  
Description  
 Introduction to functions of more than one variable, including vectors, partial differentiation, the gradient, contour diagrams and optimization. Additional topics include infinite series, convergence and Taylor series.


Course Objectives  
 The student will be able to:
 Analyze sequences and series
 Apply convergence tests to sequences and series
 Approximate functions using Taylor Polynomials
 Investigate vectors, including dot and cross products
 Demonstrate the ability to work with functions of more than one variable, which includes topics such as limits, continuity, the limit of a function at a point, computing both the equation of a plane and the equation of a tangent plane to a surface at a point, and parametric and vector equations of lines in 3space.
 Differentiate functions of more than one variable, including the directional derivative, the Gradient, the Chain Rule, and the determination of whether a function is differentiable
 Optimize functions of more than one variable for both constrained and unconstrained optimization problems; use of the Second Derivative Test to find local extrema and test for saddle points.
 Use technology such as graphing calculators and/or computer software to assist in solving problems involving any of the topics in (A) through (G) above.
 Discuss mathematical problems and write solutions in accurate mathematical language and notation.
 Interpret mathematical solutions.

Special Facilities and/or Equipment  
  Graphing Calculator.
 When taught hybrid: Four lecture hours per week in facetoface contact and one hour per week using CCC Confer. Students need internet access.

Course Content (Body of knowledge)  
  Analyze sequences and series.
 Sequences
 Convergence and divergence of sequences
 Limits
 Convergence Theorems
 Series
 Geometric series
 Alternating series
 Power series
 Differentiation of power series
 Integration of power series
 Applications to other disciplines
 Apply convergence tests to sequences and series
 Convergence and divergence of series
 Tests for convergence
 Radius of convergence
 Interval of convergence
 Approximate functions using Taylor Polynomials
 Taylor polynomials
 Applications to other disciplines
 Error in Taylor polynomial approximations
 Taylor series
 Finding and using Taylor series
 Applications to other disciplines
 Investigate vectors, including dot and cross products.
 Displacement Vectors
 Vectors operations in twospace and threespace
 Dot Product
 Cross Product
 Triple products
 Projections
 Applications to other disciplines
 Demonstrate the ability to work with functions of more than one variable
 Functions of several variables
 Graphs of Functions of several variables
 Lines in 3space
 Parametric Representations
 Vector Representations
 Contour Diagrams
 Crosssections
 Level curves
 Linear Functions
 Rectangular equation of a plane
 Equation of a tangent plane at a point
 Limits and Continuity
 Limit of a function at a point
 Differentiate functions of more than one variable, including the directional derivative, the Gradient, and the Chain Rule.
 Partial Derivatives
 Definition
 Algebraic computation
 Tangent planes
 Linear approximations
 Applications to other disciplines
 Gradients
 Applications to other disciplines
 Directional Derivatives
 Applications to other disciplines
 The Chain Rule
 Applications to other disciplines
 HigherOrder Partial Derivatives
 Differentiability
 Partial derivatives
 Directional derivatives
 Differentiability of a surface at a point
 Optimize functions of more than one variable for both constrained and unconstrained optimization problems.
 Local Extrema
 Definitions
 Second Derivative Test
 Local maximuma
 Local minima
 Saddle points
 Optimization
 Constrained Optimization
 Lagrange Multipliers
 Applications to other disciplines
 Use technology such as graphing calculators and/or computer software to assist in solving problems involving any of the topics in (A) through (F) above
 Calculator/computer utilities for solving problems involving sequences and series
 Calculator/computer utilities for constructing graphs of functions and relations in 3space, contour diagrams, and graphs of crosssections.
 Calculator/computer programs for evaluating directional derivatives.
 Discuss mathematical problems and write solutions in accurate mathematical language and notation.
 Application problems from other disciplines
 Proper notation
 Interpret mathematical solutions.
 Explain the significance of solutions to application problems.

Methods of Evaluation  
  Written homework
 Quizzes & tests
 Proctored comprehensive final examination.

Representative Text(s)  
 Stewart, James. Calculus: Concepts and Contexts. 4th ed. Belmont, CA, Brooks/Cole, Cengage Learning, 2010.

Disciplines  
 Mathematics


Method of Instruction  
  Lecture
 Discussion
 Cooperative learning exercises


Lab Content  
 Not applicable.


Types and/or Examples of Required Reading, Writing and Outside of Class Assignments  
  Homework Problems: Homework problems covering subject matter from text and related material ranging from 30  60 problems per week. Students will need to employ critical thinking in order to complete assignments.
 Lecture: Five hours per week of lecture covering subject matter from text and related material. Reading and study of the textbook, related materials and notes.
 Projects: Student projects covering subject matter from textbook and related materials. Projects will require students to discuss mathematical problems,write solutions in accurate mathematical language and notation and interpret mathematical solutions. Projects may require the use of a computer algebra system such as Mathematica or MATLAB.
 Worksheets: Problems and activities covering the subject matter. Such problems and activities will require students to think critically. Such worksheets may be completed both inside and/or outside of class.
