Survey of topics from multivariable calculus as preparation for future course work in economics or life sciences. Topics include basic techniques of integration, average value, vectors, partial derivatives, gradient, optimization of multivariable functions and constrained optimization with Lagrange multipliers. Although examples from economics and biology will be used freely, the main focus is on the underlying mathematics. Provides a very basic introduction to multivariable calculus in two or three variables. These topics, along with many others, are covered in much greater depth in MAT104, MAT201 and MAT202. Offered fall and spring. Prerequisite: MAT103 or equivalent.

# Course MAT175

## Basic Multivariable Calculus for Economics & Life Sciences

The class will begin with a discussion of vectors, dot products, equations for lines and planes, and working with curves and surfaces in 3-space. The course will focus on those topics from multivariable calculus and basic linear algebra needed for a thorough discussion of optimization: level sets, partial derivatives, linearization, the gradient, directional derivatives, implicit differentiation, the chain rule and Lagrange multipliers. Optimization is discussed mostly for functions of two or three variables, although the techniques readily extend to higher dimensions. Constrained optimization will be a major focus. The course concludes with a review of integration, a discussion of the most important techniques and applications of integration. Offered Fall and Spring.

Classes meet 3 times per week, for 50 minutes. If staffing permits, 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 assistants together, to ensure uniformity across all sections. Typically there are small quizzes in class, the same for all instructors. Homework and quizzes together usually account for about 30% of the course grade.

In order to perform well on exams, we anticipate that most students will need to spend approximately ten hours per week reading the text, reviewing class notes, solving homework problems and working through lots of extra practice problems to prepare 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.

This course usually includes optional weekly problem/review sessions, staffed by undergraduate course assistants.

- Minimally satisfies the math prerequisite for economics/finance. Prospective economics/finance students should consult the economics dept website or the finance center website for up-to-date advice on whether to take 175 or 201 or 201/202. MAT201/202 is strongly advised instead of 175 for students who wish to take ECO312, pursue graduate studies in economics or finance, or pursue a finance certificate.
- This course has substantial overlap with 201, but the latter is much more thorough. Students may not take both 175 and 201 for credit.
- Students who take 175 and do well (B or better) should be able to continue with 202 if they choose. However, 202 is a much more demanding course.

- (See Note 1 above.) Most of the students in the course will be students preparing to enter the economics department as majors, but this course provides minimal mathematical preparation for these programs. More mathematically inclined prospective economics majors or those who consider going on to graduate school in finance or economics are advised to take MAT104/201/202 instead.
- May also be suitable for some students interested in life sciences as an alternative to MAT104 or MAT201; consult your department to be sure.
- Not intended for students who will take further math courses at Princeton. Does not give adequate preparation for 300-level math courses.
- Prospective engineers should not take this course.

Very solid knowledge of differential calculus is assumed as well as basic familiarity with integral calculus (definition of definite and indefinite integrals, the fundamental theorem of calculus). 103 or equivalent is a prerequisite.

- Students deciding between 103 and 175 should probably take 103 to get a really solid foundation in calculus. You will be much better prepared both for 175 and for future quantitative courses in your major department. These two courses should be of comparable difficulty, but 175 uses and extends ideas from 103 for most of the semester.
- Students deciding between 175 and 104 have to consider the situation more carefully. By signing up for 175, you are essentially deciding that you do not want or need to take any further math courses here at Princeton. Most future science majors will need 104 instead. Similarly, prospective engineers should take 104 instead. Before you consider 175, check that all the programs or majors you consider will be satisfied with that decision. Taking 104 gives you more options in the long run, but it is definitely a more demanding course.
- Students deciding between 175 and 201 should sign up for 201 and be prepared to switch down after a couple of weeks if necessary (provided that 175 will fulfill the mathematical prerequisites for your prospective program(s) of study).