# Math Requirements

### EFFECTIVE DATE

The following math and statistics requirements have been effective as of the fall of 2003 and beyond. Students who first enrolled at Columbia in the fall of 2003 or beyond must complete the math and statistics requirements as outlined below.

Students who registered at Columbia prior to the fall of 2003 must complete the requirements listed in the bulletin that was current when they first arrived at Columbia. Note, however, that if you satisfy the new requirements listed below then you will have satisfied the earlier requirements.

### MATHEMATICS

- Students in all majors, joint majors, and the concentration in economics must take two semesters of calculus, MATH V1101 Calculus I, and MATH V1201, Calculus III. Alternatively, students may take the Honors math sequence MATH V1207 and MATH V1208.
- Students do not have to take MATH V1102 Calculus II to complete their economics requirements. Students who receive a grade of B or better in Calculus I or receive a 5 on the BC AP Placement exam will be allowed to enroll directly in Calculus III.
- Students must complete Calculus I before taking ECON W3213 Intermediate Macroeconomics and must complete Calculus III before taking ECON W3211 Intermediate Microeconomics. Alternatively, if you are taking the honors sequence in math, you may register for Intermediate Macro after completing MATH V1107 Honors I and you may register for Intermediate Micro after completing MATH V1207 Honors III.
- More information about placement in the calculus sequence can be found on the Department of Mathematics website.
- Students who initially skip MATH V1102 Calculus II may still take Calculus II after they have completed MATH V1201 Calculus III.

**Why did the department change the math requirement?**

The new math requirement is designed to better fit the needs of the economics students at Columbia. Undergraduate economics education relies heavily on the differential calculus of functions of one and several variables. Few, if any, economics courses will use integral calculus and those that do will make use of only the most basic of integrals. Calculus I is an introduction to the differential and integral calculus of functions of one variable. Calculus III extends the differential calculus to of functions of several variables. Hence, these are the two classes that teach the calculus used in the economics courses. Calculus II focuses on advanced integration techniques of functions of one variable. None of these techniques are used in any economics class, so Calculus II is not required for economics students.

The differential calculus is the study of the slope (or derivative) of a function or the rate of change of a function. The rate of change of a function (called a derivative) is an extremely useful concept in economics. For example, a cost function tells us the total cost associated with each quantity level of production of a firm. The rate of change of the cost function is called the marginal cost of the firm and it tells us how much it will cost the firm to produce one additional unit. Similarly, the revenue function of a firm tells us the total revenue generated by the sales of each quantity level and the rate of change of the revenue function, the marginal revenue, tells us how much additional revenue is generated by an additional sale. If the cost of producing one additional unit is less than the additional revenues generated by that unit, then the firm can earn more profits by producing and selling that extra unit. In other words, if the rate of change of the cost function is less than the rate of change of the revenue function, then a profit-seeking firm should produce and sell more units of the good.

The cost function is an example of a function of a single variable (the variable is the quantity produced). The study of the calculus of a single variable will teach you how to solve a wide range of economics problems, such as whether or not the firm can earn additional profits by producing and selling an additional unit. Calculus I will teach you how to solve these problems.

Economics problems often involve functions of several variables. For example, living in New York City, I can choose from a wide variety of entertainments, such as movies, concerts, plays, etc., but I don't have an infinite amount of money to spend on my entertainment (nor an infinite amount of time to enjoy all of these activities). Indeed, I must decide how to allocate my budget among these many variable choices where I must think about giving up one good (e.g., a concert) in order to get more of another (e.g., a play). In order to choose the best allocation of my budget for me, I must solve a problem of several, variables. Fortunately, we can extend the results from the calculus of one variable to that of several variables to solve these more complicated economics problems. Calculus III will teach you how to solve these types of problems.

The integral calculus begins with the study of the area under a curve. It is used sparingly (if at all) in undergraduate economics courses but extensively in engineering, physics, and statistics. The derivative of a function is in general quite straightforward to calculate, but the integral is not. In fact, the knowledge of a wide range of techniques is needed calculate many integrals. Calculus IIA teaches these many tricks of integration. Since none of these tricks are ever used in an undergraduate course, this class is not required as part of the economics major.

**STATISTICS**

Students in all majors, joint majors, and concentrations (including the premedical concentration) must take STAT W1211 Introduction to Statistics B (or a higher-level statistics course such as SIEO 4150).

**Why did the department change the statistics requirement?**

Many students and faculty members felt that students lacked the necessary training in statistics to comfortably handle the material in W3412 Introduction to Econometrics. The change to STAT W1211 will hopefully address this problem.

The department has always thought of the statistics and econometrics courses as a year-long sequence in statistical methods. Unfortunately, up until now there has been little coordination between the statistics department, which offers the statistics course, and the economics department, which offers the econometrics course, regarding the two courses. The new design of STAT W1211 was done in consultation with the econometrics professor in the economics department. The material covered in STAT W1211 was chosen in order to ease the transition from statistics to the econometrics course offered in the department.

An additional benefit of STAT W1211 is that because it has a one semester calculus requirement, the class will more accurately reflect the mathematical sophistication of the economics students than did STAT W1111. The department of statistics has been using STAT W1211 as the entry-level course to a major or concentration in statistics. Since some of the economics students are required to take higher-level statistics courses to complete their degrees, the switch to STAT W1211 will reduce the disparity in the statistics preparation of the students in W3412.

**The department would also like to strongly recommend to all students that they take the two courses W1211 and W3412 in consecutive semesters.**