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# Mathematics

## Course Catalog: Mathematics

Mathematics is one of the most intellectually challenging and academically pure subjects. Mathematical thought is a creative process of the mind that uses only logical deduction and established results that, in turn, have been derived from a few unarguable assumptions (axioms). Mathematical modeling is the process of critically investigating a given object (e.g., the functioning of a biological system) and of choosing or creating mathematical structures that explain the observed behavior and allow for prediction and manipulation of this object.

The mathematics program at Rocky Mountain College emphasizes both the pure and applied aspects of mathematics. At its core, its curriculum is designed to provide students with a solid foundation in the art of providing mathematically sound arguments and with a thorough knowledge of the most important modern mathematical tools and methods. In addition, the mathematics program offers elective courses that give students the opportunity to branch out and pursue their own interests. Many elective courses emphasize connections to other fields (such as computer science, engineering, finance, and the natural sciences).**All mathematics prerequisite courses must be completed with a grade of at least “C-”.**

### Learning Outcomes

Students who graduate with a major in mathematics will be able to:

- Establish mathematical results using a variety of proof techniques;
- Exhibit knowledge of relevant definitions, techniques, and mathematical results;
- Perform symbolic manipulation of high-level mathematical objects;
- Solve problems involving high-level mathematical objects;
- Solve computational problems algorithmically;
- Use advanced mathematical techniques to solve problems in real-world situations.

### Major in Mathematics

A minimum of 43 semester hours is required, including:

MAT 175: Calculus I

MAT 176: Calculus II

MAT 212: How to Read and Write Proofs

MAT 275: Calculus III

MAT 276: Introduction to Mathematica

MAT 310: Mathematical Statistics

MAT 311: Linear Algebra

MAT 313: Differential Equations

MAT 317: Abstract Algebra I

MAT 318: Abstract Algebra II

MAT 325: Combinatorics

MAT 481: Introduction to Real Analysis I

MAT 482: Introduction to Real Analysis II

### Major in Mathematics Education

A minimum of 41 semester hours is required, including:

MAT 175: Calculus I

MAT 176: Calculus II

MAT 212: How to Read and Write Proofs

MAT 220: Elementary Number Theory

MAT 275: Calculus III

MAT 276: Introduction to Mathematica

MAT 306: History and Philosophy of Mathematics

MAT 310: Mathematical Statistics

MAT 312: Modern Geometric Theories

MAT 317: Abstract Algebra I

MAT 318: Abstract Algebra II

MAT 422: Methods and Materials: Teaching Mathematics in the Secondary School

MAT 481: Introduction to Real Analysis I

In addition, students must complete the professional education program for secondary teaching (grades 5-12) as described in the “Education” section of the catalog.**Note:** If a student majors in both mathematics and mathematics education, he or she must complete the requirements of both majors but does not need to take additional credits within the mathematics department. Majoring in both mathematics and mathematics education requires a minimum of 46 semester hours.

### Minor in Mathematics

A minimum of 21 semester hours is required, including:

MAT 175: Calculus I

MAT 176: Calculus II

MAT 275: Calculus III

Choose three courses from the following:

Any upper division MAT course

Any course cross-listed with an upper-division MAT course

MAT 212: How to Read and Write Proofs

MAT 256: Discrete Structures and Computability

MAT 310: Mathematical Statistics

MAT 311: Linear Algebra

MAT 313: Differential Equations

MAT 317: Abstract Algebra

MAT 481: Introduction to Real Analysis I

A minimum of 6 upper-division credits is required the minor.

### Minor in Mathematics Education

A minimum of 29 semester hours is required, including:

MAT 175: Calculus I

MAT 176: Calculus II

MAT 212: How to Read and Write Proofs

MAT 220: Elementary Number Theory

MAT 310: Mathematical Statistics

MAT 312: Modern Geometric Theories

MAT 317: Abstract Algebra I

MAT 422: Methods and Materials: Teaching Mathematics in the Secondary School

Electives

In addition, students must complete the professional education program for secondary teaching (grades 5-12) as described in the “Education” section of the catalog.

### MAT 090 - Elementary Algebra

*Semester:* Fall and Spring*Semester hours:* 3

This course is designed to prepare students for college algebra. Elementary algebra topics are covered, which include solving equations and inequalities, simplifying algebraic expressions, simplifying expressions involving integers and rational numbers, and graphing equations. This course will not satisfy the mathematics core curriculum requirements and will not count toward the 120 credits required for graduation.

### MAT 100 - College Algebra

*Semester:* Fall and Spring*Semester hours:* 3

This is a basic course in intermediate and advanced algebra.

*Prerequisite:* satisfactory score on a placement exam

### MAT 103 - Mathematics for Elementary School Teachers I

*Semester:* Fall*Semester hours:* 3

This course is a survey of various mathematical topics normally taught in grades K-8, specifically numeration systems, number theory, patterns and relationships, and fractions. This course is intended for elementary education students only. Students must earn a grade of "C-" or better to fulfill requirements for the elementary education program.

*Prerequisite:* MAT 100 or satisfactory score on a placement exam

### MAT 104 - Mathematics for Elementary School Teachers II

*Semester:* Spring*Semester hours:* 3

This course is a survey of various mathematical topics normally taught in grades K-8, specifically probability and statistics, geometric basic concepts and structures, measurement, and coordinate geometry. This course is intended for elementary education students only. Students must earn a grade of "C-" or better to fulfill requirements for the elementary education program.

*Prerequisite:* MAT 103 with a grade of C- or better

### MAT 110 - Precalculus

*Semester:* Fall and Spring*Semester hours:* 3

A standard pre-calculus course emphasizing the function concept. Special attention is paid to trigonometric, exponential, and logarithmic functions.

*Prerequisite:* MAT 100 or satisfactory score on a placement exam

### MAT 131 - Trigonometry and Applied Calculus

*Semester:* Spring*Semester hours:* 3

This course is available to aeronautical science majors and aviation management majors only. This course introduces applied trigonometry, vectors, and basic differential and integral calculus to model and solve real-world problems.

*Prerequisite:* MAT 100 or satisfactory score on a placement exam

### MAT 152 - To Infinity and Beyond

*Semester:* Offered at discretion of department*Semester hours:* 3

This course is an exploration of a variety of modern mathematical topics. Topics will illustrate mathematics as a way of representing and understanding patterns and structures as an art, as a tool in other disciplines, and as a historical force. Topics may include infinity, chaos, fractals, symmetry, networks, and others.

*Prerequisite:* MAT 100 or the equivalent

### MAT 175 - Calculus I

*Semester:* Fall*Semester hours:* 5

This course is a study of the functions of one real variable and includes a brief review of circular functions. The ideas of limit, continuity, and differentiation are explained and applied to physical problems. Topics include the use of approximations and problem solving. The use of graphing calculators is required.

*Prerequisite:* satisfactory score on a placement exam or MAT 110

### MAT 176 - Calculus II

*Semester:* Spring*Semester hours:* 5

Continuing the study of the functions of one real variable, the idea of integration is applied to physical problems. This course is an introduction to sequences and series. The use of graphing calculators is required.

*Prerequisite:* MAT 175

### MAT 210 - Probability and Statistics

*Semester:* Fall, Spring, and Summer*Semester hours:* 3

This course provides a non-calculus-based study of discrete probability theory and its statistical applications. Distribution theory and its applications in hypothesis testing and setting confidence intervals are discussed.

*Prerequisite:* MAT 100 or satisfactory score on a placement exam

### MAT 212 - How to Read and Write Proofs

*Semester:* Spring; Alternate years*Semester hours:* 3

Students are introduced to the different methods of mathematical proofs. Emphasis is placed on critical reading of proofs and the ability to correct errors as well as on writing correct proofs. This course is designed as a precursor to advanced mathematics courses and should be taken during the freshman or sophomore year.

*Prerequisite:* MAT 175

### MAT 220 - Elementary Number Theory

*Semester:* Fall; Alternate years*Semester hours:* 3

Topics in this course include axiomatic development of the positive integers, construction of the real number system, and study of equations with integral solutions. Divisibility properties, prime numbers, and the analysis of congruencies will be studied.

*Prerequisite:* MAT 175 and MAT 212

### MAT 256 - Discrete Structures and Computability

*Semester:* Offered at discretion of department*Semester hours:* 3

The mathematical and theoretical underpinnings of computer science will be explored. Students will be introduced to Boolean algebra and elementary logic and their application to computer implementation and algorithm development. This course explores the historical development of computer science from its roots in mathematical models including early models of computation, such as Turing machines and other finite state machines.

*Prerequisite:* CSC 131 or CSC 143 and either MAT 110 or MAT 175

### MAT 275 - Calculus III

*Semester:* Fall*Semester hours:* 4

Vector functions of one or more real variables, scalar functions of several variables, multiple integration, and surface theory via vectors are among the topics studied. Increasing emphasis on modeling of physical problems and the analysis of geometrical problems in higher dimensional space.

*Prerequisite:* MAT 176

### MAT 276 - Introduction to Mathematica

*Semester:* Fall*Semester hours:* 2

This course provides an introduction to the basic structures and components of computer algebra systems (CAS). Lists, functions, graphics, and programming in Mathematica will be covered.

*Prerequisite:* MAT 176*Corequesite:* MAT 275

### MAT 299 - Independent Study

*Semester:* Offered at discretion of department*Semester hours:* 1-3

This course allows interested students to devise and pursue independent study in an area agreed upon in consultation with, and supervised by, a faculty member. Students should be either a major or minor in mathematics or mathematics education and have a cumulative GPA of 3.00 or greater.

### MAT 306 - History and Philosophy of Mathematics

*Semester:* Fall; Alternate years*Semester hours:* 3

This course provides a survey of mathematicians of historical note, including their motivations and studies. Students will also examine classical problems and how they were solved. Unsolved problems in mathematics will be discussed.

*Prerequisite:* MAT 175

### MAT 310 - Mathematical Statistics

*Semester:* Spring; Alternate years*Semester hours:* 3

This course is a calculus-based introduction to statistical methods and theory. The course covers basic probability rules; random variables and probability distributions; limit theorems; sampling distributions; point and interval estimation methods; hypothesis testing, including t- and chi-square tests; the simple linear regression model; and analysis of variance.

*Prerequisite:* MAT 275

### MAT 311 - Linear Algebra

*Semester:* Spring; Alternate years*Semester hours:* 3

This course introduces students to the basic structures of linear algebra, which include the following: matrices, determinants, vectors and vector spaces, inner product spaces, eigenvalues, and eigenvectors. Applications and computational aspects of these topics will be presented.

*Prerequisite:* MAT 175*Corequesite:* MAT 176

### MAT 312 - Modern Geometric Theories

*Semester:* Spring; Alternate years*Semester hours:* 3

This course provides a study of Euclidean and non-Euclidean geometries.

*Prerequisite:* MAT 175 and MAT 212

### MAT 313 - Differential Equations

*Semester:* Fall; Alternate years*Semester hours:* 3

This course examines analytic, numerical, and geometric techniques for solving first-order differential equations; bifurcations of first-order autonomous differential equations; second-order differential equations, with emphasis on the harmonic oscillator; Laplace transforms; eigenvalue/eigenvector and geometric methods for linear systems with constant coefficients; equilibrium point analysis of nonlinear systems; and analysis of limit cycles. Applications from biology, physics, and economics are presented.

*Prerequisite:* MAT 311

### MAT 316 - Complex Variables

*Semester:* Spring; Alternate years*Semester hours:* 3

This is a basic undergraduate course in complex variables. Topics will include analytic functions, Cauchy's integral formula, the residue calculus, and conformal mapping.

*Prerequisite:* MAT 275

### MAT 317 - Abstract Algebra I

*Semester:* Fall; Alternate years*Semester hours:* 3

This course provides an introduction to algebraic structures via group theory. Students explore a careful development of the concept of a group and elementary properties of groups. Emphasis is placed on creating mathematical proofs. Some applications to physical problems are included.

*Prerequisite:* MAT 275 and MAT 212

### MAT 318 - Abstract Algebra II

*Semester:* Spring; Alternate years*Semester hours:* 3

This course is an extension of MAT 317 with ring, ideal, and field theory introduced and examined. Some Galois theory may be covered.

*Prerequisite:* MAT 317

### MAT 319 - Graph Theory

*Semester:* Fall; Odd years*Semester hours:* 3

Topics covered in this course include paths, Euler and Hamiltonian problems, planar graphs, trees, directed and undirected graphs, networks, and connectedness. Applications to various disciplines will be included. Computational algorithms will be developed as appropriate.

*Prerequisite:* MAT 212 or permission from instructor

### MAT 325 - Combinatorics

*Semester:* Spring; Alternate years*Semester hours:* 3

This basic course in enumerative combinatorics emphasizes developing combinatorical reasoning skills and applying these to solve problems in various areas of math and computer science. Topics covered will include basic counting principles, generating functions, recurrence relations, the principle of inclusion/exclusion, estimation, and modeling.

### MAT 369 - Cryptography

*Semester:* Spring; Alternate years*Semester hours:* 3

This course covers both classical and modern ciphers, including mono-alphabetic substitution ciphers, periodic and block ciphers, and modern ciphers (e.g. the RSA method). Students will explore both theory and practice regarding the encryption, decryption and cryptanalysis of the ciphers.

*Prerequisite:* MAT 175

### MAT 422 - Methods and Materials: Teaching Mathematics in the Secondary School

*Semester:* Offered at discretion of department*Semester hours:* 2

This course requires focused study and consultation with a public school mathematics teacher or other acceptable professional. Hours will be arranged in consultation with the content area professor, the secondary education professor, the student, and the professional mentor. Methods of teaching mathematical content appropriate for grades 5-12 are explored. Appropriate use of technology and implications of current research in mathematics education are discussed. Current NCTM curriculum standards are used as the foundation of the course.

*Prerequisite:* EDC 040, admission to the teacher education program, senior standing

### MAT 450 - Internship

*Semester:* Offered at discretion of department*Semester hours:* 1-12

This course is a guided work experience. The student must arrange the internship in agreement with a member of the mathematics faculty and the Office of Career Services. Contract is required. The internship will not count as part of the minimum number of credits required in the major. Pass/no pass grading.

*Prerequisite:* junior or senior standing, a cumulative GPA of at least 2.00, and a major GPA of at least 2.25

### MAT 481 - Introduction to Real Analysis I

*Semester:* Fall; Alternate years*Semester hours:* 3

A proof-based course designed to establish the results in the calculus sequence from the axioms of the real-number system. Main topics include definitions and results related to limits, continuity, the intermediate value theorem, the derivative and differentiation rules, Riemann integral and Riemann-integrability, convergence of sequences and series, uniform convergence of function sequences, and power series.

*Prerequisite:* MAT 176 and MAT 212

### MAT 482 - Introduction to Real Analysis II

*Semester:* Spring; Alternate years*Semester hours:* 3

This course is a continuation of MAT 481. Topics include the Riemann-Stieltjes integral, uniform convergence, infinite series, functions of several real variables, compactness and metric space topology, and the implicit function theorem.

*Prerequisite:* MAT 481

### MAT 490 - Senior Seminar

*Semester:* Spring; Even years*Semester hours:* 2

Selected topics in mathematics are explored.

*Prerequisite:* permission from instructor

### MAT 499 - Independent Study

*Semester:* Offered at discretion of department*Semester hours:* 1-3

This course allows interested students to devise and pursue independent study in an area agreed upon in consultation with, and supervised by, a faculty member. Students should be either a major or minor in mathematics or mathematics education and have a cumulative GPA of 3.00 or greater.

*Prerequisite:* junior or senior standing

- Robyn Cummings, Professor
- Ulrich Hoensch, Professor
- Debra Wiens, Professor