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Effective: Winter 2015
MATH 1CCALCULUS5 Unit(s)

Prerequisites: Prerequisite: MATH 1B.
Advisory: Advisory: Demonstrated proficiency in English by placement as determined by score on the English placement test OR through an equivalent placement process OR completion of ESLL 25 & ESLL 249.
Grade Type: Letter Grade, the student may select Pass/No Pass
Not Repeatable.
FHGE: Communication & Analytical Thinking Transferable: CSU/UC
5 hours lecture. (60 hours total per quarter)

Student Learning Outcomes -
  • Students will be able to apply the theories and techniques of differential calculus including directional derivatives and gradient vectors to solve application problems.
  • Students will be able to apply the theories and techniques of functions and relations of many variables to solve problems.
  • Students will be able to apply the theories and techniques of sequences and series to solve application problems.
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:
  1. Analyze sequences and series
  2. Apply convergence tests to sequences and series
  3. Approximate functions using Taylor Polynomials
  4. Investigate vectors, including dot and cross products
  5. 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 3-space.
  6. 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
  7. 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.
  8. 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.
  9. Discuss mathematical problems and write solutions in accurate mathematical language and notation.
  10. Interpret mathematical solutions.
Special Facilities and/or Equipment -
  1. Graphing Calculator.
  2. When taught hybrid: Four lecture hours per week in face-to-face contact and one hour per week using CCC Confer. Students need internet access.

Course Content (Body of knowledge) -
  1. Analyze sequences and series.
    1. Sequences
      1. Convergence and divergence of sequences
      2. Limits
      3. Convergence Theorems
    2. Series
      1. Geometric series
      2. Alternating series
      3. Power series
        1. Differentiation of power series
        2. Integration of power series
      4. Applications to other disciplines
  2. Apply convergence tests to sequences and series
    1. Convergence and divergence of series
    2. Tests for convergence
    3. Radius of convergence
    4. Interval of convergence
  3. Approximate functions using Taylor Polynomials
    1. Taylor polynomials
      1. Applications to other disciplines
      2. Error in Taylor polynomial approximations
    2. Taylor series
      1. Finding and using Taylor series
      2. Applications to other disciplines
  4. Investigate vectors, including dot and cross products.
    1. Displacement Vectors
    2. Vectors operations in two-space and three-space
    3. Dot Product
    4. Cross Product
    5. Triple products
    6. Projections
    7. Applications to other disciplines
  5. Demonstrate the ability to work with functions of more than one variable
    1. Functions of several variables
    2. Graphs of Functions of several variables
    3. Lines in 3-space
      1. Parametric Representations
      2. Vector Representations
    4. Contour Diagrams
    5. Cross-sections
    6. Level curves
    7. Linear Functions
      1. Rectangular equation of a plane
      2. Equation of a tangent plane at a point
    8. Limits and Continuity
      1. Limit of a function at a point
  6. Differentiate functions of more than one variable, including the directional derivative, the Gradient, and the Chain Rule.
    1. Partial Derivatives
      1. Definition
      2. Algebraic computation
    2. Tangent planes
    3. Linear approximations
      1. Applications to other disciplines
    4. Gradients
      1. Applications to other disciplines
    5. Directional Derivatives
      1. Applications to other disciplines
    6. The Chain Rule
      1. Applications to other disciplines
    7. Higher-Order Partial Derivatives
    8. Differentiability
      1. Partial derivatives
      2. Directional derivatives
      3. Differentiability of a surface at a point
  7. Optimize functions of more than one variable for both constrained and unconstrained optimization problems.
    1. Local Extrema
      1. Definitions
      2. Second Derivative Test
        1. Local maximuma
        2. Local minima
        3. Saddle points
    2. Optimization
    3. Constrained Optimization
      1. Lagrange Multipliers
    4. Applications to other disciplines
  8. 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
    1. Calculator/computer utilities for solving problems involving sequences and series
    2. Calculator/computer utilities for constructing graphs of functions and relations in 3-space, contour diagrams, and graphs of cross-sections.
    3. Calculator/computer programs for evaluating directional derivatives.
  9. Discuss mathematical problems and write solutions in accurate mathematical language and notation.
    1. Application problems from other disciplines
    2. Proper notation
  10. Interpret mathematical solutions.
    1. Explain the significance of solutions to application problems.
Methods of Evaluation -
  1. Written homework
  2. Quizzes & tests
  3. 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 -
  1. Lecture
  2. Discussion
  3. Cooperative learning exercises
 
Lab Content -
Not applicable.
 
Types and/or Examples of Required Reading, Writing and Outside of Class Assignments -
  1. 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.
  2. 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.
  3. 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.
  4. 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.