Resources (Updated 5/11)
Stewart Textbook Website (Calculus III)
Stewart Textbook Website (Calculus I & II)
Paul's Online Math Notes - Calculus I
Paul's Online Math Notes - Calculus II
Paul's Online Math Notes - Calculus III
Paul's Online Math Notes - Differential Equations
Differential Equations Study Guide
Office Hours
Mon, Tue Period 6, Room 719D
Wed, Thu Period 7, Room 719D
You may drop by Room 719D whenever I am free (see Dr. Freeman's Daily Schedule posted above), and I will help you if I am available. If you would like to guarantee my availability, or require more than 10 minutes, I would invite you to schedule an appointment, either by email or in person.
Announcements (Updated 6/7)
Here is the Take-Home Portion of the Final Exam due by 9 AM on Sat 6/10 (you may submit it electronically - single PDF only).
Possible Times for the Complex Integration Seminar are:
Monday 6/12 8:45 AM - 10:45 AM
Monday 6/12 1:00 PM - 3:00 PM
Tuesday 6/13 currently anytime works
The Final Exam will consist of two parts: Part 1 (In-Class Problems) and Part 2 (Take-Home Problems).
Part 1 of the Final Exam will be Wednesday 6/7. AS AGREED 6/6, WE WILL MEET IN RM 719 AT 9:55 AM AND START THE EXAM AT 10:00 AM.
Part 2 of the Final Exam will be available for download by 5 PM on Wed 6/8 and is due by 9 AM on Sat 6/10 (you may submit it electronically - single PDF only).
There will be an optional Review Session Tue 6/6 from 11:00 AM to 12:00 PM in Room 632.
In all past years, the Final Exam has been part in-class and part take-home. Please consider the following two options and be ready to vote in class on Monday 6/5.
Option 1) In-Class Exam only on Wed 6/7. Approximately 10 problems.
Option 2) Part 1 In-Class Exam on Wed 6/7. Approximately 7 problems. Part 2 Take-Home Exam (posted Wed. due Sat.). Approximately 5 Problems.
The "Blank Copies of Old Exams" folder above now contains Exams from this semester.
Solutions to Exam 3 are posted below.
Exam 3 will be on Thu 5/25 and will cover HW 24-31. As usual, I will be available on Wed 5/24 8:30 - 9:00 AM to answer any question you may have. However, we will NOT be doing in-class review on Wed 5/24, but instead will be moving ahead with new material (which will not be on Exam 3).
Quiz 9 on Friday 5/19 officially covers HW 26-29.
There will not be a Quiz on Friday 5/12.
A summary of the planar set definitions is posted above under Resources (complements of Waseer).
Quiz 8 is "Take-Home" and is due by 9:00 AM on Monday 5/8. If you are absent on Monday for any reason, please submit electronically as a single PDF file by the due time.
Solutions to Exam 2 are posted below.
Exam 2 will be on Friday 4/28 and will begin at 8:35 AM sharp. It will cover through HW 23.
Quiz 8 will not take place until after Exam 2.
Quiz 7 on Friday 4/7 officially covers HW 16-20.
No meeting Wednesday 4/5. We will meet at 8:30 AM on Thu 4/6 this week to go over any homework, and will have a Quiz on Friday 4/7. I would like to have the next student presentation the first class after break on Wednesday 4/19 at 8:30 AM. Volunteer?
As agreed, two students (I forget which two) will each make their 15 minute presentation on an application of first-order equations at 8:30 AM on Wed 3/22. Please arrive so that we can begin at 8:30 AM sharp! You may be assessed on the contents of these presentations.
Quiz 4 is "Take-Home" (see HW 12 below). Due in class (by 9:00 AM) on Friday 3/17. Please note: (1) Presentation Counts! You will be held to an even higher standard of neatness and clarity for take-home assessments. (2) If you are absent on Friday 3/17, you must submit your work electronically before the same deadline; there will be no late work accepted or make-up assignments given. (3) Your work should be your own, and is subject to the BHSEC Queens Academic Honesty Policy.
Solutions to Exam 1 are posted below.
Exam 1 will be on Friday 3/10 and will begin at 8:35 AM sharp. It will cover through HW 10 & HW MFO.
Quiz 3 on Friday 3/3 will cover only the material through HW 6.
The first Quiz is postponed until Monday 2/13 due to the snow day. It will cover through HW 3A.
As of 2 PM on Feb. 1, the course meeting time has been finalized as originally scheduled - Period 1 on Mon, Wed, Thu, & Fri.
Starting Wednesday Feb. 8, we will meet every Wednesday at 8:30 AM in Room 632 to discuss homework.
Welcome to Advanced Applied Math.
Homework (Updated 6/2)
(You are responsible for all Homework posted and/or updated before 5:00 PM the school day before the due date. Due dates are subject to change, so please check them carefully.)
"ODE" denotes the Ordinary Differential Equations textbook
"FCA" denotes the Fundamentals of Complex Analysis textbook
Notes:
(1) Some of these ODE's are in differential form (i.e., the ODE has already been "multiplied" through by dx). To convert back to the more familiar Leibnitz form, just "divide" the ODE through by dx. Also consider regarding x as the dependent variable and "dividing" through by dy, which may result in an ODE which is easier to solve.
(2) Whenever possible, solve for an explicit solution (isolate the dependent variable), even though the answers in the textbook are often left in implicit form. You will generally be required to find explicit solutions on Exams and Quizzes.
(3) In the ODE Textbook (as well as in Wolfram Alpha, and many other places), "log" denotes the natural logarithm that we normally denote by "ln".
HW 34 is the last homework!
HW 34 (Complex Powers)
FCA Page 136 Problems 1, 3, 4, 5, 7, 10, 11
Also, familiarize yourself with the "log" function on Wolfram Alpha for complex arguments, which agrees with the principal value function Log z we have been discussing. For example, compute a few values and check them manually, and check that identities hold where you expect them to hold. You may need to use Wolfram Alpha on the Final Exam.
Due Mon 6/5
HW 33 (Complex Logarithms)
FCA Page 123 Problems 1, 3, 4, 5, 6, 9, 10, and these Problems
Due Fri 6/2
HW 32 (Exp, Trig, & Hyp Functions)
FCA Page 115 Problems 1, 11, 12, 13, 15, 17, 20
Due Wed 5/31
HW 31 (Harmonic Functions)
FCA Page 84 Problems 1, 2, 3, 5, 6, 14
Due Mon 5/22
HW 30 (Cauchy-Riemann Equations)
FCA Page 77 Problems 1, 3, 5, 9, 10
Due Fri 5/19
HW 29 (Analyticity)
FCA Page 70 Problems 4, 7, 9, 11, 13
Due Thu 5/18
HW 28.5 (Admissibility)
Do these problems.
Due Wed 5/17
HW 28 (Limits and Continuity)
FCA Page 63 Problems 7, 11, 12, 13, 17, 21, 25
Due Wed 5/17 (but complete as much as possible by Mon 5/15)
HW 27 (Functions of a Complex Variable)
FCA Page 56 Problems 1-5, 7c, 8c, 9, 10, 11
Also do Spring 2015 Exam 3 Problem 7
Due Mon 5/15
HW 26 (Planar Sets)
FCA Page 42 Problems 2-8, 11, 12, 15-20
Due Thu 5/11 (Extended)
HW 25 (Reduction of Order)
ODE Page 246 Problems 3, 5, 9, 15
Due Mon 5/8
HW 24 (Series Solutions Near an Ordinary Point)
ODE Page 546 Problems 3, 5, 7, 9, 11
Due Fri 5/5
HW Exam 2 Solutions Review
Review the Exam 2 Solutions and resolve any questions you may have.
Due Wed 5/3
HW PSR (Power Series Review)
Review the sections on "Power Series" & "Power Series and Functions" in Paul's Online Math Notes - Calculus II and write down any questions you may have.
Due Mon 5/1
HW 23 (Method of Variation of Parameters)
ODE Page 240 Problems 2, 3, 4, 5, 12, 15, 18, 20
Spring 2015 Exam 2 Problem 3
Due Thu 4/27
HW 22B (Complex Exponentials are Your Friends)
Do these Problems.
Due Mon 4/24
HW 22A (Method of Undetermined Coefficients)
ODE Page 231 Problems 3, 6, 7, 9, 11, 13, 15, 17, 19, 23, 30, 32
Due Mon 4/24
HW 21b (Linear Homogeneous ODE's with Constant Coefficients - No Restrictions on Roots of Characteristic Equation)
ODE Page 220 Problems 24, 25, 26, 27, 28, 29, 30, 33, 34, 35
Due Wed 4/19
HW 21a (Primitive nth Roots of Unity in Z mod p) (Updated 4/18)
(1) Classify each element of Z mod 13 (excluding 0 and 1) as a primitive nth root of unity for some n satisfying 1 < n < p.
(2) In Z mod 13 verify that, for each n for which a primitive nth root of unity exists, (1) there are exactly n roots of unity and (2) they sum to zero.
(3) Does the proof we used in C to prove that the nth roots of unity always sum to zero work in Z mod 13? Why or why not?
Due Wed 4/19
HW 20 (Powers and Roots)
FCA Page 37 Problems 4, 5, 7, 9, 10, 11, 15, 16
Due Thu 4/6
HW 19 (Return to Calculus II)
FCA Page 32 Problem 23 (answers are in the back of the book)
Due Mon 4/3
HW 18 (Return to 9th Grade Geometry)
Do this problem.
Due Mon 4/3
HW 17 (Evaluating Complex Trig & Hyperbolic Functions)
FCA Page 115 Problem 5, Spring 2015 Exam 2 Problems 1 & 2
(Note: exp(z) is simply an alternate notation for ez which avoids the need for superscripts)
Due Mon 4/3
HW 16 (Complex Exponentials)
FCA Page 31 Problems 1, 2, 3, 4, 7, 8, 11
Due Fri 3/31
HW 15 (Arg!)
FCA Page 22 Problems 11, 12, 13
Problem 11 is a very nice problem. You will need to use the fact that if z = x + i y, Arg(z) = tan-1(y/x) when x>0 (and only when x>0 !!!), together with the fact that tan-1 is odd.
Also, do:
1) Problems 1 & 3 on Spring 2014 Exam I.
Wed 3/29
HW 14 (arg!)
FCA Page 22
Problem 4 (just mimic the induction argument we used in class to prove the similar identity for conjugate)
Problem 5 (if you choose to do the absolute value last, you will do a lot of unnecessary work)
Problem 6 ("cis(theta)" is a ridiculous abbreviation for "cos(theta) + i sin(theta)")
Problem 7
Due Mon 3/27
HW 13 (Welcome to Complex Analysis II)
FCA Page 13 Problems 7, 9, 11, 13, 16
Due Thu 3/23
HW 12 (More Fun with Finite Fields - Quiz 4)
Do these problems. Also in Z mod 11, find all zeroes of the polynomial x^3 + 6 x^2 + 1, use them to factor the polynomial, and then re-multiply the factors as a check.
Due Fri 3/17 (to be turned in, as Quiz 4)
HW 11 (Welcome to Complex Analysis I)
FCA Page 5 Problems 7, 9, 11, 13, 14, 17, 19, 20, 21, 23, 24, 25
Due Wed 3/15
HW MFO (Mixed First Order ODE's)
ODE Page 104 Problems Any/All. You are not responsible for finding integrating factors and/or substitutions for types of ODE's that we did not cover.
Due Thu 3/9
HW 10 (Linear Homogeneous ODE's with Constant Coefficients - Characteristic Equation has only Real Roots, Some Possibly Multiple)
ODE Page 220 Problems 1-21 odd, 31, 32 (If you would like more practice, also do Problems 2-8 even, 12-22 even)
Due Wed 3/8
HW 9 (Even & Odd Functions)
Do these problems.
Due Mon 3/6
HW 8 (Intro to Linear ODE's)
Do these two problems.
Due Fri 3/3 (Updated)
HW 7B (Linear Independence)
For the below problems, you may either use Wronskians or the definition of Linear Dependence given in class.
1) Prove that any two non-constant functions which differ by a (non-zero) constant are Linearly Independent.
2) Prove that any set of functions that includes the (constant) zero function is Linearly Dependent.
3) Prove that the three functions: cos x, (cos x)^3, and sin x sin 2x are Linearly Dependent. (^3 means cubed)
Due Thu 3/2
HW 7A (Linear Independence)
Read ODE Lesson 63B Pages 774-776
Due Wed 3/1
HW 6 (Exact First Order ODE's)
ODE Page 79 Problems 5-17 odd. In Problem 7, "dy" should be outside the parentheses.
Due Mon 2/27
HW 5 (Bernoulli Equation)
ODE Page 97 Problems 3, 7, 8, 22 (done below as separable, do again as Bernoulli and compare), 23
Due Thu 2/16
HW 4 (Homogeneous First Order ODE's)
ODE Page 61 Problems 3, 5, 6, 8, 12-15 (solve explicitly for y in 12-15) (Hint: Problem 3 is much easier if x is regarded as the dependent variable and you let u = x/y)
Due Wed 2/15
HW 3B (More Linear First Order ODE's)
1) ODE Page 97 Problems 21, 22, 24. Solve explicitly for the solution y = f(x) and determine the interval of validity.
2) Do this problem.
Due Mon 2/13
HW 3A (Linear First Order ODE's)
ODE Page 97 Problems 1, 2, 5, 6, 9, 11, 13, 14, 17
Due Thu 2/9
HW 2B (Separable First Order ODE's)
ODE Page 55 Problems 10, 12, 18, 19, 20, 21
Due Wed 2/10
HW 2A (Separable First Order ODE's)
ODE Page 55 Problems 2, 3, 5, 7, 11, 17
Due Mon 2/8
HW 1 (Verifying Solutions to ODE's)
ODE Page 27 Problems 1-3; Page 37 Problems 1-4
Due Fri 2/5
Exam Results (Updated 5/26)
Note: All Test Statistics are before adding make-up points (if applicable).
Exam 1
Statistics
Student Work
Exam 2
Statistics
Student Work
Exam 3
Statistics
Student Work