Departmental Course Outline for Advanced Mathematics

Note: This working document describes our coverage plans, but may be revised somewhat during the year. We hope you will find it useful as an overview of the course and as an approximation of what topics will be covered.

Minimum Topic Coverage

• Functions in general
  • Definition of a function (presented multiple ways including graphically: vertical line test)
  • Zeroes, roots, x-intercepts
  • Domain and range, maxima and minima
  • Properties of functions: increasing, decreasing, continuous, even, odd
  • Function operations, including composition
  • Inverse functions
  • Graphical transformations
  • Functions as mathematical models
  • Curve fitting using various kinds of regression
  • Parametric descriptions of curves
• Polynomial and rational functions
  • Linear and quadratic functions
  • Higher-degree polynomials and powers
  • (Optional for Lv1) Theorems about real zeros and factors: Remainder Theorem, Factor Theorem, Rational Root Theorem
  • (Optional for Lv1) Theorems about complex zeros and factors: Fundamental Theorem of Algebra, Complex Conjugate Theorem
  • Rational functions including their discontinuities
• Exponential and logarithmic functions
  • Exponential functions, including e^x
  • Logarithmic functions, including ln x
  • Properties of logarithms
  • Manipulating and solving exponential and logarithmic equations
  • Exponential models of growth and decay
  • Logarithmic models
• Trigonometric functions
  • Angle measurement in radians, revolutions, and degrees
  • Right triangle trigonometry, with applications
  • Definitions of six trigonometric functions as circular functions
  • Graphs of six trigonometric functions
  • Modeling of sinusoidal waves using sine and cosine functions (amplitude, period, etc.)
  • Inverses of sine, cosine, and tangent functions
  • Basic trigonometric identities: reciprocals/quotients, Pythagorean, co-function, odd/even
  • (Optional for Lv1) Sum, difference, and double-angle identities
  • Solving trigonometric equations
  • Proving trigonometric identities
  • Law of Sines and Law of Cosines, with applications
• Complex numbers, vectors, and polar equations
  • Complex number system and properties
  • (Optional for Lv1) Vectors in the plane, including magnitude and dot product operations
  • (Optional for Lv1) Parametric equations
  • (Optional for both levels) Coordinates, vectors, and parametric equations in 3-D space
  • (Optional for Lv1) Polar equations and their graphs
  • (Optional for Lv1) Complex numbers in polar form
  • (Optional for Lv1) Powers and roots of complex numbers (including DeMoivre’s Theorem)
• (Optional for Lv1) Matrices and linear systems
  • ("") Solving linear systems in two variables using substitution, elimination, and graphs
  • ("") Review of basic matrix algebra, including identity and inverse matrices
  • ("") Solving linear systems using matrix row operations into reduced row echelon form
  • ("") Solving linear systems using multiplication by an inverse matrix
  • ("") Matrices of linear transformations
• (Optional for Lv1) Analytic geometry
  • ("") Rectangular equations of conic sections
  • ("") Parametric equations of conic sections
  • ("") Focus and focus-directrix descriptions of conic sections
  • ("") b^2 - 4ac classification for conic sections
• (Optional for Hon) Counting and probability
  • ("") Basic combinatorics: counting principles
  • ("") Permutations and combinations
  • ("") Probability concept and calculations
  • (Optional for both levels) Binomial Theorem
• (Optional for Hon) Statistics
  • ("") Random variables and probability distributions
  • ("") Measures of center (mean, median, mode)
  • ("") Measures of spread (variance, standard deviation)
  • ("") Normal distributions
• (Optional for Lv1) Sequences and series
  • ("") Sequences, including arithmetic and geometric
  • ("") Series, finite and infinite, including algebraic and geometric
• Limits
  • Limit concept; descriptions of graphs using limits
  • (Optional for Lv1) Limit properties and computations

Calculator Skills

Expected prior to Pre-Calculus

• Function graphing
• Zooming and manually setting windows
• Function values: single values and table
• Tracing graphs
• Finding zeros and intersections
• Entering lists and making scatter plots
• Graphing best fit lines
• Fixing common errors (minus vs. negative, list dimension error, etc.)
• Matrix entry and basic operations

Taught in Pre-Calculus

• Function graphing
• Maxima and minima of functions
• Regression using various function types
• Parametric graphing
• (Optional for Lv1) Polar graphing
• (Optional for Lv1) More matrix-related features
• (Optional for Lv1) Operations on lists
• Combinatorial and statistical features

Textbook Usage

Hon Advanced Mathematics 253: Precalculus: Graphical, Numerical, Algebraic by Demana, Waits, et al. (Addison Wesley Longman, 4th ed., 2001). Students are asked to review Ch. P, Prerequisites, over the preceding summer. The course generally covers the contents of Chapters 1, 2.1-2.7, 3, 4.1-4.5, 4.7-4.8, 5, 6, 7.1-7.4, 8.1-8.4, 9.1-9.4, and 10.3.

Lv1 Advanced Mathematics 252: Precalculus: A Graphing Approach (Holt, Rinehart and Winston). The course generally covers the contents of Chapters 3-8, 9.1, 10.1-10.2, 13, and 14.1. Juniors in the course will also cover additional material from at least one of the following chapters: 9, 10, 11, and/or 14.