Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.


 Karen Manning
 2 years ago
 Views:
Transcription
1 Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E: n m = n m Theorem F: 0 = Theorem G: n = n Theorem H: n/m = m n Like ll theorems, these do not come out of nowhere. They come from definition nd logicl deduction. Let us strt, then, with definition. Wht is exponentition, nywy? Wht do we men when we write number with superscript? I contend tht we men something like repeted multipliction. When we write, for exmple, 5 4, we men this simply s more convenient wy of writing (or, even more simply, 625). Formlly, let s define exponentition in the following wy 2 : n (The tripleequlssign here is used to show tht this is definition tht this eqution is not the consequence of some erlier theorem or xiom, but tht with it we re defining wht we men by n.) Let us prove these theorems. Theorem A: n+m = n m Proof: We ll strt with the left side of the eqution, pply the definition of exponentition, do some lgebr, nd eventully end up with the right side. n+m = n+ (by the definition of exponentition) = (} {{ } )( } {{ } ) (becuse multipliction is ssocitive) = n m (pplying the definition of exponentition gin) (done! this stylized A is my endofproof symbol.) Theorem B: ( n ) m = nm Proof: To prove this, we ll need to pply the definition of exponentition twice. (Well, ctully three times, but the lst time doesn t count.) For the most prt. If you re mthemticin, we could hve lengthy converstion bout everything I sy on this sheet, but for our purposes this definition is sufficient. If you disgree (nd this is in n imginry converstion between myself nd mthemticin), I will sy but this: do you believe we should tech students Dedekind cuts before we let them utter the words, rel number? 2 Note tht we hven t put ny specifictions on wht nd n cn nd/or should be must they be integers? rel numbers? complex numbers? The wy our definition works, wht with the business, it must only hold for n being positive integer, but these theorems hold for n being ny rel number. Wht we ll see in our derivtions is tht, even though we strt by considering n only s positive integer, we cn extend our ide of exponentition in very logicl wy such tht it holds for ny rel number. But this is in footnote for reson.
2 ( n ) m = (} {{ } ) m (by definition) = (} {{ } ) (} {{ } ) ( } {{ } ) (by definition, gin) } {{ } = n* (multipliction) = nm (finish by repplying the definition in reverse) Theorem C: (b) n = n b n Proof: See pttern? We ll pply the definition of exponentition, do some lgebr, nd eventully get wht we wnt. (b) n = } b b {{ b } (by definition) = (} {{ } ) (b} b {{ } b) (multipliction is commuttive we cn rerrnge) = n b n (definition) ( ) n n Theorem D: = b b n Proof: How do you think this should go? Theorem E: n m = n m. Proof: This is where things strt to get interesting. The cler first step is to write out the entire quntity: n m = But where we go from here will depend on the reltive size of n nd m. n could be bigger thn m, it could be smller, or the two could be equl. Let s strt with the first cse tht of n > m. In tht cse, both the top nd the bottom will hve t lest m of the s, nd the top will hve more it ll hve n m more (becuse m + (n m) = n). Put 2
3 differently, we cn think of our frction s being something like this: = n m = n If you re confused, count up the totl number of s on top there should be n of them, just like wht we strted with. And then wit second! There re m s on top, nd m s on the bottom! We cn cncel them out! And we just end up with: n = Which, by definition, is just n m. But wht bout these other two cses? First of ll, wht hppens in the cse tht m = n? Well, if m = n, then n will just equl m n (or n m either wy is the sme). But tht clerly is just equl to. m And I hope you will gree tht if n nd m re equl, then n m = 0, nd tht then n m = 0. So clerly, then, since we wnt n = n m, then 0 =. (Which should mke sense., fter ll, is the multiplictive m identity, so if we multiply something no times, we should end up not with zero, but with one. Think bout how if you hve frction in which everything cncels out, you hve, not 0. Sme sort of del.) Finlly. Wht if n < m? Then (by the sme rgument s the first cse) we hve something like: = n m = But wit! We cn cncel things here, too! We get: = } {{ } m = } {{ } m n m But this isn t quite wht we wnt. We wnt n m. So let s define wy, our eqution here will become: = m n = (m n) = m+n = n m stuff s being equl to stuff. Tht Excellent. This hs been long discussion, so let s review wht we ve done. We ve extended our ide of n exponent from just the positive integers to ll integers (i.e., we cn now exponentite by 0 nd negtive numbers!) But in order to do this, we hd to mke slight extension of our definition. Our originl definition only considered the cse of n exponent being positive integer; in the course of this proof, we discovered nturl wy to extend tht definition to 0 nd the negtives. Thus this hs been somewht more thn just proof it hs been prtly proof (for n > m), nd prtly redefinition. Theorem F: 0 = Proof: We lredy discussed this, in the proof of E Theorem G: n = n Proof: Agin, we lredy discussed this in the proof of E Note how my writing style is bit different in this proof thn in the previous ones I m mking n rgument in prose rther thn in nice, clen, twocolumn formt. Tht s OK. Mny mth textbooks do the sme thing, nd it s the job of the reder (nd wht job it is!) to trnslte tht into more cler, more symbolic formt if necessry.
4 Theorem H: n/m = m n Proof: So fr, we lredy know how to del with exponents when the exponents re positive integers (thnks to our definition), zero (thnks to F), or negtive integers (thnks to G). But wht if we wnt to move beyond the integers nd hve exponents tht re rtionl numbers (i.e., frctions)? This theorem will tell us how to think bout tht. First of ll, I should point out tht we don t relly hve solid definition of wht we men by stuff. (From technicl, mthemticl stndpoint, wht we re doing here is defining wht we men by such nottion.) But intuitively, I hope you gree tht the bsic ide of rdicl/root is tht if I hve n nth root of something, nd I multiply n of those nth roots, I end up with my originl thing. Mening the thing under the rdicl sign. Right? Good. So imgine we hve /m. (Cn we do this? Let s ssume we cn.) Now imgine we hve m copies of it, ll multiplied together: /m /m /m /m By the definition of exponentition, this must be ( /m ) m. But by B, we cn simplify this: /m m = m/m = In other words, whtever this /m is, if we tke m copies of it, we get. So this is just root! (The mth root, to be specific). Formlly: /m = m Wht bout the rest (the n/m prt?) Imgine we hve n/m. Becuse of Theorem B gin, we cn write this s ( n ) /m. Which, becuse of wht we just proved, is m n. So now we know how to exponentite by ny rtionl number. Hoory! Evlute ech of the following expressions. Problems ( / /2 ) 6. 6 / / 8. ( 5 0) 2/ 9. ( 7 /) /4 (. 27 ( ) / ) / ( ( /4 ) /2 ) 2/ 6. ( 2 0) / ( 0 + ) / /2 2. ( 26 ) /2 8 / ( 8 24, ) ( ) /2 27. ( + 2) / 9 /2 27 / / /2 2 0 ( 2 ) /2 (9 4 ) 27
5 Simplify ech of the following expressions so tht they re written only with positive exponents nd no rdicls y x 8 6. x x y 2 y 8. (x 2 y)(4x 5 y 4 ) 9. (2x y) ( b) 4. (6x 2 ) /2 42. (8y 6 x ) / 4. (4x )(2x) x 2 x 2 y ( ) 2 2x y 8x 2 9xy 47. (27x y 9 ) / r 8 s b x 4 y (4x + 2y) b + ( + b) b 5. (7) 2 (5b) /2 (5) /2 (7b) 4 (6) /2 b b /2 ( ) 5/9 r 2/ 55. s /5 56. (c 2/5 d 2/ )(c 6 d ) 4/ 57. (2) /2 (b) 2 (4) /5 (4) /2 (b) 2 (2) /5 58. ( x2 ) /x (b x ) x b x c (c 5/6 ) 42 (c 5 ) 2/ Prove tht ech of the following equtions re true. (How do you do this? Think of it s being like twocolumn proof in geometry. Strt with one side of the eqution nd stepbystep pply exponentition lws until you end up with the other side of the eqution.) 6. b 2 b /2 = b 62. c 2 d 6 4c d 4 = d5 4c 6. b 7 ( b) 4 = 4 5
Algebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationAddition and subtraction of rational expressions
Lecture 5. Addition nd subtrction of rtionl expressions Two rtionl expressions in generl hve different denomintors, therefore if you wnt to dd or subtrct them you need to equte the denomintors first. The
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More informationNotes for Thurs 8 Sept Calculus II Fall 2005 New York University Instructor: Tyler Neylon Scribe: Kelsey Williams
Notes for Thurs 8 Sept Clculus II Fll 00 New York University Instructor: Tyler Neylon Scribe: Kelsey Willims 8. Integrtion by Prts This section is primrily bout the formul u dv = uv v ( ) which is essentilly
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More information1.2 The Integers and Rational Numbers
.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationExponents base exponent power exponentiation
Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily
More information2 If a branch is prime, no other factors
Chpter 2 Multiples, nd primes 59 Find the prime of 50 by drwing fctor tree. b Write 50 s product of its prime. 1 Find fctor pir of the given 50 number nd begin the fctor tree (50 = 5 10). 5 10 2 If brnch
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationIn this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationIntroduction to Mathematical Reasoning, Saylor 111
Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is
More informationNumber Systems & Working With Numbers
Presenting the Mths Lectures! Your best bet for Qunt... MATHS LECTURE # 0 Number Systems & Working With Numbers System of numbers.3 0.6 π With the help of tree digrm, numbers cn be clssified s follows
More informationMechanics Cycle 1 Chapter 5. Chapter 5
Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationRational Numbers  Grade 10 [CAPS]
OpenStxCNX module: m848 Rtionl Numers  Grde 0 [CAPS] Free High School Science Texts Project Bsed on Rtionl Numers y Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More informationhas the desired form. On the other hand, its product with z is 1. So the inverse x
First homework ssignment p. 5 Exercise. Verify tht the set of complex numers of the form x + y 2, where x nd y re rtionl, is sufield of the field of complex numers. Solution: Evidently, this set contins
More informationnotation, simplifying expressions
I I. Algebr: exponents, scientific nottion, simplifying expressions  For more prctice problems nd detiled written explntions, see the
More informationScalar Line Integrals
Mth 3B Discussion Session Week 5 Notes April 6 nd 8, 06 This week we re going to define new type of integrl. For the first time, we ll be integrting long something other thn Eucliden spce R n, nd we ll
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More information1. The leves re either lbeled with sentences in ;, or with sentences of the form All X re X. 2. The interior leves hve two children drwn bove them) if
Q520 Notes on Nturl Logic Lrry Moss We hve seen exmples of wht re trditionlly clled syllogisms lredy: All men re mortl. Socrtes is mn. Socrtes is mortl. The ide gin is tht the sentences bove the line should
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationRational Expressions
C H A P T E R Rtionl Epressions nformtion is everywhere in the newsppers nd mgzines we red, the televisions we wtch, nd the computers we use. And I now people re tlking bout the Informtion Superhighwy,
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationTuring Machine Extensions
Red K & S 4.3.1, 4.4. Do Homework 19. Turing Mchine Extensions Turing Mchine Definitions An lterntive definition of Turing mchine: (K, Σ, Γ, δ, s, H): Γ is finite set of llowble tpe symbols. One of these
More informationLecture 3 Basic Probability and Statistics
Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The
More informationWell say we were dealing with a weak acid K a = 1x10, and had a formal concentration of.1m. What is the % dissociation of the acid?
Chpter 9 Buffers Problems 2, 5, 7, 8, 9, 12, 15, 17,19 A Buffer is solution tht resists chnges in ph when cids or bses re dded or when the solution is diluted. Buffers re importnt in Biochemistry becuse
More information19. The FermatEuler Prime Number Theorem
19. The FermtEuler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout
More informationSolving Linear Equations  Formulas
1. Solving Liner Equtions  Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationMatrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA
CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationWritten Homework 6 Solutions
Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f
More informationMATLAB: Mfiles; Numerical Integration Last revised : March, 2003
MATLAB: Mfiles; Numericl Integrtion Lst revised : Mrch, 00 Introduction to Mfiles In this tutoril we lern the bsics of working with Mfiles in MATLAB, so clled becuse they must use.m for their filenme
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More informationFor the Final Exam, you will need to be able to:
Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny
More informationAe2 Mathematics : Fourier Series
Ae Mthemtics : Fourier Series J. D. Gibbon (Professor J. D Gibbon, Dept of Mthemtics j.d.gibbon@ic.c.uk http://www.imperil.c.uk/ jdg These notes re not identicl wordforword with my lectures which will
More information1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE
PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the smples, work the problems, then check your nswers t the end of ech topic. If you don t get the nswer given, check your work nd look
More informationYour Thoughts. Does the moment of inertia play a part in determining the amount of time it takes an object to get to the bottom of an incline.
Your Thoughts Suppose bll rolls down rmp with coefficient of friction just big enough to keep the bll from slipping. An identicl bll rolls down n identicl rmp with coefficient of friction of. Do both blls
More informationSection 4.3. By the Mean Value Theorem, for every i = 1, 2, 3,..., n, there exists a point c i in the interval [x i 1, x i ] such that
Difference Equtions to Differentil Equtions Section 4.3 The Fundmentl Theorem of Clculus We re now redy to mke the longpromised connection between differentition nd integrtion, between res nd tngent lines.
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationfraction arithmetic. For example, consider this problem the 1995 TIMSS Trends in International Mathematics and Science Study:
In recent yers, mthemtics eductors hve begun to relize tht understnding frctions nd frctionl rithmetic is the gtewy to dvnced high school mthemtics. 1 Yet, US students continue to do poorly when rnked
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More informationRepeated multiplication is represented using exponential notation, for example:
Appedix A: The Lws of Expoets Expoets re shorthd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you
More informationTwo special Righttriangles 1. The
Mth Right Tringle Trigonometry Hndout B (length of )  c  (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Righttringles. The
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationEntry Test: Mathematics for Management and Economics
Entry Test: Mthemtics for Mngement nd Economics BM.0 Introduction BM.1 Sets BM. Assertions 4 BM. Propositionl Formuls 5 BM.4.1 Algebric Lws 6 BM.4. Equivlence of Equtions 8 BM.4. Trnsforming Equtions 9
More informationBasics of Logic Design: Boolean Algebra, Logic Gates. Administrative
Bsics of Logic Design: Boolen Alger, Logic Gtes Computer Science 104 Administrtive Homework #3 Due Sundy Midterm I Mondy in clss, closed ook, closed notes Ø Will provide IA32 instruction set hndout Ø Lst
More informationArc Length. P i 1 P i (1) L = lim. i=1
Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationON THE FRAMESTEWART ALGORITHM FOR THE TOWER OF HANOI
ON THE FRAMESTEWART ALGORITHM FOR THE TOWER OF HANOI MICHAEL RAND 1. Introduction The Tower of Hnoi puzzle ws creted over century go by the number theorist Edourd Lucs [, 4], nd it nd its vrints hve chllenged
More informationAlgorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
More informationNUMBER SYSTEMS CHAPTER 1. (A) Main Concepts and Results
CHAPTER NUMBER SYSTEMS Min Concepts nd Results Rtionl numbers Irrtionl numbers Locting irrtionl numbers on the number line Rel numbers nd their deciml expnsions Representing rel numbers on the number line
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More information2.4 Circular Waveguide
.4 Circulr Wveguide y x Figure.5: A circulr wveguide of rdius. For circulr wveguide of rdius (Fig..5, we cn perform the sme sequence of steps in cylindricl coordintes s we did in rectngulr coordintes to
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationEducation Spending (in billions of dollars) Use the distributive property.
0 CHAPTER Review of the Rel Number System 96. An pproximtion of federl spending on eduction in billions of dollrs from 200 through 2005 cn be obtined using the e xpression y = 9.0499x  8,07.87, where
More informationSample Problems. Practice Problems
Lecture Notes Comple Frctions pge Smple Problems Simplify ech of the following epressions.. +. +. + 8. b b... 7. + + 9. y + y 0. y Prctice Problems Simplify ech of the following epressions...... 8 + +
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationTriangles, Altitudes, and Area Instructor: Natalya St. Clair
Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes Tringles, ltitudes, nd re Instructor: Ntly St. lir *Note: This M session is inspired from vriety of sources, including wesomemth, reteem Mth Zoom,
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationThe ISLM Model. Underlying model due to John Maynard Keynes Model representation due to John Hicks Used to make predictions about
The SLM Model Underlying model due to John Mynrd Keynes Model representtion due to John Hicks Used to mke predictions bout nterest rtes Aggregte spending ( ggregte output) mportnt ssumption: price is
More informationThe Parallelogram Law. Objective: To take students through the process of discovery, making a conjecture, further exploration, and finally proof.
The Prllelogrm Lw Objective: To tke students through the process of discovery, mking conjecture, further explortion, nd finlly proof. I. Introduction: Use one of the following Geometer s Sketchpd demonstrtion
More informationWords Symbols Diagram. abcde. a + b + c + d + e
Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationSquare & Square Roots
Squre & Squre Roots Squre : If nuber is ultiplied by itself then the product is the squre of the nuber. Thus the squre of is x = eg. x x Squre root: The squre root of nuber is one of two equl fctors which
More information1. Inverse of a tridiagonal matrix
PréPublicções do Deprtmento de Mtemátic Universidde de Coimbr Preprint Number 05 16 ON THE EIGENVALUES OF SOME TRIDIAGONAL MATRICES CM DA FONSECA Abstrct: A solution is given for problem on eigenvlues
More informationRational Numbers and Decimal Representation
0 CHAPTER The Rel Numbers nd Their Representtions. Rtionl Numbers nd Deciml Representtion Properties nd Opertions The set of rel numbers is composed of two importnt mutully exclusive subsets: the rtionl
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationChapter 2 The Number System (Integers and Rational Numbers)
Chpter 2 The Number System (Integers nd Rtionl Numbers) In this second chpter, students extend nd formlize their understnding of the number system, including negtive rtionl numbers. Students first develop
More information