AS.110.113 - Honors Single Variable Calculus

Course Updates

Week 1
Proofs
PSet01

Proof techniques

'When I use a word,' Humpty Dumpty said, in rather a scornful tone, 'it means just what I choose it to mean, neither more nor less.'

- Through the Looking Glass

In the first week you'll learn how to write proofs. The notes are from this link. The reading material and the HWs will be handed out in class. This HW is extremely important as throughout the course you'll be doing a lot of proof writing.

Week 2
Ch 5
PSet02

Limits

  • Ch.5: Pages 90, 91, specifically read the example x.sin(1/x).
  • Ch.5: The discussion directly preceding the definition of limit on page 96.
  • Ch.5: Proof of Theorem 2 part (1).
  • Ch.5: Definitions of one sided limits, and limit as x → ∞ on page 104 and 105.
Week 3
Ch 6, 7, 8
PSet03

Continuity & Three hard theorems

  • Ch.6: Definition of continuity of f(x) at a.
  • Ch.6: Proofs of Theorem 1 and Theorem 2.
  • Ch.6: Discussion directly following Theorem 2 about continuity on intervals.
  • Ch.7: From Theorem 1 to the proof of Theorem 9, from page 120 to 124.
  • Ch.8: From the beginning till property (P13) on Pg. 133.
Week 4
Ch 9
PSet04

Derivatives

  • Ch.9: Definition of derivative and it's connection to tangents, Pg. 147 - 152.
  • Ch.9: Left-hand and right-hand derivative, Theorem 1, higher order derivatives, Pg. 152 - 161.
Week 5
Ch 10, 12
PSet05

Differentiation

  • Ch.10: Theorems 1 to 9 along with proofs.
  • Ch.12: Definitions of one-one, and inverse of a function, statements of Theorem 1, 2, 3,
  • Ch.12: Proof of Theorem 5 and the following discussion.
Week 6
Ch 11
PSet06

Importance of Derivatives

  • Ch.11: All the definitions and statements of Theorems up to Corollary 3 on Pg. 192.
  • Ch.11: Proof of Mean value and Rolle's theorems.
  • Ch.11: Read the statements of Theorem 5 and 6.
  • Ch.11: Read the proof of L'Hospital's rule
    (you can assume the Cauchy's Mean Value Theorem and don't need to read it's proof.)
Week 7
Ch 13
PSet07

Integrals

  • Ch.13: Read the definition of integral on Pg.255 and the statement of Theorem 2 on Pg.256.
  • Ch.13: Read the statements of Theorem 3 to Theorem 6.
  • Ch.13: Read the statements and proofs of Theorem 7 and 8.
Week 8
Ch 14
PSet08

Fundamental Theorem of Calculus

  • Ch.14: Theorem 1 and it's proof and the following discussion about swapping the bounds.
  • Ch.14: Theorem 2 and it's proof.
  • Ch.14: Read the discussion and examples on Pg. 289 and 290.
  • Ch.15: Definitions on Pg. 302 and Pg. 303, proof of Theorem 1 on Pg. 304.
Week 9
Ch 18, 19
PSet09

Trig. functions and exponentials

  • Ch.18: Read from the beginning of the chapter till the definition of exp on Pg.340.
  • Ch.18: Read from Pg.340 till the statement of Theorem 6 on Pg. 345.
  • Ch.19: Theorem 1 on Pg. 362 and Theorem 2 on Pg. 365.
Week 10
Ch 20
PSet10

Approximation by Polynomials

  • Ch.20: Read from the beginning of the chapter to proof of Theorem 1, Pg. 405-410.
  • Ch.20: Computation of Taylor series of arctan(x) on Pg. 414.
  • Ch.20: The statement of Theorem 4 (Taylor's theorem), computations on Pg. 421-424.
Week 11
Ch 22
PSet11

Sequences

  • Ch.22: Definition on convergence on Pg.446 and the following discussion until Pg.449.
  • Ch.22: Statements of Theorem 1 and Theorem 2.
  • Ch.22: Discussions, Theorems and Proofs on Pg.451 and Pg.452.
Week 12
Ch 23
PSet12

Infinite Series

  • Ch.23: Read the discussions on pages Pg.464-467.
  • Ch.23: Read statements and proofs of the 4 comparison tests on pages Pg.467-472.
  • Ch.23: Read the definition of Absolute Convergence and statements of Theorems 5-9.
                You can skip the proofs of these Theorems.
Break

Thanksgiving Break

Instructions for the final:
You should think of the final as an excuse to review the fundamentals. You should revise the following topics. It should be enough to go over your own HW solutions to the relevant problem sets.

Continuity: How to prove continuity/discontinuity of a function using the ε-δ definition,
Intermediate value theorem and it's proof (inf, sup etc.)
Derivatives: Basic computations, Mean value theorem
Critical points , min/max problems
Computing Taylor series, estimating errors
Integrals: Computations only (substitution tricks, by parts, trig subs etc.)

The exam will mostly consist of self contained problems (you do need to remember all the definitions and theorems) and you'll be graded on logical correctness, there will be very little credit for the 'final answer'.

Week 13
PSet13

Nowhere differentiable function

  • No reading assignment for the last week. Just solve the problem set :D.
  • Also remember that your project report is due on the day of the final Dec 13,
    finish it up now so that you can focus on other subjects during the reading period.

Projects

You have an option of doing a project in this course. If you do choose to do a project I'll reduce the weight of the final exam for you. The project will consist of some independent reading by you about anything even vaguely related to calculus. Here are a few possible topics for the projects:
  • Construction of reals, axiom of choice etc.
  • Pathological functions, nowhere differentiable function, space filling curves etc.
  • Compactness, Heine-Borel theorem etc.
  • What is a measure?
  • Fourier series
You are free to come up with your own topics, these are merely suggestions.

For the project you'll have to submit a written report at the end of the semester. The report should have a rigorous definition-theorem-proof structure.

General Information

Class : MW 1:30-2:45pm, Hodson 315
: F 1:30-2:20pm, Maryland 217
My office : Kreiger 200
Office Hours : Th 2:00-4:00pm
Help Room : M 11:00am-1:00pm
Email : anakade1 at jhu ddott edu
Textbook : Calculus, 3rd Edition, Spivak


Course Description

The official syllabus can be found here. This course covers the topics typically covered in the standard Calc 1 and Calc 2 courses, and you will also learn to write mathematical proofs, as such this is a VERY fast paced course. If you like being challenged and are curious and excited about maths this is the course for you. You should not be taking this course simply to meet your course requirements or because of the "honors" in the title of the course. This course will be taught in the mode of Inquiry Based Learning.

Inquiry Based Learning

“The only way to learn mathematics is to do mathematics.”

- Paul Halmos

In this (IBL) course there will hardly be any lectures to prove things for you. Instead, you will prove the propositions yourself and explain your proofs to each other. The IBL format fosters creativity and a deep understanding of the material. However, its success depends heavily on the students: you must be actively engaged and prepared for class every day. On one hand this mode of learning is highly rewarding and provides great insights into the subject, on the other hand it requires substantial self-study and a willingness to make lots of mistakes and not be frustrated by them.

You are also expected to listen to the ideas of other students and be respectful to them even if you think you have the "correct" solution. The environment of an IBL classroom is collaborative and not competitive and the main goal is to have loads of fun while learning some amazing mathematics.

What does a typical class look like? You come to the class having read the reading assignment and hopefully with lots of questions, we'll discuss your questions first, I'll then hand out the problem sheets which you'll solve in small groups, I'll roam around and help out wherever necessary. You'll not be allowed to use laptops or other electronic devices in class, so you need to bring either a hard copy of the book or more preferably your own handwritten notes to the class.

Homeworks

Homeworks are what make this course. Your homework will consist of three parts:
  • Reading assignments: You will be assigned parts of the book to be read BEFORE every class. These assignments are extremely crucial as there will not be any lectures and you'll be learning the subject by reading the book. You'll not be expected to have understood everything that you've read, but you'll be expected to make good notes and ask questions about it in class. If you skip this step you will likely be completely lost and almost certainly fall behind.
  • Group discussions: This is where most of the magic happens. In class instead of passively listening to a lecture you'll be actively discussing problems with other students and/or the instructor and learning the subject in a very hands-on manner.
  • Writing: You will be required to write the solutions to the problem sets by yourselves. Learning the language of mathematics is a crucial part of the learning process and like any other language you will need a lot of practice to get good at it.
I'll hand out problem sets in each class which you'll discuss in small groups, write the solutions by yourselves and submit them on the following Friday.

Grading Scheme

Class participation : 20%
Homeworks : 40%
Final Exam : 20%
Project : 20%

Each week's problem sets count as one homework, only the top 10 homeworks will be included in the final grade.