National University

MTH 215, or placement evaluation.

(Cross listed and equivalent to CSC208)
An introduction to limits and continuity. Examines differentiation and integration concepts with applications to related rates, curve sketching, engineering optimization problems and business applications. The fundamental theorem of calculus is presented with related techniques for numerical approximation. Looks at the ideas and contributions of Newton, Leibniz, Lagrange, Maria Agnesi and Riemann. Graphing calculator is required.

At the conclusion of this course, students will, by written examination, demonstrate sufficient competency so that each goal of this syllabus is realized. To this end, the student shall:

• Successfully use techniques for evaluating the limits of algebraic and trigonometric functions. As the independent variable approaches finite values or grows without bound. Invoke the definition of continuity at a point to so test prescribed functions. Be able to graphically depict three common problems that lead to discontinuity at a point. Demonstrate familiarity with, and modest algebraic capability to apply, rigorous delta-epsilon structures of mathematical nearness.
• Clearly demonstrate facility with fundamental differentiation formulas and rules. Be fully capable of employing implicit differentiation and the chain rule to elementary related-rate problems.
• Give written evidence of successful application to curve sketching, with extremal tests by first and second derivatives. Successfully recognize and perform applications of the derivative to solve optimization problems, as taken from several disciplines including business, biology, medicine, and the physical sciences.
• Demonstrate facility with each of the presented techniques of integration to derive antiderivatives. Be able to write down, and employ, the Fundamental Theorem of Calculus to an array of elementary numerical applications for each of the integrand types specified in the corresponding goal. Be able to find areas bounded by elementary functions. The student will demonstrate by simple example an understanding that continuity is sufficient, but not necessary, for Riemann integrability.

Textbook:
http://www.mbsdirect.net/national/

Course Goals:
A principal objective of the course is to instill in the student an appreciation of the underlying thoughts and historical development of the calculus, as well as a solid understanding of the power of its techniques. This course sets the foundations in differentiation and integration, coupled with their associated geometric and physical interpretations. To attain this objective, the following course goals have been established in which the student shall:

• Develop an understanding of the limit concept. The student will gain a foundation in functional definitions, and now such definitions are used to yield limits and test continuity under the concepts of nearness, both from a heuristic and mathematically rigorous approach.
• Extend limit concept to evaluate difference quotients of simple algebraic and trigonometric functions. Appreciate Leibniz geometric relevance of the difference quotient leading to instantaneous rates of change The definition of derivative of a function at A Point Will Be Presented, As Will Its Relationship with continuity at that point. Develop fundamental differentiation rules, formulas, and properties. Apply them.
• Make use of fundamental differentiation rules and properties as they apply to real-world problems of extrema, optimization, and/or related rates. Calculate near-zero values by Newton's Method.
• Employ the principal ideas and techniques of the indefinite integral to find antiderivatives leading to applications of the Fundamental Theorem of Calculus and the Mean Value Theorem. Integrands will encompass algebraic, trigonometric, and logarithmic functions.Be introduced to the uniquely imaginative process of geometrically suggested approximations leading to the historical significance of integrability as devised by Fredrich Riemann.

Course Content:
This course includes the following critical topics:

• Limits and continuity.
• Differentiation and integration concepts.
• Applications to related rates.
• Curve sketching.
• Engineering optimization problems.
• Business applications.
• Fundamental Theorem of Calculus.
• Numerical approximation.

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.