Resources
This page contains various resources that I have created or used to help me in my mathematical studies and teaching.
Worksheets
As a Graduate Student Instructor, I write my own worksheets for my students to use during class. The worksheets are listed here for the convenience of my students, and also for any other instructors who may be interested in using my worksheets for their classes.
The worksheets are numbered in the same way as chapters and sections in the textbook for their respective courses.
Math 105: The textbook is Functions Modeling Change: A Preparation for Calculus, 6th Edition by Eric Connally, et. al.
Math 115 and Math 116: The textbook is Calculus: Single Variable, 8th Edition by Deborah Hughes-Hallett, et. al.
Math 215: The textbook is Calculus: Early Transcendentals, 9th Edition by James Stewart, et. al.
Please note that I have not written a worksheet for every section in these textbooks.
Graduate students: If you would like to use my worksheets for your class, please let me know! You are of course welcome to use my worksheets; I just think it would be nice to know if anyone uses them. Credit in some form is also appreciated. I have posted PDF files and ZIP files for the convenience of any instructor who would like to use my worksheets and make changes.
Undergraduate students: I encourage you to use my worksheets as an additional learning resource if you find them helpful! (Please do let me know if you find them helpful.)
I have solutions for some worksheets, although some of them may be outdated if I revised the worksheet and did not get a chance to revise the solutions. Feel free to email me for these solutions, although I definitely recommend that you try solving them alone or with others.
All worksheets are written by Zach Deiman and released under a Creative Commons BY-NC-SA 4.0 International License.
Math 115: Calculus I
Chapter 1: Foundation for Calculus: Functions and Limits
Chapter 2: Key Concept: The Derivative
Chapter 3: Short-cuts to Differentiation
Chapter 4: Using the Derivative
Worksheet 4.1: Using First and Second Derivatives [PDF] [ZIP]
Worksheet 4.4: Families of Functions and Modeling [PDF] [ZIP]
Chapter 5: Key Concept: The Definite Integral
Worksheet 5.1: How do we measure distance traveled? [PDF] [ZIP]
Worksheet 5.3: The Fundamental Theorem and Interpretations [PDF] [ZIP]
Worksheet 5.4: Theorems about Definite Integrals [PDF] [ZIP]
Chapter 6: Constructing Antiderivatives
Worksheet 6.1: Antiderivatives Graphically and Numerically [PDF] [ZIP]
Worksheet 6.2: Constructing Antiderivatives Analytically [PDF] [ZIP]
Miscellaneous
Math 116: Calculus II
Chapter 4: Using the Derivative
Chapter 5: Key Concept: The Definite Integral
Chapter 6: Constructing Antiderivatives
Worksheet 6.2: Constructing Antiderivatives Analytically [PDF] [ZIP]
Worksheet 6.4: Second Fundamental Theorem of Calculus [PDF] [ZIP]
Chapter 7: Integration
Chapter 8: Using the Definite Integral
Chapter 9: Sequences and Series
Chapter 10: Approximating Functions Using Series
QR Exams
Recent Qualifying Review (QR) Exams and solutions can be found on the math department's website: Past Math QR Exams and Syllabi.
I took the following QR Exams during my time at the University of Michigan:
Solutions
Here are solutions that I wrote to some past Analysis QR Exams at the University of Michigan, intended to help current and future graduate students prepare for the QR Exams. Feel free to send me any questions you have about these solutions, including mistakes you have found or clarification questions that would help you better understand the material. If you have an alternate solution to one of these problems, I would be interested in seeing it. Also, if you have found these solutions to be helpful, I would greatly appreciate it if you let me know that.
Real Analysis: QR Exam Solutions (2016 – 2021) [PDF]
Complex Analysis: QR Exam Solutions (2016 – 2020) [PDF]
Applied Functional Analysis: QR Exam Solutions (2018 – 2021) [PDF]
University of Michigan Mathematics Ph.D. students Max Lahn, Ben Riley, Lukas Scheiwiller, and Ethan Zell have also compiled solutions to some of the past Topology QR Exams. This includes Differential Topology and Algebraic Topology. I have included their solutions below for convenience.
Topology: QR Exam Solutions (2016 – 2020) [PDF]
Notes for Alpha Courses
The "alpha courses" at the University of Michigan are a collection of courses designed for Mathematics Ph.D. students to gain proficiency in core mathematical areas and pass the QR Exams. Either out of interest or as a study tactic, I may sometimes write up notes from the alpha courses I have taken. If you are interested in learning more about a specific subject area, or you would like an additional resource for learning/studying purposes, feel free to take a look!
Study Strategies
To study for the Algebra I QR Exam, I worked through lots of problems, making sure to cover the material listed in the syllabus. I found it especially important to work some easy problems over and over again until I could consistently solve them without any notes. I think solving these easier problems allowed me to retain information and strategies to help me solve the harder problems.
To study for the Complex Analysis QR Exam, I wrote out a "cheat sheet," a compact list of notes covering everything listed in the syllabus. In the past, I had found this strategy helpful to study various exams where a cheat sheet was allowed on the exam.
To study for the Real Analysis QR Exam, I worked through a collection of the QR Exams that were listed on the math department's website but did not have posted solutions. I wrote up these solutions and posted them above (see "Solutions"). I got these solutions after working on the problems for a while and taking time to revise my solutions to ensure that everything was correct. For some problems, various answers on Math StackExchange were helpful. Talking with other graduate students was also helpful in compiling solutions.
To study for the Algebraic Topology QR Exam, I worked through a collection of the QR Exams that were listed on the math department's website. I wrote out another "cheat sheet," and I also typed up some more comprehensive notes with plenty of examples to help me retain the material. My notes are posted above (see "Notes for Alpha Courses").
Below are various resources I wrote to help me prepare for these exams.
GRE Mathematics Subject Test
The best way for me to study for the GRE Mathematics Subject Test was to review the necessary material and take lots of practice tests. The official GRE website has more information on what subjects are covered, and even has one practice test. The most helpful resource for me to study from was The Princeton Review's book, Cracking the GRE Mathematics Subject Test, 4th Edition. It contains a comprehensive review of material that is typically covered in the test, and also contains a couple of practice tests. Also extremely helpful was this page from Math StackExchange, which provides many more examples of practice tests and sample questions (although be aware that many of the links no longer work).
I studied for the test during Summer 2019. I particularly focused on reviewing multivariable calculus and abstract algebra, and I also taught myself the necessary topics in point-set topology and complex analysis. I took the GRE Mathematics Subject Test on October 26th 2019. I got a scaled score of 760 (on a scale of 200 to 990), which at the time was in the 70th percentile; over time, this dropped to the 66th percentile, as this metric is based on the performance of those who tested between July 1st 2018 and June 30th 2022.