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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., 11! 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.