence criteria for series, rearranging a conditionaly convergent series to get any desired sum) |

24 | March 22 |
Monday: Term Exam 4
took place in the tutorials. See the
Solution of Term Exam 4.
Homework Assignment 24 Class notes for Tuesday March 23, 2004 (absolute convergence and rearranging sums) Class notes for Thursday March 25, 2004 (the distributive law for absolutely convergent series, uniform convergence) |

25 | March 29 |
This week's office hours Wednseday (not Thursday!) 12:30-1:30.
Homework Assignment 25 Class notes for Tuesday March 30, 2004 (uniform convergence and continuity, integrability and differentiability) Handout: The Bessel J0 Function Class notes for Thursday April 1, 2004 (defining functions using power series, the Bessel example) |

26 | April 5 |
No homework assignment this week!
Handout: The Mandelbrot Set Class notes for Tuesday April 6, 2004 (the complex numbers) Transparencies: Zoom on the Mandelbrot Set (by Andy Burbanks), A Mandelbrot Set Gallery (by John Leen). Class notes for Thursday April 8, 2004 (the geometric interpretation of complex numbers, exponentials using series, Euler's formula) |

Study | April 12 |
Check out
Last Year's Sample Final Exam.
Absolutely all homework is due Friday April 16, 2PM, at the Math Aid Centre (SS 1071). |

Study | April 19 | Check out Last Year's Final Exam and its Last Year's Final Exam Solution |

Final | April 26 |
On Monday April 26 and Wednesday April 28 I will hold extra
office hours at the Math Aid Centre
(SS 1071) from 10AM until 1PM (and possibly a little beyond,
depending on demand). All graded homework assignments and term
exams will be available there.
Thursday April 29th: The Final Exam took place at CG 250, 9-12. See the Solution of the Final Exam. |

**Further resources:**

- Last year's Math 157 web pages: here.
- 2001-2002 Math 157 web pages: Fall, Winter.
- Undergraduate Information at the UofT Math Department. In particular, Other Math Courses.

September 23, 2003