I. REVIEW of COMPLEX NUMBERS ANDNEW FACTORIZATION ( 5 Days)

COURSE OUTLINE

MATH 11E

2014-2015

I.REVIEW OF COMPLEX NUMBERS ANDNEW FACTORIZATION (~5 days)

Sum and difference of cubes:

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Generalizations:

Sums of odd powers:

Differences of powers:

Special Factorizations: e.g., Factor

II.THEORY OF ALGEBRA (~25 days)

Division Algorithm; Remainder and Factor Theorems (including proofs)
Synthetic Division; application in finding the roots of an equation
Fundamental Theorem of Algebra
Complex Conjugate Theorem (including proof); Square Root Conjugate Theorem
Rational Roots Theorem (including proof), and applications
Descartes’ Rule of Signs
Using the Location Principle
Upper and Lower Bounds for Roots of a Polynomial Equation
Theorems on the relation between the roots of a polynomial equation and its coefficients
Challenge Problems using the above coefficients-roots theorems: e.g., Find a polynomial equation with integral coefficients in standard form whose roots are the squares of the roots of .
Solving Equations in Quadratic Form: e.g., Find all six roots of .
Introduction to the Graphing Calculator and Graphmatica
Graphing Polynomial Functions; significance of tangent and inflection points
Graphing Rational Functions, stressing horizontal, vertical and slant asymptotes – using a limit approach
Solving Linear Quotient Equations and Inequalities, both graphically and algebraically (representing solution sets in interval notation)
Solving Absolute Value Equations and Inequalities: e.g., Solve for all values of x:

III.PROOF BY MATHEMATICAL INDUCTION (~7 days)

Introduction to Induction Proofs using the College Algebra video (Sol Garfunkel)
Using Induction to Prove Theorems About the Sums of the Powers of Natural Numbers:

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IV.REVIEW AND EXTENSION OF BINOMIAL THEOREM (~8 days)

Sample Problems:

V.ARITHMETIC AND GEOMETRIC PROGRESSIONS (~10 days)

Arithmetic Progressions; Arithmetic Means; Arithmetic Series
Special Arithmetic Series (sum of the even natural numbers; sum of the odd natural numbers)
Solving Verbal Problems Involving Arithmetic Progressions
Geometric Progressions; Geometric Means; Geometric Series (finite and infinite)
Solving American Mathematics Competition (AMC) Problems Using Arithmetic and Geometric Progressions

VI.REVIEW AND EXTENSION OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS (~ 7 days)

VII.POLAR COORDINATES (~16 days)

Review of Trigonometric Identities and Equations
Writing a Point on the Coordinate Plane in Both Rectangular and Polar Form
Converting Complex Numbers from Rectangular to Polar Form and vice versa
De Moivre’s Theorem, and how we use it to find the powers and roots of a complex number
Converting Polar Equations to Rectangular Form
Polar Graphs: vertical and horizontal lines, circles, three types of limacons, lemniscates, roses; Symmetry Tests
Polar Distance Formula
Conics in polar coordinates; eccentricity of parabolas, ellipses, and hyperbolas

VIII. PARAMETRIC EQUATIONS AND FUNCTIONS (~12 days)

Graphing Parametric Equations; Eliminating the Parameter
Finding the Domain and Range of a Function
Composition of Functions; Inverse Functions
Special Functions: Greatest Integer Function, Even and Odd Functions, Piece-wise Functions, Absolute Value Functions
Limits of Functions (including trigonometric) and Sequences; Rules for Limits

IX.THREE-DIMENSIONAL SPACE (~7 days)

Solving Solid Geometry Problems
Coordinates in Space; Finding the Distance Between Points in Space; Reflections in the plane, in the x, y, and z axes, and in the origin; Equation of a Sphere Given its Center and Radius
Basic surfaces in and their traces in the coordinate planes and planes parallel to the coordinate planes; e.g., cylinders such as , ellipsoids such as

X. VECTORS (~28 days)

Adding and Subtracting Vectors in 2-space; Finding Resultants; Solving Physics Problems
Find the Direction Angle of a Vector in 2-space
Using the Dot Product to Find the Angle Between Two Vectors
Writing a Vector as a Linear Combination of Basis Vectors
Formula for the Distance Between a Given Point and a Given Line in the xy-coordinate plane:
Vectors in 3-space; Standard Unit Vectors ; Writing a Vector in 3-space as a Linear Combination of Basis Vectors
Finding the Three Direction Angles of a Vector in 3-space
Dot Product of Two Vectors; Orthogonal Vectors
Finding the Equation of a Plane, , given an orthogonal vector and one point on the plane, or given three points on the plane
Finding the Angle Between Two Planes
Writing the Parametric Equations of a Line in Space
Finding the Equation of the Line of Intersection of Two Planes
Finding the Angle that a Line Makes with a Plane
Formula for the Distance Between a Given Point and a Given Plane, , and its applications
Distance Between Two Parallel Planes; Distance from a Point to a Plane
(if time) Cross product and its applications

XII. TANGENT LINES TO CURVES (~3 DAYS)

Tangent lines to circles (review)
Precalculus approach to tangent lines to parabolas based on the fact that the parabola and the tangent line intersect exactly once (and the tangent line is not vertical)
Finding the horizontal tangent lines to a cubic (which will intersect the curve twice)

XIII. MATRICES AND DETERMINANTS (optional, not done in 2012-2013)

Note: The number of days listed for each unit may not be accurate and does not include days for exams.