CALCULUS IV

                                                                  MA 291

 

Catalogue Description

            As a continuation of Calculus III, this course includes the study of three-dimensional spaces, vectors, vector-valued functions, partial derivatives, multiple integration, line integrals and Green’s theorem.

 

Prerequisites:  Math 290 Calculus III or equivalent

 

Goal

            A.  To continue the traditional material covered in the 2nd year of Calculus.

            B.   To sharpen the calculus background of students.

            C.  To prepare students for more advanced math courses such as Advanced Calculus, Real Analysis, Complex Variables, Numerical Analysis, and Graduate Studies in Math.

 

Procedures

            A.  Lecture/Discussion

            B.   Use of available software (True Basic, Derive) for graphing, etc.

            C.  Possible use of graphing calculators

            D.  Computer software supplements in Math Resource Center

 

Course Content

            A.  Three-Dimensional Space; Vectors

                  1.   Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces

                  2.   Vectors

                  3.   Dot Product; Projections

                  4.   Cross Product

                  5.   Parametric Equations of Lines

                  6.   Planes in 3-Space

                  7.   Quadric Surfaces

                  8.   Spherical and Cylindrical Coordinates

 

            B.  Vector-Valued Functions

                  1.   Introduction to Vector-Valued Functions

                  2.   Limits and Derivatives of Vector-Valued Functions

                  3.   Integration of Vector-Valued Functions, Arc Length

                  4.   Unit Tangent and Normal Vectors

                  5.   Curvature

 

            C.  Partial Derivatives

                  1.   Functions of Two or More Variables

                  2.   Limits and Continuity

                  3.   Partial Derivatives

                  4.   Differentiability and Chain Rules for Functions of Two Variables

                  5.   Tangent Planes; Total Differentials for Functions of Two Variables

                  6.   Directional Derivatives and Gradients for Functions of Two Variables

                  7.   Differentiability, Directional Derivatives, and Gradients for Functions of Three Variables

                  8.   Functions of n Variables; More on the Chain Rule

                  9.   Maxima and Minima of Functions of Two Variables

               10.    Lagrange Multipliers

 

            D.  Multiple Integrals

                  1.   Double Integrals

                  2.   Double Integrals over Nonrectangular Regions

                  3.   Double Integrals in Polar Coordinates

                  4.   Surface Area

                  5.   Triple Integrals

                  6.   Centroids, Center of Gravity, Theorem of Pappus

                  7.   Triple Integrals in Cylindrical and Spherical Coordinates

 

            E.   Topics in Vector Calculus

                  1.   Line Integrals

                  2.   Line Integrals Independent of Path

                  3.   Green's Theorem

 

Evaluation Methods at the discretion of the Instructor

                  1.   Daily homework assignments.  Students are expected to do their assignments and be prepared to discuss the problems in class.

2.     Special take-home problems.  Problems will be counted collectively as an additional test.

3.     Quizzes.  Quizzes will be counted collectively as an additional test.  Quizzes will be given when necessary.

4.     Tests.  Tests will be given every three to four weeks.  The results will be discussed in class.

5.     Comprehensive final exam.  This will test whether the student has finally learned to do the problems which are representative of the course and to what extent he possesses these skills at the conclusion of the course.

 

Bibliography

 

Required Text:          

 

Larson, Hostetler & Edwards, Calculus, 6th Ed., with Expanded Solutions Manual, Houghton-Mifflin Publishing.

                                                                                                                       

Coughlin/Zitarelli, Brief Calculus with Applications, Saunders Pub., Phila., Pa., 1990.

 

Lial, Margaret L. & Miller, Charles D., Finite Math & Calculus with Applications, 3rd ed., Scott, Foremen & Co.,  1989

 

Cannon, Raymond & Williams, Gareth, Calculus Management, Social and Life Sciences, W.C. Brown Pub., Dubuque, Ia., 1988.

 

Software

 

Andrews, Richard, Student Edition of Mathcad, Addison Wesley Pub., Reading, Mass.,  1988.

 

Burgmeier, Kost, Exploration Programs in Calculus, Prentice Hall, Englewood Cliffs, N.J.  1985.

 

Derive A Mathematical Assistant, Version 1.62, Soft Warehouse Inc., Honolulu, Hawaii, 1988.

 

Finney, Hoffman, Schwartz, Wilde, The Calculus Toolkit, Addison Wesley Pub., Reading, Mass., 1986

 

Flanders, Harley, Microcalc 4.01, Software for Teaching/Learning Calculus, Ann Arbor, Mi., 1987.

 

Kemeny, Kurtz, Calculus, Tru Basic Inc., New Hampshire, 1988.

 

Meyers, Roy, Surface Plotter, Elm Software, Leechburg, Pa.,1987.

 

Rowell, James, Mathematical Modeling with Mathcad Explorations in the Calculus and Beyond, Addison Wesley Pub.,    Co., Inc., 1990.

 

Waits, Demana, Master Grapher and 3D Graphics, Addison Wesley Pub., Co., Inc., Reading, Mass., 1988.

 

 

 

 

 

 

 

 

 

                                                  


 

SAMPLE WEEKLY PROGRESS

 

 

Text:   Anton, Calculus with Analytic Geometry, John Wiley & Sons, New York, N.Y.  1988

 

Calculus IV

 

Week

 

            1.   Sec. 14.1, 14.2 (2 hours)

            2.   Sec. 14.3, 14.4, 14.5

            3.   Sec. 14.6, 14.7, 14.8

            4.   Sec. 15.1, 15.2, 15.3

            5.   Review, Exam #1, Sec. 15.4

            6.   Sec. 15.5, 16.1, 16.2

            7.   Sec. 16.3, 16.4 (2 hours)

            8.   Sec. 16.5, 16.6 (2 hours)

            9.   Sec. 16.7, Review, Exam #2

         10.    Sec. 16.8, 16.9 (2 hours)

         11.    Sec. 16.10, 17.1, 17.2

         12.    Sec. 17.2, 17.3, 17.4

         13.    Review, Exam #3, Sec. 17.5

         14.    Sec. 17.6 (optional), 17.7

         15.    Sec. 18.1, 18.2, 18.3

 

                  In actual practice, sections may take shorter or longer than time alloted.  A review lesson can be used for catching up, going ahead, etc.  If pressed for time, Chapter 15 could be deleted.  Sections 14.1, 14.2, 14.3, & 14.4 might be gone over in Calculus III, so these sections might be covered more quickly in this course.