# Curriculum and Courses

The Department of Mathematics offers courses for undergraduate and graduate students covering a wide range of mathematics. For a full listing of these courses, including course descriptions, hours, credits, prerequisites and curriculum checklists, please see the South Dakota Mines Course Catalog or visit the Mathematics Department main office. Information on course-specific learning outcomes can be found at the bottom of this page.

Students sometimes struggle with academic problems and decisions, and we always recommend that you seek help from a faculty advisor if you find yourself in this situation. Advising is an important and valued activity for faculty members and we want to assist you with the pursuit of your academic program.  Here are some other resources to help you:

## Course Learning Outcomes

### MATH 110: Survey of Mathematics

Course Learning Outcomes

A student who successfully completes this course should, at a minimum, be able to:

1. Know the major campus services important for success in college
2. Have begun the process of planning for a career in Computer Science and/or Mathematics
3. Be familiar with the ethical standards of the profession
4. Demonstrate an ability to work in teams

### MATH 114: College Algebra

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Demonstrate algebraic manipulation skills.
2. Properly use properties of exponents.
3. Solve linear equations and quadratic equations.
4. Properly manipulate rational expressions.
5. Demonstrate basic skills on building a graph and be able to interpret graphical data.
6. Calculate and interpret the inverse of a given function.

This course meets the BOR mandated GenEd Goal #5: Students will understand and apply fundamental mathematical processes and reasoning.

Student Learning Outcomes

As a result of taking a course meeting this goal, students will:

1. Use mathematical symbols and mathematical structure to model and solve real world problems.
• On a class exam, quiz, and / or homework assignment: students will correctly write and use polynomials, rational expressions, exponential expressions, logarithmic expressions, and radical expressions to solve real world problems.
2. Demonstrate appropriate communication skills related to mathematical terms.
• On a class exam, quiz, and / or homework assignment: students will identify, interpret, and use algebraic expressions and functional notation to solve real world problems. This will be demonstrated on in-class problems, labs, homework, quizzes and/or exams.
3. Demonstrate the correct use of quantifiable measurements of real world situations.
•  On a class exam, quiz, and / or homework assignment: students will evaluate functions involving polynomial, rational, exponential, logarithmic, and radical expressions.

### MATH 120: Trigonometry

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Demonstrate mastery of requisite algebraic topics, for example, rational exponents, graphing transformations, and the relationship between functions and their inverses; in particular, exponentials and logarithms.
2. Compare and calculate angle measurements in both degrees and radians.
3. Study right triangle trig and the unit circle to derive values of the six trigonometric functions.
4. Graph trigonometric functions with an understanding of their domains, ranges, period, and phase shift.
5. Identify trig identities and use to solve trig equations, reduce expressions, and manipulate trig functions.
6. Apply Law of Sines and Law of Cosines.
7. Identify domain and range of inverse trigonometric functions and apply inverse trigonometric functions to solve equations.
8. Define polar coordinates and the conversion between them and rectangular coordinates.

This course meets the BOR mandated GenEd Goal #5: Students will understand and apply fundamental mathematical processes and reasoning.

Student Learning Outcomes

As a result of taking a course meeting this goal, students will:

1. Use mathematical symbols and mathematical structure to model and solve real world problems.
• On a class exam, quiz, and / or homework assignment: students will be asked to demonstrate the use of trigonometric functions to model a real world problem.
2. Demonstrate appropriate communication skills related to mathematical terms
• On a class exam, quiz, and / or homework assignment: students will be asked to demonstrate appropriate communication skills related to the mathematical terms and concepts that are associated with trigonometric functions.
3. Demonstrate the correct use of quantifiable measurements of real world situations
• On a class exam, quiz, and / or homework assignment: students will be asked to demonstrate the correct use of quantifiable measurement for real world situations. For example, the differences between degree measure and radian measure, the differences between period and frequency, and the properties of inverse functions can each play an important role in physical applications.

### MATH 123: Calculus I

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Determine limits of various algebraic and trigonometric functions using limit laws.
2. Take derivatives of trigonometric and algebraic functions using the power rule, chain rule, product rule, and quotient rule
3. Use the derivative in applications such as velocity and acceleration, related rates, optimization, and curve sketching.
4. Integrate algebraic and trigonometric functions using the power rule and substitution.
5. Demonstrate the use of the integral in an application. Examples may include area, volume, moments, work, arc length, and surface area.
6. Use a computer algebra system to implement the solution techniques that are covered in Calculus I.

This course meets the BOR mandated GenEd Goal #5: Students will understand and apply fundamental mathematical processes and reasoning.

Student Learning Outcomes

As a result of taking courses meeting this goal, students will:

1. Use mathematical symbols and mathematical structure to model and solve real world problems.
• On a class exam, quiz, and / or homework assignment: students will be asked to demonstrate the use of trigonometric functions to model a real world problem.
2. Demonstrate appropriate communication skills related to mathematical terms
• On a class exam, quiz, and / or homework assignment: students will identify, interpret, and correctly apply standard mathematics symbols to solve problems requiring the definition and use of derivative.
• On a class exam, quiz, and / or homework assignment: students will Identify, interpret, and correctly apply standard mathematics symbols to solve problems requiring the definition and use of the integral.
3. Demonstrate the correct use of quantifiable measurements of real world situations
• On a class exam, quiz, and / or homework assignment: students will apply their knowledge of the integral in applications such as area, volume, moments, work, arc length, and surface area.
• On a class exam, quiz, and / or homework assignment: students will apply their knowledge of the derivative in applications such as related rates, linear approximations, Newton’s Method, curve sketching, optimization, velocity, and acceleration.

### MATH 125: Calculus II

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Apply the methods of calculus to a variety of transcendental functions, including logarithmic functions, exponential functions, and inverse trigonometric functions.
2. Evaluate definite and indefinite integrals using techniques including substitution, integration by parts, partial fractions, and improper integration techniques.
3. Determine convergence and divergence of infinite series using various tests, including the divergence test and the ratio test.
4. Produce Taylor series expansions for a variety of elementary functions.
5. Solve linear systems using techniques of matrix algebra.
6. Use a computer algebra system to implement the solution techniques that are covered in Calculus II.

This course meets the BOR mandated GenEd Goal #5: Students will understand and apply fundamental mathematical processes and reasoning.

Student Learning Outcomes

As a result of taking courses meeting this goal, students will:

1. Use mathematical symbols and mathematical structure to model and solve real world problems
• On a class exam, quiz, and / or homework assignment: students will identify, interpret, and correctly apply standard mathematics symbols to solve problems requiring differentiation and integration techniques.
2. Demonstrate appropriate communication skills related to mathematical terms
• On a class exam, quiz, and / or homework assignment: students will correctly use functional notation of algebra, trigonometry, and calculus.
3. Demonstrate the correct use of quantifiable measurements of real world situations
• On a class exam, quiz, and / or homework assignment: students will apply their knowledge of single-variable calculus, and infinite sequences and series, in applications such as area computation and function approximation.

### MATH 225: Calculus III

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Analyze position, velocity, and acceleration in two or three dimensions using the calculus of vector valued functions.
2. Use partial derivatives to calculate rates of change of multivariable functions and to solve multivariable optimization problems.
3. Use multiple integrals to compute volume, mass, center of mass, and related quantities for two- and three-dimensional objects.
4. Compute line integrals, including those representing work done by a variable force in a vector field.

This course meets the BOR mandated GenEd Goal #5: Students will understand and apply fundamental mathematical processes and reasoning.

Student Learning Outcomes

As a result of taking courses meeting this goal, students will:

1. Use mathematical symbols and mathematical structure to model and solve real world problems
• On a class exam, quiz, and / or homework assignment: students will identify, interpret, and correctly apply standard mathematics symbols to solve problems requiring the partial derivative.
• On a class exam, quiz, and / or homework assignment: students will identify, interpret, and correctly apply standard mathematics symbols to solve problems requiring multiple integrals.
• On a class exam, quiz, and / or homework assignment: students will identify, interpret, and correctly apply standard mathematics symbols to solve problems requiring vectors and vector functions.
2. Demonstrate appropriate communication skills related to mathematical terms
• On a class exam, quiz, and / or homework assignment: students will correctly use functional notation of algebra, trigonometry, and calculus.
3. Demonstrate the correct use of quantifiable measurements of real world situations
• On a class exam, quiz, and / or homework assignment: students will apply their knowledge of the integral in applications such as area, volume, moments, work, arc length, and surface area.
• On a class exam, quiz, and / or homework assignment: students will apply their knowledge of the derivative in applications such as rates of change, linear approximations, optimization, velocity, and acceleration.

### MATH 315: Linear Algebra

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Use the standard language and notation of linear algebra appropriately.
2. Solve a system of linear equations using Gaussian Elimination.
3. Demonstrate an understanding of the fundamental matrix concepts of singular, nonsingular, inverse, and rank.
4. Demonstrate an understanding of the fundamental concepts of vector space, subspace, basis, and dimension.
5. Compute determinants, eigenvalues, and eigenvectors, and use eigenvectors to diagonalize a matrix.
6. Use basic orthogonality principles such as least squares, the Gram-Schmidt process, and/or the QR factorization of a matrix.

### MATH 321: Differential Equations

Course Learning Outcomes

A student who successfully completes this course should be able to

1. Identify and solve first order separable ordinary differential equations.
2. Identify and solve first order linear ordinary differential equations.
3. Solve homogeneous and non-homogeneous higher order linear ordinary differential equations with constant coefficients.
4. Use Laplace Transforms to solve ordinary differential equations.
5. Solve linear systems of ordinary differential equations.
6. Use a software package to find numerical approximations of solutions to ordinary differential equations.
7. Solve applications involving the kinds of equations listed above.

### MATH 373: Introduction to Numerical Analysis

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Write finite approximations of the first and second derivative.
2. Identify the different types of error in numerical methods.
3. Implement the use of numerical approximation for integration.
4. Implement the use of Runge-Kutta to solve initial value problems.
5. Apply numerical methods to solve systems of differential equations.
6. Apply numerical methods to solve nonlinear equations.

### MATH 381: Introduction to Probability and Statistics

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Understand and produce a variety of descriptive summaries of data (e.g. numerical summary statistics, boxplots, and histograms).
2. Compute probabilities
• Using elementary counting techniques.
• Using fundamental rules of probability, including Bayes' Rule and the Law of Total Probability.
• By recognizing and using standard probability mass functions (e.g. binomial) and density functions (e.g. normal).
• Approximately, using the Central Limit Theorem.
3. Produce and understand point and interval estimates for a population proportion and a population mean.
4. Understand the fundamental logic behind a formal hypothesis test and be able to carry out such tests on a population proportion and a population mean.
5. Develop some proficiency in the use of a statistical software package.
6. Learn, and correctly use, fundamental probability and statistics language and notation.

### MATH 382: Probability and Statistics II

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Understand the principle of least-squares and determine the least-squares line for a bivariate dataset.
2. Build, using statistical software, a regression model expressing a response variable in terms of one or more predictor variables by using appropriate numerical and graphical diagnostics.
3. Carry out a variety of two- and several-sample hypothesis tests and reasonably confirm any necessary underlying assumptions. These tests include, but are not limited to, the following:
• Test of the equality of two proportions (independent samples)
• Test of the equality of two means (independent or dependent samples)
• Test of the equality of several means (independent samples)
4. Learn, and correctly use, fundamental probability and statistics language associated with building regression models and carrying out hypothesis tests.
5. Demonstrate the ability to collaborate with colleagues.

### MATH 413: Abstract Algebra

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Implement the use of modular arithmetic, greatest common divisor, and the division algorithm.
2. Prove basic facts about algebraic structures such as groups, rings, and fields.
3. Prove or disprove whether a given function between two algebraic structures is a homomorphism or an isomorphism.
4. Prove or disprove whether a given subset of an algebraic structure is a sub-structure.
5. Prove basic facts regarding quotient structures and their corresponding special sub-structures (e.g. normal subgroups, ideals).

### MATH 421: Complex Analysis

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Perform basic arithmetic operations with complex numbers.
2. Use the CR equations to test for analyticity and to compute the complex derivative.
3. Identify the similarities and differences between standard real elementary functions (such as power functions, trigonometric functions, exponential, and the logarithm) and their analytic extensions.
4. Use elementary functions (or their branches) to solve basic domain mapping problems.
5. Compute contour integrals using the integration theorems of complex analysis, such as Cauchy's Theorem, Cauchy’s Integral Formulas, or the Residue Theorem.

### MATH 423: Advanced Calculus I

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Write proofs of elementary theorems and proof-based solutions to problems related to
• Limits of sequences.
• Limits, continuity, and differentiability of functions of a single variable.
• Riemann integration theory for functions of a single variable.
2. Apply the major theorems of single variable calculus.

### MATH 424: Advanced Calculus II

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Write proofs of elementary theorems and proof-based solutions to problems related to:
• Convergence of infinite series of constant terms,
• Convergence of sequences and series of functions,
• Limits, continuity, and differentiability of functions of several variables, and
• Riemann integration theory for functions of several variables.
2. Apply the major theorems of several variable calculus.
3. Demonstrate the ability to verbally present to a technical audience.

### MATH 432: Partial Differential Equations

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Apply the solution techniques associated with the problems of the heat equation, wave equation, and Laplace's equation.
2. Write the Fourier Series for a function.
3. Use the Fourier Series to approximate the solution to partial differential equations on bounded domains.
4. Solve equations on unbounded domains using Laplace and Fourier transforms.
5. Describe the characteristics of a well-posed partial differential equation.

### MATH 443/543: Data Analysis

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. With the help of software, produce basic graphical displays to uncover structure in multidimensional datasets.
2. With the help of software, apply data reduction techniques (e.g. multidimensional scaling, principal components analysis) to analyze structure in multidimensional datasets.
3. With the help of software, apply unsupervised learning techniques (e.g. clustering) to analyze structure in multidimensional datasets.
4. With the help of software, apply supervised learning techniques (e.g. regression, logistic regression, discriminant analysis, decision trees) to make predictions in multidimensional datasets.
5. Access manage and pre-process data – including by way of a relational database management system, in a form convenient for further analysis.

Students at the 500 level will be evaluated at a higher standard.  This shall, in general, involve an additional component consisting of scholarly work related to the class.

### MATH 447: Design of Experiments

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Conduct single- and multi-factor experiments from given data.
2. Build multiple regression models in support of the design of experiments (e.g. response surface modeling).
3. Make and verify reasonable model assumptions (e.g. normality) in various design of experiments settings.
4. Demonstrate an understanding of the following principles: randomization and blocking, confounding with blocking, and aliasing (in fractional factorial designs).
5. Implement the use of software to solve problems related to the design of experiments.
6. Implement the language and notation associated with the design of experiments

### MATH 451/551: Mathematical Modeling

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Describe and interpret the components of a mathematical model
2. Construct linear system, integral equation, and/or differential equation models
3. Apply existence and uniqueness results to deterministic models
4. Analyze simple stochastic and/or probabilistic models
5. Determine model limits, analyze sensitivity, and estimate errors
6. Demonstrate the ability to collaborate with colleagues
7. Demonstrate to communicate mathematics in written form

Students at the 500 level will be evaluated at a higher standard.  This shall, in general, involve an additional component consisting of scholarly work related to the class.

Course Learning Outcomes

At a minimum, a student who completes this course should be able to:

1. Understand and apply selected topics in advanced mathematics.
2. Apply appropriate computational technology to assist in the study and investigation of mathematics.
3. Communicate mathematical ideas with colleagues through collaboration and exposition.

Students at the 500 level will be evaluated at a higher standard.  This shall, in general, involve an additional component consisting of scholarly work related to the class.

### MATH 498: Undergraduate Research and Scholarship

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Identify appropriate steps to conducting research in mathematics or applied mathematics.
2. Make use of proper notation and vocabulary as it relates to their research problem.
3. Identify proper sources that can be used for conducting research in mathematics or applied mathematics.
4. Present the research problem to a knowledgeable audience.

### MATH 402: Communicating Mathematics

Course Learning Outcomes

A student who successfully completes this course should be able to:

1. Identify the steps to writing a technical report in mathematics
2. Identify the steps to constructing a technical presentation in mathematics
3. Communicate a research problem and its solution to a knowledgeable audience