National University

1. Understanding that a sequence is a function on the integers into R1. The notion of a limit of a sequence. Pattern recognition for sequences, monotonic sequences, and convergence of monotone and bounded sequences. The Fibonacci sequence.
2. Partial sums of a series as a sequence. Convergent and divergent series, geometric, harmonic; and the Nth-Term Test as a necessary condition for convergence of a series.
3. The integral and p-series as a sequence. Deriving the p-series test, the ration and root tests, direct comparison tests. The alternating series test will be derived by the student.
4. Taylor polynomials and approximations to sufficiently smooth functions. Be able to derive the Taylor polynomials of sufficient accuracy as to approximate within a proscribed degree of accuracy.
5. Power series definitions, development and radius of convergence. Differentiation and integration of power series (converging uniformly on a contact set). Understand Ramanujan's contribution to series approximation of p.
6. Classification of ordinary and partial differential equations. Recognize a solution (general and singular, as well as a particular solution). Be able to find particular solutions matching pre-stipulated initial value conditions. Recognize ode applicable to the separation of variable methodology. Be able to perform that separability technique, and demonstrate sufficient skills in integration techniques so as to develop confidence and self-sufficiency in its approach.
7. Recognize and select an appropriate methodology to solve homogeneous ode of the first order. Exponential growth and decay applications. Exactness of first-order equations; test and solution techniques.
8. Generating integrating factors for the 2nd-order equation, if they exist. Recognizing and solving the Bernoulli nonlinear differential equation. Be able to solve generalized, and idealized, differential equations emerging from mechanical and electrical engineering applications.
9. Recognize and solve 2nd-order homogeneous linear equations. Know the concept of linear independence of functions over a subset of R1. Know the definition and conceptual considerations regarding basis of a solution space; know that all elements of that solution space can be written as linear combinations of that basis.
10. Recognize and solve 2nd-order homogeneous, linear, ordinary differential equations with forcing function. Know the methodology of undetermined coefficients. Be able to form a Wronskian from the basis functions for the corresponding homogenous equations and be able to apply it to the variation of parameters technique.
11. Demonstrate proficiency in graphing-calculator usage by investigation of Euler's technique for numerical approximation of simple initial-value problems. Recognize and perform the enhanced Euler's technique, and be able to compare rates of convergence. Be familiar with Milne's method, Runge-Kutta. Understand predictor-corrector techniques of numerical solutions of non-linear initial-value problems of ordinary differential equations.

Course Goals:
This course concludes the sequence in Elementary Calculus. Its subject matter is selected with the intent to obtain maximum utilization of student skills cultivated in the preceding 3 of this 4-course sequence. It begins with an overview of sequences and series, to include broadly employed tests for convergence: integral, p-series, comparison, alternating, ratio, and root tests. The concept of sequence broadened based on the ideas embodied in a bounded, infinite set of real numbers will be studied. Although the notion of limit point is not introduced, the ground work is presented for conceptual understanding of the Bolzano-Weierstrauss Theorem, addressed subsequently in Advanced Calculus. Taylor Polynomials and the concept of approximations to functions defined and continuous on a closed and bounded subset of R1 is explored and utilized. (The polynomials over R1 are complete in the uniform topology). Power series representations of functions follow. Taylor and Maclaurin series expansions, with
understanding of intervals of convergence conclude the initial phase of Calculus IV. The second phase of Calculus IV embraces a look at concepts and methodology of solutions to ordinary differential equations of the first order is given, which calls upon the student a full review of the several earlier-established techniques of integration. The concept of exactness, and integrating factors, for first order ode is follows an overview of linear, first order ode.

Course Content
Overview of sequences and series, to include broadly employed tests for convergence:

• Integral.
• P-series.
• Comparison.
• Alternating.
• Ratio.
• Root tests.

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.