Preliminary algebra. Polynomial equations. and three real roots altogether. Continue an investigation of its properties as follows.
|
|
- Silas Parsons
- 5 years ago
- Views:
Transcription
1 Student Solutions Manual for Mathematical Methods for Physics and Engineering: 1 Preliminary algebra Polynomial equations 1.1 It can be shown that the polynomial g(x) =4x 3 +3x 6x 1 has turning points at x = 1 and x = 1 and three real roots altogether. Continue an investigation of its properties as follows. (a) Make a table of values of g(x) for integer values of x between and. Use it and the information given above to draw a graph and so determine the roots of g(x) =0as accurately as possible. (b) Find one accurate root of g(x) =0by inspection and hence determine precise values for the other two roots. (c) Show that f(x) =4x 3 +3x 6x k =0has only one real root unless 5 k 7 4. (a) Straightforward evaluation of g(x) at integer values of x gives the following table: x g(x) (b) It is apparent from the table alone that x = 1 is an exact root of g(x) =0and so g(x) can be factorised as g(x) =(x 1)h(x) =(x 1)(b x +b 1 x+b 0 ). Equating the coefficients of x 3, x, x and the constant term gives 4 = b, b 1 b =3, b 0 b 1 = 6 and b 0 = 1, respectively, which are consistent if b 1 = 7. To find the two remaining roots we set h(x) =0: 4x +7x +1=0. 1
2 Student Solutions Manual for Mathematical Methods for Physics and Engineering: The roots of this quadratic equation are given by the standard formula as α 1, = 7 ± (c) When k = 1 (i.e. the original equation) the values of g(x) at its turning points, x = 1 andx = 1,are4and 11 4, respectively. Thus g(x) can have up to 4 subtracted from it or up to 11 4 added to it and still satisfy the condition for three (or, at the limit, two) distinct roots of g(x) = 0. It follows that for k outside the range 5 k 7 4, f(x) [=g(x)+1 k] has only one real root. 1.3 Investigate the properties of the polynomial equation by proceeding as follows. f(x) =x 7 +5x 6 + x 4 x 3 + x =0, (a) By writing the fifth-degree polynomial appearing in the expression for f (x) in the form 7x 5 +30x 4 + a(x b) + c, show that there is in fact only one positive root of f(x) =0. (b) By evaluating f(1), f(0) and f( 1), and by inspecting the form of f(x) for negative values of x, determine what you can about the positions of the real roots of f(x) =0. (a) We start by finding the derivative of f(x) and note that, because f contains no linear term, f can be written as the product of x and a fifth-degree polynomial: f(x) =x 7 +5x 6 + x 4 x 3 + x =0, f (x) =x(7x 5 +30x 4 +4x 3x +) = x[7x 5 +30x 4 +4(x 3 8 ) 4( 3 8 ) +] = x[7x 5 +30x 4 +4(x 3 8 ) ]. Since, for positive x, every term in this last expression is necessarily positive, it follows that f (x) can have no zeros in the range 0 <x<. Consequently,f(x) can have no turning points in that range and f(x) = 0 can have at most one root inthesamerange.however,f(+ ) =+ and f(0) = < 0andsof(x) =0 has at least one root in 0 <x<. Consequently it has exactly one root in the range. (b) f(1) = 5, f(0) = andf( 1)=5,andsothereisatleastonerootineach of the ranges 0 <x<1and 1 <x<0. There is no simple systematic way to examine the form of a general polynomial function for the purpose of determining where its zeros lie, but it is sometimes
3 Student Solutions Manual for Mathematical Methods for Physics and Engineering: helpful to group terms in the polynomial and determine how the sign of each group depends upon the range in which x lies. Here grouping successive pairs of terms yields some information as follows: x 7 +5x 6 is positive for x> 5, x 4 x 3 is positive for x>1 and x<0, x is positive for x> and x<. Thus, all three terms are positive in the range(s) common to these, namely 5 <x< andx>1. It follows that f(x) is positive definite in these ranges andtherecanbenorootsoff(x) = 0 within them. However, since f(x) is negative for large negative x, there must be at least one root α with α< Construct the quadratic equations that have the following pairs of roots: (a) 6, 3; (b) 0, 4; (c), ; (d) 3 + i, 3 i, wherei = 1. Starting in each case from the product of factors form of the quadratic equation, (x α 1 )(x α ) = 0, we obtain: (a) (x +6)(x +3)=x +9x +18=0; (b) (x 0)(x 4) = x 4x =0; (c) (x )(x ) = x 4x +4=0; (d) (x 3 i)(x 3+i) =x + x( 3 i 3+i) +(9 6i +6i 4i ) = x 6x +13=0. Trigonometric identities 1.7 Prove that by considering cos π 1 = 3+1 (a) the sum of the sines of π/3 and π/6, (b) the sine of the sum of π/3 and π/4. (a) Using ( ) ( ) A + B A B sin A +sinb =sin cos, 3
4 Student Solutions Manual for Mathematical Methods for Physics and Engineering: we have (b) Using, successively, the identities sin π 3 +sinπ 6 =sinπ 4 cos π 1, = 1 cos π 1, cos π 1 = 3+1. sin(a + B) =sina cos B +cosa sin B, sin(π θ) =sinθ and cos( 1 π θ) =sinθ, we obtain ( π sin 3 + π ) 4 =sin π 3 cos π 4 +cosπ 3 sin π 4, sin 7π 3 1 = sin 5π 1 = , 3+1, cos π 1 = Find the real solutions of (a) 3 sin θ 4cosθ =, (b) 4 sin θ +3cosθ =6, (c) 1 sin θ 5cosθ = 6. We use the result that if then a sin θ + b cos θ = k θ =sin 1 ( k K ) φ, where K = a + b and φ =tan 1 b a. 4
5 Student Solutions Manual for Mathematical Methods for Physics and Engineering: Recalling that the inverse sine yields two values and that the individual signs of a and b have to be taken into account, we have (a) k =,K = 3 +4 =5,φ =tan 1 ( 4/3) and so θ =sin 1 5 tan =1.339 or.66. (b) k =6,K = 4 +3 =5.Sincek>Kthere is no solution for a real angle θ. (c) k = 6, K = 1 +5 = 13, φ =tan 1 ( 5/1) and so θ =sin 13 tan 1 1 = or Find all the solutions of sin θ + sin 4θ = sin θ + sin 3θ that lie in the range π <θ π. What is the multiplicity of the solution θ =0? Using and ( ) ( ) A + B A B sin A +sinb =sin cos, ( ) ( ) A + B A B cos A cos B = sin sin, and recalling that cos( φ) =cos(φ), the equation can be written successively as sin 5θ ( cos 3θ ) =sin 5θ ( cos θ ), sin 5θ ( cos 3θ cos θ ) =0, sin 5θ sin θ sin θ =0. The first factor gives solutions for θ of 4π/5, π/5, 0, π/5 and4π/5. The second factor gives rise to solutions 0 and π, whilst the only value making the third factor zero is θ =0.Thesolutionθ = 0 appears in each of the above sets and so has multiplicity 3. 5
6 Student Solutions Manual for Mathematical Methods for Physics and Engineering: Coordinate geometry 1.13 Determine the forms of the conic sections described by the following equations: (a) x + y +6x +8y =0; (b) 9x 4y 54x 16y +9=0; (c) x +y +5xy 4x + y 6=0; (d) x + y +xy 8x +8y =0. (a) x + y +6x +8y =0. The coefficients of x and y are equal and there is no xy term; it follows that this must represent a circle. Rewriting the equation in standard circle form by completing the squares in the terms that involve x and y, each variable treated separately, we obtain (x +3) +(y +4) (3 +4 )=0. The equation is therefore that of a circle of radius 3 +4 =5centredon ( 3, 4). (b) 9x 4y 54x 16y + 9 = 0. This equation contains no xy term and so the centre of the curve will be at ( 54/( 9), 16/[ ( 4)] ) = (3, ), and in standardised form the equation is 9(x 3) 4(y +) = 0, or (x 3) (y +) = The minus sign between the terms on the LHS implies that this conic section is a hyperbola with asymptotes (the form for large x and y and obtained by ignoring the constant on the RHS) given by 3(x 3) = ±(y + ), i.e. lines of slope ± 3 passing through its centre at (3, ). (c) x +y +5xy 4x + y 6=0. As an xy term is present the equation cannot represent an ellipse or hyperbola in standard form. Whether it represents two straight lines can be most easily investigated by taking the lines in the form a i x+b i y+1 = 0, (i =1, ) and comparing the product (a 1 x+b 1 y+1)(a x+b y+1) with 1 6 (x +y +5xy 4x + y 6). The comparison produces five equations which the four constants a i, b i,(i =1, ) must satisfy: and a 1 a = 6, b 1b = 6, a 1 + a = 4 6, b 1 + b = 1 6 a 1 b + b 1 a =
7 Student Solutions Manual for Mathematical Methods for Physics and Engineering: Combining the first and third equations gives 3a 1 a 1 1 = 0 leading to a 1 and a having the values 1 and 1 3, in either order. Similarly, combining the second and fourth equations gives 6b 1 + b 1 = 0 leading to b 1 and b having the values 1 and 3, again in either order. Either of the two combinations (a 1 = 1 3, b 1 = 3, a =1,b = 1 )and(a 1 =1, b 1 = 1, a = 1 3, b = 3 ) also satisfies the fifth equation [note that the two alternative pairings do not do so]. That a consistent set can be found shows that the equation does indeed represent a pair of straight lines, x +y 3=0and x + y +=0. (d) x + y +xy 8x +8y =0. We note that the first three terms can be written as a perfect square and so the equation can be rewritten as (x + y) =8(x y). The two lines given by x + y =0andx y = 0 are orthogonal and so the equation is of the form u =4av, which, for Cartesian coordinates u, v, represents a parabola passing through the origin, symmetric about the v-axis (u =0)and defined for v 0. Thus the original equation is that of a parabola, symmetric about the line x + y = 0, passing through the origin and defined in the region x y. Partial fractions 1.15 Resolve (a) x +1 x +3x 10, (b) 4 x 3x into partial fractions using each of the following three methods: (i) Expressing the supposed expansion in a form in which all terms have the same denominator and then equating coefficients of the various powers of x. (ii) Substituting specific numerical values for x and solving the resulting simultaneous equations. (iii) Evaluation of the fraction at each of the roots of its denominator, imagining a factored denominator with the factor corresponding to the root omitted often known as the cover-up method. Verify that the decomposition obtained is independent of the method used. (a) As the denominator factorises as (x +5)(x ), the partial fraction expansion must have the form x +1 x +3x 10 = A x +5 + B x. 7
8 Student Solutions Manual for Mathematical Methods for Physics and Engineering: (i) A x +5 + B x x(a + B)+(5B A) =. (x +5)(x ) Solving A + B =and A +5B = 1 gives A = 9 7 and B = 5 7. (ii) Setting x equal to 0 and 1, say, gives the pair of equations 1 10 = A 5 + B ; 3 6 = A 6 + B 1, 1 =A 5B; 3 =A 6B, (iii) with solution A = 9 7 and B = 5 7. A = ( 5) = 9 7 ; B = () = 5 7. All three methods give the same decomposition. (b) Here the factorisation of the denominator is simply x(x 3) or, more formally, (x 0)(x 3), and the expansion takes the form 4 x 3x = A x + B x 3. (i) A x + B x(a + B) 3A = x 3 (x 0)(x 3). Solving A + B =0and 3A = 4 gives A = 4 3 and B = 4 3. (ii) Setting x equal to 1 and, say, gives the pair of equations 4 = A 1 + B ; 4 = A + B 1, 4 =A B; 4 =A B, (iii) with solution A = 4 3 and B = 4 3. A = = 4 3 ; B = = 4 3. Again, all three methods give the same decomposition. 8
9 Student Solutions Manual for Mathematical Methods for Physics and Engineering: 1.17 Rearrange the following functions in partial fraction form: x 6 (a) x 3 x +4x 4, (b) x3 +3x + x +19 x 4 +10x. +9 (a) For the function f(x) = x 6 x 3 x +4x 4 = g(x) h(x) the first task is to factorise the denominator. By inspection, h(1) = 0 and so x 1 is a factor of the denominator. Write x 3 x +4x 4=(x 1)(x + b 1 x + b 0 ). Equating coefficients: 1 =b 1 1, 4 = b 1 + b 0 and 4 = b 0, giving b 1 =0 and b 0 = 4. Thus, x 6 f(x) = (x 1)(x +4). The factor x + 4 cannot be factorised further without using complex numbers and so we include a term with this factor as the denominator, but at the price of having a linear term, and not just a number, in the numerator. f(x) = A x 1 + Bx + C x +4 = Ax +4A + Bx + Cx Bx C (x 1)(x. +4) Comparing the coefficients of the various powers of x in this numerator with those in the numerator of the original expression gives A + B =0,C B =1and 4A C = 6, which in turn yield A = 1, B =1andC =. Thus, (b) By inspection, the denominator of f(x) = 1 x 1 + x + x +4. x 3 +3x + x +19 x 4 +10x +9 factorises simply into (x +9)(x + 1), but neither factor can be broken down further. Thus, as in (a), we write f(x) = Ax + B x +9 + Cx + D x +1 = (A + C)x3 +(B + D)x +(A +9C)x +(B +9D) (x +9)(x. +1) 9
10 Student Solutions Manual for Mathematical Methods for Physics and Engineering: Equating coefficients gives A + C =1, B + D =3, A +9C =1, B +9D =19. From the first and third equations, A =1andC = 0. The second and fourth yield B =1andD =. Thus f(x) = x +1 x +9 + x +1. Binomial expansion 1.19 Evaluate those of the following that are defined: (a) 5 C 3, (b) 3 C 5, (c) 5 C 3, (d) 3 C 5. (a) 5 C 3 = 5! 3!! = 10. (b) 3 C 5. This is not defined as 5 > 3 > 0. For (c) and (d) we will need to use the identity m k m(m +1) (m + k 1) C k =( 1) =( 1) k m+k 1 C k. k! (c) 5 C 3 =( 1) C 3 = 7! 3! 4! = 35. (d) 3 C 5 =( 1) C 5 = 7! 5!! = 1. Proof by induction and contradiction 1.1 Prove by induction that n n r = 1 n(n +1) and r 3 = 1 4 n (n +1). r=1 r=1 To prove that n r=1 r = 1 n(n +1), 10
Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x)
Evaluate the function: c. (g o f )(x + 2) d. ( f ( f (x)) 1. f x = 4x! 2 a. f( 2) b. f(x 1) c. f (x + h) f (x) h 4. g x = 3x! + 1 Find g!! (x) 5. p x = 4x! + 2 Find p!! (x) 2. m x = 3x! + 2x 1 m(x + h)
More informationSolving equations UNCORRECTED PAGE PROOFS
1 Solving equations 1.1 Kick off with CAS 1. Polynomials 1.3 Trigonometric symmetry properties 1.4 Trigonometric equations and general solutions 1.5 Literal equations and simultaneous equations 1.6 Review
More information1 Solving equations 1.1 Kick off with CAS 1. Polynomials 1. Trigonometric symmetry properties 1.4 Trigonometric equations and general solutions 1.5 Literal and simultaneous equations 1.6 Review 1.1 Kick
More informationAlgebraic. techniques1
techniques Algebraic An electrician, a bank worker, a plumber and so on all have tools of their trade. Without these tools, and a good working knowledge of how to use them, it would be impossible for them
More informationa k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
More informationSKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution.
SKILL BUILDER TEN Graphs of Linear Equations with Two Variables A first degree equation is called a linear equation, since its graph is a straight line. In a linear equation, each term is a constant or
More information3. Use absolute value notation to write an inequality that represents the statement: x is within 3 units of 2 on the real line.
PreCalculus Review Review Questions 1 The following transformations are applied in the given order) to the graph of y = x I Vertical Stretch by a factor of II Horizontal shift to the right by units III
More informationCandidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.
Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;
More informationCALCULUS ASSESSMENT REVIEW
CALCULUS ASSESSMENT REVIEW DEPARTMENT OF MATHEMATICS CHRISTOPHER NEWPORT UNIVERSITY 1. Introduction and Topics The purpose of these notes is to give an idea of what to expect on the Calculus Readiness
More informationThings You Should Know Coming Into Calc I
Things You Should Know Coming Into Calc I Algebraic Rules, Properties, Formulas, Ideas and Processes: 1) Rules and Properties of Exponents. Let x and y be positive real numbers, let a and b represent real
More informationTest Codes : MIA (Objective Type) and MIB (Short Answer Type) 2007
Test Codes : MIA (Objective Type) and MIB (Short Answer Type) 007 Questions will be set on the following and related topics. Algebra: Sets, operations on sets. Prime numbers, factorisation of integers
More information10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions
Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The
More informationAS PURE MATHS REVISION NOTES
AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are
More information30 Wyner Math Academy I Fall 2015
30 Wyner Math Academy I Fall 2015 CHAPTER FOUR: QUADRATICS AND FACTORING Review November 9 Test November 16 The most common functions in math at this level are quadratic functions, whose graphs are parabolas.
More informationChapter 8B - Trigonometric Functions (the first part)
Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of
More information( ) ( ) ( ) 2 6A: Special Trig Limits! Math 400
2 6A: Special Trig Limits Math 400 This section focuses entirely on the its of 2 specific trigonometric functions. The use of Theorem and the indeterminate cases of Theorem are all considered. a The it
More informationA. Correct! These are the corresponding rectangular coordinates.
Precalculus - Problem Drill 20: Polar Coordinates No. 1 of 10 1. Find the rectangular coordinates given the point (0, π) in polar (A) (0, 0) (B) (2, 0) (C) (0, 2) (D) (2, 2) (E) (0, -2) A. Correct! These
More informationSolving Systems of Linear Equations. Classification by Number of Solutions
Solving Systems of Linear Equations Case 1: One Solution Case : No Solution Case 3: Infinite Solutions Independent System Inconsistent System Dependent System x = 4 y = Classification by Number of Solutions
More informationMath 12 Final Exam Review 1
Math 12 Final Exam Review 1 Part One Calculators are NOT PERMITTED for this part of the exam. 1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 θ 2π? 2 b) Draw these angles
More informationALGEBRAIC LONG DIVISION
QUESTIONS: 2014; 2c 2013; 1c ALGEBRAIC LONG DIVISION x + n ax 3 + bx 2 + cx +d Used to find factors and remainders of functions for instance 2x 3 + 9x 2 + 8x + p This process is useful for finding factors
More informationMath Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8
Name: Math Academy I Fall Study Guide CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 1-A Terminology natural integer rational real complex irrational imaginary term expression argument monomial degree
More informationSec 4 Maths. SET A PAPER 2 Question
S4 Maths Set A Paper Question Sec 4 Maths Exam papers with worked solutions SET A PAPER Question Compiled by THE MATHS CAFE 1 P a g e Answer all the questions S4 Maths Set A Paper Question Write in dark
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 170 Final Exam Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the function at the given value of the independent variable and
More informationThe Orchid School Weekly Syllabus Overview Std : XI Subject : Math. Expected Learning Objective Activities/ FAs Planned Remark
The Orchid School Weekly Syllabus Overview 2015-2016 Std : XI Subject : Math Month Lesson / Topic Expected Learning Objective Activities/ FAs Planned Remark March APRIL MAY Linear Inequalities (Periods
More informationREQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS
REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS The Department of Applied Mathematics administers a Math Placement test to assess fundamental skills in mathematics that are necessary to begin the study
More informationAlgebra II Introduction 1
Introduction 1 Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include logarithmic, polynomial, rational, and radical functions
More informationMark Scheme (Results) January 2007
Mark Scheme (Results) January 007 GCE GCE Mathematics Core Mathematics C (666) Edexcel Limited. Registered in England and Wales No. 96750 Registered Office: One90 High Holborn, London WCV 7BH January 007
More informationRoots 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 informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More informationCore A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document
Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Background knowledge: (a) The arithmetic of integers (including HCFs and LCMs), of fractions, and of real numbers.
More informationA Library of Functions
LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 170 Final Exam Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the function at the given value of the independent variable and
More informationMathematics 1 Lecture Notes Chapter 1 Algebra Review
Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More informationAlgebra 2 (2006) Correlation of the ALEKS Course Algebra 2 to the California Content Standards for Algebra 2
Algebra 2 (2006) Correlation of the ALEKS Course Algebra 2 to the California Content Standards for Algebra 2 Algebra II - This discipline complements and expands the mathematical content and concepts of
More information2014 Summer Review for Students Entering Algebra 2. TI-84 Plus Graphing Calculator is required for this course.
1. Solving Linear Equations 2. Solving Linear Systems of Equations 3. Multiplying Polynomials and Solving Quadratics 4. Writing the Equation of a Line 5. Laws of Exponents and Scientific Notation 6. Solving
More informationThe American School of Marrakesh. AP Calculus AB Summer Preparation Packet
The American School of Marrakesh AP Calculus AB Summer Preparation Packet Summer 2016 SKILLS NEEDED FOR CALCULUS I. Algebra: *A. Exponents (operations with integer, fractional, and negative exponents)
More informationPLC Papers. Created For:
PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use
More informationa factors The exponential 0 is a special case. If b is any nonzero real number, then
0.1 Exponents The expression x a is an exponential expression with base x and exponent a. If the exponent a is a positive integer, then the expression is simply notation that counts how many times the
More information5.4 - Quadratic Functions
Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 92 5.4 - Quadratic Functions Definition: A function is one that can be written in the form f (x) = where a, b, and c are real numbers and a 0. (What
More informationPLC Papers. Created For:
PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use
More informationfunction independent dependent domain range graph of the function The Vertical Line Test
Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding
More informationAnalytic Trigonometry. Copyright Cengage Learning. All rights reserved.
Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.1 Trigonometric Identities Copyright Cengage Learning. All rights reserved. Objectives Simplifying Trigonometric Expressions Proving
More informationb n x n + b n 1 x n b 1 x + b 0
Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)
More informationAlgebra II. Algebra II Higher Mathematics Courses 77
Algebra II Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include logarithmic, polynomial, rational, and radical functions in
More informationSince x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C)
SAT II - Math Level 2 Test #01 Solution 1. x + = 2, then x² + = Since x + = 2, by squaring both side of the equation, (A) - (B) 0 (C) 2 (D) 4 (E) -2 we get x² + 2x 1 + 1 = 4, or simplifying it, x² + 2
More informationDiagnostic quiz for M337 Complex analysis
Diagnostic quiz for M337 Complex analysis The aim of this diagnostic quiz is to help you assess how well prepared you are for M337, and to identify topics that you should revise or study before beginning
More informationMCR3U - Practice Mastery Test #6
Name: Class: Date: MCRU - Practice Mastery Test #6 Multiple Choice Identify the choice that best completes the statement or answers the question.. Factor completely: 4x 2 2x + 9 a. (2x ) 2 b. (4x )(x )
More information1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to
SAT II - Math Level Test #0 Solution SAT II - Math Level Test No. 1. The positive zero of y = x + x 3/5 is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E). 3 b b 4ac Using Quadratic
More informationComplex numbers, the exponential function, and factorization over C
Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain
More informationTopic 4 Notes Jeremy Orloff
Topic 4 Notes Jeremy Orloff 4 Complex numbers and exponentials 4.1 Goals 1. Do arithmetic with complex numbers.. Define and compute: magnitude, argument and complex conjugate of a complex number. 3. Euler
More informationReview exercise 2. 1 The equation of the line is: = 5 a The gradient of l1 is 3. y y x x. So the gradient of l2 is. The equation of line l2 is: y =
Review exercise The equation of the line is: y y x x y y x x y 8 x+ 6 8 + y 8 x+ 6 y x x + y 0 y ( ) ( x 9) y+ ( x 9) y+ x 9 x y 0 a, b, c Using points A and B: y y x x y y x x y x 0 k 0 y x k ky k x a
More informationMTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.
MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:
More informationCURRICULUM GUIDE. Honors Algebra II / Trigonometry
CURRICULUM GUIDE Honors Algebra II / Trigonometry The Honors course is fast-paced, incorporating the topics of Algebra II/ Trigonometry plus some topics of the pre-calculus course. More emphasis is placed
More informationMAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS
MAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT100 is a fast-paced and thorough tour of precalculus mathematics, where the choice of topics is primarily motivated by the conceptual and technical knowledge
More informationLearning Objectives These show clearly the purpose and extent of coverage for each topic.
Preface This book is prepared for students embarking on the study of Additional Mathematics. Topical Approach Examinable topics for Upper Secondary Mathematics are discussed in detail so students can focus
More informationSESSION CLASS-XI SUBJECT : MATHEMATICS FIRST TERM
TERMWISE SYLLABUS SESSION-2018-19 CLASS-XI SUBJECT : MATHEMATICS MONTH July, 2018 to September 2018 CONTENTS FIRST TERM Unit-1: Sets and Functions 1. Sets Sets and their representations. Empty set. Finite
More informationRotation of Axes. By: OpenStaxCollege
Rotation of Axes By: OpenStaxCollege As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions,
More informationSolving Quadratic Equations
Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic
More informationSTATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF DRAFT SYLLABUS.
STATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF 2017 - DRAFT SYLLABUS Subject :Mathematics Class : XI TOPIC CONTENT Unit 1 : Real Numbers - Revision : Rational, Irrational Numbers, Basic Algebra
More informationMath 131 Final Exam Spring 2016
Math 3 Final Exam Spring 06 Name: ID: multiple choice questions worth 5 points each. Exam is only out of 00 (so there is the possibility of getting more than 00%) Exam covers sections. through 5.4 No graphing
More informationThe University of British Columbia Midterm 1 Solutions - February 3, 2012 Mathematics 105, 2011W T2 Sections 204, 205, 206, 211.
1. a) Let The University of British Columbia Midterm 1 Solutions - February 3, 2012 Mathematics 105, 2011W T2 Sections 204, 205, 206, 211 fx, y) = x siny). If the value of x, y) changes from 0, π) to 0.1,
More informationMathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) May 2010
Link to past paper on OCR website: http://www.mei.org.uk/files/papers/c110ju_ergh.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or
More informationHow to use this Algebra II - Semester 2 Study Packet
Excellence is not an act, but a habit. Aristotle Dear Algebra II Student, First of all, Congrats! for making it this far in your math career. Passing Algebra II is a huge mile-stone Give yourself a pat
More informationMath Precalculus Blueprint Assessed Quarter 1
PO 11. Find approximate solutions for polynomial equations with or without graphing technology. MCWR-S3C2-06 Graphing polynomial functions. MCWR-S3C2-12 Theorems of polynomial functions. MCWR-S3C3-08 Polynomial
More informationPartial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.
Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,
More informationMCPS Algebra 2 and Precalculus Standards, Categories, and Indicators*
Content Standard 1.0 (HS) Patterns, Algebra and Functions Students will algebraically represent, model, analyze, and solve mathematical and real-world problems involving functional patterns and relationships.
More informationTable of Contents. Module 1
Table of Contents Module Order of operations 6 Signed Numbers Factorization of Integers 7 Further Signed Numbers 3 Fractions 8 Power Laws 4 Fractions and Decimals 9 Introduction to Algebra 5 Percentages
More informationMockTime.com. (b) (c) (d)
373 NDA Mathematics Practice Set 1. If A, B and C are any three arbitrary events then which one of the following expressions shows that both A and B occur but not C? 2. Which one of the following is an
More informationMultiplication of Polynomials
Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is
More informationPrentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Algebra II (Grades 9-12)
California Mathematics Content Standards for Algebra II (Grades 9-12) This discipline complements and expands the mathematical content and concepts of algebra I and geometry. Students who master algebra
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
Chapter 5 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 5. Overview We know that the square of a real number is always non-negative e.g. (4) 6 and ( 4) 6. Therefore, square root of 6 is ± 4. What about the square
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 1
Learning outcomes EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 1 TUTORIAL 3 - FACTORISATION AND QUADRATICS On completion of this unit a learner should: 1 Know how to use algebraic
More informationMATH 1301, Solutions to practice problems
MATH 1301, Solutions to practice problems 1. (a) (C) and (D); x = 7. In 3 years, Ann is x + 3 years old and years ago, when was x years old. We get the equation x + 3 = (x ) which is (D); (C) is obtained
More informationCHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II
CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE ALGEBRA II Course Number 5116 Department Mathematics Qualification Guidelines Successful completion of both semesters of Algebra 1 or Algebra 1
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More informationAlgebra 2. Chapter 4 Exponential and Logarithmic Functions. Chapter 1 Foundations for Functions. Chapter 3 Polynomial Functions
Algebra 2 Chapter 1 Foundations for Chapter 2 Quadratic Chapter 3 Polynomial Chapter 4 Exponential and Logarithmic Chapter 5 Rational and Radical Chapter 6 Properties and Attributes of Chapter 7 Probability
More informationGrade 11 or 12 Pre-Calculus
Grade 11 or 12 Pre-Calculus Strands 1. Polynomial, Rational, and Radical Relationships 2. Trigonometric Functions 3. Modeling with Functions Strand 1: Polynomial, Rational, and Radical Relationships Standard
More informationFurther Mathematics AS/F1/D17 AS PAPER 1
Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks Further Mathematics AS PAPER 1 CM December Mock Exam (AQA Version) Time allowed: 1 hour and 30 minutes
More information5.3 SOLVING TRIGONOMETRIC EQUATIONS
5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations
More informationMath 259 Winter Solutions to Homework # We will substitute for x and y in the linear equation and then solve for r. x + y = 9.
Math 59 Winter 9 Solutions to Homework Problems from Pages 5-5 (Section 9.) 18. We will substitute for x and y in the linear equation and then solve for r. x + y = 9 r cos(θ) + r sin(θ) = 9 r (cos(θ) +
More informationCO-ORDINATE GEOMETRY
CO-ORDINATE GEOMETRY 1 To change from Cartesian coordinates to polar coordinates, for X write r cos θ and for y write r sin θ. 2 To change from polar coordinates to cartesian coordinates, for r 2 write
More informationDuVal High School Summer Review Packet AP Calculus
DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)
Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
More informationYou should be comfortable with everything below (and if you aren t you d better brush up).
Review You should be comfortable with everything below (and if you aren t you d better brush up).. Arithmetic You should know how to add, subtract, multiply, divide, and work with the integers Z = {...,,,
More informationLone Star College-CyFair Formula Sheet
Lone Star College-CyFair Formula Sheet The following formulas are critical for success in the indicated course. Student CANNOT bring these formulas on a formula sheet or card to tests and instructors MUST
More informationTropical Polynomials
1 Tropical Arithmetic Tropical Polynomials Los Angeles Math Circle, May 15, 2016 Bryant Mathews, Azusa Pacific University In tropical arithmetic, we define new addition and multiplication operations on
More informationALGEBRA 2 X. Final Exam. Review Packet
ALGEBRA X Final Exam Review Packet Multiple Choice Match: 1) x + y = r a) equation of a line ) x = 5y 4y+ b) equation of a hyperbola ) 4) x y + = 1 64 9 c) equation of a parabola x y = 1 4 49 d) equation
More informationLearning Objectives for Math 166
Learning Objectives for Math 166 Chapter 6 Applications of Definite Integrals Section 6.1: Volumes Using Cross-Sections Draw and label both 2-dimensional perspectives and 3-dimensional sketches of the
More informationDepartment of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections 3.1, 3.3, and 3.5
Department of Mathematics, University of Wisconsin-Madison Math 11 Worksheet Sections 3.1, 3.3, and 3.5 1. For f(x) = 5x + (a) Determine the slope and the y-intercept. f(x) = 5x + is of the form y = mx
More information8.3 Partial Fraction Decomposition
8.3 partial fraction decomposition 575 8.3 Partial Fraction Decomposition Rational functions (polynomials divided by polynomials) and their integrals play important roles in mathematics and applications,
More informationExamples 2: Composite Functions, Piecewise Functions, Partial Fractions
Examples 2: Composite Functions, Piecewise Functions, Partial Fractions September 26, 206 The following are a set of examples to designed to complement a first-year calculus course. objectives are listed
More informationFor a semi-circle with radius r, its circumfrence is πr, so the radian measure of a semi-circle (a straight line) is
Radian Measure Given any circle with radius r, if θ is a central angle of the circle and s is the length of the arc sustained by θ, we define the radian measure of θ by: θ = s r For a semi-circle with
More informationA2 HW Imaginary Numbers
Name: A2 HW Imaginary Numbers Rewrite the following in terms of i and in simplest form: 1) 100 2) 289 3) 15 4) 4 81 5) 5 12 6) -8 72 Rewrite the following as a radical: 7) 12i 8) 20i Solve for x in simplest
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationCore Mathematics C1 (AS) Unit C1
Core Mathematics C1 (AS) Unit C1 Algebraic manipulation of polynomials, including expanding brackets and collecting like terms, factorisation. Graphs of functions; sketching curves defined by simple equations.
More informationDESIGN OF THE QUESTION PAPER
DESIGN OF THE QUESTION PAPER MATHEMATICS - CLASS XI Time : 3 Hours Max. Marks : 00 The weightage of marks over different dimensions of the question paper shall be as follows:. Weigtage of Type of Questions
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More information2.1 Quadratic Functions
Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.
More informationIntegers, Fractions, Decimals and Percentages. Equations and Inequations
Integers, Fractions, Decimals and Percentages Round a whole number to a specified number of significant figures Round a decimal number to a specified number of decimal places or significant figures Perform
More information