COURSE OUTLINE
MATH 11E
2013

2014
I.
REVIEW OF COMPLEX NUMBERS AND NEW FACTORIZATION
(~5 days)
Sum
and product
of the powers of
i
Theorems about complex conjugates:
and
Factorization:
Sum and difference of cubes:
Generalizations:
Sums of odd powers:
Differences of powers:
Special Factorizations: e.g., F
actor
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 Theor
em (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 F
unctions, 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 Sum
s of the Powers of Natural Numbers:
Using Induction to Prove Divisibility Theorems
Using Induction to Prove Theorems Involving Factorials: e.g.,
IV.
REVI
EW AND EXTENSION OF BINOMIAL THEOREM
(~8 days)
Sample Problems:
Find the coefficient of
in the expansion of
.
Find the constant term in the expansion
.
In the expansion of
, what is the coefficient of the
term?
How many terms are in the expansion of
?
V.
ARITHMETIC AND GEOMETRIC PROGRESSIONS
(~10 days)
Arithmetic Progressions; Arithmetic Means; Arithmetic Serie
s
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 Mathema
tics Competition (AMC) Problems Using Arithmetic and
Geometric Progressions
VI.
REVIEW AND EXTENSION OF EXPONENTIAL AND LOGARITHMIC
FUNCTIONS
(~ 7 days)
Theme: Graphing
, and other connections to transformations
Applications to T
opical Issues, such as nuclear waste and population growth
e
as a limit:
o
Related limits
:
,
VII.
POLAR COORDINATES
(~16 days)
Review of Trigonometric Identities and Equations
Writ
ing 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 Equatio
ns to Rectangular Form
Polar Graphs: vertical and horizontal lines, circles, three types of limacons, lemniscates,
roses; Symmetry Tests
Polar Distance Formula
VIII.
REVIEW OF
CONIC SECTIONS
(~3 days)
Formula for the Distance Between a Given Point
and a Given Line
on the
xy

coordinate plane
Area of an Ellipse and the Eccentricity of the Conic Sections
Optional Topic: Rotation of Axes (finding the equation of a parabola whose axis of
symmetry is diagonal)
IX.
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 F
unctions, Piece

wise
Functions, Absolute Value Functions
Limits of Functions (including trigonometric) and Sequences; Rules for Limits
X.
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
coordin
ate planes; e.g., cylinders such as
, ellipsoids such as
X
I.
VECTORS AND MATRICES
(~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
Matrix representation of vectors
Addition, multiplication of matrices
Determinants
Inverses of matrices
Systems of
equations using Cramer’s Rule
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 P
roduct 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 Equat
ions 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
Note
: The number of days listed for each unit do not include days for exams.
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