Math 131: Calculus I

Fall 2026

Department of Mathematics, Grinnell College

Books and Resources

We will follow a textbook which I am particularly fond of:

Calculus by James Stewart

(the schedule is based on the 9th edition; it is perfectly fine if you have an earlier edition, though you'll need to adjust the reading schedule accordingly. If you have trouble doing this, talk to me!)

There is an enormous amount of calculus textbooks and online resources out there, each with a slightly different point-of-view. I, for example, like Grant Sanderson's visualization of various ideas in calculus. If you are looking for supplementary resources, talk to me and I will be more than happy to recommend some!

Unsolicited advice: The textbook will be your best resource for this semester. I will assign reading for most class meeting and I strongly encourage you to complete it consistently; you can find the reading schedule here. Spending the time to digest the exposition in the textbook (highly non-trivial!) will help you glimpse the themes of calculus. This, in turn, will be your most reliable problem-solving tool. See, also, the Reading and Engagement section below.

Math Lab. The Math Lab, located in Noyce 2012, is a great place available for you to work with classmates and/or receive calculus help.

Course Goals and Basic Principles

Basic principles. The policies of this course are developed according to the following principles.

Every student can have a positive and meaningfully challenging mathematical experience.
Engaging with, and solving, meaningfully challenging mathematics problems is the best way to learn mathematics.
It is desirable and possible to evaluate students in a fair and equitable manner.

Course Goals. Topics covered in Math 131 are listed here. While covering these, I will expect students to meet the following course objectives.

Understand the core ideas of calculus. These include:

The idea of a "limit" and its applications to continuity, differentiability, and approximation.
The effective use of approximation arguments to compute quantities of interest and quantification of errors.
The basics of analytical geometry including definitions of areas, lengths, volumes, etc.

Model phenomena and quantify natural phenomena using derivatives and integrals.
Become fluent in calculus language and notation.
Explain your reasoning using words, formulas, and figures.
Begin to develop your perseverance in problem solving.

Reading and Engagement

Throughout the semester, you will be assigned reading out of the textbook. Reading this text will better position you to follow, and actively engage with, classroom discussion. The schedule for readings can be found here.

To help you develop a habit of active reading, I will assign almost daily homework (more on this in the Homework section below). Besides reading the text regularly and actively, I will expect you to attend class regularly, contribute to class discussions, and ask questions in office hours or via email.

Technology in the classroom.There is ample research to suggest that the use of technology in the classroom can be detrimental to learning. One mechanism in which this plays out is described in the following excerpt from this paper:

"The act of using a laptop migrates student focus away from participation and discussion surrounding the classroom content. Therefore, the learner begins to view classroom activities as something passive and the discussion background noise to the central focus of what is occurring on the laptop screen. The laptop computer or mobile phone may simply increase the perception of psychological distance from the discussion and reduce the sense of immediacy and involvement on the part of the student(s)."

As such, unless you have a specific accommodation, use of technology in our classroom will be restricted to note-taking technology only.

On reading mathematics. There is a wealth of information and advice on how one might read mathematics out there. This is something many of the greatest mathematicians have written about; see, for example, the end of this blog post, or point 6. in Tim Gower's blog post.

Assessment

Our course is graded using a methodology called standards-based grading, in which the exams do not have a point value or percentage. Instead, you earn your grade by showing appropriate engagement with the course and demonstrating evidence of skill on the course standards that describe the major ideas covered by each exam. These standards may be updated throughout the semester. Each problem on an exam will be clearly marked with a standard. If your work meets the standard, then you will receive full credit for it (marked as complete). Otherwise, you will get helpful feedback and, in the case of exam standards, the chance to reflect on the feedback, revise your work, and then reassess your understanding on the next exam.

To allow you to have ample opportunity to reattempt course standards, there will be five exams: 4 in-class exams and one final exam. The exam schedule can be found below.

Date	Day	Exam	# of Standards Available	Time
September 18	Friday	1	1 – 6	80 mins
October 16	Friday	2	1 – 13	80 mins
November 13	Friday	3	7 – 17	80 mins
December 11	Friday	4	14 – 22	80 mins
December 16	Wednesday	5	1 – 22	180 mins

Each exam will have questions that are directly associated with the course standards (see Course Standards below), and each standard will appear on at least two exams after the topic has been introduced in class meetings. Your goal will be to complete each standard at least once throughout the course. If you complete a standard on an earlier test, you do not need to attempt it on later tests. If you do not complete or attempt a standard on an earlier test, you will be able to attempt it again on a later test.

Caution: Exams will be long since they contain questions relevant to many standards, but midterm exams will always be allotted 80 minutes. If you are consistently completing standards as they come up, you will only need to solve a fraction of the problems on any given exam.

Final exam and re-certification. The final exam will serve two purposes:

Like a regular midterm-type exam, it is an opportunity for you to attempt to pass some standards that you are missing one last time.
A ``re-certification exam" where you will have the opportunity to re-certify core standards by solving relevant problems. To successfully re-certify a standards you must

have already have completed the standard before the final, and
complete one problem on that standard on the final exam.

Caution: The first bullet point implies that you cannot re-certify a standard if you have not passed it by the end of the semester!

Homework

General information. Homework will be assigned daily and will be due by next class period; the schedule for assignments can be found here and on our Canvas page.

Submission. Most, if not all, homework assignments will be administered via WebAssign. The calendar on Canvas shows all the homework assignment deadlines, and you can access our class-specific WebAssign page here and through the navigation links on Canvas.

Extensions. You may request a no-penalty extension on any written homework assignment or lab. If I receive your request before the assignment due date, the extension will be granted, and the new due date will be Monday of the following week by class time. Barring extenuating circumstances, no late requests will be granted. To request the extension, send me an email with the assignment name and a brief explanation for why the extension is necessary. Be sure to send your request before the original assignment deadline.

Expectations. All solutions submitted through WebAssign are expected to be your original work. That is, you did the mathematical work necessary to arrive at the result which you submit using appropriate resources (e.g. textbook, class notes, working with classmates etc. LLMs and online fora are never appropriate resources in our course.). For more on this and related academic honesty matters (e.g. citing collaborations), see the Academic Honesty section below. WebAssign allows you to store your thoughts and work on each question using the Show your Work feature. You are required to cite your collaborations using this features. I strongly encourage you to develop a system for storing your work on homework problems so that you may access it when studying later (e.g. use the "Show your Work" button, a dedicated notebook for problems with a labeling system, etc.).

On writing mathematics. The notion of well-written mathematics is subjective, but you probably have some sense of what you do and do not like to see in mathematical writing (from, say, reading your textbook). Many mathematicians have written on the subject; see these tips, this advice, or this guide, for example. A great source where many such writings are compiled is Terry Tao's blog post. Of course these sources might sometimes disagree, and that is OK. What you should ultimately focus on is the clarity of your argument and reasoning; not only for the benefit of the reader, but also for your own. The mathematical story in your own mind can be greatly clarified with good writing!

Concept Checks

Concept Checks (CC) will be given in class on Wednesdays and will be based on the previous week's material. Concept checks will be graded on a discrete scale with the following possible grades:

Satisfactory (S): The solution presented is entirely correct and is presented in a clear and organized fashion.
Probationary Satisfactory (PS):The solution presented is correct modulo some minor errors (dropped sign, arithmetic error, etc.)
Not yet (N):The solution presented contains serious errors.

Barring extenuating circumstances, there will be no make-up Concept Checks. However, an A in our course only requires completion of 9 out 10 CC.

Concept check corrections. Immediately after each CC, a PDF of the CC will be posted on Canvas (under Files). Two weeks after each CC, solutions will be posted. In the meantime, you will receive a grade for your CC on Gradescope.

If you received a (N): you will have an opportunity to reattempt the CC problem the next week. If you would like to reattempt the previous CC, after handing in this week's CC, you will receive a CC similar to the previous week's. You can receive up to a 5-minute extension to complete this second CC.
If you received a (PS): Immediately after each CC, a PDF of the CC will be posted on Canvas (under Files). A week after each CC, solutions will be posted. In the meantime, you will receive a grade for your CC on Gradescope. Before solutions are posted, and on questions where you did not get an S, you may submit your own hand-written solution and a short reflection on how this new solutions compares to the one you initially presented. I will grade your new solution and your grade may be promoted.

Expectations. Concept checks are closed-notes, closed books. No outside resources are permitted. Concept check corrections are expected to be individual; you may consult your textbook and course notes, but no other outside sources (including the Math Lab and sources listed in the Academic Honesty section below). Because solutions to the CC will be posted on Canvas at that point, no corrections will be accepted any later than a week after the CC was administered.

Grading

Your base letter grade will be based on the number of standards you pass and re-certify, and your CC completion.

Grade	Standards	Re-certification	Concept checks
A	≥ 20	≥ 7 re-certs	at least 9 S
A−	≥ 19	≥ 6 re-certs	at least 8 S
B+	≥ 18	≥ 5 re-certs	at least 8 S
B	≥ 17	≥ 4 re-certs	at least 7 S
B−	≥ 16	≥ 3 re-certs	at least 6 S
C+	≥ 15	≥ 2 re-certs	at least 5 S
C	≥ 14	≥ 2 re-certs	at least 4 S
D	≥ 12	-	-

Homework Completion.You are required to complete at least 27 homework assignment with a score => 60% by the posted deadline. Failure to do so will result in a deduction of a third of a letter grade.

Course Standards

Standards marked with "*" are core standards. If you have completed these standards by the last day of classes, you will have the opportunity to re-certify these standards on the final exam.

Definition and representation of functions: I can define and represent a function algebraically, graphically and numerically.
One-to-one functions: I can determine if a function is one-to-one given a formula, graph, or table.
Fundamental functions: I can define and use the properties of inverse, exponential and logarithmic functions.
Limits: I can define the limit of a function, demonstrate the difference between one-sided and two-sided limits. I can use the limit laws and theorems to compute limits of various functions.
Infinite limits and limits at infinity: I can compute infinite limits and limits at infinity, and distinguish between them. I can determine end behavior of functions.
Continuity*: I can determine the continuity of functions at a point. I can determine the limit of composite functions. I can define and compute continuity at endpoints and on an interval.
Differentiability*: I can define what it means for a function to be differentiable. Graph the derivative of a function. Demonstrate the relationship between differentiability and continuity. I can compute the rate of change of a function, find the slope of a tangent line to a function at a point.
Differentiating polynomials and exponentials*: I can compute the derivative of a constant, use the power rule, the constant multiple rule, compute the derivative of the sum of functions, the derivative of the natural exponential function, and compute higher order derivatives.
Product and quotient rules*: I can compute derivatives using the product rule and quotient rule.
Chain rule*: I can compute derivatives of composed functions using the chain rule.
Implicit differentiation: I can find the derivatives of relations that are defined implicitly.
Derivatives of trig, logarithmic, and inverse trig functions*: I am able to differentiate the trig, logarithmic, and inverse trig functions.
Related rates: I can use derivatives to solve problems in which the variables change with respect to time.
Using derivatives to describe the behavior of functions: I can use derivatives to find maxima and minima of functions, determine where the function is increasing and decreasing using the first derivative test, determine the concavity and inflection points using the second derivative test.
Using derivatives to graph functions*: I can use derivatives to analytically graph functions and analyze the use of technology in graphing functions.
Optimization*: I can set up and solve applied optimization problems using calculus. I can interpret my result in the context of the problem.
L'Hospital’s rule: I can determine when to apply L'Hospital’s Rule, and use it to find limits for indeterminate forms.
Definite integrals*: I can relate the definite integral to the limit of the Riemann sum. I can evaluate definite integrals using limits.
Fundamental Theorem of Calculus: I can use the Fundamental Theorem of Calculus to demonstrate the relationship between differentiation and integration, and evaluate definite integrals.
U-substitution*: I can use the substitution rule to rewrite an integral into a basic form.
Regions between Curves: I can find the area between two curves using integration.
Volume computation: I can find the volume of a three-dimensional region using the general slicing and the cylindrical shell method.

Academic Honesty

All students must adhere to Grinnell's academic honesty policy which can be found here.

Collaboration. You are encouraged to work with your classmates and to consult your textbook while brain-storming solutions and strategies to homework problems. That being said, you are expected to independently write your own solutions in your own words. If you do collaboratively work on a problem, you must indicate this on your solution. You can do this on WebAssign using the Show your Work feature. There are no formatting requirements for this; a simple "I worked on this problem with Omar" will suffice. Similarly, if you received help from a tutor at the Math Lab, you should cite this; something like "I worked with Omar from the Math Lab" will suffice.

Outside resources. It is not my intention that you will need to use outside books/papers or other resources to be able to complete any given assignment. That being said, I encourage you to consult the supplementary references listed above and other online sources (e.g. video lectures, notes, other textbooks, etc.) to help expand your understanding of material presented in class. You may not do any of the following:

consult any sources for the purpose of finding partial or full solutions to any particular problem,
post homework problems online to seek help on forums (e.g. MathStackExchange, Chegg, etc.),
Prompt an LLM (or other AI-assisted tool) or computer algebra system (such as Mathematica) to help you with the argument for a specific assigned problem unless specifically instructed to do so.

Of course these resources are very useful when conducting independent research or study, but do not serve you (or your classmates) in a structured classroom as ours. David Speyer wrote a reasonable explanation of why this policy is in place, which can be found here

Accommodations and Other Resources

Disabilities. Grinnell College makes reasonable accommodations for students with documented disabilities. Students with disabilities partner with Accessibility and Disability Resources (ADR) to make academic accommodation letters available to faculty via the accommodation portal. If you have not already worked with the Office of Disability Resources and believe you may require academic accommodations for this course, Disability Resources staff can be reached via e-mail at access@grinnell.edu, by making an appointment via the accommodation portal, or by stopping by Steiner Hall. (641-269-3124).

Medical and counseling services. Grinnell offers an array of services to help support students' physical, mental, and social well-being. If you need help, please do not hesitate to reach out to Student Health and Wellness (SHAW). I am happy to work with you in making resonable accommodations to help you engage with our classroom as fully as possible; just let me know when the need arises.

Religious holidays. I encourage students who plan to observe holy days/times that coincide with class meetings or assignment due dates to consult with me in the first three weeks of classes so that we may reach a mutual understanding of how you can meet the terms of your religious observance and also the requirements for this course.

Academic support. If you have other needs not addressed above, please let me know soon so that we can work together for the best possible learning environment. In some cases, I will recommend consulting with the Academic Success Center (ASC). They are an excellent resource for developing strategies for your learning, and they can connect you with other resources. If I notice that you are encountering difficulty, and I have reached out to you and not received a response, or if you have missed multiple class sessions or are not meeting our class objectives repeatedly, I will submit an academic alert via the ASC’s SAL portal. A message notifying you of my concern will get sent to you, along with the ASC staff and your adviser(s), so that they can reach out to you with additional offers of support

Title IX and pregnancy related conditions. Grinnell College is committed to providing learning, living, and working environments free from discrimination and harassment. For this reason, it is important to report if you or someone you know is experiencing any form of sexual harassment. Your professor is responsible to pass reports made to them to the Title IX Coordinator. If you want to make a report to a confidential resource, your professor can direct you to those spaces.

If you are a pregnant student, have pregnancy-related conditions, or are a parenting student who wishes to request reasonable related supportive measures from the College under Title IX, please email the Title IX Coordinator at titleix@grinnell.edu. The Title IX Coordinator will work with Disability Resources and your professors to provide reasonable supportive measures in support of your education while pregnant or as a parent.