Course MAT104

Calculus II (One Variable, Continued from 103)

Second semester of the standard 3-semester calculus sequence 103/104/201 for science, engineering and finance. Topics include techniques and applications of integration, convergence of infinite series and improper integrals, Taylor's theorem, introduction to differential equations and complex numbers. Emphasizes concrete computations over more theoretical considerations.  Offered both Fall and Spring.  Prerequisite: MAT103 or equivalent.


Integration by substitution, integration by parts, partial fractions, trigonometric substitution, improper integrals, convergence tests for infinite series, alternating series, absolute convergence, power series, Taylor series, complex numbers, first and second order differential equations, volume, length and surface area.

Description of classes

Classes meet 3 times per week, for 50 minutes.  Sections are generally offered MWF at 9, 10, 11 and 12:30 in both semesters.

The course is organized into small sections of 20 to 30 students. There is one course head who coordinates with all the instructors to write the exams. All students have the same homework assignments and take the same midterm and final exam. The midterm and final count for the bulk of the course grade, typically about 70%.  These exams are graded by all the instructors and graduate student AI's together to ensure uniformity across all sections. Typically there are small quizzes in class, once a week or every other week, the same quizzes for all instructors. Homework and quizzes together usually account for about 30% of the course grade.

In order to prepare adequately for future courses, like MAT201, most students will need to work lots of practice problems outside of class.  We anticipate that most students will need to spend approximately ten hours per week reading the text, reviewing class notes, solving homework problems and studying for quizzes and exams.  The course will be quite fast-paced and it is essential to work steadily throughout the semester.  Frequent feedback will be given to help students keep up and monitor progress.

  • Roughly equivalent to BC calculus or to the mandatory and optional calculus portion of Math HL in the IB program.
  • A frequent choice to fulfill the QR requirement.
  • Students who plan to major in economics may consider MAT175 instead.  However MAT104 is a better choice for more mathematically inclined economics majors, especially those who plan to go to graduate school in economics or finance.
  • Students with a strong interest in math as a major who received a 5 on the AB exam and have a SAT math score above 750 may consider an introductory proof-based course like MAT214 or MAT215 instead.  Consultation with the math placement officer is strongly advised.
Who Takes This Course

Most students in this course are incoming freshmen who consider majoring in one of the sciences or engineering.  More mathematically inclined economics majors will take this course followed by MAT201 and MAT202 instead of MAT175.  Many pre-med students take this course.  Math and physics majors do not usually take this course, but there are some exceptions. 

This is usually the best choice for students whose once strong calculus skills are rusty from disuse after a gap in their mathematical studies. Because it makes heavy use of the prerequisite material from precalculus courses and also from MAT103, it gives a thorough review of all of high school mathematics in a single semester. (This may also explain why many students consider it to be significantly harder than MAT103.)

Placement and Prerequisites


A basic knowledge of differential calculus and integration is assumed; MAT103 or its equivalent is a prerequisite. 

For those deciding between MAT103 or MAT104:

  • If your math SAT score is below 650 and/or your differential calculus is weak or rusty, you should probably start in MAT103 instead.
  • If you cannot decide between MAT103 and 104, then sign up for 104 and be prepared to switch down to MAT103 after a few weeks if necessary.  This is quite easy to arrange, but the sooner you can make a final decision the better.  The first two weeks of class and the first quiz in 104 will give you a lot of additional information and you can discuss the decision with your instructor.

For those deciding between MAT104 and MAT201

  • Keep in mind that a score of 5 on the BC calculus exam is minimally equivalent to this course, probably equivalent to a grade of C.  As a result, many students who scored a 5 on the BC calculus exam or took a similar course in high school (and did well) opt to take 104 and find it to be quite challenging.  Although many of the topics in 104 are somewhat familiar, the depth of coverage and mastery of the subject required here is much greater, with more emphasis on independent thinking.  Additional topics that are not covered in most BC calculus courses are also included.  For students who need to take further calculus (e.g.  MAT201) or for anyone planning a major where calculus will be an important analytic tool, this course provides a very thorough foundation in single variable techniques.    
  • Note that a score of 7 on the IB MathHL exam (not SL!) or an A on the British A-levels exam is treated as equivalent to a 5 on the BC exam here and at many other universities.  These scores indicate that MAT201 is a reasonable starting point for you.
  • If in doubt, sign up for MAT201 and be prepared to re-evaluate during the two weeks of classes.

For those deciding between MAT104 and MAT175:

  • you should probably take MAT104.  Taking 175 limits your options; you should not take it if you may later need further math courses at Princeton.   (Unfortunately the course numbers are misleading:  104 is significantly more demanding than 175.)

General information about placement and contact information for the placement officer can be found on the Math Placement Page.

Sample Material

Working problems from these sample exams can give you a good idea of the expectations and content in these courses as you think about which course is right for you. Just reading the questions or the solutions can be very misleading however. Try the problems yourself!

Once you have tried the problems, you can check your answers:
Midterm Exam Solutions, Final Exam Solutions.

  1. How much work is this course?
    • Most math courses require a steady time commitment.  We expect that the weekly problem sets will take at least three hours to complete, although this can vary quite a lot depending on your background and your goals.  To do well on math exams, you need to work through a lot of extra problems.  All in all, you should be ready to spend up to ten hours per week working outside of class.
  2. If I think 104 is too hard, what should I do?
    • We have designed MAT103/104 as a beginning calculus sequence.  Try an old final exam from MAT103.  If you can do most of the problems with confidence, you should be able to handle 104.  Otherwise you may want to switch into MAT103.  You can also consider MAT175, but this limits your options.  Before you switch, make sure that MAT175 will satisfy the requirements of all the programs you are considering. 
  3. If I think 104 is too easy, what should I do?
    • Entering 104 in the fall you may well find that you are already somewhat familiar with the material taught in the first few weeks, but be cautious.  Students often underestimate the difficulty of 104 because they have seen a more superficial treatment of these topics in a previous calculus course.  The first quiz should come early and help you make these decisions. Before you switch up to a 200-level math course, first try the problems on the sample final exam for 104 above.  Work through the problems -- don't just read the questions or the solutions!  If you are not able to find correct solutions to several of these old exam questions, then you should probably stay in 104. 
  4. I need both 103 and 104 for my major. After checking the math placement information, I think 104 is probably the right course for me, but I don't qualify for AP credit for MAT103. Do I have to take 103 or can I sign up for 104 instead?
    • As discussed in the previous question, be cautious.  You probably cannot accurately judge the course by looking at a list of topics or even by attending the first few lectures.  Instead, work through the problems on the sample final exam for MAT103.  If you cannot correctly solve more than half of them, then you are probably better off to start in MAT103.  Keep in mind that most students consider 104 to be significantly more difficult than MAT103, so if your calculus and precalculus background is not strong enough, it will be a real struggle to do well in 104.  On the other hand, if you can solve the problems on the old MAT103 exam, then it is reasonable for you to sign up for 104.  If you then pass it, you will automatically receive AP credit for MAT103.  Just be ready to reconsider and switch down to MAT103 early on, if necessary.  There should be an quiz early in the semester that will give you very useful information about the course expectations.
  5. If I want to switch courses, what should I do?
    • Check out drop/add/swap procedures and information about who to contact for advice about courses. But don't delay -- if you are going to switch, do it soon!
  6. I would like to switch sections within 104, what should I do?
    • If you have a time conflict, there is usually no problem. If you want to switch to another section at the same time, be prepared for a possible refusal --- the instructor may not have room for another student. Check out drop/add/swap procedures and information about who to contact for advice about courses. But don't delay -- if you are going to switch, do it soon!
  7. I have more questions that are not answered here. What should I do?
    • More information about how our courses work in general and about who to contact if you need to discuss your situation with someone from the math department can be found on the undergraduate home page.  Representatives from the math department will be available at the Academic Expo at Orientation and at freshman registration.