Rev. 5/04

NVCC COLLEGE-WIDE COURSE CONTENT SUMMARY

MTH 166 - PRECALCULUS WITH TRIGONOMETRY (5 CR.)

COURSE DESCRIPTION

Presents college algebra, analytic geometry, trigonometry, and algebraic, exponential, and logarithmic functions. (Credit cannot be awarded for both MTH 163 and MTH 166.)  Lecture 5 hours per week.

GENERAL COURSE PURPOSE

The general purpose of this one-semester course is to prepare the student for a course in a rigorous calculus sequence by providing them with the necessary competencies in algebra, functions (including polynomial, rational, exponential, logarithmic, and trigonometric functions), and analytic geometry, as well as competence in using a graphing utility. At NVCC, this course will prepare the student for the calculus sequence, MTH 173-174 – “Calculus with Analytic Geometry I-II.”

ENTRY LEVEL COMPETENCIES

The student should have completed three units of high school mathematics, including two units of algebra and one unit of geometry. The student must also obtain a satisfactory score on an entry-level skills examination, which tests for competency in the following areas:

A. Arithmetic with rational numbers (including signed numbers, fractions and mixed numerals,

     and decimal numerals)

B. Solutions of linear and quadratic equations

C. Operations with polynomials, including factoring

D. Operations with rational expressions, including simplifying

E. Use of positive integer exponents

F. Basic geometric concepts

It is desirable, but not essential, that the student have done some work with right triangle trigonometry.

COURSE OBJECTIVES

As a result of the learning experience in this course, the student should be able to:

A. Solve problems involving equations, inequalities, and systems of equations

B. Operate on functions (addition, multiplication, composition, and inverses)

C. Graph linear, quadratic, rational, exponential, logarithmic and trigonometric functions

D. Graph conic sections

E. Factor polynomials and find zeros of polynomials

F. Evaluate trigonometric and inverse trigonometric functions

G. Use trigonometric formulas to prove trigonometric identities, solve triangles and 

     trigonometric equations

H. Use a graphing utility as an aid to problem solving

MAJOR TOPICS TO BE INCLUDED

Optional Review of Algebra

A. Polynomials

B. Factoring

C. Rational Expressions

D. Rules of Exponents for positive integer exponents

E. Solution of linear equations

F. Quadratic Formula and Quadratic-type equations

G. Use of theorem:  Solutions of p=q are a subset of the solutions of p2=q2

           

Required Topics

A. Exponents and radicals

1. Definitions

                                    a. the zero exponent

                                    b. negative integer exponents

                                    c. rational exponents

                        2. Rules for rational exponents

                                    a. simplifying radicals

                                    b. rationalizing numerator and denominator

            B. Inequalities and Absolute Value

                        1. Inequalities

                                    a. definition

                                    b. interval notation

                                    c. graphing on the number line

                                    d. solution of linear, quadratic, and rational inequalities

                        2. Absolute Value

                                    a. definition

                                    b. solution of equations and inequalities containing absolute values

            C. Complex Numbers

                        1. Definition

                        2. Arithmetic operations

            D. Functions

                        1. Definitions, including domain and range

                        2. Operations

                                    a. arithmetic

                                    b. composition

                        3. Inverses with respect to composition

                        4. Difference quotient

                        5. Average rate of change of nonlinear functions

            E. Polynomial Functions

                        1. Definition

                        2. Graphs, including transformations and symmetry

                        3. Remainder Theorem and Factor Theorem

                        4. Division of Polynomials

                        5. Fundamental Theorem of Algebra

                        6. Finding zeros of polynomial functions with integral coefficients

           

F. Rational Functions

                        1. Definitions

                        2. Graphs (including asymptotes)

            G. Exponential and Logarithmic Functions

                        1. Definitions

                        2. Graphs

                        3. Finding common and natural logarithms and antilogarithms

                        4. Solution of equations involving exponentials and/or logarithms

                        5. Growth and Decay Problems and other applications

            H. Analytic Geometry

                        1. Basic Concepts

                                    a. distance between two points in the plane

                                    b. midpoint of a line segment

                        2. Linear Functions

                                    a. slope

                                    b. intercepts

                                    c. graphs

                                    d. parallel and perpendicular lines

                                    e. equation of a line

                        3. Conic sections of the form Ax2 + By2 + Cx + Dy + E = 0

                                    a. parabolas

                                                (1) finding vertex

                                                (2) graphs

                                    b. circles

                                                (1) finding center and radius by completing the square

                                                (2) graphs

                                    c. ellipses

                                                (1) finding axes and center

                                                (2) graph

                                    d. hyperbolas

                                                (1) find axes, center, and asymptotes

                                                (2) graph

            I. Solving systems of equations

            J. Trigonometric Functions

                        1. Unit Circle

                        2. Circular Functions

                                    a. definitions

                                    b. simple properties (Pythagorean, Reciprocal, Complementary)

                        3. Formulas for f(x + y), f(2x), f(x/2)

                        4. Graphs

                        5. Inverses

                        6. Proving trigonometric identities

                        7. Solution of trigonometric equations

K. Applications of Trigonometric Functions to triangles

                        1. Trigonometric functions for right triangles

                        2. Solution of right triangles

                        3. Law of Sines

                        4. Law of Cosines

EXTRA TOPICS (optional)

            A. Matrices and Determinants

                        1. Definitions of matrices and determinants

                        2. Operations on matrices

                                    a. addition

                                    b. transpose

                                    c. scalar multiplication

                                    d. matrix multiplication

                        3. Multiplicative inverses

                        4. Solving systems of linear equations

                                    a. Cramer’s Rule

                                    b. row reduction of augmented matrices

                                    c. using the multiplicative inverse matrix

            B. Sequences and series

                        1. S (sigma) notation

                        2. arithmetic

                        3. geometric

            C. Binomial Theorem

            D. Mathematical Induction

            E. Regression using a graphing utility

            F. DeMoivre’s Theorem