# Calculus FAQ

- Which calculus course should I take? There are so many courses and so many sequences, and it is hard to decide.
- I think I should take MAT100 or MAT103. How can I decide?
- I have a solid knowledge of MAT103, so I need to choose between MAT104 and MAT175. Please tell me more about MAT104.
- I have a solid knowledge of MAT103, so I need to choose between MAT104 and MAT175. Please tell me more about MAT175.
- I think I am ready for MAT201, multivariable calculus. Please tell me more about this course and who should take it.
- I think I want to be a math major maybe. Should I take MAT201? or MAT215?
- I took multivariable calculus in high school and I would like to place out of MAT201. Is this possible? How does it work?
- I took calculus in high school, but I do not have any of the test scores that might be used for placement. What should I do?
- My adviser is pushing me to take a more basic course than what feels right to me. Can I just sign up for the course that I think is best?
- I have a lot of experience with more advanced, proof-based, mathematics at the university level. Which courses are appropriate for me?

**Which calculus course should I take? There are so many courses and so many sequences, and it is hard to decide.**

We have some time to figure this out, but we need to narrow down the possibilites and make a good guess about a possible starting point. We can adjust during drop/add or pivot to a different sequence in a future semester when we have more info. There are two big factors to consider:

- what kind of math have you already learned?
- what majors are you considering?

Many people are quite undecided for several semesters, and so it makes sense to think about where your interest lie in terms of clusters of majors, and then you can think about how much math might be useful or relevant for those areas.

Mathematical reasoning and proofs are central as a tool for future theoreticians. Sequences like 215-217-300, 216-218, and courses like 214 are good choices.**Mathematics, Theoretical Computer Science or Physics:**Mathematical tools and methods are central, but proofs are optional. Sequences like 100-103-104, 201-202 or 203-204 are the usual choices for these majors.**Engineering, Natural Sciences, Economics:**Quantitative Reasoning and Data Analysis are crucial skills. The 100-103-175 sequence gives a good mathematical foundation for students in these fields.**Politics, Sociology, Psychology, non-math-track Economics:**

Which course, or sequence of courses, might be right for you is difficult to predict at the beginning of your studies. Some trial and error is usually needed to see what works for you. You should feel free to explore because the curriculum is quite flexible and many different academic paths can and do lead to the same outcome in the end.

**I think I should take MAT100 or MAT103. How can I decide?**

MAT100 gives a thorough review of precalculus topics that include graphing, solving equations and inequalities involving polynomials, rational functions, logarithms and exponentials, as well as trigonometric functions and their inverses. This is integrated with an introduction to the main topics of calculus including limits, slope of curves, tangents, areas of regions enclosed by curves and basic integration and derivative formulas.

Students in MAT103 use everything covered in MAT100 to learn how to analyze the main features of a complicated function or equation using limits and derivatives. Advanced curve-sketching and optimization in a single variable are the main applications studied, The course ends with a discussion of definite and indefinite integrals along with the Fundamental Theorem of Calculus, as preparation for studying these topics in more detail in MAT104.

*No calculators are used in our math courses. *A strong working knowledge of precalculus that is calculator-independent is needed in 103-104-201. MAT100 is designed for students with no prior calculus experience. The fast pace of 103 requires good problem solving skills together with fluency in the standard library of functions and their graphs. MAT103 does not require prior calculus experience but it is strongly recommended.

The topics in 103 are similar to those covered in AB calculus, but a 5 on the AB exam is not equivalent to a passing knowledge of the class.

**I have a solid knowledge of MAT103, so I need to choose between MAT104 and MAT175. Please tell me more about MAT104.**

MAT104 is a prerequisite for MAT201. Its topics are similar to those covered in BC calculus but a 5 on the BC exam is NOT equivalent to a passing grade in this course.

- techniques and applications of integration
- convergence and divergence of infinite series
- Taylor's theorem, polar coordinates, parametrizing curves in the plane, conic sections and a very brief intro to differential equations.

This course is very fast-paced, with a strong emphasis on sophisticated pattern recognition and algorithms. It is fundamental for further work in physics, engineering and mathematics and is required for these majors.

MAT104 is technically and conceptually demanding. It requires fluent, calculator-independent familiarity with the graphs and general properties of the standard library of functions including polynomials, rational functions, logarithms and exponentials, trigonometric functions and inverse trig functions. To do well in this class a student needs expert skills in algebraic manipulations such as factoring and completing the square, solving equations, analyzing limits in one variable, and a thorough knowledge of derivative and antiderivative formulas from MAT103.

**I have a solid knowledge of MAT103, so I need to choose between MAT104 and MAT175. Please tell me more about MAT175.**

MAT175 generalizes the main ideas of MAT103 (Limits, Derivatives, Curve Sketching, Tangent Lines, Optimization) to the multivariable setting. It is roughly equivalent to the first half of MAT201 and gives an introduction to vectors and geometry in 3-space and goes on to the study of curves and surfaces in 3-space, culminating in problems on optimization in higher dimensions and the basics of the Lagrange multiplier method.

The technical complexity of the multivariable setting, including the need to describe and visualize complex surfaces in 3-space, makes this class a challenging continuation of MAT103. It is a good compromise for students who start Princeton calculus in MAT100 or MAT103 in order to prepare for future work in more quantitative social sciences or in less quantitative natural sciences. It gives (minimal) preparation in the calculus tools needed for finance and economics for students who do not have time to take 201/202 by the second semester of the sophomore year.

It is NOT an option for prospective engineering, physics or mathematics majors. The substantial overlap between MAT175 and MAT201 means that only one of these courses can be taken for credit towards graduation.

**I think I am ready for MAT201, multivariable calculus. Please tell me more about this course and who should take it.**

MAT201 is a fast-paced and in-depth multivariable calculus course, taken primarily by prospective engineers, physics majors and mathematically-inclined students from other natural sciences. The first half of the course generalizes limits, derivatives, tangent lines and optimization (MAT103 topics) to the multivariable setting. The second half generalizes integration techniques and applications (MAT104 topics) to the several variable setting and then introduces vector fields, div, grad, curl, flux, work, circulation, line integrals, surface integrals and the higher dimensional forms of the Fundamental Theorem of Calculus (Green's theorem, Stokes' Theorem, the Divergence Theorem).

MAT201 requires very strong problem-solving skills and thorough familiarity with both precalculus and single-variable calculus. This course is lightning fast and thorough. A strong preparation is essential for success. A score of 5 on the BC calculus exam, or equivalent, is not, by itself, indicative of sufficient preparation to succeed in this course. The placement workshops at registration are designed to help you determine whether MAT104 or MAT201 is a better place to start.

**I think I want to be a math major maybe. Should I take MAT201? or MAT215?**

Prospective math majors are encouraged to take the proof-based calculus and linear algebra sequence instead. The sequence 215-217-300 and the similar, but accelerated sequence 216-218 covers calculus (single and multivariable) and linear algebra from a more theoretical/philosophical point of view. These courses may be of interest to some prospective physics, computer science or engineering students who have a very strong interest in mathematical analysis. Professor Mark McConnell can provide more information and he will be at the placement workshops at registration to answer questions about these courses.

**I took multivariable calculus in high school and I would like to place out of MAT201. Is this possible? How does it work?**

The placement officer, Professor Menezes, handles these requests for the math department. It sometimes happens that she can certify that a student's work done while in high school is equivalent to MAT201, but this is quite rare. Before contacting Professor Menezes, you should be aware that the course you took should have been offered in person, at a university. We do not accept high school or community college courses as 201 equivalent. Online courses are generally more superficial and the exams are more basic than in 201. Students must provide enough documentation to allow us to assess the depth and rigor of the course taken while in high school. A list of topics and textbook is not sufficient. The most informative documentation is the student's written graded work on exams.

Generally, the multivariable calculus courses students take in high school give a good introduction to this challenging and important topic, but they rarely cover topics with the depth and emphasis on problem-solving skills found in MAT201. The difference is usually quite apparent when we compare the written work, especially the exam questions. Students who want to do independent work in math, physics, engineering, applied mathematics, economics or finance need to be able to work creatively with functions and equations in several variables. They need strong conceptual and technical skills, and these are usually best attained by working on challenging problems. Many students in 201 and most students 203 have already taken some multivariable calculus and linear algebra in high school, but not at an equivalent level.

**I took calculus in high school, but I do not have any of the test scores that might be used for placement. What should I do?**

The good news is that you don't need to take a placement test, and in fact, the math department does not offer one. Sign up for the placement workshops at registration and we will help you figure out where to start your calculus studies at Princeton. You will get more out of them if you prepare by reviewing the information here on the math department website about the various courses offered until you can narrow down your choices to one or two possibilities.

For example, if you learned about limits, derivatives, tangent lines, critical points, extrema, and the Fundamental Theorem of Calculus (relating definite and indefinite integrals), then you are probably choosing between MAT103 and MAT104. The placement workshops will help you decide.

If you learned about integration by parts and partial fractions, volumes and area of surfaces of revolution, Taylor approximations and the convergence/divergence of infinite series, then you are most likely choosing between MAT104 and MAT201. Again the placement workshops will help you decide.

Talking to your academic advisor and/or your residence college staff will also be helpful since they know more details about your high school experience overall and they understand all the other decisions you are making in order to plan your first year course of study. The system is very flexible and it is easy to make adjustments as you get more information throughout drop/add and the beginning weeks of the semester.

**My adviser is pushing me to take a more basic course than what feels right to me. Can I just sign up for the course that I think is best?**

The final decision is yours. You are an adult, and you can gather the relevant information and decide for yourself. It is up to you how much of a work load you want to take on and how much academic risk you are willing to take.

That said, you must take responsibility for making a well-informed decision by carefully reviewing all the placement information available to you here on the math department website and by taking very seriously the issues raised in the placement workshops at the beginning of the semester.

Some issues to consider:

- The BC curriculum is not an exact match for the MAT104 course content and a 5 on the BC exam is not equivalent to passing knowledge of MAT104. Similarly MAT103 and the AB curriculum.
- 201 is extremely fast paced and intense.
- No calculators are allowed in Princeton calculus courses. If you were dependent on a calculator for graphing functions or solving equations or computing integrals or derivatives that can be done by hand, then adjusting to this will add to your workload.
- A well-prepared student will spend an average of about 10 hours per week on a math course like 104 or 201. If you are prepared to devote even more time, if you have a high tolerance for stress, and if you are willing to risk getting a C in the class, then it might not be an unreasonable gamble.
- Remember why you are taking the class in the first place. You need those skills in order to do well in more advanced courses. The ideas and techniques are fundamental tools for your future work in finance or engineering or the sciences, not boxes to be ticked off or weeder courses intended to crush your dreams.

**I have a lot of experience with more advanced, proof-based, mathematics at the university level. Which courses are appropriate for me?**

Some students enter Princeton with a very strong background in mathematics and/or physics, including rigorous proof-based courses or experience at international olympiads. If you might want to be a math major and/or you are interested in (further) proof-based courses, then you should most likely sign up for MAT216. Some students in this position opt for MAT215 instead, and a very small number begin their work at the 300 level. You should consult Professor Mark McConnell for individualized advice. He will be at the first-year orientation and he will be participating in the placement workshops there.