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
 2 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
Roots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationy 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 informationpp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64
Semester 1 Text: Chapter 1: Tools of Algebra Lesson 11: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 12: Algebraic Expressions
More informationAQA Level 2 Certificate FURTHER MATHEMATICS
AQA Qualifications AQA Level 2 Certificate FURTHER MATHEMATICS Level 2 (8360) Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the
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 informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
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 informationAble Enrichment Centre  Prep Level Curriculum
Able Enrichment Centre  Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,
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 informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More informationWASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,
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 informationTime: 1 hour 10 minutes
[C00/SQP48] Higher Time: hour 0 minutes Mathematics Units, and 3 Paper (Noncalculator) Specimen Question Paper (Revised) for use in and after 004 NATIONAL QUALIFICATIONS Read Carefully Calculators may
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,
More informationMATHEMATICS Unit Pure Core 2
General Certificate of Education June 2006 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Monday 22 May 2006 9.00 am to 10.30 am For this paper you must have: * an 8page answer book
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 informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
More informationPURE MATHEMATICS AM 27
AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and
More informationMATHEMATICS PAPER 2. Samir Daniels Mathematics Paper 2 Page 1
MATHEMATICS PAPER 2 Geometry... 2 Transformations... 3 Rotation Anticlockwise around the origin... 3 Trigonometry... 4 Graph shifts... 4 Reduction Formula... 4 Proving Identities... 6 Express in terms
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 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 informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More information106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0.
5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function
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 informationExample 1. Example 1 Plot the points whose polar coordinates are given by
Polar Coordinates A polar coordinate system, gives the coordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points
More informationQuestion: What is the quadratic formula? Question: What is the discriminant? Answer: Answer:
Question: What is the quadratic fmula? Question: What is the discriminant? Question: How do you determine if a quadratic equation has no real roots? The discriminant is negative ie Question: How do you
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 informationClass Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson
Class Notes for MATH 2 Precalculus Fall 2012 Prepared by Stephanie Sorenson Table of Contents 1.2 Graphs of Equations... 1 1.4 Functions... 9 1.5 Analyzing Graphs of Functions... 14 1.6 A Library of Parent
More informationSenior Secondary Australian Curriculum
Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero
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 informationConic Sections in Cartesian and Polar Coordinates
Conic Sections in Cartesian and Polar Coordinates The conic sections are a family of curves in the plane which have the property in common that they represent all of the possible intersections of a plane
More informationConstruction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7
About the Author v Preface to the Instructor xiii WileyPLUS xviii Acknowledgments xix Preface to the Student xxi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number
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 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. 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 informationCalculus Card Matching
Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
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 informationMathematics Chapter 8 and 10 Test Summary 10M2
Quadratic expressions and equations Expressions such as x 2 + 3x, a 2 7 and 4t 2 9t + 5 are called quadratic expressions because the highest power of the variable is 2. The word comes from the Latin quadratus
More informationAdvanced Higher Mathematics Course Assessment Specification (C747 77)
Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational
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 informationALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340
ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:
More informationPOLAR COORDINATES DEFINITION OF POLAR COORDINATES
POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand
More informationThe NotFormula Book for C1
Not The NotFormula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
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 informationg = l 2π g = bd b c d = ac bd. Therefore to write x x and as a single fraction we do the following
OCR Core 1 Module Revision Sheet The C1 exam is 1 hour 30 minutes long. You are not allowed any calculator 1. Before you go into the exam make sureyou are fully aware of the contents of theformula booklet
More informationContents. Introduction and Notes pages 23 (These are important and it s only 2 pages ~ please take the time to read them!)
Page Contents Introduction and Notes pages 23 (These are important and it s only 2 pages ~ please take the time to read them!) Systematic Search for a Change of Sign (Decimal Search) Method Explanation
More informationAlgebra 2/ Trigonometry Extended Scope and Sequence (revised )
Algebra 2/ Trigonometry Extended Scope and Sequence (revised 2012 2013) Unit 1: Operations with Radicals and Complex Numbers 9 days 1. Operations with radicals (p.88, 94, 98, 101) a. Simplifying radicals
More information29 Wyner PreCalculus Fall 2016
9 Wyner PreCalculus Fall 016 CHAPTER THREE: TRIGONOMETRIC EQUATIONS Review November 8 Test November 17 Trigonometric equations can be solved graphically or algebraically. Solving algebraically involves
More information55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim
Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationIdentify examples of field properties: commutative, associative, identity, inverse, and distributive.
Topic: Expressions and Operations ALGEBRA II  STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers
More informationAngles and Their Measure
Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two
More informationThe xintercepts of the graph are the xvalues for the points where the graph intersects the xaxis. A parabola may have one, two, or no xintercepts.
Chapter 101 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax
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 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 informationwith "a", "b" and "c" representing real numbers, and "a" is not equal to zero.
3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,
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 informationMyMathLab ecourse for Developmental Mathematics
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and
More informationTechniques of Differentiation Selected Problems. Matthew Staley
Techniques of Differentiation Selected Problems Matthew Staley September 10, 011 Techniques of Differentiation: Selected Problems 1. Find /dx: (a) y =4x 7 dx = d dx (4x7 ) = (7)4x 6 = 8x 6 (b) y = 1 (x4
More informationCHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS
CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided
More informationAdvanced Algebra 2. I. Equations and Inequalities
Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers
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 informationPrompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply?
Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply? Commentary In our foci, we are assuming that we have a
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 informationKnow the following (or better yet, learn a couple and be able to derive the rest, quickly, from your knowledge of the trigonometric functions):
OCR Core Module Revision Sheet The C exam is hour 0 minutes long. You are allowed a graphics calculator. Before you go into the exam make sureyou are fully aware of the contents of theformula booklet you
More informationFree Response Questions Compiled by Kaye Autrey for facetoface student instruction in the AP Calculus classroom
Free Response Questions 1969005 Compiled by Kaye Autrey for facetoface student instruction in the AP Calculus classroom 1 AP Calculus FreeResponse Questions 1969 AB 1 Consider the following functions
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 informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationPortable Assisted Study Sequence ALGEBRA IIA
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of
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 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 informationTest Bank Exercises in. 7. Find the intercepts, the vertical asymptote, and the slant asymptote of the graph of
Test Bank Exercises in CHAPTER 5 Exercise Set 5.1 1. Find the intercepts, the vertical asymptote, and the horizontal asymptote of the graph of 2x 1 x 1. 2. Find the intercepts, the vertical asymptote,
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 informationREVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012
REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit
More informationChapter 12. The Straight Line
302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,
More informationApplications of Integration Day 1
Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under
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 informationCork Institute of Technology. CIT Mathematics Examination, Paper 2 Sample Paper A
Cork Institute of Technology CIT Mathematics Examination, 2015 Paper 2 Sample Paper A Answer ALL FIVE questions. Each question is worth 20 marks. Total marks available: 100 marks. The standard Formulae
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 informationREVISED GCSE Scheme of Work Mathematics Higher Unit T3. For First Teaching September 2010 For First Examination Summer 2011
REVISED GCSE Scheme of Work Mathematics Higher Unit T3 For First Teaching September 2010 For First Examination Summer 2011 Version 1: 28 April 10 Version 1: 28 April 10 Unit T3 Unit T3 This is a working
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 informationChapter 10: Topics in Analytic Geometry
Chapter 10: Topics in Analytic Geometry 10.1 Parabolas V In blue we see the parabola. It may be defined as the locus of points in the plane that a equidistant from a fixed point (F, the focus) and a fixed
More informationUnit 10: Quadratic Relations
Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This
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 informationIntroduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test
Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are
More informationSpecification. GCE Mathematics
Specification GCE Mathematics Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8371) Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8372) Pearson Edexcel Level 3
More informationSECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.
(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much
More informationDefinition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =
Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a
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 informationCourse Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics
Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)
More informationMath 115 Spring 2014 Written Homework 10SOLUTIONS Due Friday, April 25
Math 115 Spring 014 Written Homework 10SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain
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 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 informationMain page. Given f ( x, y) = c we differentiate with respect to x so that
Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching  asymptotes Curve sketching the
More information1.6 Powers of 10 and standard form Write a number in standard form. Calculate with numbers in standard form.
Unit/section title 1 Number Unit objectives (Edexcel Scheme of Work Unit 1: Powers, decimals, HCF and LCM, positive and negative, roots, rounding, reciprocals, standard form, indices and surds) 1.1 Number
More information