National University

• Clearly demonstrate an ability to employ a vector function for which an inverse can and does exist. The student will demonstrate capability to produce such inverse functions, when they exist. Show evidence of solid grasp of the differentiation and integration formulas and rules associated with inverse functions.
• Show ability to formulate the functional representations of incremental components of area, volume, surface-area, work, fluid pressure, and be proficient in carrying these through to the integrand of the appropriate definite integral. The student will be successful in retrieving the antiderivative, and evaluating it, during the computation of such integrals.
• Derive antiderivatives by knowledgeable appeal to the more sophisticated methods of integration by parts, recursion, trigonometric substitution, and partial fractions. The student will demonstrate ability to set up numerical approximations to definite integrals for which closed-form solutions may be impractical to attain. The student will be able to recognize such circumstances.
• Provide evidence of recognition and understanding of behavior of infinite sequences and series. Demonstrate techniques for testing their absolute or conditional convergence, if any. The student will recognize the appropriate logarithmic or trigonometric function to which the power series converges. Similarly, for each such function, the student will be able to generate associated partial sequences of the Maclaurin and Taylor series.
• Compute with conic sections properties presented in algebraic, polar, and other parametric form. Lengths of arc, surface areas, and an understanding of applications to specific problems will be demonstrated.

Course Goals:
This course extends development of topics within Calculus I to include inverse functions, with applications to computing volume, surface area, work, centroids, and moments of inertia. Special techniques for improper integrals, conics and parametric representations are introduced. Includes techniques of integration of logarithmic, exponential, and trigonometric functions.

A principal objective of the course is to extend for the student an appreciation of the underlying thoughts and historical developed in Calculus I, as well as a solid understanding of the power of its techniques.  This course enlarges the foundations in differentiation and integration, coupled with their associated physical science applications and interpretations.

To attain this objective, the following course goals have been established in which the student shall:

• Be provided sound foundation in the existence of inverse functions of basic algebraic, trigonometric, and logarithmic expressions. Differentiation and integration of inverse functions is developed.
• Be introduced to techniques of building Riemann sums for approximation of volumes, surface areas, arc length, work, and fluid pressure. Definite integrals will extend the approximations to computational capability. Moments of inertia and centroids with intuitive insight to their applications is presented.
• Extend simpler integration techniques to more sophisticated methods of integration such as integration by parts, recursion formulas, trigonometric substitution, and partial fractions. An introduction to numerical approximation techniques and improper integrals is provided.
• Be aware of the definitions, formulation, and properties of infinite series and sequences taken from the partial sums of each series. Integral tests, and other conditions for convergence will be presented, as will the development of Maclaurin and Taylor power series.
• Become proficient in earlier-introduced, two-space conic sections; their algebraic and polar representation, properties, and computational applications. Three-dimensional surfaces in parametric form sets the stage for follow-on the vector geometry.

Course Content
This course includes the critical topics regarding differentiation and integration concepts of:

• The natural Logarithm.
• Exponential functions.
• Inverse trigonometric functions.
• Applications to volumes of revolution, work, and arc length.
• Improper integrals.

Also the student will learn appropriate logarithmic or trigonometric functions to which the power series converges. For each such function, the student will be able to generate associated partial sequences of the Maclaurin and Taylor series. Compute with conic sections properties presented in algebraic, polar, and other parametric form. Lengths of arc, surface areas, and an understanding of applications to specific problems will be studied.

Incomplete A grade given at the discretion of the instructor when a student who has completed at least two-thirds of the course class sessions and is unable to complete the requirements of the course because of uncontrollable and unforeseen circumstances. The student must convey these circumstances (preferably in writing) to the instructor prior to the final day of the course. If an instructor decides that an "Incomplete" is warranted, the instructor must convey the conditions for removal of the "Incomplete" to the student in writing. A copy must also be placed on file with the Office of the Registrar until the "Incomplete" is removed or the time limit for removal has passed. An "Incomplete" is not assigned when the only way the student could make up the work would be to attend a major portion of the class when next offered.

An "I" that is not removed within the stipulated time becomes an "F." No grade points are assigned. The "F" is calculated in the grade point average.

Withdrawal Signifies that a student has withdrawn from a course after beginning the third class session. Students who wish to withdraw must notify their admissions advisor before the beginning of the sixth class session in the case of graduate courses, or before the seventh class session in the case of undergraduate courses. Instructors are not authorized to issue a "W" grade.