This course builds on your foundation in calculus by introducing advanced integration techniques, applications of integration, improper integrals, numerical methods, calculus of parametric curves, and infinite sequences and series. These topics are core analytical tools used throughout science, engineering, and applied mathematics.
If some of this terminology is unfamiliar, that’s expected, you’re taking this course to start learning these concepts! The first section below outlines the kinds of problems and questions these concepts help us address—from modeling physical systems to analyzing complex data. By the end of the course, you’ll have a basic understanding of how calculus might help with rigorous problem-solving in STEM fields.
In addition to mastering these new topics, you’ll continue to reinforce fluency with essential skills from arithmetic, algebra, trigonometry, and Calculus I. These fundamentals remain critical across the sciences for precisely quantifying and interpreting relationships in real-world systems.
A planned schedule for the semester will be provided on D2L during the first or second week of class. This second semester course in the standard three-semester sequence will cover Chapters 6 through 9 of the textbook officially adopted by our department as described in Subsection 3.9, although almost any modern Calculus textbook will serve for this section of the course. Please refer to the schedule for important dates such as quizzes and exams.
Roughly, our semester is organized to study the following questions (in order):
How can the relationships between integrals and derivatives help us model real-world phenomena? In general, how can integration help us calculate useful quantities?
This rigorous mathematical subject consists of a basic mathematical tool kit for dealing with processes that change continuously. Part of the reason that calculus turns out to be so useful is that the details of the process turn out not to matter so much -- all that matters is that the process is continuous. In fact, if the increments are small enough, discrete processes can sometimes be approximated as continuous and treated using the techniques of calculus. More specifically, Calculus is the story of infinitely improving approximations concerned with two historical problems:
Miraculously, these two problems, which may seem barely related except for the type of line they seek to study, turn out to be deeply intertwined. This semester, we will primarily focus on the second problem, but due to the dual nature of these problems the other is necessary and a skill which you hopefully developed during Calculus I.
In order to do well in MATH122, you need to work on developing both conceptual understanding and algebraic fluency. Conceptual understanding is necessary as memorized facts are not much help if you do not know how to use them. Algebraic fluency is required as knowing the basic concepts is not very helpful if you are unable to easily use them because your knowledge is not built on a firm foundation of the governing principles.
By fluency, I mean that you can generally use them without thinking about them. Fluency is defined as the ability to speak or write a language easily and accurately. Similar to your fluency in your native language, where you likely are fluent in 10000-20000 words, you are expected to be fluent in a number of "mathematical rules" which you have learned over the last several years. In Subsection 3.12, there is a full inventory of the algebraic, trigonometric, and Calculus I skills with which you need to be fluent to be truly successful in this and future courses.
Formally, being enrolled in MATH122 requires having passed MATH121 with a ’C’ or better, or explicit consent of the instructor. Any student being found not to meet these requirements will automatically receive a failing grade at the end of the semester.
In practice, Calculus II depends heavily on algebra, analytic geometry, trigonometry and your knowledge from Calculus I. Several review worksheets are posted on D2L Brightspace to help you recall the knowledge that will help you succeed in Calculus II this semester. A list of specific material that will be used to scaffold new knowledge can be found in the Prerequisite Knowledge for Calculus II section: Subsection 3.12.
Upon successful completion of this course, students will be able to:
Demonstrate a conceptual understanding and computational proficiency in Riemann integration theory, including its specific applications such as area, volume, and physical modeling.
Analyze and work with sequences and series, including power series and Taylor expansions, demonstrating both their mathematical structure and their application to problems in calculus.
Accurately recall and apply key calculus formulas and theorems, such as derivatives and antiderivatives of standard functions, while also demonstrating the ability to derive and interpret these results.
Solve complex calculus problems by developing persistence, mathematical reasoning, and problem-solving strategies that go beyond memorization—focusing on understanding the underlying concepts of continuous change and integration techniques.
This course will be delivered in person. Active engagement is essential to your success, and students are expected to participate fully during each class session. Most class periods will include lectures punctuated by individual or small-group problem solving. Students will regularly be asked to suggest next steps or provide answers during worked examples. In some sessions, collaborative group work may fully replace lecture.
Continuous engagement with this course is essential to learning the material and achieving the course learning outcomes. You are expected to check D2L Brightspace at least once every 24 hours on weekdays. While we understand that you have commitments outside this class, to be successful in this course you should plan to work on it every school day.
Early in the semester, frequent pre-class assignments will be assigned, with additional ones distributed periodically throughout the term. These are due at the beginning of the next class period. Pre-class assignments are intended to prepare you for in-class activities, reinforce recent material, assess your progress in writing clear mathematical solutions, and support long-term goals such as the final project by breaking them into manageable stages.
There are frequent deadlines posted in D2L, and it is your responsibility to keep track of all due dates. You are encouraged to work ahead of those deadlines whenever possible to avoid last-minute issues. Regular attendance at all lecture meetings is expected and strongly contributes to your success.
You are encouraged to ask questions whenever you encounter difficulties. You may raise your hand during class, talk with peers during group work, or seek assistance outside of class. If you are more comfortable seeking help privately, you are welcome to email the instructor, attend office hours (either individually or in a small group), schedule a one-on-one meeting, or ask questions immediately after class.
If you are unable to attend class or participate due to serious illness (including COVID-19 or other health conditions), you should contact your instructor as soon as possible to make arrangements. You should also reach out to Accessibility Resources to explore appropriate accommodations.
Students enrolled in MATH 122 – Calculus II at Minnesota State University, Mankato demonstrate competencies aligned with the Minnesota teacher licensure standards, as outlined in Minnesota Rule 8710.4600. This course is part of the Bachelor of Science in Mathematics Teaching program and addresses the following standard components:
The course content includes techniques and applications of integration, improper integrals, numerical methods, parametric equations, and infinite series and sequences. These align with competencies in the following areas:
Understanding of Number Systems: Includes whole numbers, integers, rational numbers, and real numbers. Calculus specific studies analytic properties of the real number system, and sequences and series are related to both the reals and integers.
Application of Mathematical Procedures: Emphasizes recognizing patterns, analyzing variable relationships, and applying limits and boundedness in sequences and series.
This content is integrated with pedagogy through courses such as MATH484 – Technology in Secondary School Mathematics and MATH485 – Teaching Secondary School Mathematics, equipping future educators with both subject knowledge and effective instructional strategies.
By meeting the standards above, MATH122 – Calculus II plays a key role in preparing students to meet Minnesota licensure requirements and excel as secondary mathematics teachers.
MavPASS (Peer-Facilitated Academic Support System): This semester, we are offering additional support to help students to succeed in this course. Students who have taken this class in a previous semester have been hired as MavPASS Leaders to sit in on the course with you, facilitate study sessions outside of class, and hold office hours.
Note, the two Working Sessions cover the same material, so you may attend whichever is more convenient for your schedule (you do not need to attend both, but you are welcome to, however only one per week may be used to earn Learning Problem points). You can also find the schedule at www.mnsu.edu/mavpass. For additional information, please email mavpass@mnsu.edu.
To give a sense of the scale, scope, and importance of Calculus II, this is the current list of courses which require Calculus II with a ’C’ or better as an entry requirement. Note, for Computer Science, many of your classes list MATH247: Linear Algebra, which is dependent on Calculus II. This is likely true for other programs as well, and I’d like to get a similar list for MATH247 in the near future.
Finally, the degree programs for Biotechnology (BS) and Economics (BA/BS) strongly recommend Calculus II for any students planning to pursue graduate or professional school.
However, since the Learning Problems will be given as PDF files on D2L BrightSpace, almost any Calculus textbook will suffice. This includes any edition of Stewart’s texts, freely available Open Educational Resource (OER) options from sites like OpenStax, and others. The most important consideration is the order of topics. For example, Stewart’s textbooks typically present advanced integration techniques before the applications of integration, while this course takes the reverse approach—motivating techniques through early applications.
A textbook that follows a similar order to this course is Calculus by Larson and Edwards (currently in its 12th edition, though any edition will work). Students looking for an affordable physical textbook might consider an older edition of Larson and Edwards.
This book is freely available online and as a downloadable PDF. Print copies can typically be purchased for $15–$20 via Amazon. We will primarily cover content from Calculus 2, with a few sections drawn from Calculus 3. If you prefer a printed textbook, APEX Calculus Volume 2 is likely sufficient; the few Calculus 3 topics we cover can be accessed digitally.
Please be aware, that if you continue into Calculus 3, you may still be required to purchase the current edition of Stewart’s Calculus to obtain access to the digital homework system used by other instructors. These alternatives are provided in this section as a low-cost alternative to Stewart’s Calculus.
You will earn the following letter grade based on the calculation of your weighted course average based on your demonstration of course learning objections through course assessments.
For a general interpretation of grade letters and how they are designed to reflect your understanding of the material and preparedness for future courses, see Subsection 6.2.
Your final grade in this course will be based on your demonstrated understanding of the course learning outcomes, as shown through a variety of assessments. The table below shows each assessment type along with its weight toward your final grade. Descriptions for each assessment follow the table.
I am more interested in your reasoning than just your final answers. In real-world problem solving—especially in STEM—understanding the process is often more important than getting the "right" number. That’s why many out-of-class assignments include final answers: your focus should be on how to get there.
On quizzes and exams, minimal partial credit will be given unless your work includes clear, logical steps. I want to see how you’re thinking—not just what you concluded.
While not every technique we cover will apply equally to every major, the mathematical reasoning and problem-solving skills you develop will benefit you in future courses. Mastering core competencies—such as algebraic manipulation, structured reasoning, and communication—will serve you well far beyond this class.
My goal is to help you build both confidence and competence in mathematical thinking. These are lifelong skills with value across all scientific and technical fields.
This is a take-home quiz. You may use resources such as your textbook, tutoring services, office hours, or group study. It will be handed out on the first day of class and is due on Thursday of the first week.
This in-class quiz will take place on the first Friday of the semester. A sample version will be posted on D2L on the first day. Most of the quiz will closely resemble this sample, with only small variations. You are expected to have key facts and procedures memorized and ready for quick recall.
During most class meetings, you’ll work in small groups to solve problems, explore new ideas, and deepen your understanding of course concepts. Your participation grade is based on meaningful contributions to group discussions and collaborative tasks.
This includes being prepared, staying engaged, helping your teammates, and working through challenges together. Active participation benefits not only your learning but also supports the group as a whole.
Preclass Assignments are brief tasks designed to assess your mathematical writing, reinforce foundational knowledge, and support exploration of key course competencies and their connections.
A Preclass Assignment will be assigned for each class meeting during the first two weeks of the semester. These early assignments serve several purposes: to gauge your initial knowledge and skills, to help you understand course expectations, to encourage daily practice habits, to provide early and frequent feedback, and to allow both you and the instructor to make timely adjustments. These assignments also help the instructor determine whether early progress reports or additional academic support may be necessary for any students.
As the semester progresses, Preclass Assignments will occassionally be assigned on Fridays (due the following Monday) or on Tuesdays (due the following Thursday), and are generally only scheduled on days without other major assessments. These assignments are meant to sustain regular engagement with the material and ensure you are prepared for in-class activities.
You will keep a journal to document your work on the Learning Problems sets assigned on D2L for each topic of the course. While lectures will initially introduce you to key concepts and ideas, the real learning happens when you actively engage with the Learning Problems. Lectures outline what to do; the Learning Problems give you the variety, practice, and repetition needed to figure out how to actually do it.
These sets are graded for completion: 1 point for completing at least 50–70%, and 2 points for completing more than 70%. These completion levels are strong indicators of overall success in the course. If you’re aiming for a high grade (especially an A), you are strongly encouraged to complete all problems thoroughly and on time, and to carefully write out clear, organized solutions in your journal.
Your journal must be submitted for review at each quiz and exam. These are the only “real” deadlines for the Learning Problems, although suggested target dates will be posted with each set to help prevent them from piling up. Submitting your journal allows me to monitor your progress and provide feedback when necessary.
To be eligible to revise exams for partial credit, you must have mostly completed all assigned Learning Problem sets relevant to the exam. You can also earn up to two Learning Problems points per week by attending MavPASS sessions. Later in the semester, if appropriate, I may allow students to apply “excess” Learning Problem points toward limited exceptions to course policies—such as extensions on late assignments or removal of absences from the attendance record.
These will consist of 2-4 problems from the indicated sections. These problems are similar to those found on the Learning Problems sets, in-class groupworks, or preclass assignments. Often I may allow questions immediately before the quiz, with the quiz taking approximately half of the class period. If time allows, we may immediately review some or all of the quiz for the remainder of the class period. Your lowest quiz score will be dropped when calculating your final course mark at the end of the semester.
About one week prior to each chapter exam, a practice problem set will be posted. These sets will include approximately 80 problems, some of which will appear directly on the exam. If you have been consistently completing the Learning Problems sets, this packet will serve as effective review and reinforcement.
However, if you have not completed the ongoing practice, it is not realistic to expect that you can work through the entire problem set in the time available. Instead, use it strategically: scan through the problems to identify content areas where you feel uncertain or cannot immediately determine how to begin. If you find yourself unable to sketch out a general solution process for a problem, that topic likely needs further review.
To reinforce these areas, you are encouraged to bring questions from the packet to academic support resources such as MavPASS sessions, the Math and Statistics Learning Center, office hours, or the Center for Academic Success. For more information on these resources, see Section 5.
Exams will be completed in class and will follow a traditional format. Approximately 50% of each exam will consist of problems taken directly from the study packet. The remaining questions will be drawn from or modeled on problems from the Learning Problems sets, Preclass Assignments, Groupwork Assignments, and Quizzes.
Students who are meeting course expectations will have the opportunity to earn back partial credit by submitting written reflections and corrected solutions after receiving a graded exam. This revision process is designed to reinforce understanding and promote deeper learning, as well as allowing students to learn from their mistakes.
As your instructor, I fully expect you to make mistakes—that’s part of the learning process. However, it remains essential that you approach each exam with serious preparation and genuine effort. Revisions are not a substitute for studying; they are a structured opportunity to reflect on your performance, identify patterns in your understanding, and strengthen the academic habits that support long-term success.
The revision process is designed to help you engage more deeply with the material and to develop critical skills such as self-assessment, persistence, and metacognitive awareness. If one or more exams do not go as planned, revisions allow you to recover some credit while learning how to better prepare for future assessments.
That said, revisions should not be viewed as a fallback or safety net. Students who rely on revisions instead of putting forth their best effort from the outset often fall behind in the course and struggle to catch up. Use revisions as a learning tool—not a crutch. The goal is not just to improve your grade, but to grow as an independent and resilient learner.
For your final project, you will produce a creative work summarizing your learning throughout the semester. You will briefly (1-2 minutes each) present your work the last day of class. This presentation should be a brief description of the project, challenges encountered and overcome (or not), and key takeaways from your project.
In prior semesters, students have produced paintings, sculptures, short films, children’s books, board games, novellas, infographics, comic books, five-paragraph essays, and slide presentation. (Fair warning: These last two have tended to fair the worst on meeting the project goals...).
The primary goal of your final project is to summarize your learning from the semester and organize your thoughts about the material and how various topics throughout the semester are related. Your chosen format can elevate or denigrate the information you choose to include.
I consider the purpose of a final exam to be for students to revisit their learning from throughout the semester with the broad perspective they’ve gained by completing all course work. We must present information in a particular order, but information is often not strictly linear. You will often find new insights on early course material from things learned late in the term. Similarly, you may re-evaluate later knowledge having refreshed topics from early in the semester.
Given this perspective on finals, much of this is instead accomplished by your final project. Consequently, the final exam in this class is optional -- you may use it to replace one prior exam score, which is particularly helpful if you had an off-day or exceptionally busy week when one of the prior exams occured. Note, you may take the exam first and then decide wether or not it should be graded based on if you think you did better or worse than your previous lowest exam mark. If you choose to have the final exam graded, it will replace your lowest exam score, better or worse.
This document was originally created in Fall 2023 and will be updated as needed. If you have suggestions or feedback, I’d be glad to hear from you—your input is always welcome!
The purpose of this list is not to teach you the concepts from scratch. Rather, it serves as a comprehensive reminder of the topics you’ve already studied and will need to use fluently in Calculus II. Your textbook, online resources, and other references will likely assume you know these topics, which means they often won’t be explicitly explained in problems or solutions—even those found online. This list provides the names and brief descriptions of key concepts to help you identify any gaps in your understanding. It is your responsibility to ensure you’re prepared to apply them whenever they arise, both in this course and beyond.
What follows is a curated list of foundational topics that students often need in order to succeed in Calculus II. For each, I’ve noted where you likely first encountered it, though you’ve probably seen them multiple times since. This goes well beyond the general prerequisite of "Completed Calculus I with a C or better." These are the specific pieces of knowledge you’ll be expected to build upon. Struggling with these concepts is a common reason students find Calculus II challenging. You’ve invested many years into learning mathematics up to this point—you’re now expected to master and apply these ideas fluently. Math is the language of many sciences, and these foundational concepts remain critical well beyond this course.
Also, if you want to study classical physics, Newton literally wrote a book called "Conic Sections" -- because they are that important -- and completely the square is a fundamental calculation for understanding them.
Example: find \(\frac{dy}{dx}\) of \(y^2+3x^2 - \sin(xy) = 15\) or if \(g\) is also a function of \(x\text{,}\) calculate \(\frac{df}{dx}\) if \(f = g^2\ln(\sin(g))\text{.}\)
You don’t need to know the proof, but you should appreciate the reason Cauchy’s Mean Value Theorem is the key idea that makes this magnificent theorem work.
