This page contains various resources that I created to help myself and others in mathematical studies and teaching.
Contents
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. Also, while I try to update these worksheets every semester I teach a course, some do not get used or updated, so they may be somewhat outdated.
Graduate students: You are welcome to use my worksheets for your class! Please let me know when you do; it is nice to know when 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: Feel free to use my worksheets as an additional learning resource if you find them helpful! (Please do let me know if you find them helpful.)
All worksheets are written by Zach Deiman and released under a Creative Commons BY-NC-SA 4.0 International License.
This section is under construction as of Summer 2025.
Chapter 0: Algebraic Properties
Chapter 1: Linear Functions and Change
Chapter 2: Functions
Chapter 3: Quadratic Functions
Chapter 4: Exponential Functions
Worksheet 4.1: Introduction to the Family of Exponential Functions [PDF] [ZIP]
Worksheet 4.2: Comparing Exponential and Linear Functions [PDF] [ZIP]
Chapter 5: Logarithmic Functions
Worksheet 5.2: Logarithms and Exponential Models [PDF] [ZIP]
Worksheet 5.3: The Logarithmic Function and its Applications [PDF] [ZIP]
Chapter 6: Transformations of Functions and Their Graphs
Worksheet 6.1: Shifts, Reflections, and Symmetry [PDF] [ZIP]
Worksheet 6.2: Stretches and Compressions [PDF] [ZIP]
Chapter 7: Trigonometry Starting with Circles
Worksheet 7.1: Introduction to Periodic Functions [PDF] [ZIP]
Worksheet 7.2: The Sine and Cosine Functions [PDF] [ZIP]
Worksheet 7.3: Radians and Arc Length [PDF] [ZIP]
Worksheet 7.4: Graphs of Sine and Cosine Functions [PDF] [ZIP]
Worksheet 7.5: Sinusoidal Functions [PDF] [ZIP]
Worksheet 7.6: The Tangent Function [PDF] [ZIP]
Worksheet 7.8: Inverse Trigonometric Functions [PDF] [ZIP]
Chapter 8: Trigonometry Starting with Triangles
Worksheet 8.1: Trig Functions and Right Triangles [PDF] [ZIP]
Chapter 9: Trigonometric Identities, Polar Coordinates, and Complex Numbers
Worksheet 9.1: Trigonometric Equations [PDF] [ZIP]
Chapter 10: Compositions, Inverses, and Combinations of Functions
Worksheet 10.1: Revisiting Composition of Functions [PDF] [ZIP]
Worksheet 10.2: Revisiting Inverse Functions [PDF] [ZIP]
Worksheet 10.3: The Graph, Domain, and Range of an Inverse Function [PDF] [ZIP]
Worksheet 10.4: Combinations of Functions [PDF] [ZIP]
Chapter 11: Polynomial and Rational Functions
Worksheet 11.1: Power Functions and Proportionality [PDF] [ZIP]
Worksheet 11.2: Polynomial Functions and Their Behavior [PDF] [ZIP]
Worksheet 11.3: Zeros of Polynomials and Short-run Behavior [PDF] [ZIP]
Worksheet 11.4: Rational Functions [PDF] [ZIP]
Worksheet 11.5: The Short-run Behavior of Rational Functions [PDF] [ZIP]
Worksheet 11.6: Comparing Power, Exponential, and Log Functions [PDF] [ZIP]
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
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
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:
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.
QR Exam Solutions:
Real Analysis: QR Exam Solutions (2016 – 2021) [PDF]
Complex Analysis: QR Exam Solutions (2016 – 2020) [PDF]
Applied Functional Analysis: QR Exam Solutions (2018 – 2021, 2023) [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]
In past semesters, I held study groups for Real Analysis and Complex Analysis, and as part of this, I distributed weekly problem sets which you can view below. They contain most of the same problems and solutions as linked above, but are organized by topic rather than by exam.
QR Exam Study Group Problem Sets:
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 compiled notes for some of the alpha courses that 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!
Math 556: Applied Functional Analysis (Fall 2022) [PDF]
Math 592: Algebraic Topology (Winter 2023) [PDF]
Math 593: Algebra I (Rings and Modules) (Fall 2021) [PDF]
Math 597: Analysis II (Real) (Winter 2022) [PDF]
I also wrote notes for some other non-alpha courses that I took at the University of Michigan. Those notes can be found on the Writing page of my website.
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 through 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, which can be found above (see "Solutions"). I got these solutions after working on the problems on my own, revising my solutions to ensure that everything was correct, reviewing answers on Math StackExchange, and talking with other graduate students.
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.
Information on the Preliminary Exam can be found on the math department's website; see III. Preliminary Exam.
I passed the Preliminary Exam on January 7th 2025!
Here are the syllabus and transcript of my Preliminary Exam, as well as syllabi and/or transcripts of Preliminary Exams for other University of Michigan graduate students. The exact subject matter and structure will vary greatly for most graduate students, but these may be helpful for those who want examples of what a Preliminary Exam is like.
Zach Deiman (Complex Analysis and Probability): [Syllabus] [Notes] [Transcript]
Jasper Liang (Harmonic Analysis and PDE): [Syllabus] [Transcript]
Dylan Cordaro (Random Matrix Theory and Asymptotics): [Syllabus]
The following is a summary of advice that I got from others, in addition to my personal advice that I got from my own experiences.
Part of the strategy for the Preliminary Exam is understanding the purpose of the exam, just so you know what to expect. In particular:
Basically everyone passes the Preliminary Exam. It is said (by graduate students) that the Preliminary Exam is treated more as a formality required by the college. As long as you spend enough time preparing for the exam, you will pass!
You are not expected to be able to answer every question immediately. The Preliminary Exam is not only testing what you know; it is also testing what you don't know. While you should try to understand as much as you can, sometimes an examiner will ask you a question that you had not thought about before. It is normal to take some time during the exam to think about a question, and it is normal to have several minutes of back-and-forth discussion before you arrive at the answer. Part of the exam is showing the examiners how you think through questions.
The structure and syllabus of the Preliminary Exam depend on what the student and the advisor decide. Often the exam consists of recalling material from advanced courses and readings, summarizing the content of one or more research papers, giving a talk at a seminar, or a combination of these. The syllabus may be primarily constructed by the student (with advisor approval), primarily constructed by the advisor, or constructed with each other.
Here are my specific strategies on how I prepared for the exam and acted during the exam:
Before the exam: A couple of months before the exam, I wrote a summary of the research papers I was reading, as I was reading them, making sure to understand the big picture. A couple of weeks before the exam, I wrote a summarized version of that, containing only definitions, theorems, and proof sketches. A couple of days before the exam, I practiced speaking about all the topics in an empty room, both to make sure I knew the material and also to get comfortable with speaking about it at a chalkboard.
During the exam: When I did not know the answer to a question that I was asked, I talked through what I was thinking. I think this helped the examiners understand my thought process, and adjust their follow-up questions accordingly. Also, the more I talked, the more it felt like a casual conversation instead of a stressful exam. After explaining something, I would often ask things like "Should I explain this more?" Sometimes the examiners would say "No" and so I didn't have to spend too long on the details.
After the exam: I celebrated and made sure to get plenty of rest!
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 focused on reviewing multivariable calculus and abstract algebra, and also taught myself the necessary material 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.