Math 60: Analytic Geometry and Calculus


Instructor: Mark Sweeney
Office: Kassar 012
Office Hours: T 4:00-5:00 and Zoom by appointment
Pronouns: she/her, they/them

Course Description

Together, Math 50 and Math 60 cover the calculus material in Math 90, as well as review of the necessary precalculus topics. My goal is for you to leave this course with both solid mathematical knowledge in precalculus and single variable calculus, as well as a strong conceptual understanding of the material to apply it to future courses you take here.

In this second semester, we will continue to develop some of the theory of differentiation from the first semester, and also introduce more applications.

A brief list of topics we aim to cover:

  1. Implicit differentiation and related rates
  2. Differential equations
  3. Trigonometric functions
  4. L'Hopitals rule and limits
  5. Definite integrals and antiderivatives
  6. Fundamental theorem of calculus
  7. Integration techniques (substitution)
  8. Exponential and logarithmic functions

Calculus is a multifaceted subject, and we will aim to thoroughly explore it from several angles. Here are some of our guiding definitions of calculus:

  1. the study of functions by their growth
  2. the development of tools for approximating functions
  3. a mathematical framework for modeling processes in physics, chemistry, engineering, and more
In the second semester, we will focus more on the second and third definitions. Both the practical and theoretical aspects of calculus manifest in different ways in each of these. This is not a proof-based course, so we will draw heavily on examples and intuition to motivate the theory. However, where it is reasonable and helps your understanding, we may sketch some computational proofs.

Textbook: Thomas' Calculus: Early Transcendentals (14th edition). This is the same book currently used in 90/100 so it may be worth purchasin if you plan to go on to MATH100. It would appear to be available for prices between $40 and $100 dependong on whether you buy used or new, paperback or hardcover. You do not need the MathLab code so a used book is perfectly acceptable.

Course Structure

Each week there will be two 80 minute lectures (TR 2:30 - 3:50pm) and one 50 minute recitation (T 12:00 - 12:50pm).

Note: the first few lectures will be held online.

To help organize the material and your learning, most lessons are structured around a worksheet. During the lesson there will a combination of lecture, group work, and class discussion revolving around the worksheet, guided by the instructor.

The format of recitations is similar, but it will have no lecture and include no new material.

You should expect to work at least 12 hours a week for this course, roughly broken down as follows:


Homework is crucial to success in this course. All assignments are mandatory.
It is nearly impossible to learn and master any course without spending time working on problems. The only way to learn math is by doing math!! Homework will usually be slightly different than problems discussed in class; presenting the material in different contexts will help you develop a more flexible understanding.

Each homework set is short and two are due each week to ensure you are continuously engaging with the material without overwhelming your schedule. Problem sets and their due dates will be posted to the calendar on this course website (see below). Homework will typically be released Fridays and Mondays, and due during class on Tuesday and Thursday (sometimes moved slightly to accommodate holidays and exams). In the calendar, each assignment appears under the day it is due.

Homework grading will emphasize conceptual understanding and a willingness to put effort into the assignment.

Late homework will be accepted with no penalty but we ask that you email the instructor in advance (when possible) to let them know and arrange an alternative time of way to submit your assignment (ideally within a week of the original due date).


There will be a take home midterm exam and a take home final exam. The exams are not cumulative. There will be opportunities to submit corrections to the exams as well; my priority is your learning, not assessments.


Math 50 is only offered on an S/NC basis. Assignments are graded on a combination of effort and correctness, and we will always provide ample opportunities for revision if you are concerned about a particular grade. The breakdown of your grade will be as follows:


Brown University is committed to full inclusion of all students. Please inform me early in the term if you may require accommodations or modification of any of course procedures. You may speak with me after class, during office hours, or by appointment. If you need accommodations around online learning or in classroom accommodations, please be sure to reach out to Student Accessibility Services (SAS) for their assistance (, 401-863-9588). Undergraduates in need of short-term academic advice or support can contact an academic dean in the College by emailing Graduate students may contact one of the deans in the Graduate School by emailing

Please let me know if you have any related questions and are not sure if they fall within the scope of SEAS. I am happy to work with you to the extent that I am able to improve you learning.

Academic Integrity and Collaboration Policy

Our goal, above all else, is that you work hard and learn the material as well as you are able.

You are strongly encouraged to work together on your homework assignments, and may also consult other sources, including the internet, but with three small stipulations:

  1. Make a reasonable attempt at the problem on your own first
  2. Mention who you worked with and cite your sources
  3. Always write up the solutions in your own words, and be sure you understand and internalize any steps that were explained to you by someone else
For purely computational problems, I ask that you do not use symbolic calulators like WolframAlpha or Mathematica, simply because that defeats the purpose. You are more than welcome to compare with your classmates or use those tools only to check your answers.

Another good resource is the Math Resource Center, available four nights a week with TAs who can answer your questions.

Calendar and Files

Here is our course calendar and all the files (class notes, worksheets, homework assignments, and solutions). Each day also has a brief description of the topics covered.

In this calendar, the homework file is located at day it is due. For instance, the homework file in Thursday of Week 2 is due on that Thursday, 9/15.

Homework due Tuesday will be posted the Friday before, and homeworks due on Thursday will be posted on the Monday before. Typically, homework due on Thursday will include concepts from Tuesday's class, so skimming it ahead of time might be useful!

Topics, especially for later weeks, are subject to change. Some of the unmarked days are specifically for extra topics chosen by student interest.

Week 2: 1/31, 2/2
Recitation Tuesday Thursday
Implicit Differentiation More practice with implicit differentation; optimization
worksheet solutions worksheet solutions worksheet solutions
Read 3.7 Read 3.7

Week 3: 2/7, 2/9
Recitation Tuesday Thursday
Related rates More practice with related rates
worksheet solutions worksheet solutions worksheet solutions
Read Section 4.5
Section 3.7: 3, 4, 6, 8, 22, 24, 26, 28, 50*
solutions solutions

Week 4: 2/14, 2/16
Recitation Tuesday Thursday
none l'Hôpital's Rule and "indeterminate" forms Trigonometric functions
worksheet solutions worksheet solutions
solutions solutions

Week 5: 2/21, 2/23
Recitation Tuesday Thursday
holiday holiday Estimating area
worksheet solutions

Week 6: 2/28, 3/2
Recitation Tuesday Thursday
Integration I and FTC Discrete approximation of integrals, integrating polynomials
worksheet solutions worksheet solutions worksheet solutions
solutions solutions

Week 7: 3/7, 3/9
Recitation Tuesday Thursday
none Discrete approximation with summation notation, integral as area Integration IIa (evenly spaced Riemann sums)
worksheet solutions worksheet solutions
solutions solutions

Week 8: 3/14, 3/16
Recitation Tuesday Thursday
Brief return to LH Integration IIb (Riemann sums over partitions)
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Week 9: 3/21, 3/23
Recitation Tuesday Thursday
Midterm review, u-substitution u-substitution
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solutions solutions

Week 10: 3/28, 3/30
Recitation Tuesday Thursday
Spring Break Spring Break Spring Break

Week 11: 4/4, 4/6
Recitation Tuesday Thursday
Go over midterm Computing integrals
worksheet solutions worksheet solutions worksheet solutions
solutions solutions

Week 12: 4/18, 4/20
Recitation Tuesday Thursday
Volumes of revolution Volumes of revolution
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Week 13: 4/25, 4/27
Recitation Tuesday Thursday
Applications Applications
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solutions solutions


Most of my past students have enjoyed seeing my bunnies, Cujo & Lula (brown and grey, respectively), so a few photos are attached here. They make a welcome break from mathematics for me, and I hope for you as well!!

The bunnies occasionally visit my office - think of them as extra TAs.

Cujo and Lula napping together in front of a brick wall.
Cujo's friend Peter, a different rabbit

This is Cujo & Lula's friend Peter, whose friend David contributed to the design of this website. Thanks!