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4 Semester Credits Text: Stewart, Calculus: Early Transcend MATH 1061 (Calculus I) 4 semester credits Text: Stewart, Calculus: Early Transcendentals, 7th Ed., Chapters 2-6 Prerequisite: MATH 1026 (Precalculus) or at least 750 on the Math Placement Test Course Description: The first part of a three-semester sequence of courses on calculus (MATH 1061, 1062, 2063) for students in engineering and science. Topics covered include functions, limits and continuity, differentiation, applications of the derivative, optimization, antiderivatives, fundamental theorem of calculus, definite and indefinite integrals. Learning Objectives: The successful Calculus I student should be able to: 1. Compute the derivative of a function using the limit definition and derivative theorems. 2. Use the derivative to find tangent lines to a graph, find the slope of a graph at a point, and compute the rate of change of a dependent variable with respect to an independent variable. 3. Determine absolute extrema on a closed interval for continuous functions and to use the first and second derivatives to analyze and sketch the graph of a function. 4. Compute indefinite and definite integrals using the Fundamental Theorem of Calculus and the method of substitution. 5. Use definite integrals to find areas of planar regions. 6. Apply the competencies above to a wide range of functions, including polynomial, rational, algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, hyperbolic, and inverse hyperbolic. Content Detail: • Limits and Continuity: §§2.2, 2.3, 2.5 • Derivatives: §§2.1, 2.6-2.8 • Differentiation Rules and Derivatives of Polynomial, Exponential, Logarithmic, Trigonometric, and Hyperbolic Functions: §§3.1-3.4, 3.6, 3.11 • Implicit Differentiation: §3.5 • Applications of the Derivative and Related Rates: §§3.8-3.10 • Maximum and Minimum Values: §4.1 • Mean Value Theorem: §4.2 • Curve Sketching: §§4.3, 4.5 • l’Hospital’s Rule: §4.4 • Optimization Problems: §4.7 • Antiderivatives: §4.9 • Area, Definite Integrals, and the Fundamental Theorem of Calculus: §§5.1-5.3 • Indefinite Integrals: §5.4 • Integration by Substitution: §5.5 • Applications of Integration (Area, Volume, Work, Average Value): §§6.1-6.5 MATH 1062 (Calculus II) 4 semester credits Text: Stewart, Calculus: Early Transcendentals, 7th Ed., Chapters 7, 8, 10-12 Prerequisite: MATH 1061 (Calculus I) Course Description: The second part oF a three semester sequence oF courses on calculus (MATH 1061, 1062, 2063) for students in engineering and science. Topics covered include techniques of integration, applications of the integral, sequences and series, and vectors. Learning Objectives: The successful Calculus II student should be able to: 1. Employ a variety oF integration techniques to evaluate integrals, including integration by parts, trigonometric integrals and substitutions, and partial Fraction decomposition. Use L’Hôpital’s Rule to evaluate limits oF certain types oF indeterminate Forms. 2. Evaluate improper integrals, including integrals over inFinite intervals, as well as integrals in which the integrand becomes inFinite within the interval oF integration. 3. Determine whether a sequence or series converges or diverges and the sum oF a convergent geometric series. 4. Derive, differentiate and integrate a power series for a function. 5. Analyze curves given parametrically, graph polar equations and Find the area oF polar regions. 6. PerForm vector operations in the plane and space and calculate and apply the dot and cross product of vectors. Content Detail: • Techniques oF Integration (Integration by Parts, Trigonometric Integrals, Trigonometric Substitution, Partial Fractions): §§7.1-7.5 • Applications oF Integration (Arc Length, Area oF a SurFace oF Revolution, Centroid, Hydrostatic Force): §§8.1-8.3 • Parametric Curves: §§10.1-10.2 • Polar Coordinates, including Area and Arc Length: §§10.3-10.4 • Sequences and Series: §11.1-11.2 • Convergence Tests: §§11.3-11.7 • Power Series; Taylor and Maclaurin Series: §§11.8-11.11 • Vectors and Dot and Cross Products: §§12.1-12.4 • Equations of Lines and Planes in 3-space: §12.5 MATH 2063 (Multivariable Calculus) 4 semester credits Text: Stewart, Calculus: Early Transcendentals, 7th Ed., Chapters 13-16 Prerequisite: MATH 1062 (Calculus II) Course Description: Study of lines and planes, vector-valued functions, partial derivatives and their applications, multiple integrals, and calculus of vector fields. Learning Objectives: The successful Multivariable Calculus student should be able to: 1. Graph and find the equations of a line, plane and surfaces in space. 2. Find partial derivatives, directional derivatives, gradients and differentials of functions of several variables and use them to solve and interpret solutions of applied problems. 3. Find extrema of functions of several variables using the second partials test and Lagrange multipliers, and solve applied problems. 4. Evaluate iterated integrals and use them to find the area of plane regions. 5. Evaluate multiple integrals in rectangular, polar, cylindrical and spherical coordinates and use them to solve applications involving volume, surface area, density, moment, and centroids. Content Detail: • Equations of Lines and Planes in 3-space (Review): §12.5 • Calculus of Vector-Valued Functions: §§13.1-13.2 • Arc Length and Curvature: §13.3 • Velocity and Acceleration of Motion in 3-space: §13.4 • Limits and Continuity of Functions of Several Variables: §§14.1-14.2 • Partial Derivatives; Tangent Planes; the Chain Rule: §§14.3-14.5 • Directional Derivatives and the Gradient Vector: §14.6 • Maximum and Minimum Values and Lagrange Multipliers: §§14.7-14.8 • Double Integrals in Rectangular and Polar Coordinates: §§15.1-15.4 • Applications of Double Integrals: §§15.5-15.6 • Triple integrals in Rectangular, Cylindrical, and Spherical Coordinates: §§15.7-15.9 • Change of Variable in Multiple Integrals: §15.10 • Vector Fields: §16.1 • Line Integrals and the Fundamental Theorem of Line Integrals: §16.2-16.3 • Green’s Theorem: §16.4 • Curl and Divergence: §16.5 • Parametric Surfaces and Surface Integrals: §16.6-16.7 • Stokes’ and Divergence Theorems: §16.8-16.9 MATH 2076 (Linear Algebra) 3 semester credits Text: Lay, Linear Algebra and Its Applications, 4th Ed., Chapters 1-7 Prerequisite: MATH 1062 (Calculus II) Course Description: Study of linear equations, matrices, Euclidean n-space and its subspaces, bases, dimension, coordinates, orthogonality, linear transformations, determinants, eigenvalues and eigenvectors, diagonalization. Learning Objectives: The successful Linear Algebra student should be able to: 1. Solve linear systems of equations using Gauss-Jordan elimination. 2. Perform common matrix operations with matrices containing real and complex elements. 3. Evaluate determinants, explain their properties, and use them in applications. 4. Use both algebraic and geometric representations of Euclidean vectors in basic computations. 5. Use relationships between systems of linear equations, subspaces, linear transformations, and matrices. 6. Work with abstract vector spaces. 7. Compute eigenvalues and eigenspaces for matrices and, when possible, to use them to (orthogonally) diagonalize the matrix. 8. Work with bases, including verifying a set is a basis, constructing a basis, orthogonalizing a basis, making a change of basis and finding the dimension of a vector subspace. 9. Work with linear transformations, including finding the range, kernel, rank, nullity and a matrix representation. 10. Prove elementary theorems from linear algebra. Content Detail: • Systems of linear equations: §§1.1, 1.2, 1.3 • Matrix form of equations, linear independence: §§1.4, 1.5, 1.7 • Linear transformations: §§1.8, 1.9 [1.10 (optional)] • Matrix operations: §§2.1, 2.2, 2.3 • [Partitioned matrices: §2.4 (optional)] • Determinants: §§3.1, 3.2 • Vector spaces: §§4.1, 4.2 • Null spaces, column spaces, bases, coordinates: §§4.3, 4.4 • Dimension, rank: §§4.5, 4.6 • Eigenvectors, eigenvalues, diagonalization: §§5.1, 5.2, 5.3 • Orthogonality: §§6.1, 6.2, 6.3 • [Gram-Schmidt: §6.4 or Least squares: §6.5 (optional)] • Diagonalization: §7.1 .
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