MATH 2212– Content Standards

Title: Calculus of one Variable II

Course Description: Prerequisite: Math 2211
Applications and techniques of integration; transcendental and trigonometric functions; polar coordinates; infinite sequences and series; indeterminate forms; improper Integrals.

Goals.

Course Content Standards. The following content standards are offered as guidelines for assessing student progress, judging the effectiveness of instructional programs, and developing curricular units. These standards describe what a student should be able to demonstrate at the completion of the course.

The content standards are labeled CS1 through CS10 and carry a further designation such as CS1A, CS4B, or both. The “A” designation indicates topics that are used explicitly in the course, but introduced in earlier courses. Although these topics are reinforced in Math 2212, they may be considered prerequisite material. The “B” designation indicates that topics are introduced in Math 2212.

CS 1A. Quantitative Reasoning

Students will use quantitative reasoning in problem solving situations including:

  • Geometric, symbolic, algebraic, and analytic representation and manipulation of quantitative information;
  • Pattern recognition.

CS 2A. The Real Number System

Students will use algebraic and order properties of the real number system and subsystems of the set of real numbers.

CS 3A. Functions.

Students will use and investigate functions and related concepts including:

  • Representations of functions using formulas, graphs, and parameters;
  • Operations on functions defined by arithmetic operations, composition, and inversion;
  • Types of elementary functions such as polynomial, rational, radical, absolute value, trigonometric, and piecewise-defined functions.
  • Properties of functions and their graphs involving monotonicity, extrema, concavity, and other salient features.

CS 4A. Limits and Continuity.

Students will demonstrate knowledge of and be able to use concepts and techniques related to limits and continuity including:

  • Performing analytic and graphical interpretations of concepts;
  • Evaluating limits;
  • Determining points of continuity/discontinuity of functions;
  • Applying properties of limits and continuity related to operations on functions.

CS 4B. Limits and Continuity.

Students will evaluate limits of indeterminate form.

CS 5A. Analytic Geometry.

Students will demonstrate knowledge of and be able to use analytic geometry concepts and related techniques including conic sections.

CS 5B. Analytic Geometry.

Students will demonstrate knowledge of and be able to use representations and transformations involving rectangular and polar coordinate systems.

CS 6A. Differentiation.

Students will demonstrate an understanding of the derivative at a point, derivative functions, and related concepts including:

  • Interpretation of the derivative at a point in terms of difference quotients, slopes of tangent lines and (instantaneous and average) rates of change;
  • The Mean Value Theorem for derivatives and related results;
  • Applying properties of differentiation related to elementary functions and operations on functions;
  • Application of the derivative to investigating properties of functions;
  • Implicit differentiation and differentials.

CS 6B. Differentiation.

Students will use the derivatives of exponential/logarithmic functions, and apply the technique of logarithmic differentiation.

CS 7A. Integration.

Students will demonstrate an understanding of integration and related concepts including:

  • The definite integral as an accumulation of small quantities;
  • The Fundamental Theorem of Calculus and antiderivatives;
  • The Mean Value Theorem for integrals;
  • Applying properties of integration related to elementary functions, operations on functions, and elementary substitutions;
  • Applications of integration in a variety of contexts.

CS 7B. Integration.

Students will demonstrate an understanding of integration and related concepts including:

  • Integrals involving exponential and logarithmic functions;
  • Integration by parts and other techniques of integration;
  • Evaluation of improper integrals.

CS 8B. Sequences and Series.

Students will demonstrate an understanding of sequences, series, and related concepts including:

  • Limits of sequences, sums of series, and radii of convergence;
  • Geometric series, alternating series, power series, and Taylor polynomials;
  • Tests of convergence and absolute convergence.

CS 9A. Applications.

While applying analytic, algebraic, geometric, and algorithmic techniques to solving applied problems students will:

  • Use appropriate technology;
  • Communicate how the problem is modeled by a mathematical formulation, and how to interpret the result of the mathematical analysis.

CS 10A. Mathematical Proof.

Students will demonstrate an understanding of mathematical proof and related concepts including:

  • Analysis of the logical structure of mathematical proofs and derivations;
  • Use contradictions and counter examples appropriately;
  • Use mathematical induction.
  • The rudiments of e,d- proofs.