Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE The absolute value of the complex number z a bi is


 Kathryn Bradford
 1 years ago
 Views:
Transcription
1 0_0605.qxd /5/05 0:45 AM Page Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values of complex umbers. Write the trigoometric forms of complex umbers. Multiply ad divide complex umbers writte i trigoometric form. Use DeMoivre s Theorem to fid powers of complex umbers. Fid th roots of complex umbers. Why you should lear it You ca use the trigoometric form of a complex umber to perform operatios with complex umbers. For istace, i Exercises 05 o page 480, you ca use the trigoometric forms of complex umbers to help you solve polyomial equatios. The Complex Plae Just as real umbers ca be represeted by poits o the real umber lie, you ca represet a complex umber z a bi as the poit a, b i a coordiate plae (the complex plae). The horizotal is called the real ad the vertical is called the imagiary, as show i Figure FIGURE 6.44 (, ) or i The absolute value of the complex umber a bi is defied as the distace betwee the origi 0, 0 ad the poit a, b. (, ) or + i Defiitio of the Absolute Value of a Complex Number The absolute value of the complex umber z a bi is a bi a b. (, 5) FIGURE 6.45 If the complex umber a bi is a real umber (that is, if b 0), the this defiitio agrees with that give for the absolute value of a real umber a 0i a 0 a. Example Fidig the Absolute Value of a Complex Number Plot z 5i ad fid its absolute value. Solutio The umber is plotted i Figure It has a absolute value of z 5 9. Now try Exercise.
2 0_0605.qxd /5/05 0:45 AM Page 47 Sectio 6.5 Trigoometric Form of a Complex Number 47 Trigoometric Form of a Complex Number (a,b) b a r θ I Sectio.4, you leared how to add, subtract, multiply, ad divide complex umbers. To work effectively with powers ad roots of complex umbers, it is helpful to write complex umbers i trigoometric form. I Figure 6.46, cosider the ozero complex umber a bi. By lettig be the agle from the positive real (measured couterclockwise) to the lie segmet coectig the origi ad the poit a, b, you ca write a r cos ad b r si where r a b. Cosequetly, you have a bi r cos r si i FIGURE 6.46 from which you ca obtai the trigoometric form of a complex umber. Trigoometric Form of a Complex Number The trigoometric form of the complex umber z a bi is z rcos i si where a r cos, b r si, r a b, ad ta ba. The umber r is the modulus of z, ad is called a argumet of z. The trigoometric form of a complex umber is also called the polar form. Because there are ifiitely may choices for, the trigoometric form of a complex umber is ot uique. Normally, is restricted to the iterval 0 <, although o occasio it is coveiet to use < 0. Example Writig a Complex Number i Trigoometric Form Write the complex umber z i i trigoometric form. Solutio The absolute value of z is r i 6 4 z = 4 4π ad the referece agle ta b a is give by Because ta ad because z i lies i Quadrat III, you choose to be So, the trigoometric form is 4. z rcos i si. z = i 4 4 cos 4 4 i si. See Figure FIGURE 6.47 Now try Exercise.
3 0_0605.qxd /5/05 0:45 AM Page Chapter 6 Additioal Topics i Trigoometry Example Writig a Complex Number i Stadard Form Techology A graphig utility ca be used to covert a complex umber i trigoometric (or polar) form to stadard form. For specific keystrokes, see the user s maual for your graphig utility. Write the complex umber i stadard form a bi. z 8 cos i si Solutio Because cos ad si, you ca write z 8 cos i si i 6i. Now try Exercise 5. Multiplicatio ad Divisio of Complex Numbers The trigoometric form adapts icely to multiplicatio ad divisio of complex umbers. Suppose you are give two complex umbers z ad z r cos i si r cos i si. The product of z ad is give by z z z r r cos i si cos i si r r cos cos si si isi cos cos si. Usig the sum ad differece formulas for cosie ad sie, you ca rewrite this equatio as z z r r cos i si. This establishes the first part of the followig rule. The secod part is left for you to verify (see Exercise 7). Product ad Quotiet of Two Complex Numbers Let z r cos i si ad z r cos i si be complex umbers. z z r r cos i si Product z r cos z r i si, z 0 Quotiet Note that this rule says that to multiply two complex umbers you multiply moduli ad add argumets, whereas to divide two complex umbers you divide moduli ad subtract argumets.
4 0_0605.qxd /5/05 0:45 AM Page 47 Sectio 6.5 Trigoometric Form of a Complex Number 47 Example 4 Multiplyig Complex Numbers Techology Some graphig utilities ca multiply ad divide complex umbers i trigoometric form. If you have access to such a graphig utility, use it to fid z z ad z z i Examples 4 ad 5. Fid the product z z of the complex umbers. z cos i si Solutio z z cos i si 6 cos 6 i si 6 6 cos 5 5 i si 6 cos i si 60 i 6i You ca check this result by first covertig the complex umbers to the stadard forms z i ad z 4 4i ad the multiplyig algebraically, as i Sectio.4. z z i4 4i 4 4i i 4 6i Now try Exercise 47. z 8 cos i si cos 6 i si 6 Multiply moduli ad add argumets. Example 5 Dividig Complex Numbers Fid the quotiet z z of the complex umbers. z 4cos 00 i si 00 Solutio z 4cos 00 i si 00 z 8cos 75 i si 75 4 cos00 75 i si cos 5 i si 5 i i Now try Exercise 5. z 8cos 75 i si 75 Divide moduli ad subtract argumets.
5 0_0605.qxd /5/05 0:45 AM Page Chapter 6 Additioal Topics i Trigoometry Powers of Complex Numbers The trigoometric form of a complex umber is used to raise a complex umber to a power. To accomplish this, cosider repeated use of the multiplicatio rule. z rcos i si z rcos i si rcos i si r cos i si z r cos i si rcos i si r cos i si z 4 r 4 cos 4 i si 4 z 5 r 5 cos 5 i si 5. This patter leads to DeMoivre s Theorem, which is amed after the Frech mathematicia Abraham DeMoivre ( ). The Grager Collectio Historical Note Abraham DeMoivre ( ) is remembered for his work i probability theory ad DeMoivre s Theorem. His book The Doctrie of Chaces (published i 78) icludes the theory of recurrig series ad the theory of partial fractios. DeMoivre s Theorem If z rcos i si is a complex umber ad is a positive iteger, the z rcos i si Example 6 r cos i si. Fidig Powers of a Complex Number Use DeMoivre s Theorem to fid i. Solutio First covert the complex umber to trigoometric form usig r So, the trigoometric form is ad z i cos i si. The, by DeMoivre s Theorem, you have i cos i si cos i si 4096cos 8 i si Now try Exercise 75. arcta.
6 0_0605.qxd /5/05 0:45 AM Page 475 Roots of Complex Numbers Recall that a cosequece of the Fudametal Theorem of Algebra is that a polyomial equatio of degree has solutios i the complex umber system. So, the equatio x 6 has six solutios, ad i this particular case you ca fid the six solutios by factorig ad usig the Quadratic Formula. x 6 x x Cosequetly, the solutios are x ±, Sectio 6.5 Trigoometric Form of a Complex Number 475 x x x x x x 0 x ± i, Each of these umbers is a sixth root of. I geeral, the th root of a complex umber is defied as follows. ad x ± i. Defiitio of the th Root of a Complex Number The complex umber u a bi is a th root of the complex umber z if z u a bi. Exploratio The th roots of a complex umber are useful for solvig some polyomial equatios. For istace, explai how you ca use DeMoivre s Theorem to solve the polyomial equatio x [Hit: Write 6 as 6cos i si. ] To fid a formula for a th root of a complex umber, let u be a th root of z, where u scos i si ad z rcos i si. By DeMoivre s Theorem ad the fact that u z, you have s cos i si rcos i si. Takig the absolute value of each side of this equatio, it follows that s r. Substitutig back ito the previous equatio ad dividig by r, you get cos i si cos i si. So, it follows that cos cos ad si si. Because both sie ad cosie have a period of, these last two equatios have solutios if ad oly if the agles differ by a multiple of. Cosequetly, there must exist a iteger k such that k k. By substitutig this value of ito the trigoometric form of u, you get the result stated o the followig page.
7 0_0605.qxd /5/05 0:45 AM Page Chapter 6 Additioal Topics i Trigoometry Fidig th Roots of a Complex Number For a positive iteger, the complex umber z rcos i si has exactly distict th roots give by r cos k i si where k 0,,,...,. k r FIGURE 6.48 π π Whe k exceeds, the roots begi to repeat. For istace, if k, the agle is cotermial with, which is also obtaied whe k 0. The formula for the th roots of a complex umber z has a ice geometrical iterpretatio, as show i Figure Note that because the th roots of z all have the same magitude r, they all lie o a circle of radius r with ceter at the origi. Furthermore, because successive th roots have argumets that differ by, the roots are equally spaced aroud the circle. You have already foud the sixth roots of by factorig ad by usig the Quadratic Formula. Example 7 shows how you ca solve the same problem with the formula for th roots. Example 7 Fidig the th Roots of a Number + i + 0i FIGURE 6.49 i + + 0i i i Fid all the sixth roots of. Solutio First write i the trigoometric form cos 0 i si 0. The, by the th root formula, with 6 ad r, the roots have the form 6 cos 0 k 6 So, for k 0,,,, 4, ad 5, the sixth roots are as follows. (See Figure 6.49.) cos 0 i si 0 cos i si i cos i si i cos i si cos 4 4 i si i cos 5 5 i si i i si 0 k 6 cos k i si k. Now try Exercise 97. Icremet by 6
8 0_0605.qxd /5/05 0:45 AM Page 477 Sectio 6.5 Trigoometric Form of a Complex Number 477 Activities. Use DeMoivre s Theorem to fid i. Aswer: 64cos 4 i si Fid the cube roots of 8 i 8 cos i si. Aswer: cos i si 9 9 cos 8 8 i si 9 9 cos 4 4 i si 9 9. Fid all of the solutios of the equatio x 4 0. Aswer: cos 4 i si 4 i cos i si 4 4 i cos 5 5 i si 4 4 i cos 7 7 i si 4 4 i i FIGURE i i Note i Example 8 that the absolute value of z is r i 8 ad the agle is give by ta b a. I Figure 8.49, otice that the roots obtaied i Example 7 all have a magitude of ad are equally spaced aroud the uit circle. Also otice that the complex roots occur i cojugate pairs, as discussed i Sectio.5. The distict th roots of are called the th roots of uity. Example 8 Fidig the th Roots of a Complex Number Fid the three cube roots of z i. Solutio Because z lies i Quadrat II, the trigoometric form of z is z i 8 cos 5 i si 5. By the formula for th roots, the cube roots have the form 68 cos Fially, for k 0,, ad, you obtai the roots 68 cos 5600 i si 5600 cos 45i si cos 560 i si 560 cos 65i si cos 560 i si 560 cos 85i si 85 See Figure k i si Now try Exercise 0. 5º 60k. W RITING ABOUT MATHEMATICS A Famous Mathematical Formula The famous formula e a bi e a cos b i si b arcta 5 i i i. is called Euler s Formula, after the Swiss mathematicia Leohard Euler (707 78). Although the iterpretatio of this formula is beyod the scope of this text, we decided to iclude it because it gives rise to oe of the most woderful equatios i mathematics. e i 0 This elegat equatio relates the five most famous umbers i mathematics 0,,, e, ad i i a sigle equatio. Show how Euler s Formula ca be used to derive this equatio.
9 0_0605.qxd /8/05 0:09 AM Page Chapter 6 Additioal Topics i Trigoometry 6.5 Exercises VOCABULARY CHECK: Fill i the blaks.. The of a complex umber a bi is the distace betwee the origi 0, 0 ad the poit a, b.. The of a complex umber z a bi is give by z rcos i si, where r is the of z ad is the of z.. Theorem states that if z rcos i si is a complex umber ad is a positive iteger, the z r cos i si. 4. The complex umber u a bi is a of the complex umber z if z u a bi. PREREQUISITE SKILLS REVIEW: Practice ad review algebra skills eeded for this sectio at I Exercises 6, plot the complex umber ad fid its absolute value.. 7i i 4. 5 i i 6. 8 i I Exercises 7 0, write the complex umber i trigoometric form z = i z = i 4 z = 6 4 z = + i I Exercises 0, represet the complex umber graphically, ad fid the trigoometric form of the umber.. i. i. i i 5. i 5 6. i 7. 5i 8. 4i i 0. i i 4. i 4 5. i 6. i 7. 5 i 8. 8 i i i I Exercises 40, represet the complex umber graphically, ad fid the stadard form of the umber.. cos 0 i si 0. 5cos 5 i si 5. cos 00 i si cos 5 i si cos i si cos 5 5 i si 7. 8 cos i si 8. 7cos 0 i si 0 9. cos8 45 i si cos0º 0 i si0º 0 I Exercises 4 44, use a graphig utility to represet the complex umber i stadard form cos 4. 9 i si 4. cos 65.5 i si cos 58º i si 58º 9 0 cos i si 5 5 I Exercises 45 ad 46, represet the powers z, z, z, ad graphically. Describe the patter. 45. z i 46. z i z 4
10 0_0605.qxd /8/05 0:09 AM Page 479 Sectio 6.5 Trigoometric Form of a Complex Number 479 I Exercises 47 58, perform the operatio ad leave the result i trigoometric form i si 4 6 cos i si cos i si 4 cos i si cos 40i si 40 cos 60i si cos 00i si cos 00 i si cos 0i si cos 00 i si cos 5 i si 5cos 0 i si 0 cos 50 i si cos 0 i si 0 cos 0 i si cos 40 i si 40 cos5 i si5 55. cos i si 5cos 4. i si cos. i si. cos 5 i si cos 0 i si 0 6cos 40 i si cos 00 i si 00 I Exercises 59 66, (a) write the trigoometric forms of the complex umbers, (b) perform the idicated operatio usig the trigoometric forms, ad (c) perform the idicated operatio usig the stadard forms, ad check your result with that of part (b). 59. i i 60. i i 6. i i 6. 4 i 6. 4i i 64. i 6 i i 4i 4 i I Exercises 67 70, sketch the graph of all complex umbers z satisfyig the give coditio cos z z I Exercises 7 88, use DeMoivre s Theorem to fid the idicated power of the complex umber.write the result i stadard form. 7. i 5 7. i 6 7. i i i i 77. 5cos 0 i si cos 50 i si cos 4 i si 80. i si 8. 5cos. i si cos 0 i si i i 85. cos 5 i si cos 0 i si cos 0 i si cos cos 8 i si I Exercises 89 04, (a) use the theorem o page 476 to fid the idicated roots of the complex umber, (b) represet each of the roots graphically, ad (c) write each of the roots i stadard form. 89. Square roots of 5cos 0 i si Square roots of 6cos 60 i si Cube roots of 9. Fifth roots of 8 cos i si cos 5 5 i si Square roots of 5i 94. Fourth roots of 65i 95. Cube roots of 5 i 96. Cube roots of 4 i 97. Fourth roots of Fourth roots of i 99. Fifth roots of 00. Cube roots of Cube roots of 5 0. Fourth roots of 4 0. Fifth roots of 8 i 04. Sixth roots of 64i
11 0_0605.qxd /8/05 0:09 AM Page Chapter 6 Additioal Topics i Trigoometry I Exercises 05, use the theorem o page 476 to fid all the solutios of the equatio ad represet the solutios graphically. 05. x 4 i x x x x 4 6i 0 0. x 6 64i 0. x i 0. x 4 i 0 Sythesis True or False? I Exercises 6, determie whether the statemet is true or false. Justify your aswer.. Although the square of the complex umber bi is give by bi b, the absolute value of the complex umber z a bi is defied as a bi a b. 4. Geometrically, the th roots of ay complex umber z are all equally spaced aroud the uit circle cetered at the origi. 5. The product of two complex umbers z r cos i si ad z r cos i si. is zero oly whe r 0 ad/or r By DeMoivre s Theorem, 4 6i 8 cos i si Give two complex umbers z r cos i si ad z r cos i si, z 0, show that z r cos z r i si. 8. Show that z rcos i si is the complex cojugate of z rcos i si. 9. Use the trigoometric forms of z ad z i Exercise 8 to fid (a) zz ad (b) zz, z Show that the egative of z rcos i si is z rcos i si.. Show that i is a sixth root of.. Show that 4 i is a fourth root of. Graphical Reasoig I Exercises ad 4, use the graph of the roots of a complex umber. (a) Write each of the roots i trigoometric form. (b) Idetify the complex umber whose roots are give. (c) Use a graphig utility to verify the results of part (b) Skills Review I Exercises 5 0, solve the right triagle show i the figure. Roud your aswers to two decimal places. B a C 5. A, a A 0, b A 4 5, c. 0. Harmoic Motio I Exercises 4, for the simple harmoic motio described by the trigoometric fuctio, fid the maximum displacemet ad the least positive value of t for which d d 6 cos. 4 t. d 4. 6 si 5 4t B 66, a.5 B 6, b. B 8 0, c 6.8 d cos t 8 d si 60t I Exercises 5 ad 6, write the product as a sum or differece si 8 cos 6. cos 5 si c b A
8.3 POLAR FORM AND DEMOIVRE S THEOREM
SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,
More informationThe Field of Complex Numbers
The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationOnestep equations. Vocabulary
Review solvig oestep equatios with itegers, fractios, ad decimals. Oestep equatios Vocabulary equatio solve solutio iverse operatio isolate the variable Additio Property of Equality Subtractio Property
More informationSection 8.3 : De Moivre s Theorem and Applications
The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =
More informationGeometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4
3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial
More informationArithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...
3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationSection 6.1 Radicals and Rational Exponents
Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More information7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b
Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationSection 9.2 Series and Convergence
Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationGrade 7. Strand: Number Specific Learning Outcomes It is expected that students will:
Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationAlgebra Work Sheets. Contents
The work sheets are grouped accordig to math skill. Each skill is the arraged i a sequece of work sheets that build from simple to complex. Choose the work sheets that best fit the studet s eed ad will
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationGCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4
GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook Alevel Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5
More informationEquation of a line. Line in coordinate geometry. Slopeintercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Pointslope form ( 點 斜 式 )
Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationf(x + T ) = f(x), for all x. The period of the function f(t) is the interval between two successive repetitions.
Fourier Series. Itroductio Whe the Frech mathematicia Joseph Fourier (76883) was tryig to study the flow of heat i a metal plate, he had the idea of expressig the heat source as a ifiite series of sie
More informationLiteral Equations and Formulas
. Literal Equatios ad Formulas. OBJECTIVE 1. Solve a literal equatio for a specified variable May problems i algebra require the use of formulas for their solutio. Formulas are simply equatios that express
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationMath 114 Intermediate Algebra Integral Exponents & Fractional Exponents (10 )
Math 4 Math 4 Itermediate Algebra Itegral Epoets & Fractioal Epoets (0 ) Epoetial Fuctios Epoetial Fuctios ad Graphs I. Epoetial Fuctios The fuctio f ( ) a, where is a real umber, a 0, ad a, is called
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationLesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationMath 475, Problem Set #6: Solutions
Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b oegative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),
More informationMESSAGE TO TEACHERS: NOTE TO EDUCATORS:
MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please fid suggested lesso plas for term 1 of MATHEMATICS Grade 12. Please ote that these lesso plas are to be used oly as a guide ad teachers
More informationM06/5/MATME/SP2/ENG/TZ2/XX MATHEMATICS STANDARD LEVEL PAPER 2. Thursday 4 May 2006 (morning) 1 hour 30 minutes INSTRUCTIONS TO CANDIDATES
IB MATHEMATICS STANDARD LEVEL PAPER 2 DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI 22067304 Thursday 4 May 2006 (morig) 1 hour 30 miutes INSTRUCTIONS TO CANDIDATES Do ot ope
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
More informationFigure 40.1. Figure 40.2
40 Regular Polygos Covex ad Cocave Shapes A plae figure is said to be covex if every lie segmet draw betwee ay two poits iside the figure lies etirely iside the figure. A figure that is ot covex is called
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (1226) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationMATH 083 Final Exam Review
MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationCounting II 3, 7 3, 2 3, 9 7, 2 7, 9 2, 9
Coutig II Sometimes we will wat to choose objects from a set of objects, ad we wo t be iterested i orderig them. For example, if you are leavig for vacatio ad you wat to pac your suitcase with three of
More informationMath 105: Review for Final Exam, Part II  SOLUTIONS
Math 5: Review for Fial Exam, Part II  SOLUTIONS. Cosider the fuctio fx) =x 3 l x o the iterval [/e, e ]. a) Fid the x ad ycoordiates of ay ad all local extrema ad classify each as a local maximum or
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 informationDivide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015
CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationMath 152 Final Exam Review
Math 5 Fial Eam Review Problems Math 5 Fial Eam Review Problems appearig o your iclass fial will be similar to those here but will have umbers ad fuctios chaged. Here is a eample of the way problems selected
More information8.5 Alternating infinite series
65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 4 diagram sheets ad a page formula sheet. Please tur over Mathematics/P DoE/Exemplar
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationREVISION SHEET FP2 (AQA) CALCULUS. x x π π. Standard Calculus of Inverse Trig and Hyperbolic Trig Functions = + arcsin x = + ar sinh x
the Further Mathematics etwork www.fmetwork.org.uk V 07 REVISION SHEET FP (AQA) CALCULUS The mai ideas are: Calculus usig iverse trig fuctios & hperbolic trig fuctios ad their iverses. Calculatig arc legths.
More information3.2 Introduction to Infinite Series
3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are
More informationUnderstanding Rational Exponents and Radicals
x Locker LESSON. Uderstadig Ratioal Expoets ad Radicals Name Class Date. Uderstadig Ratioal Expoets ad Radicals Essetial Questio: How are radicals ad ratioal expoets related? A..A simplify umerical radical
More informationSolving DivideandConquer Recurrences
Solvig DivideadCoquer Recurreces Victor Adamchik A divideadcoquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More informationMath Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:
Math 355  Discrete Math 4.14.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let
More informationTangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem
116 Taget circles i the ratio 2 : 1 Hiroshi Okumura ad Masayuki Wataabe I this article we cosider the followig old Japaese geometry problem (see Figure 1), whose statemet i [1, p. 39] is missig the coditio
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationAlgebra Vocabulary List (Definitions for Middle School Teachers)
Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 http://www.math.tamu.edu/~stecher/171/f02/absolutevaluefuctio.pdf
More informationTAYLOR SERIES, POWER SERIES
TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationhttp://www.webassign.net/v4cgijeff.downs@wnc/control.pl
Assigmet Previewer http://www.webassig.et/vcgijeff.dows@wc/cotrol.pl of // : PM Practice Eam () Questio Descriptio Eam over chapter.. Questio DetailsLarCalc... [] Fid the geeral solutio of the differetial
More informationSUMS OF nth POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif.
SUMS OF th OWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION N.A. Draim, Vetura, Calif., ad Marjorie Bickell Wilcox High School, Sata Clara, Calif. The quadratic equatio whose roots a r e the sum or differece
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More information