Resources (Updated 2/6)
Dr. Freeman's Daily Schedule (TBA)
Office Hours
Period 3 - Tuesday
Period 7 - Monday, Wednesday
Room 719D
You may also 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.
Wolfram Alpha Calculus Examples
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
Veitch Calculus 2 Final Exam Study Guide
Differential Equations Study Guide
Announcements (Updated 6/5)
Here is the Take-Home Portion of the Final Exam.
The Final Exam will consist of two parts: Part 1 (Take-Home Problems) and Part 2 (In-Class Problems)
Part 1 of the Final Exam will be available for download before 3:20 PM on Tue 6/5 and is due at 1:35 PM on Wednesday 6/6.
Part 2 of the Final Exam will be on Wednesday 6/6 from 1:35 PM to 3:20 PM (Periods 6 & 7) in Room 632.
Solutions to Exam 3 are posted below.
Problems Set Due Fri 6/1 9:00 AM
Remaining Presentations
Thu 5/31 8:35 AM - P.M. - Application of First ODE's - Movement of Liquids
Fri 6/1 8:35 AM - P.S. - Application of First ODE's - Geometric Problems
Please make sure you have attended the required number of Presentations.
As agreed by all, Exam 3 will start at 8:35 AM sharp. Please plan to arrive at or before 8:30 AM, as late arrivals will not be given extra time. Good Luck!
Exam 3 will be on Mon 5/21 and will cover through HW 31.
The Final Exam will be on Wednesday, June 6 (tentatively).
Please see updates to Presentation Assignments below.
Quiz 8 (Take Home) Due Mon 5/7 at 9:00 AM
If you are absent on Monday for any reason, please submit electronically as a single PDF file by the due time.
Quiz 7 will be on Mon 4/30 will cover VoP (but you may need to know MUC as well).
Solutions to Exam 2 are posted below.
Exam 2 has been rescheduled for Monday 4/23 and will cover through HW 23B.
Quiz 6 Solutions. Please make sure you can do these types of problems before Exam 2!
Officially, Quiz 6 will cover HW 16 - HW 20. However, you may find the following two skills to be particularly useful:
1) Knowing how to calculate sin z, cos z, cosh z, and sinh z where z is a given complex number.
2) Knowing how to use complex exponentials to efficiently evaluate certain real integrals.
Quiz 6 will be on Friday 4/13.
To satisfy the Presentation component of your final grade (15% - see Syllabus), you have two options:
Option 1) Present a 20-minute oral presentation on a mutually-agreed topic, and attend a number of other student presentations (this number will be determined by the number of students electing this option, but will be no greater than three).
Option 2) Complete a formal write-up of 3 Problem Set problems, as described below.
If you would like to elect Option 1) you are required to visit me in my office before 11:30 AM Thursday, March 29th to discuss. Otherwise, you will be deemed to have elected Option 2) by default.
Quiz 5 will be on Wed 3/28 and cover HW 13 - HW 15.
Quiz 4 (Take Home)
You are expected to use your brain and class notes only. No collaboration. No internet resources. Due Mon 3/19 in class. It is strongly suggested that you write a multiplication table first.
Solutions to Exam 1 are posted below.
As agreed by all, Exam 1 will start at 8:35 AM sharp. Please plan to arrive at or before 8:30 AM, as late arrivals will not be given extra time. Good Luck!
Exam 1 will be on Fri 3/9 and will cover through HW 10 & HW MFO.
Thu 3/8 will be student-led review. Everyone should be prepared to present and discuss 2-3 problems from old Exams and/or HW. You should decide among yourselves who is presenting which problems to minimize overlap, and so that all topics are covered.
Quiz 1 and Quiz 2 will be on Thu 2/8 and Thu 2/15.
The Exams are tentatively scheduled for Fri 3/9, Fri 4/20, and Mon 5/21 (Fri 5/18 is Symposium Day).
Starting Wed 2/7, we will meet every Wednesday at 8:30 AM in Room 632 to discuss homework.
Welcome to Advanced Applied Math.
Presentation Assignments (Updated 5/9)
Problem Set Option (Problem Set)
1) Complete Problem 1
2) Complete Problem 2 (added on 5/9)
3) Attend a Presentation
Presentation Option
1) Give a 20-minute Presentation (including Q & A).
2) Attend a Presentation
3) Attend a second Presentation
Below is the Presentation schedule. Presentations will start exactly at the scheduled times. You must be on time and present for the entire Presentation to receive credit for attending.
Thu 5/17 8:35 AM Rm 632 - C. R. - Dilution and Accretion Problems
Fri 5/18 1:35 PM Rm 630 - A. B. - The Basel Problem (Symposium Day)
Thu 5/24 8:35 AM Rm 632 - P. S.
Fri 5/25 8:35 AM Rm 632 - T. S.
Thu 5/31 8:35 AM Rm 632 - P. M. - The Application of ODEs in the Movement of Liquids
Homework (Updated 6/1)
(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 (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 - recommended as part of you Final Exam studying
HW 33 (Complex Logarithms)
FCA Page 123 Problems 1, 3, 4, 5, 6, 9, 10, and these Problems
Due Wed 5/30
HW 32 (Exp, Trig, & Hyp Functions)
FCA Page 115 Problems 1, 11, 12, 13, 15, 17, 20
Due Fri 5/25
HW 31 (Harmonic Functions)
FCA Page 84 Problems 1, 2, 3, 5, 6, 14
Due Thu 5/17
HW 30 (Cauchy-Riemann Equations)
FCA Page 77 Problems 1, 3, 5, 9, 10
Due Wed 5/16
HW 29 (Analyticity)
FCA Page 70 Problems 4, 7, 9, 11, 13
Due Mon 5/14
HW 28.5 (Admissibility)
Do these problems.
Due Fri 5/11
HW 28 (Limits and Continuity)
FCA Page 63 Problems 7, 11, 12, 13, 17, 21, 25
Due Thu 5/10
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 and Spring 2017 Exam 3 Problem 7
Due Wed 5/9
HW 26 (Planar Sets)
FCA Page 42 Problems 2-8, 11, 12, 15-20
Also do Spring 2015 Exam 3 Problem 1 and Spring 2017 Exam 3 Problem 1
Due Mon 5/7
HW 25 (Reduction of Order)
ODE Page 246 Problems 3, 5, 9, 15
Due Thu 5/3
HW 24 (Series Solutions Near an Ordinary Point)
ODE Page 546 Problems 3, 5, 7, 9, 11
Due Mon 4/30
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 Fri 4/27
HW 24 (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 4/26
HW 23B (Complex Exponentials are Your Friends)
Do these Problems.
Due Thu 4/19
HW 23A (Method of Undetermined Coefficients)
ODE Page 231 Problems 3, 6, 7, 9, 11, 13, 15, 17, 19, 23, 30, 32
Due Wed 4/18
HW 22 (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 Mon 4/16
HW 21 (Primitive nth Roots of Unity in Z mod p)
(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? If not, why not?
Due Fri 4/13
HW 20 (Powers and Roots)
FCA Page 37 Problems 4, 5, 7, 9, 10, 11, 15, 16
Due Wed 4/11
HW 19 (Return to Calculus II)
FCA Page 32 Problem 23 (answers are in the back of the book)
Due Mon 4/9
HW 18 (Return to 9th Grade Geometry)
Do this problem.
Due Mon 4/9
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 Thu 3/28
HW 16 (Complex Exponentials)
FCA Page 31 Problems 1, 2, 3, 4, 7, 8, 9, 10, 11
Due Wed 3/28
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.
Mon 3/26
HW 14 (arg!)
FCA Page 22
Problem 6 ("cis(theta)" is a ridiculous abbreviation for "cos(theta) + i sin(theta)"), Problem 7
Due Fri 3/23
HW 13 (Welcome to Complex Analysis II)
FCA Page 13 Problems 7, 9, 11, 13, 16
FCA Page 22 Problem 4 (just mimic the induction argument we used in class to prove the similar identity for conjugate), Problem 5
Due Thu 3/22
HW 12 (Fun with Finite Fields)
Prove that the set {0, 1, 2, 3} with addition and multiplication defined "modulo 4" is, in fact, NOT a Field. What goes wrong? (Hint: make addition and multiplication tables)
However, there is a Finite Field with 4 elements with a different addition and multiplication table. Type "finite field of order 4" into Wolfram Alpha to see it. How do you find the additive inverse of an element in this field? Which elements have square roots in this field? Solve "a x^2 + b x + 1 = 0" in this field, where {0, 1, a, b} are the field elements.
Due Fri 3/16
HW 11b (Welcome to Complex Analysis Ib)
FCA Page 5 Problems 19, 20, 21, 23, 24, 25
Due Thu 3/15
HW Exam 1 Redux
Review the solutions to Exam 1. Find your mistakes. If you made algebra and/or calculus errors, come up with strategies to avoid making them in the future. If you got stuck, figure out why. If there is anything you do not understand, get is resolved ASAP!
Due Wed 3/14
HW 11a (Welcome to Complex Analysis Ia)
FCA Page 5 Problems 7, 9, 11, 13, 14, 17
Due Wed 3/14
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 3/8
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 3/7
HW 9 (Even & Odd Functions)
Do these problems.
This HW is optional. It is similar to the HW given for this topic in Calc II. You will not be tested directly on these types of problems, but as a AA Math student, you should know how to do them!
HW 8 (Intro to Linear ODE's)
Do these two problems.
Due Mon 3/5
HW 7 (Linear Independence)
Read ODE Lesson 63B Pages 774-776
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 the three functions: cos x, (cos x)^3, and sin x sin 2x are Linearly Dependent. (^3 means cubed)
Due Thu 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 2/26
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 2/15
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 2/14
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 2/12
HW 3A (Linear First Order ODE's)
ODE Page 97 Problems 1, 2, 5, 6, 9, 11, 13, 14, 17
Due Fri 2/9
HW 2B (Separable First Order ODE's - Part 2)
ODE Page 55 Problems 10, 12, 18, 19, 20, 21
Due Wed 2/7
HW 2A (Separable First Order ODE's - Part1)
ODE Page 55 Problems 2, 3, 5, 7, 11, 17
Due Wed 2/7
HW 1 (Verifying Solutions to ODE's)
ODE Page 27 Problems 1-3; Page 37 Problems 1-4
Due Mon 2/5
Exam Results (Updated 6/1)
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