Honors assignments

Standard Honors

Standard Honors requires you to complete extra problems on every homework assignment.  These are listed on the "Homework" page.

Project Honors

For Project honors, you are required to complete 5 projects in Fall semester, 6 Honors projects in second semester, from the list below.  There are three general categories of Honors projects: Additional Content (going over additional Calculus material that we didn't cover in class), Additional Applications, and Miscellaneous.  You can do a maximum of 3 projects from any one category in each semester.   

Additional Content

• Read Chapter 1 of Spivak "Calculus" See here.  Then do the following:  (1) on a separate sheet of paper, write out (in detail) the proof that a negative times a negative is positive which is given in the book.  Clearly indicate how the distributive property is used to prove this fact.  (2) on the same sheet of paper, complete Problems 1(i), 2, and 3(i)-(v).  Be very careful to justify each step you use based on the axioms in the text!  This project explains the modern point of view on an old question: why is a negative times a negative equal to a positive?  Why is the "flip-and-multiply" method of dividing fractions the correct one?

• Read Chapter 2 of Spivak "Calculus" See here, then do the following on a separate sheet of paper: (1) explain, in your own words, the method of induction for proving mathematical facts.  (2)  Summarize in your own words the proof that the square root of 2 is irrational.  (3) Complete Problems 1 and 13 from the chapter.  Feel free to ask for help if you're stuck!  This project introduces some of the basic concepts of number theory, a branch of mathematics that is very important in the computer age.

• Read Section 1.4 in your textbook and complete Problems 7,11,22,23,25,29,32.  Make sure you label the axes and the scales in your graphs.

• Read the last part of Section 1.6 in your textbook on Inverse Trigonometric Functions, then complete Problems 59,61,63,64,65,67,68,69,71,72 in this section.  

• Read Section 2.4 in your textbook (The Precise Definition of a Limit), then complete Problems 2.4-1,3,7,11,12,19,25.  This is a difficult, theoretical section.  But it might be covered in a college Calculus class, so if you plan to take college Calculus this might be a good project!

• Read Section 3.11 in your textbook (Hyperbolic functions), then complete Problems 3.11-1,5,9,13,21,31,33,41,51.

• Read Section 4.5 in your textbook and complete the problems listed on the Homework page for Section 4.5.  Do not use Desmos to draw the graphs - draw them using Calculus, then CONFIRM your graphs using Desmos.

Applications

• Do the Applied Project "Where Should a Pilot Start Descent?" on Pages 206-207 of your textbook.

• Do the Applied Project "The Calculus of Rainbows" on Pages 279-280 of your textbook.  WARNING: This one is quite difficult; if you can convincingly show me that you understood the whole project, then I'll give you credit for 2 Honors assignments.

• Do the Applied Project "The Shape of a Can" on Pages 333-334 of your textbook. 

• Do Problems 3.7-11,13,21,23,28,29,34,35 on a variety of applications of derivatives.

• Read Section 4.8 in your textbook (Newton's Method) then do Problems 4.8-1,5,7,11,13,29,31.

• Read Section 6.4 in your textbook (Work) then do Problems 6.4-1,5,13,17,19,29

• Prepare for and compete in a Mathematical Modeling contest.  Worth 1 project if you participate, and an additional project if you attend meetings before-hand and make substantive efforts to prepare.

• Watch the 3Blue1Brown series on Neural Networks. There are three episodes; the first one is here.  You probably won't understand the whole thing, but write a summary of what you learned and what was confusing from each episode.

• Watch the 3Blue1Brown move on the Central Limit Theorem. It is linked here. The Central Limit Theorem is a foundational fact in mathematics; it underlies the "bell curve" that you might have heard about.

Miscellaneous

• Watch the film "Dimensions" (See here).  Then write a 3-4 page summary of what you watched; about a paragraph or two per chapter.  Some things to definitely discuss in your summary: 1.  How can we understand 4-dimensional objects by analogy to lower-dimensional concepts?  Give several examples from the movie.  Which were the most enlightening to you?  2.  How are complex numbers related to rotations?  Did this movie give you a different perspective on complex numbers?  3.  How are complex numbers related to fractals?  Explain in as much detail as you can.  4.  What is the Hopf Fibration?  (You might need to look this up on Wikipedia after the movie; it is rather tough to understand!)

• Watch the film "Chaos" (See here).   Then write a 3-4 page summary of what you watched; about a paragraph or two per chapter.  Some point to definitely touch on in your summary:  1.  How would you define chaos (in its scientific/mathematical sense)?  2.  What does the Poincare-Bendixson Theorem say?  What are some examples discussed in the movie regarding this theorem?  3. What is the "Butterfly Effect" and what is the relation to Lorenz's "Strange Attractor"?  How does the Strange Attractor exhibit both chaos and order?

• Pick an unsolved problem in mathematics, and prepare a 10-15 minute presentation to the class explaining the problem and any work or progress that has been made on it.  I highly encourage you to look on Google or Wikipedia for interesting problems; believe it or not, there are many, many unsolved math problems out there that are easy to state and intrinsically interesting!

• Pick a mathematician and prepare a 10-15 minute presentation on their life, including a brief description of the mathematics they worked on (I understand that this is hard for modern-day mathematicians).  I'm happy to give a variety of suggestions here - there are lots of very interesting mathematicians to learn about!!

• Prepare a 10-15 minute class presentation on how the Mandelbrot set is defined (obviously, start by looking up what the Mandelbrot set is!).  Include visuals and at least some mathematical details.  For example, if your presentation doesn't mention complex numbers then something has gone wrong!

• Research the RSA Cryptosystem, which is the way that information is transferred securely over the Internet.  Prepare a 10-15 minute presentation explaining to the class what a public key cryptosystem does.  This can be very hands-on and fun!

• THIS ONE CAN COUNT FOR UP TO 3 PROJECTS IF YOU DO EXCELLENT WORK!!!  Interested in how mathematicians are fighting for social justice?  Order a copy of the book Radical Equations by Robert P. Moses, then read it and answer the questions linked here.  In order to get credit for the 3 projects, it must be clear from your answers that you read the book carefully and thought long and hard about its message.  Click here for the Amazon webpage for the book.

• Propose a book to read!  I have lots of options in my room, and know of many other great possibilities. This could potentially be a full-year project is the book is of high mathematical quality (I can help you in deciding which books meet this criterion). Anyone looking to make a long-term book option should be prepared to schedule regular times to meet in office hours to discuss what you have read, as I would prefer that to just a general book report on what you read.