ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates


 Jody Ross
 4 years ago
 Views:
Transcription
1 Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates Once the value α + β and αβ have been found, new quadratic equations can be formed with roots : Roots α and β α 3 and β 3 a and b Sum of roots (a + b) ab Sum of roots (a + b) 3 3ab(a + b) Sum of roots a + b ab Product of roots (ab) Product of roots (ab) 3 Product of roots ab The new equation becomes x (sum of new roots)x + (product of new roots) = 0 The questions often ask for integer coefficients! Don t forget the = 0 Example The roots of the quadratic equation 3x + 4x = 0 are α and β. Determine a quadratic equation with integer coefficients which has roots a 3 b and ab 3 Step : a + b = 4 3 ab = 3 Step : Sum of new roots a 3 b + ab 3 = ab(a + b ) = ((a + b) 3 ab) = 3 = 7 æ ç 6 è 9 + ö 3 ø
2 Step 3 : Product of roots Step 4 : Form the new equation a 3 b ab 3 = a 4 b 4 = (ab) 4 = 8 x + x = 0 8x + 66x + = 0. Summation of Series These are given in the formula booklet REMEMBER : n å = n r = S 5r = 5 S r Always multiply brackets before attempting to evaluate summations of series Look carefully at the limits for the summation 0 å = å 0  å 6 n å = å n  å n r =7 r = r = r =n+ r = r = Summation of ODD / EVEN numbers Example : Find the sum of the odd square numbers from to 49 Sum of odd square numbers = Sum of all square numbers Sum of even square numbers Sum of even square numbers = = ( ) 4 =4å r r = 49 Sum of odd numbers between and 49 is å r r å r = = æ ç è 6 = = 085 ö 4 æ ø è ç ö ø
3 3. Matrices ORDER a ù ë b ú û a ë c b ù d û ú x x Addition and Subtraction must have the same order a ë c b ù d û ú ± e ë g f ù h ú = a ± e û ë c ± g b ± f ù d ± h û ú Multiplication a ù 3 ë ú b û = ë 3a ù 3b û ú 3 ë 5 4 ù û ú ë 3 0 ù 0 ú = 3 x + 4 x 3 û ë 5 x + x 3 3 x x 0 ù 5 x 0 + x 0 ú = 5 û ë 30 ù 50 û ú NB : Order matters Do not assume that AB = BA Do not assume that A B = (AB)(A+B) Identity Matrix I = 0 ù AI = IA = A ú ë 0 û 4. Transformations Make sure you know the exact trig ratios Angle θ sin θ cos θ tan θ ½ ½ Undefined To calculate the coordinates of a point after a transformation Multiply the Transformation Matrix by the coordinate Find the position of point (,) after a stretch of Scale factor 5 parallel to the xaxis ë 5 0 ù 0 ú û ë ù û ú = 0 ë ú ù û (0,)
4 To Identify a transformation from its matrix Consider the points (,0) and (0,) ë ù û ú (,0) ð (4,0) (0,) ð (0,) Stretch Scale factor 4 parallel to the xaxis and scale factor parallel to the yaxis Standard Transformations REFLECTIONS 0 ù 0 ù in the yaxis in the xaxis ú ú ë 0 û ë 0 û Reflection in y = x 0 ù Reflection in the line y= (tan θ)x ú ë 0 û ENLARGEMENT k ë 0 0 ù k û ú Scale factor k Centre (0,0) STRETCH cos q ë sin q If all elements have the same magnitude then look at θ = 45 (reflection in y = (tan.5)x ) as one of the transformations sin q ù cos q û ú In the formula booklet a 0 ù ë 0 b û ú Scale factor a parallel to the xaxis Scale factor b parallel to the yaxis In the formula booklet ROTATION cos q ë sin q sinq ù ú cosq û cos q sinq ë si n q co sq û ú ù Rotation through θ anticlockwise about origin (0,0) Rotation through θ Clockwise about origin (0,0) If all elements have the same magnitude then a rotation through 45 is likely to be one of the transformations (usually the second) ORDER MATTERS!!!! make sure you multiply the matrices in the correct order A figure is transformed by M followed by M Multiply M M
5 5. Graphs of Rational Functions Linear numerator and linear denominator y = 4x 8 x + 3 horizontal asymptote vertical asymptote y = (x 3)(x 5) (x + )(x + ) distinct linear factors in the denominator quadratic numerator vertical asymptotes horizontal asymptote The curve will usually cross the horizontal asymptote distinct linear factors in the denominator linear numerator vertical asymptotes y = x 9 3x x + 6 horizontal asymptote horizontal asymptote is y = 0 y = (x 3)(x + 3) (x ) Quadratic numerator quadratic denominator with equal factors vertical asymptote horizontal asymptote y = x + x 3 x + x + 6 Quadratic numerator with no real roots for denominator (irreducible) The curve does not have a vertical asymptote Vertical Asymptotes Solve denominator = 0 to find x = a, x = b etc Horizontal Asymptotes multiply out any brackets look for highest power of x in the denominator and divide all terms by this as x goes to infinity majority of terms will disappear to leave either y = 0 or y = a To find stationary points k = x + x 3 rearrange to form a quadratic ax + b x x + c = 0 + x + 6 * b 4ac < 0 b 4ac = 0 b 4ac > 0 the line(s) y = k stationary point(s) the line(s) y = k do not intersect occur when y = k intersect the curve the curve subs into * to find x subs into * to find x coordinate coordinate
6 INEQUALITIES The questions are unlikely to lead to simple or single solutions such as x > 5 so Sketch the graph (often done already in a previous part of the question) Solve the inequality (x + )(x + 4) (x )(x ) The shaded area is where y < < So the solution is x < 0, < x <, x > 6. Conics and transformations You must learn the standard equations and the key features of each graph type Mark on relevant coordinates on any sketch graph Parabola Standard equations are given in the formula booklet but NOT graphs Ellipse x y a + y b = Hyperbola x a y b = = 4ax æ ç x ö è a ø + æ ç x ö è a ø Rectangular Hyperbola xy = c æ è ç y ö b ø = æ è ç y ö b ø = You may need to complete the square x 4x + y 6y = (x ) 4 + (y 3) 9 = (x ) + (y 3) =
7 Transformations Translation a ù Replace x with (x a) Circle radius centre (,.3) ú ë b û Replace y with (y  b) (x  ) + (y  3) = Reflection in the line y = x Replace x with y and vice versa Stretch Parallel to the xaxis scale factor a Stretch Parallel to the yaxis scale factor b Replace x with a x Replace y with b y Describe a geometrical transformation that maps the curve y =8x onto the curve y =8x6 x has been replaced by (x) to give y ù = 8(x) Translation ë 0 û ú 7. Complex Numbers real z = a + ib imaginary i = i = Addition and Subtraction ( + 3i) + (5 i) = 7 + i (add/subtract real part then imaginary part) Multiplication  multiply out the same way you would (x)(x+4) ( 3i)(6 + i) = + 4i 8i 6i = 4i + 6 = 8 4i Complex Conjugate z* If z = a + ib then its complex conjugate is z* = a ib  always collect the real and imaginary parts before looking for the conjugate Solving Equations  if two complex numbers are equal, their real parts are equal and their imaginary parts are equal. Find z when 5z z* = 3 4i Let z = x + iy and so z*= x iy 5(x + iy) (x  iy) = 3 4i 3x + 7iy = 3 4i Equating real : 3x = 3 so x = Equating imaginary : 7y = 4 so y=  z = i
8 8. Calculus Differentiating from first principles Gradient of curve or tangent at x is f (x) = You may need to use the binomial expansion Differentiate from first principles to find the gradient of the curve y = x 4 at the point (,6) f(x) = 4 f( + h) = ( + h) 4 = 4 + 4( 3 h) + 6( h )+ 4(h 3 )+ h 4 = 6 + 3h + 4h + 8h 3 + h 4 f( + h) f() h = 6 + 3h + 4h + 8h 3 + h 4 6 h = 3 + 4h + 8h + h 3 As h approaches zero Gradient = 3 You may need to give the equation of the tangent/normal to the curve easy to do once you know the gradient and have the coordinates of the point Improper Integrals Improper if one or both of the limits is infinity Very important to include these statements the integrand is undefined at one of the limits or somewhere in between the limits Very important to include these statements
9 9. Trigonometry GENERAL SOLUTION don t just give one answer there should be an n somewhere!! SKETCH the graph of the basic Trig function before you start Check the question for Degrees or Radians MARK the first solution (from your calculator/knowledge) on your graph mark a few more to see the pattern Find the general solution before rearranging to get x or θ on it s own. Example Find the general solution, in radians, of the equation cos x=3sin x ( sin x) = 3sin x (Using cos x + sin x =) sin x + 3 sin x = 0 (sin x + )(sin x ) = 0 no solutions for sin x =  sinx = ½ General Solutions p 5p x = pn + p 6, x = pn + 5p p + p 6 p + 5p 6 You may need to use the fact that tan q = sin q to solve equations of the form cos q sin (x 0.) = cos (x 0.) 0. Numerical solution of equations Rearrange into the form f(x) = 0 To show the root lies within a given interval evaluate f(x) for the upper and lower interval bounds One should be positive and one negative change of sign indicates a root within the interval Interval Bisection  Determine the nature of f(lower) and f(upper) sketch the graph of the interval  Investigate f(midpoint) positive or negative?  Continue investigating new midpoints until you have an interval to the degree of accuracy required Linear Interpolation  Determine the Value of f(lower) and f(upper) sketch the graph of the interval  Join the Lower and Upper points together with a straight line   Mark p the approximate root   Use similar triangles to calculate p (equal ratios)
10 Newton Raphson Method  given in formula book as x n + = x n f (x n ) f ' (x n ) Given in formula book When working with Trig functions you probably need radians check carefully! value of new approximation value of previous approximation  you may be required to draw a diagram to illustrate your method tangent to the curve at x n Gradient of the tangent = f (x n ) f(xn f (x n ) = ) x n x n + NB : When the initial approximation is not close to f(x) the method may fail! DIFFERENTIAL EQUATIONS  looking to find y when dy/dx is given x n+  EULER s FORMULA y n+ = y n + hf(x n )  allows us to find an approximate value for y close to a given point x n Given in formula book dy = dx f(x) h = step size Example dy = e cos x, given that when y = 3 when x =, use the Euler Formula with step size dx 0. to find an approximation for y when x =.4 x = y = 3 h =0. f(x) = e cos θ y = (e cos ) = (approximate value of y when x =.) y 3 = (e cos. ) = 3.63 (approximate value of y when x =.4). Linear Laws  using straight line graphs to determine equations involving two variables  remember the equation of a straight line is y = mx + c where m is the gradient c is the point of interception with the yaxis  Logarithms needed when y = ax n or y = ab x Remember : Log ab = Log a + Log b Log a x = x Log a
11  equations must be rearranged/substitutions made to a linear form y 3 =ax + b plot y 3 against x y 3 =ax 5 + bx ( x ) y 3 x = ax 3 + b plot y 3 x against x 3 y = ax n (taking logs) log y = log a + n log x plot log y against log x y=ab x (taking logs) log y = log a + x log b plot log y against x if working in logs remember the inverse of log x is 0 x EXAMPLE It is thought that V and x are connected by the equation V = ax b The equation is reduced to linear from by taking logs Log V = Log a + b log x Using data given Log V is plotted against Log x The gradient b is. 50 = The intercept on the log V axis is.3 So Log a =.3 a =0.3 =9.95 The relationship between V and x is therefore V = 0x 3
y intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract 
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationAdvanced Math Study Guide
Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular
More informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationAlgebra 2: Themes for the Big Final Exam
Algebra : Themes for the Big Final Exam Final will cover the whole year, focusing on the big main ideas. Graphing: Overall: x and y intercepts, fct vs relation, fct vs inverse, x, y and origin symmetries,
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationEL9650/9600c/9450/9400 Handbook Vol. 1
Graphing Calculator EL9650/9600c/9450/9400 Handbook Vol. Algebra EL9650 EL9450 Contents. Linear Equations  Slope and Intercept of Linear Equations 2 Parallel and Perpendicular Lines 2. Quadratic Equations
More informationMSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions
MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial
More informationSection 1: How will you be tested? This section will give you information about the different types of examination papers that are available.
REVISION CHECKLIST for IGCSE Mathematics 0580 A guide for students How to use this guide This guide describes what topics and skills you need to know for your IGCSE Mathematics examination. It will help
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationMATHEMATICS Unit Pure Core 2
General Certificate of Education January 2008 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Wednesday 9 January 2008 1.30 pm to 3.00 pm For this paper you must have: an 8page answer
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More information10 Polar Coordinates, Parametric Equations
Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationFunction Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015
Harold s s Cheat Sheet 8 December 05 Algebra Constant Linear Identity f(x) c f(x) x Range: [c, c] Undefined (asymptote) Restrictions: c is a real number Ay + B 0 g(x) x Restrictions: m 0 General Fms: Ax
More informationDear Accelerated PreCalculus Student:
Dear Accelerated PreCalculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, collegepreparatory mathematics course that will also
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (1,3), (3,3), (2,3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the xcomponent of a point in the form (x,y). Range refers to the set of possible values of the ycomponent of a point in
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationNational 5 Mathematics Course Assessment Specification (C747 75)
National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for
More informationMathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)
( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More informationSome Lecture Notes and InClass Examples for PreCalculus:
Some Lecture Notes and InClass Examples for PreCalculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
More informationGRAPHING IN POLAR COORDINATES SYMMETRY
GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry  yaxis,
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationA Level Further Mathematics
A Level Further Mathematics Contents For courses in Year 13 starting from September 2014 Course Overview... 2 Schemes of Work... 3 Further Pure 1... 3 Further Pure 2... 5 Further Pure 3... 7 Decision 2...
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationMATHEMATICS Unit Pure Core 1
General Certificate of Education June 2009 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 1 MPC1 Wednesday 20 May 2009 1.30 pm to 3.00 pm For this paper you must have: an 8page answer book
More informationSOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen
SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen DEFINITION. A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationCOMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i
COMPLEX NUMBERS _4+i _i FIGURE Complex numbers as points in the Arg plane i _i +i i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 115 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationTHE COMPLEX EXPONENTIAL FUNCTION
Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula
More information2013 MBA Jump Start Program
2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of
More informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationMathematics PreTest Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics PreTest Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {1, 1} III. {1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More informationList the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated
MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationMathematics I, II and III (9465, 9470, and 9475)
Mathematics I, II and III (9465, 9470, and 9475) General Introduction There are two syllabuses, one for Mathematics I and Mathematics II, the other for Mathematics III. The syllabus for Mathematics I and
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationComplex Algebra. What is the identity, the number such that it times any number leaves that number alone?
Complex Algebra When the idea of negative numbers was broached a couple of thousand years ago, they were considered suspect, in some sense not real. Later, when probably one of the students of Pythagoras
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationSPECIFICATION. Mathematics 6360 2014. General Certificate of Education
Version 1.0: 0913 General Certificate of Education Mathematics 6360 014 Material accompanying this Specification Specimen and Past Papers and Mark Schemes Reports on the Examination Teachers Guide SPECIFICATION
More informationThe Mathematics Diagnostic Test
The Mathematics iagnostic Test Mock Test and Further Information 010 In welcome week, students will be asked to sit a short test in order to determine the appropriate lecture course, tutorial group, whether
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More information2 Session Two  Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision  Session 2: Complex Numbers and Vectors 1 2 Session Two  Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More informationTRIGONOMETRY Compound & Double angle formulae
TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae
More informationOne advantage of this algebraic approach is that we can write down
. Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the xaxis points out
More informationHow to Graph Trigonometric Functions
How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle
More informationGraphs of Polar Equations
Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate
More informationMath Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.
Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that
More information4.1. COMPLEX NUMBERS
4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationGraphing Quadratic Functions
Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x value and L be the yvalues for a graph. 1. How are the x and yvalues related? What pattern do you see? To enter the
More informationNew HigherProposed OrderCombined Approach. Block 1. Lines 1.1 App. Vectors 1.4 EF. Quadratics 1.1 RC. Polynomials 1.1 RC
New HigherProposed OrderCombined Approach Block 1 Lines 1.1 App Vectors 1.4 EF Quadratics 1.1 RC Polynomials 1.1 RC Differentiationbut not optimisation 1.3 RC Block 2 Functions and graphs 1.3 EF Logs
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationAlgebra II. Weeks 13 TEKS
Algebra II Pacing Guide Weeks 13: Equations and Inequalities: Solve Linear Equations, Solve Linear Inequalities, Solve Absolute Value Equations and Inequalities. Weeks 46: Linear Equations and Functions:
More informationMAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:
MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex
More informationExpression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds
Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationFriday, January 29, 2016 9:15 a.m. to 12:15 p.m., only
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: AAPR.3: Identify zeros of polynomials
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More information