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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.