# UNIVERSITY OF NORTH ALABAMA MA 110 FINITE MATHEMATICS

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1 MA 110 FINITE MATHEMATICS Course Description. This course is intended to give an overview of topics in finite mathematics together with their applications and is taken primarily by students who are not majoring in science, engineering, commerce, or mathematics (i.e., students who are not required to take calculus). The course includes sets, counting, permutations, combinations, basic probability (including Bayes Theorem), an introduction to statistics (including work with Binominal Distributions and Normal Distributions), matrices and their applications to Markov chains and decision theory. Additional topics may include symbolic logic, linear models, linear programming, the simplex method and applications. Credit Hours. 3 Goal: The student shall demonstrate knowledge of counting techniques, including permutations and combinations, basic probability, including Bayes Theorem, basic statistics, including work with binomial and normal distributions, matrices and their applications to Markov chains and decision theory. I. Set Theory 1. Introduction to sets 2. Subsets 3. Complement, union and intersection of sets 4. Venn diagrams 5. Applications II. Probability 1. Basic concepts 2. Conditional probability; Independence 3. Bayes theorem III. Counting Principles 1. Permutation and combinations 2. Applications of counting 3. Binomial probability 4. Probability distribution and expected value IV. Statistics 1. Frequency distribution 2. Measures of central tendency 3. Measures of variation 4. Normal distribution 5. Binomial distribution

2 MA 110 continued page 2 UNA V. Matrices 1. Addition and scalar multiplication 2. Matrix multiplication VI. Application 1. Markov chains and application 2. Decision making (game theory) Requirements: Regular attendance. Successful completion of exams. Each student is required to have his/her own calculator. Evaluation of course: There will be a minimum of 3 hourly examinations at 100 points each and a final examination. Revised 6/06

3 MA 112 PRECALCULUS ALGEBRA Course Description. This course emphasizes the algebra of functions including polynomial, rational, exponential, and logarithmic functions. The course also covers systems of equations and inequalities, quadratic inequalities, and the binomial theorem. Additional topics may include matrices, Cramer s rule, and mathematical induction. Credit Hours. 3 Course Objective: The student shall demonstrate knowledge of 1. The basic concepts of arithmetic. 2. The basic concepts of algebra. 3. Elementary functions. 4. The sequential nature of mathematics and the interrelated nature of the various branches of mathematics 5. Problem-solving strategies which shall include reading and interpreting the problem, devising a plan to solve the problem, carrying out that plan, and reflecting on the reasonableness of the answer. Working problems backwards. 6. Estimation, prediction, and an ability to check answers. 7. Spatial relationships. 8. Standard mathematics vocabulary and symbols, and demonstrate the ability to use the language and symbols of mathematics accurately in communication. 9. Use fundamental mathematical operations, algorithms and measurements. 10. Present and interpret data in graphical form. 11. Select or create appropriate mathematical models to solve problems in mathematics and in other disciplines and integrate problem solving strategies learned in mathematics into the solution of problems encountered in daily living. Algebra review including factoring, rational expressions, the binomial theorem, rational exponents, and radicals. 1. Linear and quadratic equations and inequalities, applications 2. Relations and their graphs including linear relations, the parabola, the circle, and inequalities. 3. Functions, algebra of functions, graphing basic functions and their variations, inverse functions, exponential and logarithmic functions.

4 MA 112 continued page 2 UNA 4. Polynomial and rational functions including graphing, polynomial division, zeros of polynomial functions, rational zeros and real zeros of polynomial functions. 5. Systems of equations and inequalities including solution of two and three variable linear systems of equations, solution of linear system by matrices, nonlinear systems of equations, and systems of inequalities. 6. Matrices. Course Requirements: Regular attendance. Scientific calculator. Course Evaluation: There will be 3 or 4 tests and a final exam. Revised 6/06

5 MA 113 PRE-CALCULUS TRIGONOMETRY Course Description. This course is a continuation of Pre-Calculus Algebra. It includes the study of trigonometric and inverse trigonometric functions and includes extensive work with trigonometric identities and trigonometric equations. The course also covers vectors, complex numbers, DeMoivre s Theorem, and polar coordinates. Additional topics may include conic sections, sequences and using matrices to solve linear systems. Credit Hours. 3 Course Objectives: The student shall demonstrate knowledge of elementary and trigonometric functions, standard mathematics vocabulary and symbols. The student shall demonstrate ability to use fundamental mathematical operations, algorithms, measures, to use language and symbols of mathematics accurately in communication, to present and interpret data in graphical form and to select or create appropriate mathematical models to solve problems in mathematics and in other disciplines. I. Trigonometric Functions 1. Angles and Angular Measure 2. Right Triangle Trigonometry 3. Trigonometric Functions of Angles 4. Graphs of Trigonometric Functions II. Trigonometric Identities 1. Sums and Differences 2. Double Angle 3. Half Angle 4. Product-to-Sum and Sum-to-Product III. Analytic Trigonometry 1. Trigonometric Equations 2. Inverse Trigonometric Functions IV. Applications 1. Law of Sines 5. Polar Coordinates 2. Law of Cosines 3. Complex Numbers 4. Vectors Course Requirements: Course Evaluation: The student will be responsible for regularly attending class, tests and completing assigned homework. There will be at least 3 major tests and a final exam. Revised 6/06

6 MA 115 PRE-CALCULUS ALGEBRA AND TRIGNOMETRY Course Description: This course is a one semester combination of Pre-calculus Algebra and Pre-calculus Trigonometry intended for superior students. The course covers the following topics: algebra of functions (including polynomial, rational, exponential, and logarithmic functions); systems of equations and inequalities; quadratic inequalities; the binomial theorem; the study of trigonometric and inverse trigonometric functions including extensive work with trigonometric identities and trigonometric equations; vectors; complex numbers; DeMoivre s Theorem; polar coordinates. Credit Hours. 3 Course Textbook: Algebra and Trigonometry with Analytic Geometry, 10 th ed., by Swokowski Course Objectives: 1. The student shall demonstrate knowledge of functions, including polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions. 2. The student shall demonstrate knowledge of the sequential nature of mathematics and the interrelated nature of the various branches of mathematics. 3. The student shall demonstrate a knowledge of problem solving by reading and interpreting the problem, using mathematical models and reflecting on the reasonableness of the answer. 4. The student shall demonstrate a knowledge of working problems backwards. 5. The student shall demonstrate a knowledge of estimating making predictions and checking. 6. The student shall demonstrate a knowledge of spatial relationships, including the use of vectors and polar coordinates. 7. The student shall demonstrate a knowledge of standard mathematics vocabulary and symbols; and demonstrate an ability to use the vocabulary and symbols accurately in communication. 8. The student shall demonstrate ability to use fundamental mathematics operations, algorithms, and measurements.

7 MA 115 continued page 2 UNA 9. The student shall demonstrate an ability to present and interpret data in graphical form. 10. The student shall demonstrate an ability to select or create appropriate mathematical models to solve problems in mathematics and in other disciplines. 11. The student shall demonstrate ability to integrate problem solving strategies learned in mathematics into the solution of problems encountered in daily living. 12. The student shall demonstrate an understanding of the polar representation of a complex number and DeMoivre s Theorem. I. Solving Equations and Inequalities 1. Linear Equations 2. Quadratic Equations and Complex numbers 3. Other types of equations 4. Inequalities II. Functions and Graphs 1. Rectangles Coordinates System 2. Lines 3. Functions 4. Quadratic Functions 5. Operations on Functions 6. Inverse Function III. Polynomial and Rational Functions 1. Division of Polynomials 2. Zeros of Polynomials 3. Complex and Rational Zeros of Polynomials 4. Rational Functions, Properties & Graphs IV. Exponential and Logarithmic 1. Definition 2. Properties 3. Equations involving exponential and logarithmic expressions V. Trigonometric Functions 1. Angles 2. Definition 3. Applications with right triangles 4. Graphs VI. Analytic Trigonometry 1. Verifying Identities 2. Solving equations 3. Special formulas 4. Inverse functions

8 MA 115 continued. Page 3 UNA VII. VIII. IX. Applications of Trigonometry 1. Law of Sines 2. Law of Cosines 3. Trigonometric form of complex number 4. DeMoivre s Theorem 5. Polar Coordinates Systems of Equations 1. Substitution Method 2. Type of Systems 3. Linear Systems The Binomial Theorem Course Requirements: Regular attendance. Course Evaluation: There will be at least four 50 minute in-class test and a Comprehensive final examination. Revised 6/06

9 MA 125 CALCULUS I Course Description. This is the first of three courses in the basic calculus sequence taken primarily by students in science, engineering and mathematics. Topics include the limit of a function; the derivative of algebraic, trigonometric, exponential, and logarithmic functions; and the definite integral and its basic applications to area problems. Applications of the derivatives are covered in detail, including approximations of error using differentials, maximum and minimum problems, and curve sketching using calculus. Credit Hours. 4 Course Objectives: The student shall demonstrate knowledge of: 1. Various problem-solving strategies including, but not limited to, reading the problem, interpreting the problem, selecting an appropriate mathematical model, solving the problem and reflecting on the reasonableness of the answer. 2. Resources available to enhance personal knowledge of mathematics. The student shall demonstrate ability to: 3. Use estimation and approximation skills, and assess the reasonableness of solutions to problems. 4. Explain the role, nature and limitation of current and emerging technology. 5. Use current technology in problem solving and in exploring mathematical concepts. 6. Present and interpret data in graphical form. 7. Select or create appropriate mathematical models to solve problems in mathematics and in other disciplines. I. Functions 1. Functions and the Analysis of Graphical Information 2. Properties of Functions 3. Graphing Functions on Calculators and Computers; Computer Algebra Systems 4. New Functions from Old

10 MA 125 continued page 2 UNA II. III. IV. 5. Mathematical Models; Linear Models 6. Families of Functions 7. Mathematical Models 8. Parametric Equations Limits and Continuity 1. Limits (An Intuitive Introduction) 2. Limits (Computation Techniques) 3. Limits (Discussed More Rigorously) 4. Continuity 5. Limits and Continuity of Trigonometric Functions The Derivative 1. Tangent Lines and Rates of Change 2. The Derivative 3. Techniques of Differentiation 4. Derivatives of Trigonometric Functions 5. The Chain Rule 6. Implicit Differentiation 7. Related Rates 8. Local Linear Approximation; Differentials Analysis of Functions and Their Graphs 1. Analysis of Functions I: Increase, Decrease, and Concavity 2. Analysis of Function II: Relative Extrema; First and Second Derivatives Tests 3. Analysis of Function III: Applying Technology and the Tools of Calculus 4. Rectilinear Motion (Motion Along a Line) 5. Absolute Maxima and Minima 6. Applied Maximum and Minimum Problems 7. Newton s Method 8. Rolle s Theorem, Mean-Value Theorem V. Integration 1. An Overview of the Area Problem 2. The Indefinite Integral; Integral Curve and Direction Fields 3. Integration of Substitution 4. Sigma Notation 5. The Definite Integral 6. The fundamental Theorem of Calculus 7. Rectilinear Motion Revisited; Average Value 8. Evaluating Definite Integrals by Substitution VI. Applications of the Definite Integrals 1. Area Between Two Curves 2. Volumes by Slicing; Disks and Washers 3. Volumes by Cylindrical Shells 4. Length of a Plane Curve 5. Area of a Surface of Revolution

11 MA 125 continued page 3 UNA Course Requirement: Course Evaluation: Regular attendance. Each student must have a TI-85 or TI-86 Graphing Calculator. Tests and quizzes will be given for the purpose of evaluation. Revised 6/06

12 MA 126 CALCULUS II Course Description: This is the second of three courses in the basic calculus sequence. Topics include vectors in the plane and in space, lines and planes in space, applications of integration (such as volume, arc length, work and average value), techniques of integration, infinite series, polar coordinates, and parametric equations. Credit Hours. 4 Course Objectives: The student shall demonstrate knowledge of: 1. Various problem-solving strategies including, but not limited, to reading the problem, interpreting the problem, selecting an appropriate mathematical model, solving the problem and reflecting on the reasonableness of the answer. 2. Resources available to enhance personal knowledge of mathematics. The student shall demonstrate ability to: 3. Use estimation and approximation skills, and assess the reasonableness of solutions to problems. 4. Explain the role, nature, and limitations of current and emerging technology. 5. Use current technology in problem solving and in exploring mathematical concepts. 6. Present and interpret data in graphical form. 7. Select or create appropriate mathematical models to solve problems in mathematics and in other disciplines. I. Applications of the Definite Integral In Geometry, Science, and Engineering 1. Area Between Two Curves 2. Volumes by Slicing; Disks and Washers 3. Volumes by Cylindrical Shells 4. Length of a Plane Curve 5. Area of a Surface of Revolution 6. Work 7. Fluid Pressure and Force 8. Hyperbolic Functions and Hanging Cables

13 MA 126 continued page 2 UNA II. III. IV. Principles of Integral Evaluation 1. An Overview of Integration Methods 2. Integration by Parts 3. Trigonometric Integrals 4. Trigonometric Substitutions 5. Integrating Rational Functions by Partial Fractions 6. Using Tables of Integrals and Computer Algebra Systems 7. Numerical Integration; Simpson s Rule 8. Improper Integrals Infinite Series 1. Sequences 2. Monotone Sequences 3. Infinite Series 4. Convergence Tests 5. Taylor and Maclaurin Series 6. The Comparison, Ratio, and Root Tests 7. Alternating Series; Conditional Convergence 8. Power Series 9. Convergence of Taylor Series; Computational Methods 10. Differentiating and Integrating Power Series; Modeling with Taylor Series Analytic Geometry in Calculus 1. Polar Coordinates 2. Tangent Lines and Arc Length for Parametric and Polar Curves 3. Area in Polar Coordinates 4. Conic Sections in Calculus 5. Conic Sections in Polar Coordinates Course Requirements: Regular attendance. IT-85 or IT-86 Graphing Calculator. Course Evaluation: There will be a minimum of 4 hourly examinations at 100 pts. each and a comprehensive final examination. Revised 6/06

14 MA 227 CALCULUS III Course Description: This is the third of three courses in the basic calculus sequence. Topics include vector functions, functions of two or more variables, partial derivatives (including applications), quadratic surfaces, multiple integration, and vector calculus including Green s Theorem, Curl and Divergence, surface integrals, and Stoke s Theorem. Credit Hours. 4 Course Objectives: The student shall demonstrate knowledge of: 1. Various problem-solving strategies including, but not limited to, reading the problem, interpreting the problem, selecting an appropriate mathematical model, solving the problem and reflecting on the reasonableness of the answer. 2. Resources available to enhance personal knowledge of mathematics. The student shall demonstrate ability to: 3. Use estimation and approximation skills, and assess the reasonableness of solutions to problems. 4. Explain the role, nature, and limitations of current and emerging technology. 5. Use current technology in problem solving and in exploring mathematical concepts. 6. Present and interpret data in graphical form. 7. Select or create appropriate mathematical models to solve problems in mathematics and in other disciplines. I. Three-Dimensional Space; Vectors II. Vector-Valued Functions III. Partial Derivatives IV. Multiple Integrals V. Topics in Vector Calculus Course Requirements: Course Evaluation: Regular attendance. IT-85 or IT-86 Graphing Calculator. There will be 4 or 5 major tests and a final examination. In addition, there will be several unannounced short quizzes.

15 MA 237 Linear Algebra Course Description. This course introduces the basic theory of linear equations and matrices, real vector spaces, bases and dimensions, linear transformations and matrices, determinants, eigenvalues and eigenvectors, inner product spaces, and the diagonalization of symmetric matrices. Additional topics may include quadratic forms and the use of matrix methods to solve systems of linear differential equations. Credit Hours. 3 Course Objectives: The student shall demonstrate knowledge of: 1. Linear algebra. 2. The sequential nature of mathematics and the interrelated nature of the various branches of mathematics. The student shall demonstrate ability to: 3. Use language and symbols of mathematics accurately in communication. 4. Select or create appropriate mathematical models to solve problems in mathematics and in other disciplines. Linear equations and matrices including systems of linear equations, matrix operations, algebraic properties of matrix operations, special types of matrices and partitioned matrices, echelon form of a matrix, elementary matrices finding A t, equivalent matrices, and LU-factorization. Real vector spaces including vectors in the plane and in 3-space, vector spaces, subspaces, linear independence, basis and dimension, coordinates and isomorphisms, homogeneous systems, and rank of a matrix. Inner product spaces including standard inner product on R 2 and R 3, inner product spaces, Gram-Schmidt process, orthogonal complements.

16 MA 237 continued page 2 UNA Linear transformations and matrices including kernel and range of a linear transformation, matrix of a linear transformation, and similarity. Determinants including properties of determinants, cofactor expansion, inverse of a matrix, and other applications of determinants. Eigenvalues and eigenvectors, diagonalization, and diagonalization of symmetric matrices. Course Requirements: Regular attendance. Course Evaluation: There will be a minimum of 3 hourly examinations at 100 points each and a final comprehensive examination at 150 points. Revised 6/06

17 MA 238 APPLIED DIFFERENTIAL EQUATIONS I Course Description. An introduction to numerical methods, qualitative behavior of first-order differential equations, techniques for solving separable and linear equations analytically, and applications to various models (e.g., population, motion, chemical mixtures, etc.); techniques for solving higher-order linear differential equations with constant coefficients (general theory, undetermined coefficients, reduction of order, and the method of variation of parameters), with emphasis on interpreting the behavior of the solutions, and applications to physical models whose governing equations are of higher order; the Laplace transform as a tool for the solution of initial-value problems whose inhomogeneous terms are discontinuous. Credit Hours: 3 Course Objectives: The course is intended to strengthen the student s understanding of calculus by using it to solve differential equations. It also provides the student with a knowledge of how differential equations can be used to model and solve applied problems, especially in science. Terminology, methods of solving first-order differential equations, differential equations of higher order, modeling with differential equations, the Laplace transform. Course Requirements: Regular class attendance is expected. Students are required to apply techniques learned in calculus to the solution of differential equations. They are expected to derive and solve differential equations which serve to model a physical process where changes occur as time progresses. Course Evaluation: There will be at least 3 major tests and a final exam.

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