Download Math 11E

COURSE OUTLINE
MATH 11E
2014-2015
I.
REVIEW OF COMPLEX NUMBERS AND NEW FACTORIZATION (~5 days)
 Sum ( ) and product ( ) of the powers of i
 Theorems about complex conjugates: z1  z2  z1  z2 and z1  z2  z1  z2
 Factorization:
 Sum and difference of cubes:
 a 3  b3   a  b   a 2  ab  b 2 
 a 3  b3   a  b   a 2  ab  b 2 
 Generalizations:
 Sums of odd powers: a 5  b5   a  b   a 4  a 3b  a 2b 2  ab3  b 4 
 Differences of powers: a 5  b5   a  b   a 4  a 3b  a 2b 2  ab3  b 4 
 Special Factorizations: e.g., Factor a 4  4b 4
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 x3  4 x 2  3x  2 .
 Solving Equations in Quadratic Form: e.g., Find all six roots of x 6  7 x3  8 .
 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:
3x  2  x  5  19
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:
n n 1
 1  2  3  ...  n   2   n  N
 12  22  32  ...  n2 
n n 1 2 n 1
6
2
n n 1
2
 n N
  n N
 13  23  33  ...  n3  


 Using Induction to Prove Divisibility Theorems
 Using Induction to Prove Theorems Involving Factorials: e.g.,
11!  2  2!  3  3!  ...  n  n!   n  1!1
IV. REVIEW AND EXTENSION OF BINOMIAL THEOREM (~8 days)
Sample Problems:
 Find the coefficient of x8 in the expansion of  x 2  4 x  3  x  3 10 .
 Find the constant term in the expansion  2 x 2  4x  .
6
 In the expansion of  a  b  c  d  , what is the coefficient of the a 3b 2 c 6 d term?
12
 How many terms are in the expansion of  a  b  c  d  ?
12
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)
 Theme: Graphing y  f  x  a  , and other connections to transformations
n
n
1
 1
 e as a limit: Define e  lim 1    lim 
n 
 n  n k 0 k !
n
n
1 

 2
o Related limits: lim 1   , lim 1  
n 
 2n  n   n 
o Use Excel to compare how quickly each limit converges to e
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 3 and their traces in the coordinate planes and planes parallel to the
coordinate planes; e.g., cylinders such as x 2  z 2  36 , ellipsoids such as
x 2  4 y 2  9 z 2  144
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  x0 , y0  and a Given Line
Ax  By  C  0 in the xy-coordinate plane: d 

Ax0  By0  C

A2  B 2
 Vectors in 3-space; Standard Unit Vectors i , j , k ; 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, Ax  By  Cz  D  0 , 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  x0 , y0 , z0  and a Given Plane,
Ax  By  Cz  D  0 , 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 y  a ( x  r )( x  s )( x  t ) (which will
intersect the curve twice)
XIII. MATRICES AND DETERMINANTS (optional, not done in 2012-2013)
 Definition of a matrix and its dimensions;  aij  notation




Addition, multiplication of matrices
Determinants
Inverses of square matrices
Systems of equations using Gaussian Elimination and Cramer’s Rule
Note: The number of days listed for each unit may not be accurate and does not include days for
exams.