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Mathematics (MATH)

Chair: K. Valente

Students study mathematics for many reasons: preparation for a career, use in another field, or the beauty of the subject itself. Colgate math majors go on to careers in mathematics, finance, medicine, law, teaching, business administration and areas of industry and education with a mathematics and science orientation. Non-majors find mathematics courses interesting and useful too. Mathematics uses a universal language that assists precise expression, logical reasoning and expression of abstract concepts. Mathematics is also an art form, to be studied for its own intrinsic beauty.

Typical entry-level mathematics courses include the Calculus sequence (MATH 161, 162, and 163). All mathematics courses are open to qualified students. Entering first-year students who have successfully completed at least three years of secondary school mathematics, including trigonometry, should be adequately prepared for MATH 161. Students who have studied calculus in secondary school are typically ready to enter MATH 162 or 163.

Placement Guidelines

If you have been exposed to calculus in high school, whether you have received AP credit or not, you should carefully consider your placement within the calculus sequence. You need to ensure you are appropriately challenged so that you stay engaged and, at the same time, do not find yourself struggling to catch up. The department does not assign students to any particular course; rather you have the opportunity to place yourself based on your background and comfort with the material as well as the guidance provided here. Please refer to the list of topics under MATH 161, 162, and 163 for additional information that you might find useful in considering your placement.

Anyone who has had a full year of study in calculus is strongly encouraged to register for MATH 162 or 163. Anyone who has completed a course preparing for the AP Calculus AB exam (but did not take or successfully complete the exam) can enroll in MATH 162 or 163. Anyone who has completed a course preparing for the AP Calculus BC exam (but did not take or successfully complete the exam) should enroll in MATH 163.

Special consideration should be given to borderline cases, and the following information might help you to decide on an appropriate placement. Students are allowed to drop back from MATH 162 to MATH 161 at any time during the first three weeks of classes. Students who choose to enroll in MATH 163 should note that MATH 162 is a prerequisite for several 300-level courses that students in Economics, Mathematical Economics, Physics, Chemistry, and other majors often choose to take as electives. Permission can be given in order to override this prerequisite when it’s absent from a student’s record, but permission typically requires evidence that the student has completed a course with appropriate content coverage for MATH 162. Additionally, Colgate policy prevents students from taking a course that serves as a prerequisite for another once the more advanced course has been successfully completed. This means that students receiving credit for MATH 163 at Colgate cannot enroll in MATH 162 some time later.

Students with appropriate backgrounds and interests in pursuing mathematics as a major or minor may also consider enrolling in MATH 214, Linear Algebra, or MATH 250, Number Theory & Mathematical Reasoning. The department site link (at the top of the page) will take you to prerequisite information for these courses, and students with prerequisite credit may still require the permission of the instructor in order to enroll.

Professor Dan Schult is available by email ( to address placement questions over the summer. Members of the mathematics faculty will also be available during Orientation and the first week of classes to advise students about placement within the calculus sequence and the mathematics program more generally.

Advanced Placement

Colgate course credit is awarded to students receiving a score of 4 or 5 on the AP Calculus AB exam; a score of 3, 4, or 5 on the AP Calculus BC exam; or a score of 4 or 5 on the AP Statistics exam. Students receiving a score of 4 or 5 on the AP Calculus AB exam will receive credit for MATH 161. Students receiving a score of 3 on the AP Calculus BC exam will receive credit for MATH 161. Students receiving a score of 4 or 5 on the AP Calculus BC exam will receive credit for MATH 161 and MATH 162. Students receiving a score of 4 or 5 on the AP Statistics exam will receive credit for MATH 105. Please note, however, if a student enrolls in a course or drops back to a lower level course for which he or she has received AP credit, the AP credit will be excluded from the student’s academic record and the course will be noted on the transcript as a repeated course. There are no other circumstances under which a student will receive mathematics credit at Colgate University for a course taken in high school.


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MATH 105, Introduction to Statistics
An introduction to the basic concepts of statistics. Topics include experimental design, descriptive statistics, correlation, regression, basic probability, mean tendencies, the central limit theorem, point estimation with errors, hypothesis testing for means, proportions, paired data, and the chi-squared test for independence. Emphasis is on statistical reasoning rather than computation, although computation is done via software.

MATH 161, Calculus I
An introduction to the basic concepts of differential and integral calculus including limits and continuity; differentiation of algebraic, trigonometric, exponential, and logarithmic functions; applications of the derivative to curve sketching, related rates, and maximum-minimum problems; Riemann sums and the definite integral; and the fundamental theorem of calculus.

MATH 162, Calculus II
A continuation of the study of calculus begun in MATH 161. Topics covered include the calculus of inverse trigonometric functions, techniques of integration, improper integrals, l'Hôpital's rule and indeterminate forms, applications of integration, and Taylor series. Note: MATH 161 may not be taken after credit is earned for MATH 162.

MATH 163, Calculus III
The content from MATH 162 and MATH 163 is extended to several variables. Among the topics considered are surfaces in three-dimensional space, partial derivatives, maxima and minima, and multiple integrals. Note: MATH 162 may not be taken after credit is earned for MATH 163.

MATH 214, Linear Algebra
A study of sets of linear equations and their transformation properties over vector spaces. Topics include: systems of linear equations, matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and diagonalization.

MATH 250, Number Theory & Math Reasoning
Questions about the positive integers 1, 2, 3 . . . have fascinated people for thousands of years. The ancient Greeks noted the existence of right triangles with sides of integral length, corresponding to equations such as 32 + 42 = 52 and 52 + 122 = 132. Is there a way of describing all such “Pythagorean Triples”? As another example, 5 = 12 + 22, 13 = 22 + 32, 17 = 12 + 42, while none of the primes 7, 11, or 19 can be expressed as the sum of two squares. Is there a pattern? Does it continue forever? This course focuses on such equations as a means for introducing students to the spirit and methods of modern mathematics. The emphasis throughout is on developing the ability to construct logically sound mathematical arguments and communicate these arguments in writing.