Department of Mathematics The Ohio State University 2016-2017 Mathematics Courses Course Number 1050 1075 1116 1118 1125 1126 1130 1131 1135 1136 1148 1149 1150 1151 1152 1156 1157 1161.01 1161.02 1165 1166 1172 1181H 1187H 1295 2137 2138 2153 2162.01 2162.02 2167 2168 2173 Course Title Precollege Mathematics I Precollege Mathematics II Excursions in Mathematics Mathematics for Architects Mathematics for Elementary Teachers I Mathematics for Elementary Teachers II College Algebra for Business Calculus for Business Number and Operations for Teachers Measurement & Geometry for Teachers College Algebra Trigonometry (Autumn 2016) Trigonometry (Spring 2017) Pre-Calculus Calculus I Calculus II Calculus for the Biological Sciences Mathematical Modeling for the Biological Sciences Accelerated Calculus I Accelerated Calculus I for Honors Engineers Math for Middle School Teachers I Math for Middle School Teachers II Engineering Mathematics A Honors Calculus I Honors Problem Solving Introductory Seminar Algebra and Coordinate Geometry for Teachers Calculus and its History for Teachers Calculus III Accelerated Calculus II Accelerated Calculus II for Honors Engineers Calculus for Middle School Teachers History of Mathematics for Middle School Teachers Engineering Mathematics B Course Number 2174 2177 2182H 2255 2366 2415 2568 3345 3350 3532 3588 3589 3607 3618 4181H 4182H 4350 4504 4507 4512 4530 4545 4547 4548 4551 4552 4556 4557 4568 4573 4575 4578 4580 4581 5520H 5522H 5529H 5530H 5540H 5576H Course Title Linear Algebra & Differential Equations for Engineers Mathematical Topics for Engineers Honors Calculus II Differential Equations and Their Applications Introduction to Discrete Mathematics Ordinary and Partial Differential Equations Linear Algebra Foundations of Higher Mathematics Introduction to Mathematical Biology Mathematical Foundations of Actuarial Science Practicum in Actuarial Science Introduction to Financial Mathematics Beginning Scientific Computing Theory of Interest Honors Analysis I Honors Analysis II Quantitative Neuroscience History of Mathematics Geometry Partial Differential Equations for Science & Engineering Probability Analysis Overview Introductory Analysis I Introductory Analysis II Vector Analysis Complex Analysis Dynamical Systems Partial Differential Equations Linear Algebra for Engineering Graduate Students Elementary Number Theory Combinatorial Mathematics Discrete Mathematical Models Abstract Algebra I Abstract Algebra II Honors Linear Algebra and Differential Equations Honors Complex Analysis Honors Combinatorics Honors Probability Honors Differential Geometry Honors Number Theory Course Number 5590H 5591H 5630 5631 5632 5633 5634 5756 5757 Course Title Honors Abstract Algebra I Honors Abstract Algebra II Life Contingencies I Life Contingencies II Financial Economics for Actuaries Loss Models I Loss Models II Mathematical Methods in Relativity Theory I Mathematical Methods in Relativity Theory II Mathematics 1050 Precollege Mathematics I Autumn, Spring 5 credits Catalog Description: Fractions and decimals, basic algebra, graphing lines, factoring, systems of equations. Credit for this course will not count toward graduation in any degree program. Prerequisite: Math Placement Level T; or Math 1040 or 40 or 50; or permission of department. Exclusions: Not open to students with credit for any Math course above 1050 (050). Purpose of Course: Mathematics 1050 is designed to meet the needs of the students entering The Ohio State University at the lowest placement, course code T. This course will prepare students for Math 1075. Follow-up Course: Math 1075 Sequencing Chart: Page 1 2016-2017 Math 1050 Mathematics 1050 Precollege Mathematics I Autumn, Spring 5 credits Text: Beginning Algebra, 8th edition, by Aufmann & Lockwood, Cengage, ISBN: 9781305942653 Topics List: 1. Review: real numbers, number line, integral exponents, scientiﬁc notation, negative numbers, fractions, order of operations, basic geometry. 2. Linear equations and inequalities in one variable. Applications: word problems. 3. Graphical representations, straight lines as graphs, slope, intercepts, slope-intercept form, and point-slope form. Linear inequalities in two variables. 4. Systems of two linear equations in two unknowns. 5. Polynomials: addition, subtraction, multiplication, factoring, division. 6. Solving quadratic equations by factoring. Applications. 7. Introduction to function notation. Page 2 2016-2017 Math 1050 Mathematics 1075 Precollege Mathematics II Autumn, Spring 4 credits Catalog Description: Algebraic, rational, and radical expressions; functions and graphs; quadratic equations; absolute value; inequalities; and applications. Prerequisite: Math Placement Level S, a grade of C– or better in Math 1050, or credit for Math 75 or 1074. Exclusions: Not open to students with credit for any higher numbered math class, or for any quarter math class numbered higher than 75. Text: Intermediate Algebra: Math 1075, OSU Custom Edition, Miller, O'Neill & Hyde, McGraw-Hill, ISBN 9781269577980 Follow-up Courses: Math 1116 for students in liberal arts or students in the precertification programs on regional campuses. Math 1125 for students intending to pursue a M.Ed. in early or middle childhood. Math 1130 College Algebra for Business Math 1148 Traditional College Algebra Sequencing Chart: Page 1 2016-2017 Math 1075 Mathematics 1075 Precollege Mathematics II Autumn, Spring 4 credits Topics List: Ch. 4 Linear Inequalities 4.1 Solving linear inequalities using addition & subtraction 4.2 Solving linear inequalities using multiplication & division 4.3 Solving compound inequalities 4.4 Solving absolute value equations & inequalities 4.5 Graphing systems of inequalities in two variables Ch. 6 Factoring Polynomials 6.1 Introduction to factoring polynomials 6.2 Factoring trinomials of the form x 2 bx c 6.3 Factoring trinomials of the form ax 2 bx c 6.4 Factoring special binomials 6.5 Factoring by grouping; General strategies for factoring 6.6 Solving equations by factoring Ch. 9 Rational Functions 9.1 Graphs of rational functions 9.2 Reducing rational expressions; Multiplying and dividing rational expressions 9.3 Adding and subtracting rational expressions 9.4 Combining operations; Complex rational expressions 9.5 Solving equations containing rational expressions 9.6 Inverse and joint variation; Other applications yielding equations with fractions Ch. 7 Solving Quadratic Equations 7.1 Extraction of roots and properties of square roots 7.2 Solving quadratic equations by completing the square 7.3 The quadratic formula 7.4 Applications of quadratic equations 7.5 Complex numbers; Solving quadratic equations with complex solutions Ch. 8 Functions: Linear, Absolute Value, and Quadratic 8.1 Functions and representations of functions 8.2 Linear Functions 8.3 Absolute value functions 8.4 Quadratic functions Ch. 10 Square Root & Cube Root Functions and Rational Exponents 10.1 Evaluating radical expressions 10.2 Adding & subtracting radical expressions 10.3 Multiplying & dividing radical expressions 10.4 Solving equations containing radical expressions 10.5 Rational exponents & radicals Page 2 2016-2017 Math 1075 Mathematics 1116 Excursions in Mathematics Autumn, Spring 3 credits Catalog Description: Critical thinking and problem solving, with relevant topics met in everyday life. Appropriate for non-science majors. Prerequisite: Math Placement Level R or higher; or credit for 1075, 75, 104, 1073 or 1074; or ACT math score ≥ 22 or SAT math score ≥ 520 (scores must be less than 2 years old). Exclusions: Not open to students with credit for Math 1152 or higher, or for quarter math class numbered 153 or higher. Purpose of Course: The emphasis in this course is on intuitive understanding and developing some facility for applying mathematical ideas to problem solving. Follow-up Courses: None. Math 1116 is a terminal course. Text: Excursions in Modern Mathematics, 8th edition, by Tannenbaum, Pearson, ISBN 9780321825735 Topics List: 1. Graph theory: graphs, Euler and Hamilton circuits, algorithms for Traveling Salesman Problem, spanning trees, etc. 2. Voting & apportionment: preference ballots; apportionment paradoxes; Congressional apportionment; methods of Jeﬀerson, Adams, and Webster. 3. Patterns & growth: Fibonacci and recursive sequences, golden ratio, population growth models: linear, exponential, and logistic. 4. Symmetry: Rigid motions, rosettes, friezes, rudiments of group theory. 5. Counting & probability: counting principles, permutations and combinations, multiplication rule, randomness, probability. 6. Fractals: recursive deﬁnitions, standard examples (Koch snowﬂake, Sierpinski gasket etc.), self-similarity, fractional dimension. 7. Linear programming: mixture problems, examples in low dimension, corner point principle, algorithms. Page 1 2016-2017 Math 1116 Mathematics 1118 Mathematics for Architects Autumn, Spring 3 credits Catalog Description: Topics in geometry for architecture majors. Prerequisite: Enrollment in the School of Architecture and: Math Placement Level L or M; C- or better in 1148 or 1150; or credit for 148 or 150. Purpose of Course: The majority of the audience is made up of Architecture majors for whom the course is a requirement. The intent of the course is to introduce these students to the mathematics inherent in 2D and 3D design. Moreover, there is an emphasis on similar figures and the issues that arise when scaling lengths, areas, and volumes. Follow-up Courses: There are really no follow-up courses. To start any other mathematics sequence will probably involve beginning at an appropriate entry level course. Students interested in further course work in mathematics should consult the Mathematics Advisors in 250 Mathematics Bldg. Text: Course Notes, by Snapp Topics List: 1. 2. 3. 4. 5. 6. 7. 8. Page 1 Geometric models, transformations, matrices. Plane tessellations, concepts of symmetry. Polyhedra: Platonic and Archimedean. Invariants: area, volume, Euler characteristic. Mathematics of perspective drawing. String art: curves deﬁned as envelopes of tangent lines, ruled surfaces. Discrete curvature: Descartes theorem and beyond. Higher dimensions: tesseracts and other 4-D polyhedra. 2016-2017 Math 1118 Mathematics 1125 Mathematics for Elementary Teachers I Autumn, Spring 5 credits Catalog Description: Math 1125 involves numbers, operations, geometry, measurement, and mathematical reasoning for prospective elementary school teachers. Prerequisite: A grade of C- or above in 1075; or credit for 1074, 75, or 104; or Math Placement Level R or above; or ACT math score ≥ 22 or SAT math score ≥ 520 (scores must be less than 2 years old) and enrollment in Early Childhood or Special Education major, or in Middle Childhood major or pre-major with area of concentration different than Math. Exclusions: Not open to students with credit for 106. Purpose of Course: To develop an appreciation of, and basic competency in, the use of analytical thought in the development of a cohesive body of useful mathematical knowledge, with special emphasis on topics encountered in elementary and middle school mathematics programs. Math 1125 addresses the meaning of whole numbers, integers, rational numbers, and operations with these, number theory, and algebraic thinking. Appropriate only for those preparing to become early childhood educators and for those preparing to teach subjects other than math in middle school. Follow-up Courses: Math 1126. Text: Mathematics for Elementary Teachers, with Activity Manual, 4rd Edition, by Sybilla Beckmann, Pearson, ISBN for the package is 9780321836715 (loose-leaf) and Student Packet. Topics List: 1. Counting and the decimal system. 2. Fractions and integers and their meaning. 3. Addition and subtraction of fractions, decimals, and integers. 4. Multiplication of fractions, decimals, and integers. 5. Division of fractions, decimals, and integers. 6. Ratios and proportional reasoning. 7. Number theory: factors and multiples, LCM, GCF, divisibility tests, prime numbers, unique factorization, notations for fractions and decimals. 8. Algebraic thinking: writing expressions, solving equations, sequences. 9. Problem solving and justification are themes of the course. *Currently taught in either lecture/recitation or workshop format. Page 1 2016-2017 Math 1125 Mathematics 1126 Mathematics for Elementary Teachers II Autumn, Spring 5 credits Catalog Description: Continuation of 1125: Math 1126 involves geometry, measurement, number theory, algebraic thinking, counting techniques, probability, a mathematical reasoning for prospective elementary school teachers. Prerequisite: A grade of C- or above in 1125; and enrollment in Early Childhood or Special Education major, or in Middle Childhood major or pre-major with area of concentration different than Math. Exclusions: Not open to students with credit for 107. Purpose of Course: To develop an appreciation of, and basic competency in, the use of analytical thought in the development of a cohesive body of useful mathematical knowledge, with special emphasis on topics encountered in elementary and middle school mathematics programs. Math 1126 addresses basic geometric concepts and measurement, symmetry and rigid motions, congruence, similarity and scaling, coordinate geometry, algebraic thinking, linear functions, counting techniques and probability. Appropriate for those preparing to become early childhood educators and for those preparing to teach subjects other than math in middle school. Text: Mathematics for Elementary Teachers, with Activity Manual, 4rd Edition, by Sybilla Beckmann, Pearson, ISBN for the package is 9780321836715 (loose-leaf) and Student Packet. Topics List: 1. Spatial visualization and basic geometric concepts: angles, 2- and 3-D shapes and their properties. 2. Measurement: meaning of length, area, volume, measurement techniques, unit conversion, actions preserving area/volume, and scaling. 3. Transformations: symmetry, congruence, similarity. 4. Geometric constructions with various tools (compass, paper folding). 5. Algebraic thinking: expressions, measurement formulas, scaling, functions, use of formulas, graphs, and tables, sequences, and coordinate geometry. 6. Counting: inclusion/exclusion, fundamental counting principle, tree diagrams, permutations and combinations, Pascal’s triangle. 7. Basic ideas of probability: Law of Large Numbers, sample and event spaces, use of tree diagrams, simulations, and discussion of common misconceptions. 8. Problem solving and justifications at multiple levels are themes of the course. *Currently taught in either lecture/recitation or workshop format. Page 1 2016-2017 Math 1126 Mathematics 1130 College Algebra for Business Autumn, Spring 4 credits Catalog Description: Algebraic, exponential, and logarithmic functions. Matrix algebra. Applications to business. Prerequisite: Math Placement Level N; C- or better in 1075; or credit for 104; or ACT math score ≥ 22 or SAT math score ≥ 520 (scores must be less than 2 years old). Exclusions: Not open to students with credit for 1131 or for any math course numbered 1149 or higher, or a quarter-system math course numbered 150 or higher. Purpose of Course: Math 1130 is a pre-calculus course with a finance section slanted toward a business program. The applications are business related. Follow-up Course: Math 1131 Text: Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences, 13th Edition; by Haeussler, Paul, and Wood; published by Pearson; ISBN 9781256966096 Topics List: Review of Algebra: 0.7: Equations, In Particular Linear Equations. 0.8: Quadratic Equations. 1.2: Linear Inequalities. 1.3: Applications of Inequalities. 1.6: Sequences Functions: 2.1: Functions. 2.2: Special functions. 2.3: Combinations of functions. 2.4: Inverse Functions. 2.5: Graphs of functions Page 1 2016-2017 Math 1130 Mathematics 1130 College Algebra for Business Autumn, Spring 4 credits 3. Linear and Quadratic Functions: 3.1: Lines 3.2: Applications and Linear Functions. 3.3: Quadratic Functions 3.4: Systems of Linear Equations. 3.6: Applications of Systems of Equations. 4. Exponential and Logarithmic Functions: 4.1: Exponential Functions. 4.2: Logarithmic functions. 4.3: Properties of Logarithms. 4.4: Logarithmic and Exponential Equations. 5. Mathematics of Finance: 5.1: Compound Interest. 5.2: Present Value. 5.3: Interest Compounded Continuously. 5.4: Annuities 5.5: Amortization of Loans. 6. Matrix Algebra: 6.1: Matrices. 6.2: Matrix Addition and Scalar Multiplication. 6.3: Matrix Multiplication. 6.4: Solving Systems by Reducing Matrices. 6.6: Inverses Page 2 2016-2017 Math 1130 Mathematics 1131 Calculus for Business Autumn, Spring 5 credits Catalog Description: Survey of calculus of one and several variables; applications to business. Prerequisite: Math Placement Level L; C- or better in 1130, 1148, 1144, or 1150; credit for 130 or 148. Exclusions: Not open to students with credit for a math course numbered 1151 (151.xx) or higher, or for 132 or 1134. Text: Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences, 13th Edition, by Haeussler, Paul, Wood, published by Pearson: ISBN-10: 1-256-96609-6, ISBN-13: 978-1-256-96609-8. Topics List: 1. Idea of limits, continuity, and derivative. Interpret derivative as a limit, slope, and rate of change. 2. Calculate derivatives of algebraic, exponential, and logarithmic functions. 3. Monotonicity, concavity, extrema of functions, second derivative tests, applications to graphing. 4. Applications: marginal analysis in business, optimization. 5. Anti-derivatives, separable ﬁrst-order ODEs. 6. Riemann integral, substitution, Fundamental Theorem, area, applications. 7. Partial derivatives, extrema and second derivative test for two-variable functions, Lagrange multipliers. Applications to business. Page 1 2016-2017 Math 1131 Mathematics 1135 Number and Operations for Teachers Autumn, Spring 5 credits Catalog Description: This course is the first in a two semester sequence for teachers of elementary and middle grade students. This course focuses on concepts of numbers and arithmetic operations, including modern and historical perspectives. Prerequisite: A grade of C- or above in 1075; or credit for 1074, 75, or 104; or Math Placement Level R or above; or ACT math subscore of 22 or higher that is less than 2 years old. Exclusions: Not open to students with credit for 106. Text: Mathematics for Elementary Teachers, with Activity Manual, 4th Edition, by Sybilla Beckmann, Pearson, ISBN for the package is 9780321836715 (loose-leaf). Purpose: This course covers the concepts of whole numbers (positive and negative), place value (base-ten and alternate bases), decimals, and fractions. Some content on irrational numbers appears at the end, and this is extended in Algebra and coordinate geometry for teachers (2137). The four arithmetic operations are covered both conceptually and algorithmically. Attention is given to ensuring that students can perform the algorithms correctly and explain why they give accurate answers. Lastly, the course covers the concepts of proportions and how they are related both to multiplication/division and to fractions. Factors, divisibility, and some elementary number theory complete the course. Topics List: 1. Counting numbers, decimals 2. Meaning of fractions 3. Meaning of addition and subtraction 4. Meaning of multiplication 5. Multiplying fractions, decimals, integers 6. Meaning of division 7. Dividing fractions, decimals, integers 8. Meaning of ratios, rates, proportions 9. Greatest common divisor, least common multiple 10. Rational and irrational numbers Page 1 2016-2017 Math 1135 Mathematics 1136 Measurement & Geometry for Teachers Autumn, Spring 5 credits Catalog Description: This course is the second in a two semester sequence for teachers of elementary and middle grade students. This course focuses on concepts of measurement and geometry, including modern and historical perspectives. Prerequisite: A grade of C- or above in “Number and Operations for Teachers” (Math 1135) Text: Mathematics for Elementary Teachers, with Activity Manual, 4th Edition, by Sybilla Beckmann, Pearson, ISBN for the package is 9780321836715 (loose-leaf). Recommended Supplemental Texts: Geometric Structures: An Inquiry-Based Approach for Prospective Elementary and Middle School Teachers, by Douglas Aichele and John Wolfe, Pearson, ISBN 9780131483927 Elementary Geometry for Teachers, by Thomas Parker and Scott Baldridge, Sefton-Ash Publishing, ISBN 9780974814056 Purpose: The course consists of fundamental topics in Euclidean geometry starting with measurement. This includes the concepts of length, area, volume, angles, units of measurement, precision and error. The basic properties of two and three dimensional geometric shapes and their relationships are a central part of the course. Special emphasis is put on geometric reasoning through problem solving, including unknown angle, length, area, and volume. The course also covers topics on transformations in the plane, symmetries, congruence, and similarity. Some geometric constructions and basic geometric proofs are included. Additional topics include an introduction to functions and equations, primarily in the linear case, and a brief introduction to probability. Topics List: 1. Measurement 2. Planar shapes 3. Polyhedra 4. Plane geometry 5. Transformations in the plane, congruence, symmetry 6. Linear equations and graphs 7. Algebra and linear equations 8. Probability Page 1 2016-2017 Math 1136 Mathematics 1148 College Algebra Autumn, Spring, Summer 4 credits Catalog Description: Functions: polynomial, rational, radical, exponential, and logarithmic. Introduction to rightangle trigonometry. Applications. Prerequisite: Math Placement Level N; C- or better in 1075; or credit for 104 or 148; or ACT math score ≥ 22 or SAT math score ≥ 520 (scores must be less than 2 years old). Exclusions: Not open to students with credit for 1144 or 1150 or higher, or for a quarter-system math course numbered 150 or higher. Purpose: College Algebra provides students a college level academic experience that emphasizes the use of algebra and functions in problem solving and modeling, where solutions to problems in realworld situations are formulated, validated, and analyzed using mental, paper-and-pencil, algebraic and technology-based techniques as appropriate using a variety of mathematical notation. Students should develop a framework of problem-solving techniques (e.g., read the problem at least twice; define variables; sketch and label a diagram; list what is given; restate the question asked; identify variables and parameters; use analytical, numerical and graphical solution methods as appropriate; determine the plausibility of and interpret solutions). – Adapted from the MAA/CUPM CRAFTY 2007 College Algebra Guidelines. This course is intended to satisfy the requirements of the Ohio Board of Regents TMM001 College Algebra course with learning outcomes specified in: http://regents.ohio.gov/transfer/otm/otm-learning-outcomes.php Text: College Algebra & Trigonometry, 1st Edition, by Miller and Gerken, published by McGraw-Hill. ISBN: 9781259976612 Technology: All students are required to have a graphing calculator, TI-83 or TI-84. Note: Any calculators (including TI-89 and TI-92) that use a Computer Algebra System (CAS) are not permitted. Page 1 2016-2017 Math 1148 Mathematics 1148 College Algebra Autumn, Spring, Summer 4 credits Sequencing Chart: Topics List: Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9 Week 10 Week 11 Week 12 Week 13 Week 14 Page 2 Section 1.7 – Inequalities Section 2.3 – Functions and Relations Section 2.4 – Linear Equations in Two Variables Section 2.5 – Applications of Linear Equations Section 9.1 – Systems of Linear Equations in Two Variables Section 9.2 – Systems of Linear Equations in Three Variables Section 2.6 – Transformations of Graphs Section 2.7 – Analyzing Graphs of Functions Test 1 Section 2.8 – Algebra of Functions and Composition Section 3.1 – Quadratic Functions and Applications Section 3.2 – Polynomial Functions Section 3.3 – Division of Polynomials Section 3.5 – Rational Functions Section 3.5 – Rational Functions Section 3.6 – Polynomial and Rational Inequalities Test 2 Section 4.1 – Inverse Functions Section 4.2 – Exponential Functions Section 4.2 – Exponential Functions Section 4.3 – Logarithmic Functions Section 4.3 – Logarithmic Functions Section 4.4 – Properties of Logarithms Section 4.4 – Properties of Logarithms Section 4.5 – Exponential and Logarithmic Equations Section 4.6 – Modeling with Exponential and Logarithmic Functions Comprehensive review, Final Exam 2016-2017 Math 1148 Mathematics 1149 Trigonometry Autumn, Spring, Summer 3 credits Catalog Description: Trigonometric functions and their properties. Vectors, polar coordinates and complex numbers. Prerequisite: C- or better in 1148, or permission of department. Exclusions: Not open to students with credit for 1144, or for any math course numbered 1150 or higher. Text: Precalculus: Mathematics for Calculus, 6th Edition, by J. Stewart, L. Redlin, and S. Watson, published by Cengage. ISBN Loose-leaf: 9781133904489 Hardback: 9780840068077 Technology: A graphing calculator is a required component in this course. It is recommended that you use a TI83, TI-83 plus, or a TI-84. Note that the TI-89, TI-92, and calculators that use a Computer Algebra System are not permitted. Topics List: 6.1 6.2 6.3 5.1 5.2 5.3 5.4 5.5 Angle Measure Trigonometry of Right Triangles Trigonometric Functions of Angles The Unit Circle Trigonometric Functions of Real Numbers Trigonometric Graphs More Trigonometric Graphs Inverse Trigonometric Functions and Their Graphs Midterm 1 Page 1 Autumn 2016 Math 1149 Mathematics 1149 Trigonometry Autumn, Spring, Summer 3 credits 6.4 6.5 6.6 7.1 7.2 7.3 7.4 7.5 8.3 Right Triangles The Law of Sines The Law of Cosines Trig. Identities Addition and Subtraction Formulas Double/Half Angle and Product‐Sum Formulas Trig. Equations More Trig. Equations Polar Forms of Complex Numbers; De Moivre’s Theorem Midterm 2 9.1 9.2 11.1 11.2 11.3 Page 2 Vectors in Two Dimensions The Dot Product Parabolas Ellipses Hyperbolas Autumn 2016 Math 1149 Mathematics 1149 Trigonometry Autumn, Spring, Summer 3 credits Catalog Description: Trigonometric functions and their properties. Vectors, polar coordinates and complex numbers. Prerequisite: C- or better in 1148, or permission of department. Exclusions: Not open to students with credit for 1144, or for any math course numbered 1150 or higher. Text: College Algebra & Trigonometry Mathematics 1e, by Miller and Gerken, ISBN 9781259976612. This textbook is packaged with an access code to Connect Math for a period of 720 days. It may be purchased at the bookstore or online via Carmen/Canvas. Technology: A graphing calculator is a required component in this course. It is recommended that you use a TI83, TI-83 plus, or a TI-84. Note that the TI-89, TI-92, and calculators that use a Computer Algebra System are not permitted. Topics List: 5.1 5.2 5.3 5.4 5.5 5.6 Angles and Their Measure Right Triangle Trigonometry Trigonometric Functions of any Angle Trigonometric Functions and the Unit Circle Graphs of Sine and Cosine Functions. Omit sinusoidal behavior. Graphs of Other Trigonometric Functions. Midterm 1 5.7 6.1 6.2 6.3 6.5 7.2 7.3 Inverse Trigonometric Functions. Omit inverse cot(t), sec(t), and csc(t).\ Fundamental Trigonometric Identities Sum and Difference Formulas Double-Angle and Half-Angle Formulas Trigonometric Functions. Solving graphically is optional. The Law of Sines The Law of Cosines Midterm 2 Page 1 Spring 2017 Math 1149 Mathematics 1149 Trigonometry Autumn, Spring, Summer 3 credits 8.3 8.4 8.5 11.1 11.2 11.3 Page 2 Complex Numbers in Polar Form. Omit nth roots of complex numbers. Vectors Dot Product The Ellipse. Applications is optional. The Hyperbola. Applications is optional. The Parabola. Applications is optional. Spring 2017 Math 1149 Mathematics 1150 Precalculus Autumn, Spring 5 credits Catalog Description: Functions: polynomial, rational, radical, exponential, logarithmic, trigonometric, and inverse trigonometric. Applications. Prerequisite: Math Placement Level M. Exclusions: Not open to students with credit for 1144, 1148, 1149, for any higher numbered math course, or for any quarter-system math course 150 or higher. Text: College Algebra & Trigonometry Mathematics 1e, by Miller and Gerken, ISBN 9781259976612. This textbook is packaged with an access code to Connect Math for a period of 720 days. It may be purchased at the bookstore or online via Carmen/Canvas. Technology: Every student is required to have a graphing calculator comparable in capability to a TI-83 or TI-84. However, calculators with symbolic algebra capabilities are not allowed during exams or quizzes. Topics List: 2.3 2.4 2.6 2.7 2.8 3.1 3.2 3.3 Functions and Relations. Linear Functions. Cover the average rate of change only. Transformations of Graphs. Analyzing Graphs of Functions. Omit step functions. Algebra and Composition of Functions. Quadratic Functions. Omit models using regression. Introduction to Polynomial Functions. Division of Polynomials and The Remainder and Factor Theorems. Omit Synthetic Division. 3.4 Zeros of Polynomials. Cover only paragraph 2 (“Apply the fundamental theorem of algebra”). 3.5 3.6 4.1 Rational Functions. Polynomial and Rational Inequalities. Omit applications. Inverse functions. Midterm 1 Page 1 2016-2017 Math 1150 Mathematics 1150 Precalculus Autumn, Spring 5 credits 4.2 4.3 4.4 4.5 4.6 5.1 5.2 5.3 5.4 5.5 5.6 5.7 6.1 Exponential Functions. Logarithmic Functions. Properties of Logarithms. Exponential and Logarithmic Equations. Modeling with Exponential and Logarithmic Functions. Omit logistic growth and models using regression. Angles and Their Measure. Right Triangle Trigonometry. Trigonometric Functions of any Angle. Trigonometric Functions and The Unit Circle. Graphs of Sine and Cosine Functions. Omit sinusoidal behavior. Graphs of Other Trigonometric Functions. Inverse Trigonometric Functions. Fundamental Trigonometric Identities. Midterm 2 6.2 6.3 6.5 7.1 7.2 7.3 8.3 8.4 8.5 9.1 9.2 11.1 11.2 11.3 Sum and Difference Formulas. Double-Angle and Half-Angle Formulas. Trigonometric Equations. Solving graphically is optional. Applications of Right Triangles. Omit the bearing of an object. The Law of Sines. The Law of Cosines. Complex Numbers in Polar Form. Omit nth roots of complex numbers. Vectors. Dot Product. Systems of Linear Equations in Two Variables. Cover briefly. Systems of Linear Equations in Three Variables. Omit modeling. The Ellipse. Applications is optional. The Hyperbola. Applications is optional The Parabola. Applications is optional Midterm 3 12.1 12.2 12.3 Page 2 Sequences and Series. Arithmetic Sequences and Series. Geometric Sequences. 2016-2017 Math 1150 Mathematics 1151 Calculus I Autumn, Spring, Summer 5 credits Catalog Description: Differential and integral calculus of one real variable. Prerequisite: Math Placement Level L, or C- or better in: 1150, or in both 1148 & 1149; or in 150 or 1144. Exclusions: Not open to students with credit for any higher numbered math class. Text: Calculus for Scientists and Engineers: Early Transcendentals 2nd edition, by Briggs, Cochran, and Gillett, published by Pearson. ISBN: 9781256776468 Topics List: 1.1 Review of Functions 1.2 Representing Functions 1.3 Inverse, Exponential, and Logarithmic Functions 1.4 Trigonometric Functions and Their Inverses 2.1 The Idea of Limits 2.2 Definitions of Limits 2.3 Techniques for Computing Limits 2.4 Infinite Limits 2.5 Limits at Infinity 2.6 Continuity 3.1 Introducing the Derivative Midterm 1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 4.1 4.2 Page 1 Working with Derivatives Rules of Differentiation The Product and Quotient Rules Derivatives of Trigonometric Functions Derivative as Rates of Change The Chain Rule Implicit Differentiation Derivatives of Logarithmic and Exponential Functions Derivatives of Inverse Trigonometric Functions Related Rates Maxima and Minima What Derivatives Tell Us 2016-2017 Math 1151 Mathematics 1151 Calculus I Autumn, Spring, Summer 5 credits Midterm 2 4.3 4.4 4.5 4.6 4.7 4.9 5.1 5.2 5.3 5.4 Graphing Functions Optimization Problems Linear Approximations and Differentials Mean Value Theorem L’Hospital’s Rule Antiderivatives Approximating Areas under Curves Definite Integrals Fundamental Theorem of Calculus Working with Integrals Midterm 3 5.5 6.1 Substitution Rule Velocity and Net Change Final Page 2 2016-2017 Math 1151 Mathematics 1152 Calculus II Autumn, Spring, Summer 5 credits Catalog Description: Integral calculus, sequences and series, parametric curves, polar coordinates, vectors. Prerequisite: C- or better in 1151, 1156, 152.xx, 161.xx or 161.01H; or 114 or 1114. Exclusions: Not open to students with credit for any higher numbered math class or with credit for quartersystem Math courses numbered 153.xx or above. Text: Calculus for Scientists and Engineers: Early Transcendentals, 2nd OSU custom edition, by Briggs, Cochran, Gillett, published by Pearson, ISBN: 9781256776468 Topics: 7.1 6.2 6.3 6.4 6.5 6.6 6.7 7.2 Basic Approaches to Integration Regions Between Curves Volume by Slicing Volume by Shells Length of Curves Surface Area Physical Applications Integration by Parts Midterm 1 7.3 7.4 7.5 7.8 9.1 9.2 9.3 9.4 9.5 9.6 Trigonometric Integrals Trigonometric Substitution Partial Fractions Improper Integrals Overview of Sequences and Series Sequences Series Divergence and Integral Tests Ratio, Root, and Comparison Tests Alternating Series Page 1 Midterm 2 10.1 10.2 10.3 10.4 8.1 8.2 8.3 11.1 11.2 Approximating Functions with Polyn Properties of Power Series Taylor Series Working with Taylor Series Basic Ideas of Differential Equations Direction Fields and Euler’s Method Separable Differential Equations Parametric Equations Polar Equations Midterm 3 11.3 12.1 12.2 12.3 12.4 12.5 Calculus in Polar Coordinates Vectors in the Plane Vectors in Three Dimensions Dot Products Cross Products Lines and Curves in Space Final 2016-2017 Math 1152 Mathematics 1156 Calculus for Biological Sciences Autumn 5 credits Catalog Description: Limits, continuity, derivatives, mean value theorem, extrema, curve sketching, related rates, differentiation of the trig, log, and exponential functions, basic integration techniques, with particular motivations from and application to the Biological Sciences. Prerequisite: A grade of C- or above in 1148 and 1149, or a grade of C- or above in 1150, or credit for 150, or Math Placement Level L. Not open to students with credit for 1151 (151.xx) or above. GE quant reason math and logical analysis course. Exclusions: Not open to students with credit for 1151, or with credit for any higher numbered math class. Purpose of Course: To provide students with a solid foundation in one-variable differential calculus, to model and analyze phenomena in the Biological Sciences. Follow-up Course: Math 1157 Text: Calculus for Biology and Medicine, 3rd Edition, by Claudia Neuhauser, Pearson, ISBN 9780321644688 Topics List: 1.2 1.3/2.1 2.2 3.1-3.4 3.5 4.1 4.2-4.3 4.4 4.5-4.7 5.1-5.3 5.4 5.5 5.8 6.1 6.2 6.3 7.1-7.2 Page 1 Elementary Functions Graphing/Exponential Growth and Decay Sequences Limits and Continuity Properties of Continuous Functions Derivatives Rules of Differentiation, Product and Quotient Rules Chain Rule and Higher Derivatives Derivatives of Special Functions and Inverse Functions Extrema, Mean Value Theorem, Monotonicity, Concavity, Inflection Points Optimization L'Hospital's Rule Antiderivatives The Definite Integral The Fundamental Theorem of Calculus Applications of Integration Integration Techniques Also: Small-group Projects 2016-2017 Math 1156 Mathematics 1157 Mathematical Modeling for the Biological Sciences Spring 5 Credits 5 credits Catalog Description: Models in life sciences using multivariable calculus, linear algebra, dynamical systems, and ordinary differential equations. Prerequisite: C- or better in: 1156, 1151, 1161.xx or 1181H; or credit for 152.xx. Exclusions: Not open to students with credit for 1152 or with credit for any higher numbered math class. Purpose of Course: To provide students with a solid foundation in one-variable calculus, to introduce multi-variable tools in a biological setting, to model and analyze phenomena in the life sciences. Text: Calculus for Biology and Medicine, 3rd Edition, by Claudia Neuhauser, Pearson, ISBN 9780321644688 Topics List: 7.3 7.5 7.6 8.1 8.2 9.1-9.2 9.3 10.3 10.4 11.1-11.2 11.3-11.4 Also: Page 1 Rational Functions and Partial Fractions Numerical Integration Taylor Approximation Solving Differential Equations Equilibria and Their Stability Linear Systems and Matrices Linear Maps, Eigenvectors, an Eigenvalues Multivariable functions & Partial Derivatives Tangent Planes, Differentiability, and Linearization Linear Systems: Theory and Applications Nonlinear Autonomous Systems: Theory and Applications Small-group Projects 2016-2017 Math 1157 Mathematics 1161.01, 1161.02 Accelerated Calculus I Accelerated Calculus I for Honors Engineer Autumn 5 Credits Catalog Description: Differential and integral calculus of one real variable. Autumn, Spring 5Credits %5 5 credits Prerequisites: Math Placement Level L and previous calculus experience. Exclusions: For 1161.01: Not open to students with credit for any math course numbered 1152 or higher, or for the quarter-system math courses 151.xx and 152.xx, or for any quarter-system course numbered 162.xx or higher. For 1161.02: Intended for students in Freshman Engineering Honors. Text: Calculus for Scientists and Engineers: Early Transcendentals, 2nd OSU custom edition, by Briggs, Cochran, Gillett, Person, ISBN: 9781269753449 Topics: 2.1; 2.2 2.2; 2.3 2.4; 2.5 2.5; 2.6 2.7 3.1 3.2; 3.3 3.4; 3.5 3.5; 3.6 3.7 The Idea of Limits; Definition of Limits Definition of Limits; Limit Laws Infinite Limits; Limits at Infinity Limits at Infinity; Continuity, the Intermediate Value Theorem Precise Definition of Limits Introducing the Derivative Rules of Differentiation; Product and Quotient Rules Derivatives of Trig Functions; Derivatives as Rate of Change Derivatives as Rate of Change; The Chain Rule Implicit Differentiation Midterm 1 Page 1 2016-2017 Math 1161.01, 1161.02 3.8; 3.9 3.10 4.1 4.2; 4.3 4.4 4.5; 4.6 4.6; 4.7 4.9 5.1 5.2 Mathematics 1161.01, 1161.02 Accelerated Calculus I Accelerated Calculus I for Honors Engineer Autumn 5 Credits Derivatives of Logarithms and Exponential Functions; Derivatives of Inverse Autumn, Spring 5Credits Functions %5 Related Rates 5 credits Maxima and Minima What derivatives Tell Us; Graphing Optimization Problems Linear Approximations and Differentials; Mean Value Theorem Mean Value Theorem; L'Hopital's Rule Antiderivatives Approximating Areas under Curves, Sigma Notation Definite Integrals Midterm 2 5.3 5.4; 5.5 5.5; 6.1 6.2 6.3 6.4 6.5; 6.6 11.5 6.7 6.8; 6.9 7.1; 7.2 Fundamental Theorem of Calculus Working with Integrals; Substitution Rule Substitution Rule; Velocity and Net Change Regions between Curves Volumes by Slicing Volumes by Shells Lengths of Curves; Surface Area Physical Applications: Density & Mass, Work, Lifting Problems, Force & Pressure Log and Exponential Functions Again; Exponential Growth and Decay Integration: Basic Approaches; Integration by Parts Midterm 3 7.3 7.4 7.5 7.8 Trig Integrals Trig Substitutions Partial Fractions Improper Integrals Final Page 2 2016-2017 Math 1161.01, 1161.02 Mathematics 1165 Math for Middle School Teachers Autumn 5 Credits Catalog Description: Algebra and reasoning for middle school teachers. Autumn, Spring 5Credits %5 5 credits Prerequisite: A grade of C- or above in 1148 and 1149, or grade of C- or above in 1150, or credit for 150, or Math Placement Level L; and enrollment in Middle Childhood Education major within either College of Arts and Sciences or College of Education and Human Ecology with Math as an Area of Concentration. Purpose of Course: The purpose of the course is to prepare teachers of middle school students. In particular, it intends to deepen and extend the prospective teachers’ content knowledge of the mathematics they will teach as well as their ability to reason with and communicate that knowledge. Follow-up Courses: Math 1166, Math 2167, and Math 2168 Text: Course Notes, by B. Snapp Recommended Text: Algebra Connections, by Papick, published by Pearson, ISBN 9780131449282 Topics List: 1. Number systems (whole, integer, rational, real): uses, notations (including place value), and comparison of size. Addition and Subtraction 2. Division algorithm, Euclidean algorithm, Diophantine equations, counting techniques. 3. Algebra: polynomials, their structure and arithmetic, division algorithm. 4. Solving equations: linear, quadratic, etc., using complex numbers. 5. Introduction to mathematical induction. 6. Applications: modeling real-world topics. 7. Problem solving (a theme throughout the course). Page 1 2016-2017 Math 1165 Mathematics 1166 Math for Middle School Teachers II Spring 5 Credits Catalog Description: Geometry for middle school teachers. Autumn, Spring 5Credits %5 5 credits Purpose of Course: The purpose of the course is to prepare teachers of middle school students. In particular, it intends to deepen and extend the prospective teachers’ content knowledge of the mathematics they will teach as well as their ability to reason with and communicate that knowledge. Prerequisite: C- or better in 1165, and enrollment in Middle Childhood Education major with Math as an Area of Concentration. Follow-up Courses: Math 2167 and Math 2168 Text: Course Notes. Topics List: 1. Visual reasoning via “proofs without words.” Measurement (also teaching measurement in middle school) 2. Geometric constructions, congruence, similarity, and problem solving. 3. Coordinate geometry with emphasis on solving equations. 4. Non-Euclidean geometries. 5. Geometric transformations coordinate geometry, complex numbers. 6. Scaling and relationship between perimeter and area. 7. Measurement issues. 8. Modeling real-world situations. Page 1 2016-2017 Math 1166 Mathematics 1172 Engineering Mathematics A Autumn, Spring, Summer 5 Credits Catalog Description: Autumn, Spring Techniques of integration, Taylor series, differential calculus of several variables. 5Credits %5 5 credits Prerequisites: C- or better in 1151, 152.xx, 1156, 1161.xx, 161.01H, 161.xx, 1114 or 114. Exclusions: Not open to students in math, pre-actuarial science, or actuarial science. Not open to students with credit for any higher numbered math class, or for 1152; or for 254.xx or higher numbered math class. Text: Calculus for Scientists and Engineers: Early Transcendentals, 2nd OSU custom edition, by Briggs, Cochran, Gillett, published by Pearson, ISBN: 9781256776468 Topics: 6.2 6.3 6.4 6.5 6.7 7.1 7.2 7.3 Regions between Curves Volume by Slicing Volume by Shells Lengths of Curves Physical Applications Basic Approaches to Integration Integration by Parts Trigonometric Integrals Midterm 1 7.4 7.5 7.8 9.1 9.2 9.3 9.4 9.5 10.1 10.2 10.3, 10.4 Trig Substitution Partial Fractions Improper Integrals Overview of Sequences and Series Sequences Series (and Idea of Convergence) Divergence Test (and Properties of Convergent Series only) Ratio Test (only) Approx functions with Polynomials Properties of Power Series Taylor Series Midterm 2 Page 1 2016-2017 Math 1172 Mathematics 1172 Engineering Mathematics A Autumn, Spring, Summer 5 Credits 11.1 11.2 11.3 11.4 12.1, 12.2 12.3, 12.4 12.5 12.6 12.7 12.8 Parametric Equations Polar Equations Calculus in Polar Coordinates Conic Sections (Conic Sections in Polar optional) Vectors in the Plane and 3-Space Dot Products, Cross Products Lines and Curves in Space Calculus of Vector-Valued Functions Motion in Space Lengths of Curves Autumn, Spring 5Credits %5 5 credits Midterm 3 13.1 13.2 13.3 13.4 13.5 13.6 Planes and Surfaces Graphs and Level Curves Limits and Continuity Partial Derivatives The Chain Rule Directional Derivatives, Gradient Final Page 2 2016-2017 Math 1172 Mathematics 1181H Honors Calculus I Autumn 5 Credits Catalog Description: Single variable calculus treated in depth. Autumn, Spring 5Credits %5 5 credits Prerequisites: 1151 or 151.xx, and permission of department. Exclusions: Not open to students with credit for any higher numbered math class. Text: Calculus with Analytic Geometry, 2nd edition, by Simmons, published by McGraw-Hill, ISBN: 9780070576424 Topics: 2.1 2.2 2.3 2.4 2.5 A2 3.1 3.2 3.3 3.4 3.5 3.6 12.2 4.1 4.2 4.3 4.4 4.5 2.6 A4 The Problem of Tangents How to Calculate the Slope of the Tangent The definition of the Derivative Velocity and Rate of Change The Concept of a Limit; Two Trigonometric Limits Theorems about Limits; Derivatives of Polynomials The Product and Quotient Rules Composite Functions and the Chain Rule Some trig Derivatives; Implicit Functions and Fractional Exponents Derivatives of Higher Order Indeterminate Form 0/0, L'Hopital's Rule Increasing and Decreasing Functions, Maxima and Minima Concavity and Points of Inflection Applied Maxima and Minima Problems Reflection & Refraction Related Rates Continuous Functions The Mean Value Theorem Midterm I 5.2 5.3 5.4 6.1 6.2 6.3 6.4 6.5 6.6 Page 1 Differentials and Tangent Line Approx'n Indefinite Integrals, Integration by Substitution; Differential Equations, Separation of Variables Introduction The Problem of Areas. The Sigma Notation and Certain Special Sums Area under a Curve, Definite Integrals, Riemann The Computation of Areas as Limits; The Fundamental Theorem of Calculus 2016-2017 Math 1181H Mathematics 1181H Honors Calculus I Autumn 5 Credits 6.7 7.2 7.3 7.4 7.5; 7.6 7.7 8.2 8.3 8.4 8.5 9.1; 9.2 9.3; 9.4 9.5 9.6; 9.7 Algebraic and Geometric Areas Autumn, Spring Area between Two Curves; 5Credits %5 Volumes: The Disk Method 5 credits Volumes: The Method of Cylindrical Shells Arc Length; Area of a Surface of Revolution Work and Energy Review of Exponents and Logarithms; The number e and the function y=e^x The Natural Logarithm Function, Euler Applications, Population Growth and Radioactive Decay Review of Trig.; Der've and Integrals of Sin and Cos Der've of the Other Four Fns The Inverse Trig Functions Simple Harmonic Motion; (“Optional”) Hyperbolic Functions MIDTERM 2 10.1; 10.2 10.3: 10.4 10.5; 10.6 10.7 10.8 12.2; 12.3 12.3; 12.4 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 Basic Formulas; Method of Substitution Certain Trig Integrals; Trig Substitutions; Completing the Square Integration by Parts Strategy for Dealing with Integrals Indeterminate Form 0/0, L'Hopital's Rule; Other Indeterminate Forms Improper Integrals What is an Infinite Series?; Convergent Sequences Convergent and Divergent Series General Properties of Convergent Series Series of Non-negative Terms, Compar. Tests Integral Test, Euler's Constant Ratio and Root Test Alternating Series Test, Absolute Convergence MIDTERM 3 14.2 14.3 14.4 14.5 14.6 14.8 The Interval of Convergence Differentiation and Integration of Power Series Taylor' Series and Taylor's Formula Computations Using Taylor's Formula Applications to Differential Equations Operations on Power Series FINAL Page 2 2016-2017 Math 1181H Mathematics 1187H Problem Solving Autumn 1-2 Credits each Prerequisite: Permission of Department. Catalog Description: An advanced enrichment course for interested and capable students. Purpose of Course: To offer an experience in problem solving in mathematics for interested and talented students beyond what they would encounter in a standard program. It is preparation for the National Putnam Mathematics Exam. This course is repeatable to a maximum of 6 credit hours, and is graded S/U. This course may not be counted in a major or minor program in Mathematics. Topics: Interesting special problems as chosen by the instructor. Page 1 2016-2017 Math 1187H Mathematics 1295 Introductory Seminar 1 credit Catalog Description: Seminar on mathematical topics for beginning math and actuarial science majors. Prerequisite: Enrollment in math or actuarial science major, or permission of department. Topics List: 1. Overview of the scope of mathematics, its subfields, and its applications. 2. Discussion of the OSU math major and differences among the tracks. 3. Outline of programs and activities that can benefit math majors. 4. Presentation of various different sorts of career opportunities for math majors. Page 1 2016-2017 Math 1295 Mathematics 2137 Algebra and Coordinate Geometry for Teachers Autumn, Spring 3 credits Catalog Description: This is one of two independent courses which follow Measurement and geome-try for teachers to provide necessary content for middle grade teachers. This course focuses on algebra, coordinate geometry, and their connections through equations in one or more unknowns. Modern and historical perspectives are woven throughout. Prerequisite: A grade of C- or above in “Measurement and Geometry for Teachers” (Math 1136). A grade of C-or above in Math 1149 or 1150, or credit for 150, or math placement level L. Text: Basic Mathematics, by Serge Lang, Springer, ISBN 9780387967875 Purpose: This course integrates the various types of numbers introduced in the previous course to present them as members of a single (real) number system. The notion that new numbers are discovered as solutions to equations is promoted, and motivated by connecting various equations with mathematical models. Matrices are introduced and used as linear transformations, mainly in the plane. The complex numbers are introduced as general solutions to quadratic equations and the relationship between complex arithmetic and transformations in the plane is explored. The course ﬁnishes with several weeks of geometry content for middle grade teachers, including more material on proofs, triangle congruence, and non-Euclidean geometry. The main example is “Taxicab geometry”, based on the l1 norm. Topics List: 1. Polynomial arithmetic as “base-x” and binomial theorem 2. Real number system 3. Polynomial equations and their roots 4. Exponential and logarithm functions 5. Complex numbers 6. Matrices 7. Complex arithmetic and linear transformations in the plane 8. Geometry proofs 9. Taxicab geometry Page 1 2016-2017 Math 2137 Mathematics 2138 Calculus and its History for Teachers Autumn, Spring 3 credits Catalog Description: This is one of two independent courses which follow “Measurement and Geometry for Teachers” (Math 1136) to provide necessary content for middle grade teachers. This course focuses on functions and calculus, including modern and historical perspectives. Prerequisite: A grade of C- or above in “Measurement and Geometry for Teachers” (Math 1136). A grade of C- or above in Math 1149 or 1150, or credit for 150, or math placement level L. Text: Calculus, by Frank Morgan, CreateSpace Independent Publishing Platform, ISBN 9781478356882 Purpose: This course serves to introduce students to the key ideas of calculus and to important historical developments in the subject. A thorough introduction to functions as mappings is given, and the trigonometric functions are used throughout the course as a key example of functions not given by algebraic expressions. The essential concepts of limit, derivative, integral, and the fundamental theorem are emphasized, together with core applications. An introduction to Taylor series, especially the Taylor expansions for sine and cosine, completes the class. Topics List: 1. Functions 2. Elementary approach to how functions change 3. Defining rate of change 4. Concept of limit 5. Derivatives 6. Interpretations of first and second derivatives 7. Sine, cosine and logarithm functions 8. Product rule and chain rule 9. Applications of derivatives 10. Antiderivatives 11. Riemann sums 12. Fundamental theorem of calculus 13. Applications of integration 14. Taylor approximations, infinite sequences 15. Series Page 1 2016-2017 Math 2138 Mathematics 2153 Calculus III Autumn, Spring, Summer 4 credits Catalog Description: Multivariable differential and integral calculus. Prerequisites: C- or better in 1152, 1172, 1534, 1544, 1181H, or 4181H; or credit for 153.xx, 154, 162.xx, or 162.01H. Exclusions: Not open to students with credit for any higher numbered math class, or for any quarter math class numbered 254 or higher. Text: Calculus for Scientists and Engineers: Early Transcendentals, 2nd OSU custom edition, by Briggs, Cochran, Gillett, published by Pearson, ISBN: 9781256776468 Topics: CHAPTER 12. Vectors and Vector-Valued Functions Section 1. Vectors in the Plane Section 2. Vectors in Three Dimensions Section 3. Dot Products Section 4. Cross Products Section 5. Lines and Curves in Space Section 6. Calculus of Vector-Valued Functions Section 7. Motion in Space Section 8. Length of Curves Section 9. Curvature and Normal Vectors CHAPTER 13. Functions of Several Variables Section 1. Planes and Surfaces, and brief conic section review via pages 761 to 766 of Section 11.4 Section 2. Graphs and Level Curves Section 3. Limits and Continuity Section 4. Partial Derivatives Section 5. The Chain Rule Section 6. Directional Derivatives and the Gradient Section 7. Tangent Planes and Linear Approximation Section 8. Maximum/Minimum Problems Section 9. Lagrange Multipliers CHAPTER 14. Multiple Integration Page 1 2016-2017 Math 2153 Mathematics 2153 Calculus III Autumn, Spring, Summer 4 credits Section 1. Double Integrals over Rectangular Regions Section 2. Double Integrals over General Regions Section 3. Double Integrals in Polar Coordinates Section 4. Triple Integrals Section 5. Triple Integrals in Cylindrical and Spherical Coordinates Section 6. Integrals for Mass Calculations Section 7. Change of Variables in Multiple Integrals CHAPTER 15. Vector Calculus Section 1. Vector Fields Section 2. Line Integrals Section 3. Conservative Vector Fields Section 4. Green’s Theorem Section 5. Divergence and Curl Section 6. Surface Integrals Section 7. Stokes’ Theorem Section 8. Divergence Theorem Page 2 2016-2017 Math 2153 Catalog Description: Multivariable calculus; introduction to Taylor series. Mathematics 2162.01, 2162.02 Accelerated Calculus II Accelerated Calculus II for Engineers 2162.01(Sp) 2162.02(Au, Sp) 5 credits 5 credits Prerequisites: C- or better in 1161.xx or 1181H. Exclusions: For 2162.01: Not open to students with credit for any higher numbered math class numbered 2162 or higher. For 2162.02: Intended for students in Freshman Engineering Honors and not open to students with credit for any higher numbered math class numbered 2162 or higher. Text: Calculus for Scientists and Engineers: Early Transcendentals, 2nd OSU custom edition, by Briggs, Cochran, Gillett, Person, published by Pearson, ISBN: 9781269753449 Topics: 9.1; 9.2 9.2; 9.3 9.4 9.5 9.5; 9.6 10.1; 10.2 10.3; 10.4 An Overview; Sequences Sequences; Infinite Series Divergence and Integral Tests Ratio, root, and Comparison Tests Ratio, root, and Comparison Tests; Alternating Series; Approximating Functions with Polynomials; Properties of power Series Taylor Series; Working with Taylor Series 11.1; 11.2 11.2; 11.3 Parametric Equations; Polar Coordinates Polar Coordinates; Calculus in Polar Coordinates Midterm 1 Page 1 2016-2017 Math 2162.01, 2162.02 12.1; 12.2 12.3;12.4 12.5 12.6 12.7; 12.8 12.9 13.1; 13.2 13.3 13.4;13.5 13.6 Mathematics 2162.01, 2162.02 Accelerated Calculus II Accelerated Calculus II for Engineers 2162.01(Sp) 2162.02(Au, Sp) 5 credits 5 credits Vectors in the Plane; Vectors in Three Dimensions Dot Products; Cross Products Lines and Curves in Space Calculus of Vector-Valued Functions Motion in Space; Length of Curves Curvature and Normal Vectors Planes and Surfaces; Graphs and Level Curves Limits and Continuity Partial Derivatives; Chain Rule Directional derivative and the Gradient MIDTERM 2 13.7 13.8 13.9 14.1; 14.2 14.2; 14.3 14.4; 14.5 14.5; 14.6 15.1 15.2 15.3 Tangent Plane and Linear Approximation Maximum/Minimum Problems Lagrange Multipliers Double Integral over Rectangular Regions; Double Integrals over General Regions Double Integrals over General Regions; Double integrals in Polar Coordinates Triple Integrals; Triple Integrals in Cylindricals and Sphericals Triple Integrals in Cylindricals and Sphericals; Integrals for Mass Calculations Vector Fields Line Integrals Conservative Vector Fields MIDTERM 3 15.4 15.5 15.6 15.7 15.8 Page 2 Green's Theorem Divergence and Curl Surface Integrals Stokes' Theorem Divergence Theorem 2016-2017 Math 2162.01, 2162.02 Mathematics 2167 Calculus for Middle School Teachers Autumn 3 credits Catalog Description: Concepts of Calculus for Middle School Math teachers. Prerequisite: A grade of C- or above in 1165, or credit for 1164 or 110; and enrollment in Middle Childhood Education major or pre-major with Math as area of concentration. Exclusions: Not open to students with credit for 111. Purpose of Course: The purpose of the course is to prepare teachers of middle school students. In particular, it intends to deepen and extend the prospective teachers’ content knowledge of the mathematics and mathematical reasoning that they will teach as well as their ability to reason with and communicate that knowledge. Follow-up Courses: Math 2168 Text: Under Consideration. Optional text: Calculus Connections: Mathematics for Middle School Teachers. By Asma Harcharras and Dorina Mitrea (2007). Published by Prentice Hall. ISBN: 0-13-144923-0 Supplementary Text: Course Notes Topics List: 1. Rates described pictorially, in writing, and with symbols. 2. Informal and formal measurement of (instantaneous) rates and their connection to middle school mathematics. 3. Informal and formal measurement of (accumulated) areas and their connection to middle school mathematics. 4. The Fundamental Theorem of Calculus. 5. Applications of differential calculus. 6. Applications of integral calculus. Page 1 2016-2017 Math 2167 Mathematics 2168 History of Mathematics for Middle School Teachers Spring 3 credits Catalog Description: Historical and mathematical discussion of topics in the middle school math curriculum. Prerequisite: C- or better in 2167; or credit for 111. Limited to Middle Childhood majors and pre-majors with Math as an Area of Concentration. Purpose of Course: The purpose of the course is to prepare prospective middle school math teachers. In particular, it intends to deepen and extend prospective teachers’ connections among topics in mathematics through the study of the history of mathematics, as well as continuing to develop their ability to reason with and communicate that knowledge. Follow-up Courses: None Text: Math through the Ages: A Gentle History for Teachers & Others, Expanded Edition, by Berlinghoff & Bouvea, published by Mathematical Association, ISBN: 9780883857366 Course Packet Topics List: 1. History of Mathematics, from ancient to modern times. 2. Development of number systems, operations, geometry, trigonometry, algebra, calculus, statistics, and probability. 3. Applications: modeling real-world topics. 4. Problem solving (a theme throughout the course). Page 1 2016-2017 Math 2168 Mathematics 2173 Engineering Mathematics B Autumn, Spring 3 credits Catalog Description: Multiple integrals, line integrals, vector fields, second order ordinary differential equations. Prerequisite: Math 1172, 1544, or 154. Exclusions: Not open to students with credit for 1152, 2153, or for any higher numbered math class, or for any quarter-system math class numbered 254 or higher. Text: Calculus for Scientists and Engineers: Early Transcendentals, 2nd OSU custom edition, by Briggs, Cochran, Gillett, published by Pearson, ISBN: 9781269753449 Topics List: 13.6 13.8 13.9 14.1 14.2 14.3 (Review of) Directional Derivatives and the Gradient Vector Maximum and Minimum Values Lagrange Multipliers Double Integrals over Rectangular Regions Double Integrals over General Regions Double Integrals in Polar Coordinates Midterm 1 14.4 14.5 14.7 15.1 15.2 15.3 Triple Integrals Triple Integrals in Cylindrical & Spherical Coordinates Change of Variables in Multiple Integrals Vector Fields Line Integrals Conservative Vector Fields 16.1 Appendix C 16.2 16.3 16.4 Midterm 2 Basic Ideas of Second Order ODE’s Complex Arithmetic Linear Homogeneous Equations Linear Homogeneous Equations Applications; Complex Forcing Functions Final Page 1 2016-2017 Math 2173 Mathematics 2174 Linear Algebra and Differential Equations Autumn, Spring 3 credits Catalog Description: Matrix theory, eigenvectors and eigenvalues, ordinary and partial differential equations. Prerequisite: 2173 and either major in ENG, Physics, or Chemistry or permission of math department. Exclusions: Not open to students with credit for both (i) 2415 (415) or 2255 (255) and (ii) 2568 (568 or 571). Text: Part II: Elementary Ordinary & Partial Differential Equations, OSU custom edition, by Boyce, published by Wiley, ISBN: 9781119934462 Introduction to Linear Algebra, 5th edition, by Johnson, Riess and Arnold, published by Pearson, ISBN: 9780321628217 Topics List: Part One = Matrix Algebra Textbook sections from Arnold, Riess, and Johnson’s Introduction to Linear Algebra, 5th edition Chapter 1: Matrices and Linear Systems of Equations Chapter 3: The Vector Space ℝn Chapter 4: The Eigenvalue Problem 1.1 1.2 1.3 1.5 1.6 1.7 1.9 3.1-3.2 3.3 3.4 Introduction and Gaussian Elimination and Systems of Linear Equations Echelon Form and Gauss-Jordan Elimination Consistent Systems of Linear Equations Matrix Operations Algebraic Properties of Matrix operations Linear Independence and Nonsingular Matrices Matrix Inverses and the Properties Review and Vector Space Properties in ℝn Examples of Subspaces Basis for Subspaces; Dimension Midterm I Page 1 2016-2017 Math 2174 Mathematics 2174 Linear Algebra and Differential Equations Autumn, Spring 3 credits 4.1 4.2 4.4 4.5 4.6 4.7 The Eigenvalue Problem for 2x2 Matrices Determinants and the eigenvalue Problem Eigenvalues and characteristic Polynomial Eigenvectors and Eigenspaces Complex Eigenvalues and Eigenvectors Similarity Xformations and Diagonalization Part Two = Systems of Linear Differential Equations Textbook Sections from Boyce & DiPrima’s Part II: Elementary Ordinary & Partial Differential Equations Ch. 7: Systems of First Order Linear Equations (no lectures, but assigned as an independent class project) Midterm 2 Part Three = Partial Differential Equations and Fourier Series Textbook Sections from Boyce & DiPrima’s Part II: Elementary Ordinary & Partial Differential Equations 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 Page 2 Two point Boundary Value Problem Fourier Series The Fourier Convergence Theorem Even and Odd Functions Separation of Variables; Heat Conduction Equation Other Heat Conduction Problems The Wave Equation; Vibrations of an Elastic String Laplace's Equation (optional) 2016-2017 Math 2174 Mathematics 2177 Mathematical Topics for Engineers Autumn, Spring 4 credits Catalog Description: Multiple integrals, line integrals; matrix algebra; linear (ordinary and partial) differential equations. Prerequisite: Major, pre-major, or minor in BiomedE, CEEGS, FABEng, MatScEn, CBE, or WeldEn; and: 1172, 2153, 1544 (154), 254.xx, 263.xx, 263.01H, or 264H Exclusions: Not open to students with credit for 2174 or 5520H; or with credit for both (i) 2415 (415), 2255 (255) or 4512 (512) and (ii) 2568 (568 or 571). Text: Math 2177, Custom Edition for OSU, Pearson, ISBN-13 978-1-256-82676-7 or ISBN-10 1-25682676-6 –OR– the textbooks listed below. Topics List: PART ONE: Multivariable Integral Calculus Textbook Sections from Calculus for Scientists and Engineers: Early Transcendentals, by Briggs, Cochran, Gillett and Shulz, Chapters 13-15 2177 Custom 1.8 1.9 2.1 2.2 2.3 2.4 2.5 2.7 3.1 3.2 3.3 Original Text 13.8 13.9 14.1 14.2 14.3 14.4 14.5 14.7 15.1 15.2 15.3 Topic Maximum/Minimum Problems Lagrange Multipliers Double Integrals over Rectangular Regions Double Integrals over General Regions Double Integrals in Polar Coordinates Triple Integrals Triple Integrals in Cylindrical and Spherical Coordinates Change of Variables in Multiple Integrals Vector Fields Line Integrals Conservative Vector Field Midterm 1 Page 1 2016-2017 Math 2177 Mathematics 2177 Mathematical Topics for Engineers Autumn, Spring 4 credits PART TWO: Matrices and Linear Systems of Equations Textbook Sections from Introduction to Linear Algebra, by Johnson, Riess, and Arnold, 5th edition, Chapter 1: Matrices and Systems of Linear Equations 2177 Original Custom Text 4.1 1.1 4.2 1.2 4.3 1.3 4.4 4.4 4.5 1.5 4.6 1.6 4.7 1.7 4.8 1.8 Midterm 2 Topic Introduction to Matrices and Systems of Linear Equations Echelon Form and Gauss-Jordan Elimination Consistent Systems of Linear Equations Applications (optional) Matrix Operations Algebraic Properties of Matrix operations Linear Independence and Nonsingular Matrices Data Fitting, Numerical Integration and Numerical Differentiation PART THREE: 2nd Order Constant Coefficient O.D.E.' s Textbook Sections from Calculus for Scientists and Engineers: Early Transcendentals, by Briggs, Cochran, Gillett and Shulz, Chapter 16 and Appendix C 2177 Original Custom Text 5.1 16.1 Appx C Appx C 5.2 16.2 5.3 16.3 5.4 16.4 Midterm 3 Topic Basic Ideas Complex Numbers Linear Homogeneous Equations Linear Nonhomogeneous Equations Applications PART FOUR: Fourier Series & Partial Differential Equations Textbook Sections from Fundamentals of Differential Equations and Boundary Value Problems, by Nagle, Saff and Snider, 8th Edition, Chapter 10 2177 Custom 6.1 6.2 6.3 6.4 6.5 6.6 Page 2 Original Text 10.1 10.2 10.3 10.4 10.5 10.6 Topic Introduction: A Model for Heat Flow Method of Separation of Variables Fourier Series Fourier Cosine and Sine Series The Heat Equation The Wave Equation 2016-2017 Math 2177 Mathematics 2182H Honors Calculus II Spring 5 credits Catalog Description: Multivariable calculus treated in depth. Prerequisite: C or better in 1181H or 4181H. Exclusions: Not open to students with credit for both 162.01H and 263.01H Text: Calculus with Analytic Geometry, 2nd Edition, by George F. Simmons, published by McGrawHill, ISBN: 0070576424 Topics List: 15.1; 15.2 15.3; 15.4 16.1 16.2 16.3 16.3; 16.4 16.5 17.1 17.2 17.3 17.4 17.5 17.6 17.7 Conic sections: Ellipse, Parabola, Hyperbola Conic sections: Ellipse, Parabola, Hyperbola Polar coordinate system Graphs of polar equations Polar Equations of conics and spirals Polar Equations of conics and spirals; Arc length and tangent lines Areas in polar coordinates Parametric Equations of Curves Cycloids and other similar Figures Vector Algebra, the Unit Vectors i and j; Derivatives of Vector Functions, Velocity and Acceleration Curvature and the Unit Normal Vector Tangential and Normal Components of Acceleration Kepler's Laws and Newton's Law of Universal Gravitation Midterm 1 Page 1 2016-2017 Math 2182H Mathematics 2182H Honors Calculus II Spring 5 credits 18.1 18.2 18.3 18.4 18.5 18.6 18.7 19.1 19.2 19.3 19.4 19.5 19.6 19.7; 19.8 19.10 Coordinates and Vectors in 3-D Space The Cross Product of Two Vectors The Dot Product of Two Vectors Lines and Planes Cylinders and Surfaces of Revolution Quadratic Surfaces; Cylindrical and Spherical Coordinates Function of Several Variables Partial Derivatives The Plane Tangent to a Surface Increments and Differentials, the Fundamental Lemma Directional Derivatives and the Gradient The Chain Rule for Partial Derivatives Maximum and Minimum Problems Implicit Functions Midterm 2 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 Volumes as Iterated Integrals Double Integrals and Iterated Integrals Physical Applications of Double Integrals Double Integrals in Polar Coordinates Triple Integrals Cylindrical Coordinates Spherical Coordinates, Gravitational Attraction Area of Curved Surfaces Midterm 3 21.1 21.2 21.3 21.4 21.5 Page 2 Line Integrals in the Plane Independence of Path, Conservative Fields Green's Theorem Surface Integrals and Gauss' Theorem Stokes' Theorem 2016-2017 Math 2182H Mathematics 2255 Differential Equations and Their Applications Autumn, Spring, Summer 3 credits Catalog Description: Ordinary differential equations, their series solutions, numerical methods, Laplace transforms, physical applications. Prerequisite: C- or better in 2153, 2162.xx, 2173, 2182H, or 4182H; or credit for 254.xx, 263.xx, 263.01H, or 264H. Text: Ordinary Differential Equations and their Applications, OSU custom edition, by Boyce, published by Wiley, ISBN 9781119934455 Topics List: 1.3 2.1 INTRODUCTION Classification of Differential Equations Linear Equations with Variable Coefficients 2.2 2.4 2.5 2.6 2.7 2.8 2.9 FIRST ORDER DIFFERENTIAL EQUATIONS Separable Equations Differences between Linear and Nonlinear Equations Autonomous Equations and Population Dynamics Exact Equations and Integrating Factors Numerical Approximations: Euler’s Method The Existence and Uniqueness Theorem First Order Difference Equations 3.1 3.3 3.2 3.4 SECOND ORDER LINEAR EQUATIONS Homogeneous Equations with Constant Coefficients Complex Roots of the Characteristic Equation Solutions of Linear Homogeneous Equations; the Wronkian Repeated Roots; Reduction of Order Midterm 1 3.5 3.6 3.7 3.8 Page 1 Nonhomogeneous Equations; Method of Undetermined Coefficients Variation of Parameters Mechanical and Electrical Vibrations Forced Vibrations 2016-2017 Math 2255 Mathematics 2255 Differential Equations and Their Applications Autumn, Spring, Summer 3 credits 6.1 6.3 6.2 6.4 6.5 6.6 THE LAPLACE TRANSFORM Definition of the Laplace Transform Step Functions Solution of Initial Value Problems Differential Equations with Discontinuous Forcing Functions Impulse Functions The Convolution Integral Midterm 2 4.1 4.2 4.3 4.4 HIGHER ORDER LINEAR EQUATIONS General Theory of nth Order Equations Homogeneous Equations with Constant Coefficients The Method of Undetermined Coefficients Material The Method of Variation of Parameters 5.1 5.2 5.3 5.4 SERIES SOLUTIONS OF SECOND ORDER LINEAR EQUATIONS Review of Power Series Series Solutions near an Ordinary Point, Part I Series Solutions near an Ordinary Point, Part II Euler's Equation; Regular Singular Points Midterm 3 5.5 5.6 5.7 Page 2 Series Solutions near a Regular Singular Point, Part I Series Solutions near a Regular Singular Point, Part II Bessel's Equation 2016-2017 Math 2255 Mathematics 2366 Introduction to Discrete Mathematics Spring 2 credits Catalog Description: Mathematical reasoning, logic, sets, functions, recursive definitions, elementary counting principles. Prerequisites: C- or better in 1131 or 1151; credit for 132 or 152.xx; or permission of department. Exclusions: Open only to majors in MIS (Management Information Systems). Not open to students with credit for 366. Text: Discrete Mathematics with Applications, 4th edition, by Epp, published by Cengage, ISBN: 9780495391326, Lecture Notes by G. Baker. Topics List: Topics for this discrete math course depend on future discussions with colleagues in Management Information Systems. Page 1 2016-2017 Math 2366 Mathematics 2415 Ordinary and Partial Differential Equations Autumn, Spring, Summer 3 credits Catalog Description: Ordinary and partial differential equations: Fourier series, boundary and initial value problems. Prerequisite: 2153, 2162.xx, 2173, 2182H, 4182H, both (1172 or 1544 or 154) and 2568, 254.xx, 263.xx, 263.01H, or 264H. Exclusions: Not open to students with credit for Math 2255, 5520H, 2174, 255, 415.xx, or 521H. Textbook: Elementary Differential Equations and Boundary Value Problems, 10th Edition, by W. Boyce and R. DiPrima, ISBN 978-1-118-15738-1 -or- Lectures Notes, by Greg Baker, published by Zip Printing. Topics List: 1.1 Some Basic Mathematical Models & Direction Fields 1.3 Classification of Differential Equations 1.2 Solutions to some Differential Equations 2.2 Separable Equations 2.1 Linear Equations with Variable Coefficients 2.3 Modeling with First Order Differential Equations 2.4 Difference between Linear and Nonlinear Equations 2.5 Autonomous Equations and Population Dynamics 3.1 Homogeneous Equations with Constant Coefficients; 3.3 Complex Roots of the Characteristic Equation 3.4 Repeated Roots Midterm 1 3.2 3.4 4.5 3.7 3.9 10.1 Solutions of Linear Homogeneous Equations; the Wronskian Reduction of Order Non-homogeneous Equations; Method of Undetermined Coefficients Mechanical and Electrical vibrations Forced Vibrations Two-point Boundary Value Problem Midterm 2 Page 1 2016-2017 Math 2415 Mathematics 2415 Ordinary and Partial Differential Equations Autumn, Spring, Summer 3 credits 10.2 10.3 10.4 10.5 10.7 7.1 7.3 7.5 7.6 7.4 Page 2 Fourier Series The Fourier Convergence Theorem Even and Odd Functions Separation of Variables; Heat Conduction in a Rod Wave Equation: Vibrations of an Elastic String Introduction Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors Homogeneous Linear Systems with Constant Coefficients Complex Eigenvalues Basic Theory of Systems of 1st Order Linear Equations 2016-2017 Math 2415 Mathematics 2568 Linear Algebra Autumn, Spring, Summer 3 credits Catalog Description: Matrix algebra, vector spaces and linear maps, bases and dimension, eigenvalues and eigenvectors, applications. Prerequisite: C- or above in 1172, 1544, 2153, 2162.xx, 2182H, or 4182H; or C- or above in both 1152 and CSE 2321; or credit for 154, 254.xx, 263.xx, 263.01H, or 264H. Exclusions: Not open to students with credit for 4568 (568), 5520H (520H), or 572. Text: Introduction to Linear Algebra, 5th edition, by L.W. Johnson, R.D. Riess, and J.T. Arnold, published by Pearson, ISBN Softcover: 0321628217, Hardcover: 0201658593 Topics List: 1.1 1.2 1.3 1.5 1.6 1.7 1.9 Part I Introduction to Matrices and Systems of linear equations Echelon Form and Gaussian-Jordan Elimination Consistent Systems of linear Equations Matrix Operations Algebraic Properties of Matrix operations Linear Independence and Nonsingular Matrices Matrix Inverses and Their Properties Midterm 1 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 5.2 5.3 5.4 3.6 3.7 Page 1 Part II Vectors in The Plane (Review only because it was done in 1152) Vectors in Space (Review only because it was done in 1152) The Dot Product and The Cross Introduction Vector Space Properties of ℝn Examples of Subspaces Bases for Subspaces Dimension Vector Spaces Subspaces Linear Independence, Bases, and Coordinates Orthogonal Bases for Subspaces Linear Transformation from ℝn to ℝm 2016-2017 Math 2568 Mathematics 2568 Linear Algebra Autumn, Spring, Summer 3 credits Midterm 2 4.1 4.2 4.4 4.5 4.6 4.7 Part III The Eigenvalue Problem for 2x2 Matrices Determinants and the Eigenvalue Problem Eigenvalues and Characteristic Polynomial Eigenvectors and Eigenspaces Complex Eigenvalues and Eigenvectors Similarity Transformations and Diagonalization Final Page 2 2016-2017 Math 2568 Mathematics 3345 Foundations of Higher Mathematics Autumn, Spring 3 credits Catalog Description: Introduction to logic, proof techniques, set theory, number theory, real numbers. Prerequisite: Major or minor in Math, CSE, or CIS. Math: C- or better in 2153, 2162.xx, 2173, or 2182H; or credit for 254.xx, 263.xx or 263.01H. CIS or CSE: C- or better in both CSE 2321 and: C- or better in 1152, 1161.xx, 1172, 1181H, 1534, or 1544; or credit for 153.xx, 154, 162.xx, or 162.01H. Exclusions: Not open to students with credit for 345. Text: Lecture Notes, by N. Falkner. Topics List: 1. Propositional calculus; quantifiers. 2. Simple examples of mathematical proofs. 3. Mathematical induction. 4. Sets and functions: surjections, injections, bijections. 5. Infinite sets: countable and uncountable. Page 1 2016-2017 Math 3345 Mathematics 3350 Introduction to Mathematical Biology Spring 3 credits 3 credits Catalog Description: Introduction to quantitative and qualitative analysis of several mathematical models for biological systems. Purpose: This course provides students with experience and knowledge in mathematical analysis of differential equations models, as well as with numerical tools for simulating those models. Prerequisite: C- or better in Math 2255, 2415, 5520H; or credit for 255, 415.xx, or 521H. Text: Lecture Notes Topics List: 1. Population dynamics: Logistic growth. 2. Population dynamics: Lotka-Volterra predator-prey model. 3. Modeling specific diseases (e.g. HIV, cancer). 4. Competition models. 5. Dynamics of neurons. 6. Bifurcution theory. 7. Enzyme kinetics. 8. Cells proliferation and death. Page 1 2016-2017 Math 3350 Mathematics 3532 Mathematical Foundations of Actuarial Science Spring 3 credits Catalog Description: Problem workshop for applications of calculus and probability to actuarial science and risk management. Prerequisite: C- or better in Math 4530, 5530H, or Stat 4201; or credit for 530, 531H, or Stat 420. Topics List: 1. Random variables. 2. Discrete distributions. 3. Continuous distributions. 4. Central Limit Theorem and law of large numbers. 5. Risk models. Page 1 2016-2017 Math 3532 Mathematics 3588 Practicum in Actuarial Science Spring 3 credits Catalog Description: Presentations by practicing actuaries on topics drawn from their fields of expertise; oral presentations by students on selected topics in actuarial science. Prerequisite: 3rd year standing and completion of second writing course. Exclusions: Open only to actuarial science majors. Text: None. Topics List: 1. Business communication. 2. Problems in life insurance. 3. Problems in property and casualty insurance. 4. Problems in pension consulting. 5. Problems in health care consulting. 6. Risk management. Page 1 2016-2017 Math 3588 Mathematics 3589 Introduction to Financial Mathematics Autumn 3 credits Catalog Description: Introduction to mathematics used in financial asset pricing, based on the binomial asset pricing model. This course prepares students for further study of stochastic calculus in continuous time. Purpose: This course is designed as an introduction to the concepts encountered in financial mathematics for students who don’t have a background in continuous-time stochastic calculus. Prerequisite: {C- or better in 3345 or credit for 345}; and {C- or better in 4530, 5530H, or Stat 4201, or credit for 530, 531H, 345 or Stat 420}; or permission of department. Text: Stochastic Calculus for Finance I, The Binomial Asset Pricing Model, by Shreve, published by Springer, ISBN: 9780387249681 Topics List: 1. First principles; assumptions about stock behavior and description of basic financial instruments; put and call options. 2. Arbitrage, and no-arbitrage pricing. 3. One-period and multi-period models; replication and hedging. 4. Conditional expectations. 5. Martingales and Markov processes. 6. Change of measure. 7. Utility functions and the capital asset pricing model. 8. Stopping times and American derivatives. 9. Random walks and passage times. Page 1 2016-2017 Math 3589 Mathematics 3607 Beginning Scientific Computing Autumn, Spring 3 credits Catalog Description: 3 credits Introduction to uses of computers to solve problems arising in the physical and biological sciences, and in engineering and finance. Prerequisite: {C- or better in 2255, 2415, or 5520H; and C- or better in 2568 or 5520H}; or: {credit for 255, 415.xx, or 521H; and credit for 568, 571, or 520H}. Purpose: Math 3607 is a course which has three main goals: it introduces students to MATLAB (or improves their knowledge of MATLAB); it uses MATLAB to solve practical problems from various areas of mathematics, physics, engineering, business, and finance; and it presents the numerical analysis needed to use MATLAB effectively. The principle underlying this course is that the way to learn MATLAB and numerical analysis is by doing it, not by reading about it. This course is taught in a computer lab and MATLAB will be used to some extent every class. Textbook: Learning MATLAB, Problem Solving, and Numerical Analysis through Examples, by Ed Overman (downloadable e-book). Topics List: 1. MATLAB as a scalar calculator, round-off errors, debugging. 2. Arrays in MATLAB, probability theory, Markov processes. 3. Graphics in MATLAB, applications of probability theory, histograms. 4. Programming in MATLAB, more probability theory, mathematical biology. 5. Function m-files in MATLAB, more Markov processes, chaos. 6. More about functions, randomness. 7. Solving linear systems of equations. 8. Interpolation and approximation. 9. The solution of nonlinear equations and unconstrained optimization. 10. Numerical differentiation and integration 11. Time-evolution ordinary differential equations, boundary-value ordinary differential equations, stochastic differential equations, examples from many disciplines. 12. Eigenvalues, Fourier series. Page 1 2016-2017 Math 3607 Mathematics 3618 Theory of Interest Autumn, Spring, Summer 3 credits Catalog Description: 3 credits Financial transactions involving interest: measurement of interest, force of interest, annuitiescertain, introduction to financial derivatives. Prerequisite: C- or better in 1152, 2162.xx, 1172, 2182H, 4181H; or credit for 153, 162, 162H, or 191H. Exclusions: Open only to actuarial science majors and pre-majors, and to math majors. Text: Mathematics of Investment & Credit, 5th edition, by Broverman, published by Actex. ISBN: 9781566987677 Derivatives Markets, 2nd edition, by McDonald, published by Pearson, ISBN: 9780321280305 Topics List: 1. Compound and simple rates of interest and discount, force of interest. 2. Annuity certain and annuity due. 3. Mortgage amortizations. 4. Evaluation of bonds. 5. Durations. 6. Asset and liability matching. 7. Introduction to options, futures, and other derivatives. Page 1 2016-2017 Math 3618 Mathematics 4181H (Au), 4182H (Sp) Honors Analysis I, Honors Analysis II 5 credits each 3 credits Catalog Description: 4181H and 4182H is an enriched honors sequence introducing students to mathematical underpinnings of calculus. 4181H is the first of the calculus sequence designed to introduce students to the mathematical underpinnings of analysis. 4182H is a continuation with a rigorous treatment of multivariable calculus including gradients, multiple integrals, line and surface integrals, Green’s theorem, the divergence theorem, and Stokes’ Theorem. Prerequisite: 4181H: Permission of department. 4182H: B- or better in 4181H, or permission of department. Purpose of Course: This two-semester sequence comprises the most intensive first year honors track in mathematics. It is designed to challenge talented, highly motivated students, regardless of their chosen major area of study. The courses introduce students to the mathematical underpinnings of calculus and stimulate the development of mathematical thinking, in addition to covering the material of the traditional calculus sequence. 4181H and 4182H will fulfill the analysis requirement for a Math major. The sequence is taught by faculty members in small sections with considerable teacherstudent interaction. Text: 4181H: Calculus, 4th edition, by Spivak, published by Publish or Perish, ISBN: 9780914098918 4182H: Advanced Calculus, by Folland, published by Pearson, ISBN: 9780130652652 Topics List: 4181H: 1. Properties of real numbers 2. Mathematical induction 3. Definition of integral 4. Integrals of polynomials and trigonometric functions. 5. Applications 6. Continuity, limits, derivatives and applications 7. Fundamental Theorem of Calculus and integration techniques 8. Taylor series 9. Sequences and series of numbers and functions 10. Uniform convergence 11. Power series 12. If time permits, some differential equations or complex-valued functions. Page 1 2016-2017 Math 4181H, 4182H Mathematics 4181H (Au), 4182H (Sp) Honors Analysis I, Honors Analysis II 5 credits each 3 credits 4182H: 1. Multivariable calculus (vector approach) 2. Gradients 3. Multiple integrals 4. Line and surface integrals 5. Green's Theorem 6. Divergence theorem 7. Stokes' Theorem. Page 2 2016-2017 Math 4181H, 4182H Mathematics 4350 Quantitative Neuroscience Spring 3 credits Catalog Description: Introduction to mathematical modeling and computational analysis of neuronal systems, HodgkinHuxley model, dynamical systems methods, neuronal networks, models for neurological disease. Prerequisite: Math 1152 (152) or 1157 or permission of instructor. Purpose: Mathematical models and computational methods have been very useful in understanding biological mechanisms underlying neuronal behavior. The Hodgkin‐Huxley model, for example, has formed the basis for our understanding of how action potentials are generated and how they propagate along a nerve axon. More recently, mathematical models have been used to help understand cellular processes responsible for both normal and pathological firing patterns that arise in a wide range of neuronal systems. Examples include models for sleep rhythms, sensory processing, Parkinsonian tremor and working memory. This course provides a detailed introduction to how mathematical and computational methods have been used to both develop and analyze models that arise in neuroscience. We begin by deriving the Hodgkin--‐Huxley model and then describe dynamical system methods for analyzing models. After discussing the dynamics of single neurons, we consider neuronal networks and describe how different types of population firing patterns depend on biological details, such as the intrinsic properties of individual neurons and synaptic coupling. We conclude by considering specific systems, including models for sleep rhythms, olfaction, working memory and neurological disease. Text: Foundations of Mathematical Neuroscience, by G. Bard Ermentrout and David H. Terman Topics List: 1. Overview: Neurons, synapses, neuronal firing patterns 2. Hodgkin‐Huxley Model: Resting potential, Nernst equation, Goldman‐Hodgkin‐Katz equation, Cable equation, action potential 3. Dynamics I: Introduction to differential equations; phase‐planes; oscillations 4. Dynamics II: Stability analysis, bifurcation theory, numerical methods 5. Single cell dynamics I: Propagating action potentials; rhythmic behavior 6. Single cell dynamics II: Variety of channels, bursting oscillations; dendrites - multi‐ compartment models 7. Synapses: Simple networks 8. Networks: Classification of network behavior; synchrony, role of different types of channels and coupling 9. Models for sleep: Sleep/wake cycle, Thalamocortical oscillations 10. Parkinson’s disease: Basal ganglia, origin of pathological firing patterns, Deep brain stimulation 11. Olfaction 12. Vision 13. Stroke 14. Presentation of projects Page 1 2016-2017 Math 4350 Mathematics 4504 History of Mathematics Spring 3 credits Catalog Description: Development of mathematics from primitive origins to present forms. development of arithmetic, algebra, geometry, trigonometry, and calculus. Topics include: Prerequisite: C- or better in 2568, 4507, or 5520H; or credit for 568, 571, 507, 580, or 520H; or permission of department. Exclusions: Open only to math majors, or students with graduate standing in Ed T&L. Not open to students with credit for 504. Text: Journey through Genius, by William Dunham, published by Wiley, ISBN: 0471500305 Topics List: 1. Development of arithmetic; Babylonian tablets and Egyptian papyri. 2. Development of geometry: Pythagoras, Thales, Euclid, Archimedes, Ptolemy, and nonEuclidean geometry. 3. Development of algebra and calculus. Page 1 2016-2017 Math 4504 Mathematics 4507 Geometry Autumn, Spring 3 credits Catalog Description: Topics in Euclidean, spherical, and hyperbolic geometries. Connections to high school mathematics, calculus, and the theory of groups are emphasized. Prerequisite: C- or better in 3345 and in C- or better in 2568 or 5520H; or credit for 345, and credit for 568, 571, or 520H; or graduate standing. Purpose: This course treats Euclidean, spherical, and hyperbolic geometry from a unified point of view. Moreover, in this course students essentially write their own “textbook” with the proofs of a majority of the theorems left to the student. With this in mind, we hope to encourage the student to become a “do-er” of mathematics. Text: Course notes. Topics List: 1. Neutral geometry. 2. Euclidean geometry. 3. Spherical geometry. 4. Hyperbolic geometry. Page 1 2016-2017 Math 4507 Mathematics 4512 Partial Differential Equations for Science and Engineering Autumn, Spring, Summer 3 credits Catalog Description: Second-order PDEs; boundary value problems; Fourier series; wave, heat and Laplace equations; applications. Prerequisite: C- or better in 2173, 2153, 2162.xx, 2182H, or 4182H; or credit for 254.xx, 263.xx, 263.01H, or 264H. Intended for undergraduate and master degree students in Engineering and Science. Exclusions: Not open to students with credit for 4557, 512, or 557. Not open to students with a math major, math minor or actuarial science major. Purpose of Course: This course develops problem solving skills with little emphasis on theory. Students should be able to solve the PDE’s and ODE’s and interpret the solution. Text: Partial Differential Equations & Boundary Value Problems, 9th OSU custom edition, by Boyce, published by Wiley, ISBN: 9781119935148 Topics List: Part I: ODE's via The Laplace Transform (Chapter 6); Euler's and Bessel's Equation (½ of Chapter 5) 6.1 Definition of the Laplace Transform 6.2 Solution of Initial Value Problems 6.3 Step Functions 6.4 Differential Equations with Discontinuous Forcing Functions 6.5 Impulse Functions 6.6 Convolution Integral 5.4 Euler's Equation; Regular Singular Points 5.5 Series Solution near a Singular Point: Part I 5.6 Series Solution near a Singular Point: Part II 5.7 Bessel's Equation Midterm I Page 1 2016-2017 Math 4512 Mathematics 4512 Partial Differential Equations for Science and Engineering Autumn, Spring, Summer 3 credits Part II: Partial Differential Equations and Fourier Series (Chapter 10) 10.1 The Two-Point Boundary Value Problem 10.2 Fourier Series 10.3 Fourier Convergence Theorem 10.4 Even and Odd Functions App A Heat Conduction Equation: Motivation via Derivation 10.5 Separation of Variables; Heat Conduction in a Rod 10.6 Other Heat Conduction Problems: Nonhomogeneous, Mixed Boundary Conditions App B Wave Equation: Motivation via Derivation; 10.7 Vibrations of an Elastic String 10.8 Laplace's Equation: Separation in Cartesian Coordinates Dirichelet vs. Neumann Boundary Conditions 10.8 Separation and Solution in Polar and Cylindrical Coordinates Midterm II 11.1 11.2 11.2 11.3 11.4 11.5 11.6 Page 2 Part III: Boundary Value Problems (Chapter 11) Two-point Boundary Value Problems Sturm-Liouville Boundary Value Problems I Sturm-Liouville Boundary Value Problems II Nonhomogeneous Boundary Value Problems Singular Sturm-Liouville Problems Bessel Series Expansion: Vibrating Drum (If time permits: Series of Orthogonal Functions: Mean Convergence) 2016-2017 Math 4512 Mathematics 4530 Probability Autumn, Spring, Summer 3 credits Catalog Description: Combinatorial probability, random variables, independence, expectation, variance. Purpose: This is an introductory probability course designed to give students a firm grasp of random variables, where they occur, and how they are used, and to develop the computational tools necessary to work with them. Prerequisite: C- or better in 2153, 2162.xx, 2173, 2177, 2182H, 4182H; or credit for 254.xx, 263.xx, 263.01H, or 264H. Exclusions: Not open to students with credit for any of 530, 5530H (531H), or Stat 4201 or 420. Follow-up Courses: Math 3532 (actuarial science majors), Math 3589 (financial math), Stat 4202. Text: Probability, by Pitman, published by Springer, ISBN: 9780387979748 Topics List: I. Discrete probability. 1. First principles: outcome spaces, basic counting techniques, and partitions. 2. Venn diagrams and the inclusion-exclusion principle. 3. Conditional probability and independence; decision trees and Bayes’ Theorem. 4. Discrete random variables; mass and generating functions; joint distributions. 5. Binomial, hypergeometric, geometric, negative binomial, and Poisson variables; applications and relationships. 6. Statistics on discrete variables. II. Continuous probability 7. First principles: density functions, calculation of probabilities and statistics. 8. Moments and moment-generating functions. 9. Common distributions and their applications; exponential, gamma, uniform, normal. 10. The central limit theorem and normal approximation to the binomial distribution. 11. Relationships between the exponential, gamma, and Poisson distributions. 12. Hazard rates and survival functions. 13. Cumulative distribution functions, percentiles, and change of variables. 14. Joint distribution of continuous variables; independence and marginal distributions; density of a function of two variables Page 1 2016-2017 Math 4530 Mathematics 4545 Analysis Overview Autumn 4 credits Catalog Description: Topics in calculus and analysis. Prerequisites: Either C- or better in 2153, 2162.xx, 2173, 2182H, or 4182H; or credit for 254, 263.xx, 263.01H, 264H, or equivalent; -and- C- or better in Math 2568, 5520H, or equivalent. Exclusions: Entry to this course is restricted to graduate students in Statistics or Biostatistics who have permission from the Departments of Statistics or Biostatistics. Text: Introduction to Real Analysis, by William F. Trench, Edition1.03, published by Library of Congress Cataloging-in-Publication Data, ISBN: 0-13-045786-8 Topics List: 1. Limits and continuity of functions. 2. Derivative, mean value theorem, optimization. 3. Sequences and series, uniform convergence, power series, Taylor's theorem. 4. Riemann integral, substitution, bounded variation, limit properties, Rieman-Stieltjes integral. 5. Multivariable functions, directional derivatives, chain rule, Taylor's theorem. 6. Inverse and implicit function theorems, Lagrange multipliers, multiple integrals, Jacobians, differentiation under the integral sign. Page 1 2016-2017 Math 4545 Mathematics 4547 Introductory Analysis I Autumn, Spring 3 credits Catalog Description: 4547-4548 involved advanced calculus covering: sequences, limits, continuity, differentiation, Riemann integral, sequences and series of functions, Taylor series, and improper integrals. Prerequisite: C- or better in 3345, or credit for 345. Text: Introduction to Real Analysis, 4th edition, by Bartle & Sherbert, published by Wiley, ISBN: 9780471433316 Topics List: 1. Sequences and their limits. 2. Bolzano-Weierstrass Theorem and Cauchy’s criterion. 3. Convergence and absolute convergence of series. Tests for convergence. 4. Power series. 5. Continuous functions. Page 1 2016-2017 Math 4547 Mathematics 4548 Introductory Analysis II Autumn, Spring 3 credits Catalog Description: 4547-4548 involved advanced calculus covering: sequences, limits, continuity, differentiation, Riemann integral, sequences and series of functions, Taylor series, and improper integrals. Prerequisite: C- or better in 4547, or credit for 548. Text: Introduction to Real Analysis, 4th edition, by Bartle & Sherbert, published by Wiley, ISBN: 9780471433316 Topics List: 1. Uniform continuity. 2. Derivatives. 3. Mean Value Theorem, L’Hopital’s rule. 4. Taylor series. 5. Riemann integral. 6. Exponential and logarithmic functions. 7. Sequences and series of functions. Page 1 2016-2017 Math 4548 Mathematics 4551 Vector Analysis Autumn, Spring 3 credits Catalog Description: Vector operations; Jacobian and change of variables; div, grad and curl; Green's Stokes', and divergence theorems; applications. Prerequisite: C- or better in 2153, 2162.xx, 2173, or 2182H; or credit for 254.xx, 263.xx or 263.01H. Exclusions: Not open to students with credit for Math 4182H, 264H, 513 or 551. Text: Advanced Calculus, 5th edition, by Kaplan, published by Pearson, ISBN: 9780201799378 Topics List: 1. Vector operations, multiple integrals, line and surface integrals. 2. Vector operators: div, grad, and curl. 3. Jacobians and change of variables. 4. Green's Theorem, Stokes theorem. 5. Divergence Theorem. 6. Applications. Page 1 2016-2017 Math 4551 Mathematics 4552 Complex Analysis Spring, Summer 3 credits Catalog Description: Introduction to analytic functions of a complex variable, integral theorems, power series, residues, conformal mapping. Prerequisite: C- or better in 2153, 2162.xx, 2173, 2182H, or 4182H; or credit for 254.xx, 263.xx, 263.01H, or 264H. Exclusions: Not open to students with credit for Math 5522H, 552 or 514. Purpose: This course provides a comprehensive introduction to complex analysis, emphasizing applications that are useful in science and engineering. Text: Complex Variables and Applications, 8th edition, by Brown & Churchill, published by McGrawHill, ISBN: 0073051942 Topics List: Complex numbers, polar form (Ch. 1) Analyticity, Cauchy-Riemann equations (Ch. 2) Elementary functions (Ch. 3) Cauchy integral theorem and consequences (Ch. 4) Midterm 1 Power series (Ch. 5) Residues and poles (Ch. 6) Applications of residues (Ch. 7) Mapping by elementary functions (Ch. 8) Conformal mapping (Ch. 9) Midterm 2 Applications of conformal mapping (Ch. 10) Schwarz-Christoffel transformation (Ch. 11) Poisson integral, Dirichlet problem (Ch. 12) Page 1 2016-2017 Math 4552 Mathematics 4556 Dynamical Systems Autumn 3 credits 5 credits Catalog Description: Systems of linear, first-order differential equations; existence and uniqueness theorems; numerical methods; qualitative theory (phase plane analysis, linearization, stability, limit cycles); and physical applications. Prerequisite: C- or better in 2153, 2162.xx, 2173, 2182H, or 4182H; or credit for 254.xx, 263, 263H, or 264H. Text: Nonlinear Dynamics and Chaos, Steven H. Strogatz, published by Westview Press, ISBN 9780738204536 Topics List: 1. One-dimensional flows: geometric way of thinking; fixed points and stability; population growth and other applications. 2. Bifurcations in one-dimensional flows: saddle-node, transcritical and pitchfork bifurcations; imperfect bifurcations. 3. Theory: existence; uniqueness; continuous dependence. 4. Phase planes: phase portraits; vector fields; nullclines; fixed points; stability; linearization. 5. Linear systems: classification of linear systems; what does the linear system say about the nonlinear system? 6. Limit cycles; introduction; Poincare-Bendixson theorem; conservative systems. 7. Bifurcations of two-dimensional flows; saddle-node, transcritical, and pitchfork bifurcations; Hopf bifurcation theorem. 8. XPPAUT: phase planes; bifurcations; applications. 9. Global bifurcations: homoclinic orbits; Poincare map; stability of periodic orbits. 10. Singular perturbations: Relaxation oscillator; averaging. 11. Applications: (e.g., Neurons). 12. One-dimensional maps: Logistic map. 13. Smale horseshoe: symbolic dynamics. 14. Applications. Page 1 2016-2017 Math 4556 Mathematics 4557 Partial Differential Equations Autumn, Spring 3 credits 5 credits Catalog Description: First and second-order PDE’s; existence and uniqueness, initial and boundary value problems, Fourier series; Green's functions; wave, heat and Laplace equations; nonlinear PDE’s; applications. Prerequisite: C- or better in 2255, 2415, 4556, or 5520H; or credit for 255, 415.xx, or 521H. Exclusions: Not open to students with credit for Math 4512 or 512. Text: Partial Differential Equations, an Introduction, 2nd edition, Walter A.Strauss, published by Wiley, ISBN: 0471548685. Topics List: 1. Definition of a PDE, linearity; solution of first-order linear (transport) equation; modeling with PDEs. 2. Well-posed problems, initial- and boundary conditions; secondorder equations; classification into types; the wave equation. 3. Causality and energy; diffusion equation; diffusion on the whole line. 4. Solution of the wave and diffusion equations on a half-line; diffusion and waves with sources. 5. Separation of variables for the wave equation, Dirichlet, Neumann and Robin conditions. 6. Fourier series; sine and cosine series; orthogonality and general Fourier series; completeness and convergence. 7. Midterm. Gibbs phenomenon. 8. Laplace’s equation; maximum principle; rectangular coordinates. 9. Poisson’s formula; Laplace’s equation in circular coordinates. 10. Green’s identities; maximum principle; Dirichlet principle; Green’s second identity. 11. Green’s functions; symmetry; half-space and sphere. 12. Wave equation in two and three dimensions; energy; causality; Huyghens’ principle. 13. Rays and characteristics; relativistic geometry; sources; the diffusion equation. 14. The Schrödinger equation; the hydrogen atom. Page 1 2016-2017 Math 4557 Mathematics 4568 Linear Algebra for Engineering Graduate Students Autumn, Spring, Summer 3 credits Catalog Description: Matrix algebra, vector spaces and linear maps, bases and dimension, eigenvalues and eigenvectors, applications. Prerequisite: C- or better in 2153, 2162.xx, 1172, 2182H or 4182H; or credit for 254, 263.xx, 263.01H or 264H Exclusions: Restricted to graduate students in engineering. Not open to students with credit for 2568 (568), 5101 (601), 5520H (520H) or 572. Text: Introduction to Linear Algebra, 5th edition, by L.W. Johnson, R.D. Riess, and J.T. Arnold, published by Pearson, ISBN: 9780321628217. Topics List: 1.1 1.2 1.3 1.5 1.6 1.7 1.9 Part I Introduction to Matrices and Systems of linear equations Echelon Form and Gaussian-Jordan Elimination Consistent Systems of linear Equations Matrix Operations Algebraic Properties of Matrix operations Linear Independence and Nonsingular Matrices Matrix Inverses and Their Properties Midterm 1 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 5.2 5.3 5.4 3.6 3.7 Page 1 Part II Vectors in The Plane (Review only because it was done in 1152) Vectors in Space (Review only because it was done in 1152) The Dot Product and The Cross Introduction Vector Space Properties of ℝn Examples of Subspaces Bases for Subspaces Dimension Vector Spaces Subspaces Linear Independence, Bases, and Coordinates Orthogonal Bases for Subspaces Linear Transformation from ℝn to ℝm 2016-2017 Math 4568 Mathematics 4568 Linear Algebra for Engineering Graduate Students Autumn, Spring, Summer 3 credits Midterm 2 4.1 4.2 4.4 4.5 4.6 4.7 Part III The Eigenvalue Problem for 2x2 Matrices Determinants and the Eigenvalue Problem Eigenvalues and Characteristic Polynomial Eigenvectors and Eigenspaces Complex Eigenvalues and Eigenvectors Similarity Transformations and Diagonalization Final Page 2 2016-2017 Math 4568 Mathematics 4573 Elementary Number Theory Spring (odd years) 3 credits Catalog Description: Prime numbers, modular arithmetic, Diophantine equations, combinatorial analysis; introduction to concepts of abstract algebra. Purpose: To introduce students to some topics in number theory at the upper undergraduate level and make connections to other areas of mathematics, such as combinatorics and abstract algebra. Prerequisite: C- or better in 3345 or 4181H; or credit for 345 or 264H. Exclusions: Not open to students with credit for Math 5576H, 576H or 573. Follow-up Courses: 4580-4581; or for students with an honors background, 5590H-5591H Text: An Introduction to the Theory of Numbers, 5th edition, by Niven, Zuckerman & Montgomery, published by TBS, ISBN: 9780471625469 Topics List: 1. Prime numbers and factorization 2. Congruences and modular arithmetic; the Euler phi-function ø(n). 3. Fermat’s “Little” Theorem. Primitive roots. 4. Quadratic Reciprocity 5. Numerical Functions of number theory; multiplicative functions and Moebius inversion 6. Diophantine equations. 7. Number theory from an algebraic viewpoint; groups, rings and fields. 8. Possible additional topics: continued fractions, Pell’s equation, and elliptic curves. Page 1 2016-2017 Math 4573 Mathematics 4575 Combinatorial Mathematics Spring (even years) 3 credits Catalog Description: Enumerative techniques, combinatorial identities, graph theory, algorithms, error correcting codes Purpose for the Course: Combinatorics and discrete mathematics are increasingly important, particularly for their applications in computer science. This course will give a brief overview of this subject. Prerequisite: C- or better in 2568 or 5520H; credit for 568, 571, or 520H. Exclusions: Not open to students with credit for Math 5529H. Topics List: 1. Counting principles: factorials, permutations and combinations, binomial coefficients, Stirling numbers, double counting. 2. Combinatorial identities: bijections, binomial theorem, generating functions. 3. Graph theory: bridges of Konigsberg, Eulerian circuits, trees, edge coloring, vertex coloring, planar graphs, Kempe's proof of the 5-color theorem 4. Error correcting codes: sphere packing bound, Hamming codes 5. (Optional.) Algorithms: Djkstra's algorithm for minimum spanning tree, depth first and breadth first algorithms for trees, greedy algorithm for graph coloring. Page 1 2016-2017 Math 4575 Mathematics 4578 Discrete Mathematical Models Spring 4 credits Catalog Description: Homogeneous and non-homogeneous difference equations of one or several variables, Markov chains, graph theory, network flows. Prerequisite: {C- or better in 2568 or 5520H; and C- or better in 4530, 5530H or Stat 420}; or {credit for 568, 571, or 520H; and credit for 530, 531H, or Stat 420}. Exclusions: Not open to students with credit for 578. Topics List: 1. Homogenous and non-homogeneous difference equations. 2. Application in finance, genetics, economics. 3. Matrix methods, nonlinear equations, stability, bifurcation, harvesting. 4. Application of Markov chains with absorbing and non-absorbing states, limiting behavior. 5. Graph theoretical algorithms, network flows, applications. 6. Linear/integer programming. Comment: This course requires the student to use a programming language chosen by the instructor to complete required course work. Page 1 2016-2017 Math 4578 Mathematics 4580 Abstract Algebra I Autumn, Spring 3 credits Catalog Description: 4580-4581 includes elementary number theory, group theory, vector spaces and linear transformation, and field theory. Prerequisite: {C- or better in 3345, and C- or {credit for 345; and credit for 568, 571, or 520H}. better in 2568 or 5520H} or Exclusions: Not open to students with credit for 581 or 591H. Purpose: Math 4580-4581 constitutes a two-semester sequence on abstract algebra, intended to familiarize students with the principal concepts, mode of thinking, and important theorems in this subject area. Considerable emphasis is placed on connections between this material and the traditional topics of high school mathematics - Euclidean geometry, polynomial equations, and trigonometry. Math 4580 begins with a brief review of elementary number theory. Groups are introduced in the context of geometrical symmetry. There is a study of number systems along with an introduction to the theory of polynomial equations. Finally, this is all applied to the constructibility problem for regular polygons. Text: An Introduction to Abstract Algebra, by Ronald Solomon (in-house notes). Topics List: 0. Equivalence Relations and the Integers 1. Isometrics 2. Congruences and Groups 3. The Isometrics of ℝ2 and Symmetry Groups 4. The Integers, Polynomials, and their Generalizations 5. Rational Numbers, Real Numbers and Decimals 6. Roots and the Complex Numbers 7. The Cyclotomic Polynomials 8. Fermat and the Gaussian Integers 9. Constructible Numbers 10. Some Linear Algebra and a Nonconstructibility Criterion 11. The Method of Mr. Gauss Page 1 2016-2017 Math 4580 Mathematics 4581 Abstract Algebra II Autumn, Spring 3 credits Catalog Description: 4580-4581 includes elementary number theory, group theory, vector spaces and linear transformation, and field theory. Prerequisite: C- or better in Math 4580; or credit for 5590H or 581. Exclusions: Not open to students with credit for 582 or 592H. Purpose: Math 4580-4581 constitutes a two-semester sequence on abstract algebra, intended to familiarize students with the principal concepts, mode of thinking, and important theorems in this subject area. Considerable emphasis is placed on connections between this material and the traditional topics of high school mathematics - Euclidean geometry, polynomial equations, and trigonometry. Math 4581 deepens the theory of groups and of polynomial equations, culminating in Galois' theory of equations and the classification of finite symmetry groups in ℝ3. Text: Notes on Abstract Algebra, by Ron Solomon (in-house notes) Topics List: 1. Permutation groups, orbits, and Lagrange's Theorem 2. The Orbit Counting Formula 3. Imaginaries and Galois fields 4. Gaussian integers and Fermat's two squares theorem 5. Review and Midterm 1 6. Symmetric polynomials and the Fundamental Theorem of Algebra 7. Nonconstructibility and a Lagrange Theorem for fields 8. Galois' Theory of Equations 9. The Galois Correspondence 10. Review and Midterm 2 11. The isometry group of ℝ2 12. Linear algebra in ℝ3 13. The Platonic solids and their symmetries 14. The finite subgroups of SO(3) Page 1 2016-2017 Math 4581 Mathematics 5520H Honors Linear Algebra and Differential Equations Autumn 5 credits Catalog Description: Linear transformations and matrices, spectral theorem, ordinary differential equations, existence and uniqueness theorems, phase space, stability, oscillations. Prerequisite: C or better in 4182H or in both 2182H and 3345; or C or better in 264H or in both 263H and 345; or permission of department. Text: Linear Algebra: An Introductory Approach, revised 4th edition, by Curtis, published by Springer, ISBN: 9780387909929 Introduction to Ordinary Differential Equations, by Coddington, published by Dover, ISBN: 9780486659428 Topics List: 1. Vector spaces and linear transformations. 2. Systems of equations, determinants. 3. Spectral theorem. 4. Ordinary, linear and nonlinear differential equations. 5. Existence and uniqueness theorems. 6. Phase space, stability, and periodic points. Page 1 2016-2017 Math 5520H Mathematics 5522H Honors Complex Analysis Spring 5 credits Catalog Description: Theoretical treatment of complex analysis. Prerequisite: C or better in 5520H or permission of department. Text: Vary, for example: An Introduction to Complex Function Theory, by B.P. Palka Elementary Theory of Analytic Functions of One or Several Complex Variables, by H. Cartan Complex Analysis, 2nd edition, by Bak-Newman Complex Analysis with Applications, by Silverman Topics List: 1. Complex numbers, Riemann's sphere. Complex functions, elementary functions, Möbius transformations. 2. Holomorphic functions, Cauchy-Riemann equations. 3. Line integrals. Cauchy’s integration theorem and its consequences. 4. Harmonic functions. 5. Sequences and series of holomorphic functions. Power series, analytic functions. 6. Isolated singularities, meromorphic functions, the calculus of residues. 7. Conformal mappings, the Riemann mapping theorem. 8. Geometric principles. 9. Mittag-Leffler's and Weierstrass's expansions of meromorphic functions. 10. Analytic continuation, Riemann surfaces. 11. Applications to number theory, geometry, physics. Page 1 2016-2017 Math 5522H Mathematics 5529H Honors Combinatorics Autumn (even numbered years) 5 credits Catalog Description: Techniques of combinatorial mathematics; connections with geometry, algebra, analysis, and probability. Prerequisite: C or better in 4182H, or in both 2182H and 3345; or credit for 264H, or for both 263H and 345; or permission of department. Text: Vary, for example: Discrete Mathematics, by Lovasz, Pelican & Vestergombi, published by Springer, ISBN: 9780387955858 Proofs from the Book, 4th edition, by Aigner, Ziegler & Hofmann, published by Springer, ISBN: 9783642008559 Combinatorics: Topics, Techniques, Algorithms, by P. Cameron, published by Cambridge University Press, ISBN: 9780521338936 Topics List: 1. Counting principles. 2. Generating functions. 3. Combinatorial probability. 4. Finite fields and applications. 5. Theory of partitions. 6. Famous graphs. 7. Ramsey theory. 8. Permutation groups. Page 1 2016-2017 Math 5529H Mathematics 5530H Honors Probability Spring (odd numbered years) 5 credits Catalog Description: Theoretical treatment of probability, with applications within and outside mathematics. Prerequisite: C or better in 5529H, or permission of department. Purpose of Course: The acquaintance with rigorous probability theory, its history and its multiple connections, will better prepare honor students for graduate studies and will help them get involved in research at earlier stages of their careers. Text: Vary, for example: Heads and Tails. An Introduction to Limit Theorems in Probability, E. Lesigne, AMS, ISBN 0821837141 Elementary Probability Theory with Stochastic Processes, Kai Lai Chung Elementary Probability Theory, Kai Lai Chung and Farid AitSahlia,, 4th Edition, Springer, 2003 Topics List: 1. Historical origins of probability. 2. Diverse ways of sampling, allocation, models. 3. Random variables, expectation, moments. 4. Important distributions. 5. Limit theorems: law of large numbers, central limit theorem. 6. Random walks and Markov chains. 7. Statistical independence in analysis and number theory. Page 1 2016-2017 Math 5530H Mathematics 5540H Honors Differential Geometry Spring (even numbered years) 5 credits Catalog Description: Geometry of curves and surfaces in 3-dimensional space, curvature, geodesics, Gauss-Bonnet Theorem, Riemannian metrics. Prerequisite: C or better in 5520H, or in both 2182H and 2568; or credit for 520H, or in both 263.01H and 568; or permission of department. Text: Text vary, for example: Differential Geometry of Curves and Surfaces, DoCarmo Elements of Differential Geometry, R. Milman and G. Rarker Topics List: 1. Geometry of curves; Frenet-Serret equations. 2. Curvature of surfaces, First Fundamental Form, Gauss's Theorema Egregium. 3. Geodesics, exponential map. 4. Isometries, conformal mappings; mapmaking. 5. Gauss-Bonnet Theorem. 6. Riemannian metrics, non-Euclidean geometry. Page 1 2016-2017 Math 5540H Mathematics 5576H Honors Number Theory Autumn (odd numbered years) 5 credits Catalog Description: Elementary analytic and algebraic number theory, tracing its unifying role in the development of mathematics through history. Prerequisite: C or better in 4182H, or in both 2182H and 3345; or credit for 264H, or for both 263H and 345; or permission of department. Purpose of Course: The intention of this course is to present number theory, the "Queen of Mathematics" through its historical development. Being one of the oldest mathematical disciplines, number theory, in the course of its history, both benefited from and contributed to such major mathematical areas as geometry, algebra and analysis. These courses will be especially beneficial for honor students planning to pursue careers in mathematics, physics, computer science and education, but may be of interest to engineering students as well. Text: Vary, for example: An Introduction to the Theory of Numbers, 6th edition, by Hardy, Wright, Heath & Brown, published by Oxford, ISBN: 9780199219865. An Introduction to the Theory of Numbers, I. Niven, H.S. Zukkerman, H.L. Montgomery Number Theory: An Introduction to Mathematics, Parts A and B, by William A. Coppel, Springer-Velag. Topics List: 1. Review of Egyptian and Mesopotamian Mathematics. Greek tradition. Three classical Greek problems (cube doubling, angle trisection, circle quadrature). 2. Famous irrationalities. 3. Continued fractions and applications thereof (quadratic surds, Pell’s equation, Diophantine approximations, etc.) 4. More on diophantine approximation. Algebraic numbers. Liouville numbers. A glimpse into the Thue-Siegel-Roth Theorem. 5. Uniform distribution modulo one. Weyl criterion. Some important sequences. PisotVijayaraghavan numbers. Formulation and discussion of Margulis’ solution of Oppenheimer’s conjecture. 6. Normal numbers. Champernoun’s example. Almost every number is normal. LevyKhinchine Theorem on normality of continued fractions. 7. Infinitude of primes. Euler’s identity. Chebyshev’s Theorem. Bertrand’s Postulate. Dirichlet’s Theorem on primes in progressions. Average rate of growth of classical number-theoretical functions. 8. Finite fields. Wedderburn’s Theorem. Applications: Latin Squares and Cryptography. 9. Quadratic reciprocity. Page 1 2016-2017 Math 5576H Mathematics 5576H Honors Number Theory Autumn (odd numbered years) 5 credits 10. Pythagorean triangles. Representation of integers as sums of squares. Quaternions, Cayley’s octavas. Hurwitz’ Theorem. Minkowski’s geometry of numbers. 11. p-adic numbers, their construction and axiomatic characterization (Ostrowski’s Theorem). Minkowski-Hasse principle. 12. Fermat’s last theorem. Some easy cases. A glimpse into modern developments (elliptic curves, Mordell-Weil Theorem, etc.). Page 2 2016-2017 Math 5576H Mathematics 5590H (Au), 5591H, (Sp) Honors Abstract Algebra I Honors Abstract Algebra II 5 credits each Catalog Description: 5590H and 5591H includes elementary number theory, group theory, ring theory, abstract linear algebra, field theory, and Galois theory. Prerequisite: 5590H: C or better in 5520H or in 520H, or permission of department. 5591H: C or better in 5590H or permission of department. Text: Vary, for example: Abstract Algebra, 3rd edition, by Dummit & Foote, published by Wiley, ISBN: 9780471433349 Algebra, by M. Artin Topics in Algebra, by I. Herstein Topics List: 5590H: 1. Integers, unique factorization; congruences, Euler function. 2. Groups, subgroups, homomorphisms and isomorphisms, normal subgroups, quotient groups, permutation groups, cyclic groups, Cauchy Theorems, Sylow's Theorems; direct products, fundamental theorem for finite Abelian group; G-sets. 3. Rings, subrings, ideals, morphisms, polynomial rings, prime and maximal ideals. 4. Commutative rings, factorization theory, Euclidean rings, principal ideal rings, unique factorization domains, Gauss' lemma; illustrations in the integers of quadratic number fields. 5. Modules over commutative rings, submodules, quotients and direct sums; fundamental theorem for modules over principal ideal domains. 5591H: 1. Vector spaces (as a special case of modules); linear maps and matrices, canonical forms, dual spaces. 2. The theory of determinants. 3. Bilinear and quadratic forms; inner product and unitary spaces; principal axis theorem. 4. Fields, algebraic and transcendental (extensions), existence of closure (over countable fields), tests for polynomial irreducibility; normality, separability, field automorphisms. 5. Galois theory, the subgroup-subfield correspondence theorem, group theory interrelations; extensions of finite fields, cyclotomic extensions. 6. Solvable groups and solvability by radicals. Page 1 2016-2017 Math 5590H, 5591H Mathematics 5630 Life Contingencies I Autumn 3 credits Catalog Description: 5630-5631 introduces students to the mathematical theory of contingencies. Includes material from examinations by the Society of Actuaries and the Casualty Actuarial Society. Prerequisite: C- or better in 3618 and C- or better in 4530, 5530H, or Stat 4201; or credit for 530, 531H, or Stat 420; or permission of department. Exclusions: Open only to actuarial science majors. Text: Actuarial Mathematics for Life Contingent Risks, by Dickson, Hardy & Waters, published by Cambridge, ISBN: 9780521118255 Topics List: 1. Survival distributions. 2. Individual risk models. 3. Life tables. 4. Topics from life insurance. 5. Life annuities. 6. Benefit premiums. Page 1 2016-2017 Math 5630 Mathematics 5631 Life Contingencies II Spring 3 credits Catalog Description: 5630-5631 introduces students to the mathematical theory of contingencies. Includes material from examinations by the Society of Actuaries and the Casualty Actuarial Society. Prerequisite: C- or better in Math 5630, or credit for 630. Exclusions: Open only to actuarial science majors, and to MMS students specializing in Financial Math. Text: Actuarial Mathematics for Life Contingent Risks, by Dickson, Hardy & Waters, published by Cambridge, ISBN: 9780521118255. Topics List: 1. Benefit reserve. 2. Multiple life functions. 3. Multiple decrement models. 4. Random and deterministic survivorship group. 5. Valuation of pension plans. 6. Applications. Page 1 2016-2017 Math 5631 Mathematics 5632 Financial Economics Autumn, Spring, Summer 3 credits Catalog Description: Introduction to the evaluation of options, futures, and other derivatives, interest models and risk management techniques. Includes material from examinations by the Society of Actuaries and the Casualty Actuarial Society. Prerequisite: {C- or better in 3618, or credit for 618, or permission of department} –and– {C- or better in 4530 or Stat 4201 or credit for 530 or Stat 420}. Exclusions: Restricted to actuarial science majors, math majors, and students with graduate standing. Text: Derivatives Markets, 2nd ISBN: 9780321280305 edition, by McDonald, published by Addison-Wesley, Topics List: 1. Option relationships. 2. Binomial option pricing. 3. Black-Scholes formula. 4. Market making and delta hedging. 5. Exotic options. 6. Brownian motions and Ito's Lemma. 7. Interest rate models. Page 1 2016-2017 Math 5632 Mathematics 5633 Loss Models I Autumn 3 credits Catalog Description: 5633-5634 introduces students to the construction and evaluation of actuarial models, with topics covered by examinations of the Society of Actuaries and the Casualty Actuarial Society. Prerequisite: {C- or better in 4530, 5530H, Stat 4201, or credit for 530, 531H, or Stat 420}; and {C- or better in Stat 4202 or credit for Stat 421}. Exclusions: Open only to actuarial science majors and to MMS students specializing in Financial Math. Text: Loss Models: From Data to Decisions, 4th edition, by Klugman, Panjer and Willmot, published by Wiley, ISBN: 9781118315323. Topics List: 1. Measures of risk. 2. Characteristics of actuarial models. 3. Severity models. 4. Frequency models. 5. Aggregate loss models. Page 1 2016-2017 Math 5633 Mathematics 5634 Loss Models II Spring 3 credits Catalog Description: 5633-5634 introduces students to the construction and evaluation of actuarial models, with topics covered by examinations of the Society of Actuaries and the Casualty Actuarial Society. Prerequisite: C- or better in 5633. Exclusions: Open only to actuarial science majors and to MMS students specializing in Financial Math. Topics List: 1. Estimation of data. 2. Parameter estimation. 3. Model selection. 4. Simulation. 5. Credibility. Page 1 2016-2017 Math 5634 Mathematics 5756, 5757 Modern Mathematical Methods of Relativity I, II Autumn (even years), Spring (odd years) 3 credits Catalog Description: Special relativity as moving frames; tensors, exterior algebra and exterior calculus; differentiable manifolds and space time structures; parallel transport, torsion and curvatures, metric compatibility; structure equations of differential geometry. Prerequisite: Multivariable differential calculus and linear algebra (e.g. Math 2568 and/or 5101). A physics course (e.g. Physics 133 or higher). No prior knowledge of tensor calculus is assumed. However, we do assume a mature attitude towards mathematics and physics. Purpose: Develop from the bottom up the fundamental mathematical concepts and methods responsible for the successes in 20th century physics, mathematics, and theoretical engineering. Thus Math 5756 concretizes these developments in terms of: a) Special Relativity as the cognitive bridge to 20th century geometry b) Multilinear algebra as a source of geometrical structures, c) Linear algebra’s marriage to multi-variable calculus d) differential geometry as a three level hierarchy characterized by its Differential structure Parallel transport structure (a.k.a. covariant derivative) Metric structure e) The exterior calculus f) Cartan’s two structural equations for the various flavors of differential geometry, and their application to g) The Cartan-Misner calculus Text: a) b) c) d) Page 1 Gravitation by C. W. Misner, K. S. Thorne, and J. A. Wheeler. Selections from Mathematical Methods of Classical Mechanics by V.I. Arnold. Selections from Lecture Notes on Elementary Topology and Geometry by I. M. Singer. Selections from Spacetime Physics, 2nd edition, by E. Taylor and J.A. Wheeler 2016-2017 Math 5756, 5757 Mathematics 5756, 5757 Modern Mathematical Methods of Relativity I, II Autumn (even years), Spring (odd years) 3 credits Topics List: Math 5756 (Autumn): A rapid course in special relativity: spacetime geometry, event horizons and accelerated frames; • tensors, metric geometry vs symplectic geometry; • exterior calculus, Maxwell field equations; • manifolds, Lie derivatives, and Hamiltonian dynamics in phase space; • parallel transport, torsion, tensor calculus; • curvature and Jacobi’s equation of geodesic deviation; • Cartan’s two structural equations, metric induced properties, and Cartan-Misner curvature calculus. Math 5757 (Spring): • • • • • Page 2 Geodesics: Hamilton-Jacobi theory, the principle of constructive interference; stress-energy tensor: hydrodynamics in curved spacetime and Einstein field equations; The conservation laws and the Bianch identities mathematized in terms of the “Boundary of a Boundary is zero (@ @ = 0)” Principle. Solutions to the Einstein’s field equations: stars, black holes, gravitational collapse, geometry and dynamics of the universe; vector harmonics, tensor harmonics, acoustic and gravitational waves in violent relativistic backgrounds. 2016-2017 Math 5756, 5757

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