### Math 112

Introduction to Analysis, Spring 2024 (Section S01)

Current week   |   Moodle   |   Course information
Instructor: David Perkinson (schedule)
Course lecture notes: Math 112 Lecture Notes, by David Perkinson. (To be revised throughout the semester. Last update: 4/7/24.)
Supplemental text: Introduction to Analysis, by Irena Swanson.
LaTeX Overleaf pointers
For details on office hours and drop-in tutoring, see the top of our Moodle page.
Final exam: We will have an in-class final exam, May 8, 1–4 pm, E314.
Week 1: January 25 - 29
Monday: Welcome. Preliminaries, LaTeX.
To do list for Wednesday.

Wednesday: Induction.
Reading for Wednesday, Week 1. Supplemental reading: sections 1.4 and 1.5 in Swanson.
Lecture/quiz. (Do the reading before watching this video lecture.).
Group problems. Solutions.
Friday: Sets.
Lecture/quiz.
Turn in: HW1F (overleaf template).
Group problems. Solutions.

Week 2: January 29 - February 2
Monday: More sets. Cartesian products.
Reading for Monday, Week 2. Also read Mathematical Writing. Supplemental reading: Sections 2.1 and 2.2 in Swanson.
Lecture/quiz.
Group problems. Solutions.
Wednesday: Relations and equivalence relations.
Lecture/quiz. (The lecture makes reference to the notation $$[n]:=\{1,\dots,n\}$$, which conflicts with our use of square brackets to denote equivalence classes. Context should make the meaning clear.)
Group problems: Solutions.
Friday: Equivalence classes, $$\mathbb{Z}/n\mathbb{Z}$$.
Lecture/quiz.
Turn in: HW2F, (overleaf template).

Week 3: February 5 - 9
Monday: Functions.
Lecture/quiz.
Group problems. Solutions.
Wednesday: More functions.
Reading for Wednesday, Week 3. Note: This reading is a bit longer then the others. Supplemental reading: Section 2.4 in Swanson.
Lecture/quiz.
Group problems: Solutions.
Friday: Modular arithmetic.
Lecture/quiz.
Turn in: HW3F, (overleaf template). Here are a few pointers for typesetting functions: typesetting functions.
Group problems: Solutions.

Week 4: February 12 - 16
Monday: Something awesome (cardinality).
Exam distributed today (review sheet). No reading or lecture/quiz for today.
Group problems. Solutions.
Wednesday: More awesome (more cardinality).
Exam due by noon today.
No reading or lecture/quiz for today.
Group problems: Solutions.
Friday: Field axioms.
Lecture/quiz.
Turn in: HW4F, (overleaf template).
Group problems: Solutions.

Week 5: February 19 - 23
Monday: Order axioms.
Reading for Monday, Week 5. Also read Addendum to Friday, Week 4 (more examples of consequences of the field axioms). Supplemental reading: Sections 2.6 and 2.7 in Swanson.
Lecture/quiz,
Group problems. Solutions.
Wednesday: Completeness.
Lecture/quiz,
Exam 1 resubmissions due today via Gradescope.
Group problems: Solutions.
Friday: Extrema.
Lecture/quiz.
Turn in: HW5, (overleaf template).
Group problems. Solutions.

Week 6: February 26 - March 1
Monday: Complex numbers I.
Lecture/quiz.
Group problems. Solutions.
Wednesday: Complex numbers II.
Reading for Wednesday, Week 6. Supplemental reading: Sections 3.2 and 3.3 in Swanson.
Lecture/quiz.
Group problems. Solutions.
Friday: Complex numbers III.
Lecture/quiz.
Turn in: HW6, (overleaf template).
Group problems. Solutions.

Week 7: March 4 - 8
Monday: Topology.
Lecture/quiz.
Group problems. Solutions.
Wednesday: Sequences I.
Lecture/quiz.
Group problems. Solutions.
Friday: Sequences II.
Lecture/quiz.
Turn in: HW7, (overleaf template).

Week 8: March 18 - 22
Monday: In class (easy) exam (review sheet) .
No reading or lecture/quiz for today.

Wednesday: Something awesome.
Exam~2 distributed by noon today (review sheet). [Change: the Exam 2 was distributed at the beginning of Spring Break.]
No reading or lecture/quiz for today.
Group problems. Solutions.
Friday: More awesome.
Exam 2 due by noon today.
No reading or lecture/quiz for today.
Related videos: The Mandelbrot Set and Filled Julia Set. For experimentation: Julia set web app.
Group problems. Solutions.

Week 9: March 25 - 29
Monday: New-from-old limit theorem.
Lecture/quiz.
Group problems. Solutions.

Wednesday: Monotone Convergence Theorem.
Lecture/quiz.
Group problems. Solutions.

Friday: Sequences III.
Reading for Friday, Week 9. Supplemental reading: Sections 8.3, 8.4, 8.7, and 8.8 in Swanson. Recommended reading: Discrete dynamical systems.
Lecture/quiz.
Turn in: HW9 (overleaf template).

Week 10: April 1 - 5
Monday: Series. Geometric series.
Lecture/quiz.
Note: There will be a short quiz at the beginning of class this Friday. It will just ask you for the definition of the limit of a sequence as it appears on page 1 of the lecture for Wednesday, Week 7.
Group problems. Solutions.

Wednesday: Series tests I.
Lecture/quiz.
Exam 2 resubmissions due today via Gradescope.

Friday: Series tests II.
Lecture/quiz.
Quiz today. Class will begin with a short quiz asking you for the definition of the limit of a sequence as it appears on page 1 of the lecture for Wednesday, Week 7.
Turn in: HW10 (overleaf template).
Group problems. Solutions.

Week 11: April 8 - 12
Monday: Series tests III.
Lecture/quiz
Group problems. Solutions.
Wednesday: Limits of functions.
Reading for Wednesday, Week 11. Supplemental reading: Sections 4.1 and 4.2 in Swanson.
Lecture/quiz
Group problems. Solutions.
Friday: Continuity and derivatives.
Reading for Friday, Week 11. Supplemental reading: Sections 5.1, 6.1, and 6.2 in Swanson.
Lecture/quiz
Group problems. Solutions.
Turn in: HW11F (overleaf template).

Week 12: April 15 - 19
Monday: Power series I.
Lecture/quiz

Wednesday: Power series II.
Reading for Wednesday, Week 12. Supplemental reading: Sections 9.3 and 9.4 in Swanson.
Lecture/quiz

Friday: Taylor series I.
Reading for Friday, Week 12. Supplemental reading: Sections 6.5 and 9.7 in Swanson.
Lecture/quiz
Turn in: HW12 (overleaf template).

Week 13: April 22 - 26
Monday: Taylor series II. The complex exponential function.
Reading for Monday, Week 13. Supplemental reading: Sections 6.5 and 9.7 in Swanson.
Lecture/quiz

Wednesday: Two theorems from calculus.
Reading for Wednesday, Week 13. Supplemental reading: Sections 5.2 and 5.3 in Swanson.
Lecture/quiz

Friday: Three theorems by Euler.
The $$\LaTeX$$ document preparation system