Description of Course Goals and Curriculum
Continuation of MAT103. Topics include techniques of integration, arclength, area, volume, convergence of series and improper integrals, L'Hopital's rule, power series and Taylor's theorem, introduction to differential equations and complex numbers.
MAT 104 is designed not only to familiarize students with the mathematical processes of Calculus II, but also more significantly to develop students’ problem-solving abilities by asking them to apply the covered mathematical principles in often complex and unfamiliar problems on problem sets, quizzes and exams.
The course covers integration, infinite series and an introduction to differential equations. The organization of content within these units is highly cumulative and builds in complexity. Though the complexity does develop throughout a unit, because the pace of the course is rather quick it is helpful to attempt to anticipate and predict how the complexity might progress or be added to fundamental principles or concepts as they are introduced either by glancing at the readings for the rest of the unit or looking at some of the challenge problems before you will be expected to know how to solve them.
Students are expected to have background knowledge of concepts covered in Calculus I (MAT 103 or equivalent). These concepts and skills include limits, continuity, integration, definite integrals, the Fundamental Theorem of Calculus and some basic integration techniques. Because MAT 103 and 104 use the same textbook, it is easy to review these concepts at any point in the course if necessary. Another less obvious expectation or demand of the course is the ability to do mental math efficiently as calculators are not permitted on quizzes or exams. Calculators may be used on problem sets, but whenever possible it is beneficial to attempt calculations mentally in order to practice this skill for quizzes and exams. It is also expected that students will show their work on all problem sets, quizzes and exams.
Learning From Classroom Instruction
Lecture-The purpose of the lecture is, according to the syllabus, to “provid[e] an intuition for the main results, usually via examples, instead of focusing on proofs.” This description is very accurate though exactly what this looks like may depend on the particular instructor’s style. It is true across sections that proofs are not a focus of the course. However, some instructors may approach “providing an intuition for main results” in different ways. Some instructors prefer to do fewer examples for the benefit of really understanding the problem-solving technique. Others prefer a breadth of examples and therefore go through many different examples in class at a fast pace.
As a student it is important to respond to and supplement these different styles. Generally, because lecture is focused on examples in both cases, it is helpful to clarify concepts by doing the assigned reading before lecture. Also, if the instructor does fewer examples in more depth, a student might supplement lectures with more practice problems from the book or practice exams (see external resources below) after or outside of lecture. Conversely, if the instructor goes through many examples at a faster pace, it is helpful to go through lecture notes following lecture with a focus on understanding the general techniques rather than the specifics of the problem itself. In regard to lecture notes, it is often helpful to write down every step that the instructor writes on the board so that they are understandable and analyzable when you return to them.
Assigned readings-Instructors recommend completing the assigned readings from the textbook before coming to lecture and doing so is definitely useful as explained above. However, it is important to be strategic about how this is done. Because the lectures are heavily focused on examples, it is beneficial to focus reading of the text mainly on concepts and general applications in the text’s examples before lecture. Although reading the text before lecture is important it can be sometimes confusing or overwhelming if the student is completely unfamiliar with the content as many are. In that case, the concepts often become much clearer through person-to-person explanation in lecture. Therefore, it is often helpful to read the text for general familiarity with the concepts before lecture but then also to return to it and the examples more in depth following lecture when the concepts are likely to have been clarified. Because the quizzes and exams present unfamiliar problems, it is often most useful to read and study the book’s examples more for an understanding of general problem-solving techniques than for the specifics of the particular problems.
Office Hours-Students are able to attend any instructor’s office hours so it may be beneficial to try out a few of the instructor’s office hours at the beginning of the semester so they can get an idea of which instructor’s style best fits their learning style. It is best to at least attempt the problem set before attending office hours so students have particular questions in mind. Another source of particular questions could be difficult problems from the previous problem sets, quizzes or exams. Office hours are also a great time to get more in-depth explanations of difficult concepts.
Learning For and From Assignments
Problem sets- The main function of problem sets in the course is to develop the student’s ability to apply mathematical principles to complex and unfamiliar problems. Students are expected to take away primarily the ability to independently apply concepts to new and difficult problems. Though collaboration on problem sets is allowed, it is often helpful to attempt the entire problem set, if only just the beginning of each problem, independently because it allows one to build up problem-solving skills and confidence for quizzes and exams.
Some of the lecture examples will closely resemble the problem set problems if not in format then at least in technique. They are often higher in difficulty level though. Similarly, the difficulty level of the exam questions is often higher than that of the pset problems. An effective way to prepare for and tackle a pset is to maintain a focus on general problem-solving techniques throughout lecture and readings as they will be most useful when it comes to attempting a problem set. A great way to learn from the problem set is to do an overview of the problem set in its entirety if not directly after completing it then at least before exams to get a mental index of techniques used that can be readily and easily referenced for quizzes, exams, and future problem sets.
Tests-Quiz questions require applications to more complex and unfamiliar problems. In many cases, exam questions require synthesis of concepts in addition to application to complex and unfamiliar problems. This synthesis most often manifests in the multiple steps of the problem. In order to begin the problem a student may need to apply one principle, but then to complete it he or she may need to apply another one.
When studying for quizzes and exams it is beneficial to review lecture notes and assigned readings primarily for an understanding of the different problem-solving techniques that unify different example problems. It is helpful to create an actual index of these techniques that you can review and reference leading up to the exam and then immediately before it for confidence and self-assurance as you head into the test itself.
One of the most helpful ways to utilize the practice exams and other practice problems (see external resources below) posted by instructors is to attempt them on your own and to try to get as far along in the problem as you can. If you get stuck or are unable to do a problem, it is alright to reference the exam key to see how to complete it. However, because exams often have relatively few questions with multiple parts, it is beneficial to return to the same exam or problem the next day or the next week to attempt to do it again and see how far you get. The hope is that you will eventually get to a point where you are able to do all the steps without referencing the key.
External Resources
One great external resource is the Ed Nelson site (accessible by typing MAT 104 Ed Nelson into google if the link is not working). Ed Nelson was an instructor for MAT 104 at Princeton years ago and therefore his website contains older exams that instructors may not post to Blackboard.
The most effective way to use this resource is to do practice problems. As described above, it is best to attempt them on your own and to try to get as far along in the problem as you can. If you get stuck or are unable to do a problem, it is alright to reference the exam key to see how to complete it. However, because exams often have relatively few questions with multiple parts, it is beneficial to return to the same exam or problem the next day or the next week to attempt to do it again and see how far you get this time. The hope is that you will eventually get to a point where you are able to do all the steps without referencing the key.
What Students Should Know About This Course For Purposes Of Course Selection
Students can expect to develop highly valuable problem-solving skills as well as to learn central concepts of Calculus II. The problem-solving skills developed are invaluable in any course or independent work undertaken at Princeton. Additionally, knowledge of Calculus II will be used in many upper-level or even introductory level science courses like PHY 103 and PHY 104. The time demand of the course is mostly what the student makes of it beyond the commitment of the weekly problem set. Students who are eligible for placement into MAT 104 (see the department’s >website for information on placement) should not be deterred from taking it in favor of a lower level math course because of the difficulty level since the difficulty of courses in the math department stem primarily from the need to apply concepts to complex and unfamiliar problems, which is relatively constant across the course levels.