MAT 157 Analysis I, 2023-24

Tentative Schedule:

September 7-8. First lecture (Spivak Section 1)

F: What is this course about?

Practice problems: Try your hand at some parts of Problems 1-6 in Section 1 (no hand-in).

September 11-15. Numbers of various sorts (Spivak Sections 1-2 and 25)

M: Addition, multiplication, and order relation: 12 postulates

W: Natural numbers. The principle of induction

F: Rational, irrational, real, and complex numbers

Assignment 1 (due Thursday, Sept. 21, 11:59pm)   Corrections and comments
Solutions

September 18-22. More about the integers. Construction of the real numbers (Spivak Sections 2, 29)

M: Notation (sets and their elements). Recursive definitions: sums, products, factorials, and binomial coefficients

W: Combinatorial meaning of the binomial coefficients, Pascal's triangle

F: Archimedean property and completeness. Maximum and supremum. Dedekind cuts

Assignment 2 (due Thursday, Sept. 28, 11:59pm)   Corrections and comments (last updated Sunday 12:05pm)
Solutions   Picture of Pascal's triangle modulo 2

September 25-29. Construction of the real numbers, cont'd. Functions and their graphs (Spivak Sections 29-30; 3)

M: Addition, multiplication, and order relation of Dedekind cuts   Rich Schwartz' notes on Dedekind cuts

W: The range, domain, and graph of a function; composition of functions. Permutations (Spivak Section 3)

F: Examples: Sequences as functions whose domain is N

Assignment 3 (due October 5, 11:59pm)
Solutions  

October 2-6. Excursion: Sequences and their limits (Spivak Section 22, Theorem 2)

M: Bounded sequences, monotone sequences. Limits

W: Convergence theorem for bounded monotone sequences. The Newton-Heron algorithm for computing square roots

F: How can sequences behave? Tail of a sequence; subsequences; divergent sequences. Examples

2022 Midterm 1, previous Midterm 1   (Style sample: from MAT 351)

October 9-13.

M: Thanksgiving (no lecture, no tutorials)

W: The Bolzano-Weierstrass theorem

R: Midterm 1, 7-9pm, Exam Center Room 100   List of topics   Solutions

F: Limits

Assignment 4 (due October 19)   Corrections and comments
Solutions

October 16-20. Continuity (Spivak Sections 6-8)

M: Continuous functions. Compositions

W: Intermediate value theorem

F: Intermediate value theorem (two proofs)

Assignment 5 (due October 26)   Corrections and comments
Solutions

October 23-27. Differentiability (Spivak Section 9)

M: Extreme value theorem

W: Extreme Value Theorem (two proofs)

F: Two definitions of the derivative (slope of the tangent line = limit of the differential quotient)

Assignment 6 (due Nov. 5)   Corrections and comments
Solutions

October 30 - November 3. The significance of the derivative (Spivak Sections 10-11)

M: Local maxima and minima.

W: Leibniz vs. Newton notation. The Mean value theorem and its consequences.

F: Rolle's theorem, and proof of the Mean Value Theorem. Product, quotient, and chain rule.

2022 Midterm 2, previous Midterm 2  

November 6-10 -- Reading week

November 13-17. Monotonicity and convexity. (Spivak Sections 11-12)

M: The second derivative. Convex and concave functions

W: Necessary vs. sufficient conditions for extrema

R: Midterm 2, 7-9pm McLennan Physical Laboratories, Rooms 202/203   List of topics   Solutions (Thursday, Friday)

F: Examples. How to precisely graph a function (with zeroes, local max/mins, inflection points, and asymptotes)

Assignment 7 (due November 23; Problems 1-4 only)   Corrections and comments
Solutions

November 20-24. Inverse functions (Spivak Section 12)

M: Inverse functions

W: Continuity of inverse functions (A7 Problem 5). L'Hopital's rule

F: Inverse Function Theorem

Assignment 8 (due November 30)   Corrections and comments

November 27-December 1. Integration (Spivak Section 13)

M: L'Hopital's rule for limits of type infinity/infinity

W: Upper and lower sums

F: Integrability and (uniform) continuity

Assignment 9 (due Monday, December 11)   Corrections and comments

December 4-6. The Fundamental Theorem of Calculus (Spivak Section 14)

M: Uniform continuity. Continuous => integrable. Statement of the FTC

W: Proof of the Fundamental Theorem of Calculus. Natural logarithm and exponential

Practice problems on logarithm and exponential (no hand-in; to be discussed January 8-11)

January 8-12. Techniques of integration
Practice problems on integration techniques (no hand-in)

January 15-19. Logarithm and Exponential (Spivak Sections 18-19)

R: Midterm 3, 7-9pm

Assignment 10 (due January 25)

January 22-26. Trigonometric Functions (Spivak Section 15)
Assignent 11 (due February 1)

January 29 - February 2. Approximation by Polynomials. Taylor's theorem (Spivak Section 20)
Assignment 12 (due February 8)

February 19-23 -- Reading Week

February 26 - March 1.

R: Midterm 4, 7-9pm

Assignment 14 (due March 7)

March 25-29.
Assignment 18 (due April 4)

April 1-5.
Practice Problems (no hand-in)

Final exam (to be scheduled by the registrar)

---

The University of Toronto is committed to accessibility. If you require accommodations for a disability, or have any accessibility concerns about the course, the classroom or course materials, please contact Accessibility Services as soon as possible: disability.services@utoronto.ca, or http://studentlife.utoronto.ca/accessibility