# Math 11E - Math123.net

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10 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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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.