MAT 295 comprises the first four credits of this eight-credit sequence. The development of scientific calculators with graphics capability has made possible some significant changes in the way this material is taught, and many colleges and universities are now incorporating them in their calculus sequence. In sections of calculus offered off-campus through Project Advance and in selected campus sections, design changes have been made to integrate graphing calculators into the learning process. The course design allows for some variations in pacing, as determined by site instructors and the supervising faculty.
Students who elect to enroll in the course sequence should have completed four years of high school mathematics.
The mathematical content of this program is typical of most traditional first-year calculus courses. The concepts of limit, continuity, derivative, anti-derivative, and definite integral are developed in the usual way, and they are then applied to the traditional collection of functions: polynomial, rational, trigonometric, and exponential, together with their inverses, compositions, and algebraic combinations.
The results are then applied to a wide variety of problems from geometry, physics, and other sciences. These include maximum and minimum problems, related rates, areas, volumes and surfaces of revolution, arc length, work, fluid pressure, velocity and acceleration, and exponential growth and decay.
Curve sketching is introduced at the very beginning and emphasized throughout, as we believe strongly that this is an important skill for any calculus student to acquire. Graphing calculators are a help here, since they contribute substantially to an understanding of the functions being sketched. They are only a help, however; the calculators are not used as a substitute for the skill itself.
During the course, students are introduced to progressively more sophisticated programming techniques for the calculator. They are shown how to write programs first for the evaluation and tabulation of functions and then for numerical evaluation of limits, derivatives, and roots (the last by Newton’s Method). Students then learn to do finite sums, Riemann sums, and finally numerical integration (by Simpson’s Rule). Programs are stored in the calculator as they are written and are used throughout the course.
- Review of Pre-Calculus: a) trigonometry, b) graphing of functions, c) special functions, including 1×1, sgn x, and [x].
- Limits (including one-sided and at ± ∞): a) definitions (intuitive and formal), b) techniques of evaluation.
- Continuity: a) definitions (at a point and on an interval), b) the Intermediate Value Theorem, c) use of IVT for numerical approximation of roots.
- Derivatives: a) definition, b) geometric and physical interpretation, c) formulas for xn, sin x, and cos x, d) product, quotient, and chain rules, e) implicit differentiation, f) higher derivatives, g) Rolle’s Theorem and the Mean Value Theorem for derivatives, h) differentials, i) anti-derivatives.
- Applications of Derivatives: a) increasing and decreasing functions, b) critical points and extreme values, c) max-min problems, d) related rate problems, e) concavity and inflection points, f) linear approximation, g) error estimates, h) Newton’s Method.
- Brief Review of Conic Sections.
- Definite Integral: a) definition (area under a curve, Riemann sum), b) average value of a function, c) Mean Value Theorem for integrals, d) Fundamental Theorem of Calculus (two versions), e) integrals of xn, sin x, and cos x, f) substitution in an integral.
- Applications of the Definite Integral: a) areas between curves, b) volumes and surface areas of solids of revolution, c) arc lengths of curves, d) work done by a force, e) force due to fluid pressure.