# APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line

Save this PDF as:

Size: px
Start display at page:

Download "APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line"

## Transcription

1 APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente y coorinte system clle the rel numer line or -is (see Figure D.1). The rel numer corresponing to point on the rel numer line is the coorinte of the point. As Figure D.1 shows, it is customry to ientify those points whose coorintes re integers The rel numer line Figure D Rtionl numers Figure D Irrtionl numers Figure D.3 e π The point on the rel numer line corresponing to zero is the origin n is enote y. The positive irection (to the right) is enote y n rrowhe n is the irection of incresing vlues of. Numers to the right of the origin re positive. Numers to the left of the origin re negtive. The term nonnegtive escries numer tht is either positive or zero. The term nonpositive escries numer tht is either negtive or zero. Ech point on the rel numer line correspons to one n only one rel numer, n ech rel numer correspons to one n only one point on the rel numer line. This type of reltionship is clle one-to-one-corresponence. Ech of the four points in Figure D. correspons to rtionl numer one tht cn e written s the rtio of two integers. Note tht 4. 9 n Rtionl numers cn e represente either y terminting ecimls such s.4, 1 or y repeting ecimls such s Rel numers tht re not rtionl re irrtionl. Irrtionl numers cnnot e represente s terminting or repeting ecimls. In computtions, irrtionl numers re represente y eciml pproimtions. Here re three fmilir emples e (See Figure D.3.) D1

2 D APPENDIX D Preclculus Review 1 < if n only if lies to the left of. Figure D.4 Orer n Inequlities One importnt property of rel numers is tht they re orere. If n re rel numers, is less thn if is positive. This orer is enote y the inequlity <. This reltionship cn lso e escrie y sying tht is greter thn n writing >. When three rel numers,, n c re orere such tht < n < c, you sy tht is etween n c n < < c. Geometriclly, < if n only if lies to the left of on the rel numer line (see Figure D.4). For emple, 1 < ecuse 1 lies to the left of on the rel numer line. The following properties re use in working with inequlities. Similr properties re otine if < is replce y n > is replce y. (The symols n men less thn or equl to n greter thn or equl to, respectively.) Properties of Inequlities Let,, c,, n k e rel numers. 1. If < n < c, then < c. Trnsitive Property. If < n c <, then c <. A inequlities. 3. If <, then k < k. A constnt. 4. If < n k >, then k < k. Multiply y positive constnt.. If < n k <, then k > k. Multiply y negtive constnt. NOTE Note tht you reverse the inequlity when you multiply the inequlity y negtive numer. For emple, if < 3, then 4 > 1. This lso pplies to ivision y negtive numer. So, if > 4, then <. A set is collection of elements. Two common sets re the set of rel numers n the set of points on the rel numer line. Mny prolems in clculus involve susets of one of these two sets. In such cses, it is convenient to use set nottion of the form {: conition on }, which is re s follows. The set of ll such tht certin conition is true. { : conition on } For emple, you cn escrie the set of positive rel numers s : >. Set of positive rel numers Similrly, you cn escrie the set of nonnegtive rel numers s :. Set of nonnegtive rel numers The union of two sets A n B, enote y A B, is the set of elements tht re memers of A or B or oth. The intersection of two sets A n B, enote y A B, is the set of elements tht re memers of A n B. Two sets re isjoint if they hve no elements in common.

3 APPENDIX D.1 Rel Numers n the Rel Numer Line D3 The most commonly use susets re intervls on the rel numer line. For emple, the open intervl, : < < Open intervl is the set of ll rel numers greter thn n less thn, where n re the enpoints of the intervl. Note tht the enpoints re not inclue in n open intervl. Intervls tht inclue their enpoints re close n re enote y, :. Close intervl The nine sic types of intervls on the rel numer line re shown in the tle elow. The first four re oune intervls n the remining five re unoune intervls. Unoune intervls re lso clssifie s open or close. The intervls, n, re open, the intervls, n, re close, n the intervl, is consiere to e oth open n close. Intervls on the Rel Numer Line Intervl Nottion Set Nottion Grph Boune open intervl, : < < Boune close intervl, : Boune intervls (neither open nor close),, : < : < Unoune open intervls,, : < : > Unoune close intervls,, : : Entire rel line, : is rel numer NOTE The symols n refer to positive n negtive infinity, respectively. These symols o not enote rel numers. They simply enle you to escrie unoune conitions more concisely. For instnce, the intervl, is unoune to the right ecuse it inclues ll rel numers tht re greter thn or equl to.

4 D4 APPENDIX D Preclculus Review EXAMPLE 1 Liqui n Gseous Sttes of Wter Descrie the intervls on the rel numer line tht correspon to the temperture (in egrees Celsius) for wter in. liqui stte.. gseous stte. Solution. Wter is in liqui stte t tempertures greter thn C n less thn 1 C, s shown in Figure D.()., 1 : < < 1. Wter is in gseous stte (stem) t tempertures greter thn or equl to s shown in Figure D.(). 1, : 1 1 C, () Temperture rnge of wter (in egrees Celsius) () Temperture rnge of stem (in egrees Celsius) Figure D. A rel numer is solution of n inequlity if the inequlity is stisfie (is true) when is sustitute for. The set of ll solutions is the solution set of the inequlity. EXAMPLE Solving n Inequlity Solve < 7. Solution < 7 Write originl inequlity. < 7 A to ech sie. < 1 < 1 < 6 Divie ech sie y. The solution set is, 6. If, then ( ) If, then () NOTE In Emple, ll five inequlities liste s steps in the solution re clle equivlent ecuse they hve the sme solution set. If , then ( 7) Checking solutions of Figure D.6 9 < Once you hve solve n inequlity, check some -vlues in your solution set to verify tht they stisfy the originl inequlity. You shoul lso check some vlues outsie your solution set to verify tht they o not stisfy the inequlity. For emple, Figure D.6 shows tht when or the inequlity < 7 is stisfie, ut when 7 the inequlity < 7 is not stisfie.

5 APPENDIX D.1 Rel Numers n the Rel Numer Line D EXAMPLE 3 Solving Doule Inequlity Solve 3 1. Solution [, 1] Solution set of 3 1 Figure D Write originl inequlity. 3 1 Sutrct from ech prt. 1 1 Divie ech prt y n reverse oth inequlities. 1 The solution set is, 1, s shown in Figure D.7. The inequlities in Emples n 3 re liner inequlities tht is, they involve first-egree polynomils. To solve inequlities involving polynomils of higher egree, use the fct tht polynomil cn chnge signs only t its rel zeros (the -vlues tht mke the polynomil equl to zero). Between two consecutive rel zeros, polynomil must e either entirely positive or entirely negtive. This mens tht when the rel zeros of polynomil re put in orer, they ivie the rel numer line into test intervls in which the polynomil hs no sign chnges. So, if polynomil hs the fctore form r r 1 < r < r 3 <... 1 r... r n, < r n the test intervls re, r 1, r 1, r,..., r n 1, r n, n r n,. To etermine the sign of the polynomil in ech test intervl, you nee to test only one vlue from the intervl. EXAMPLE 4 Solving Qurtic Inequlity Solve < 6. Choose ( 3)( 3. ) Choose 4. ( 3( ) ) Solution < 6 6 < 3 < Write originl inequlity. Write in generl form. Fctor. 3 Co h ose ( 3) ( Testing n intervl Figure D.8 1. ) 3 4 The polynomil 6 hs n 3 s its zeros. So, you cn solve the inequlity y testing the sign of 6 in ech of the test intervls,,, 3, n 3,. To test n intervl, choose ny numer in the intervl n compute the sign of 6. After oing this, you will fin tht the polynomil is positive for ll rel numers in the first n thir intervls n negtive for ll rel numers in the secon intervl. The solution of the originl inequlity is therefore, 3, s shown in Figure D.8.

6 D6 APPENDIX D Preclculus Review Asolute Vlue n Distnce If is rel numer, the solute vlue of is,, The solute vlue of numer cnnot e negtive. For emple, let 4. Then, ecuse 4 <, you hve 4 if if < Rememer tht the symol oes not necessrily men tht is negtive. Opertions with Asolute Vlue Let n e rel numers n let n e positive integer. 1.., n n NOTE You re ske to prove these properties in Eercises 73, 7, 76, n 77. Properties of Inequlities n Asolute Vlue Let n e rel numers n let k e positive rel numer. 1.. k if n only if k k. 3. k if n only if k or k. 4. Tringle Inequlity: Properties n 3 re lso true if is replce y <. EXAMPLE Solving n Asolute Vlue Inequlity Solve 3. 1 units units 4 Solution set of 3 Figure D Solution Using the secon property of inequlities n solute vlue, you cn rewrite the originl inequlity s oule inequlity. 3 Write s oule inequlity A 3 to ech prt. 1 The solution set is 1,, s shown in Figure D.9.

7 APPENDIX D.1 Rel Numers n the Rel Numer Line D7 EXAMPLE 6 A Two-Intervl Solution Set (, ) (1, ) Solution set of Figure D.1 > 3 + Solution set of + Solution set of Figure D.11 Solve > 3. Solution Using the thir property of inequlities n solute vlue, you cn rewrite the originl inequlity s two liner inequlities. < 3 or > 3 < > 1 The solution set is the union of the isjoint intervls, n 1,, s shown in Figure D.1. Emples n 6 illustrte the generl results shown in Figure D.11. Note tht if >, the solution set for the inequlity is single intervl, wheres the solution set for the inequlity is the union of two isjoint intervls. The istnce etween two points n on the rel numer line is given y. The irecte istnce from to is n the irecte istnce from to is, s shown in Figure D.1. Distnce etween n Directe istnce from to Directe istnce from to Figure D.1 EXAMPLE 7 Distnce on the Rel Numer Line Distnce = Figure D The istnce etween 3 n 4 is (See Figure D.13.). The irecte istnce from 3 to 4 is c. The irecte istnce from 4 to 3 is or The mipoint of n intervl with enpoints n is the verge vlue of n. Tht is, Mipoint of intervl,. To show tht this is the mipoint, you nee only show tht is equiistnt from n.

8 D8 APPENDIX D Preclculus Review EXERCISES FOR APPENDIX D.1 In Eercises 1 1, etermine whether the rel numer is rtionl or irrtionl In Eercises 11 14, write the repeting eciml s rtio of two integers using the following proceure. If , then Sutrcting the first eqution from the secon prouces or Given <, etermine which of the following re true. () < (c) > (e) > () < () 1 < 1 (f) < 16. Complete the tle with the pproprite intervl nottion, set nottion, n grph on the rel numer line. Intervl Nottion, 4 1, 7 In Eercises 17, verlly escrie the suset of rel numers represente y the inequlity. Sketch the suset on the rel numer line, n stte whether the intervl is oune or unoune < < < 8 In Eercises 1 4, use inequlity n intervl nottion to escrie the set. 1. y is t lest 4.. q is nonnegtive. Set Nottion : 3 11 Grph 3. The interest rte r on lons is epecte to e greter thn 3% n no more thn 7%. 4. The temperture T is forecst to e ove 9 F toy. < In Eercises 44, solve the inequlity n grph the solution on the rel numer line < 3 < < 9. 3 > 3. > < > > 3 3. <, > < < 9 3 In Eercises 4 48, fin the irecte istnce from to, the irecte istnce from to, n the istnce etween n () 16, 7 () 16, () 9.34,.6 () 16 11, 7 In Eercises 49 4, use solute vlue nottion to efine the intervl or pir of intervls on the rel numer line = = = = = = = 3 = = = = =

9 APPENDIX D.1 Rel Numers n the Rel Numer Line D9 3. () All numers tht re t most 1 units from 1. () All numers tht re t lest 1 units from () y is t most two units from. () y is less thn units from c. In Eercises 8, fin the mipoint of the intervl () 7, 1 () 8.6, () 6.8, 9.3 () 4.6, Profit The revenue R from selling units of prouct is R 11.9 n the cost C of proucing units is C 9 7. = = = = To mke (positive) profit, R must e greter thn C. For wht vlues of will the prouct return profit? 6. Fleet Costs A utility compny hs fleet of vns. The nnul operting cost C (in ollrs) of ech vn is estimte to e C.3m 3 where m is mesure in miles. The compny wnts the nnul operting cost of ech vn to e less thn \$1,. To o this, m must e less thn wht vlue? 61. Fir Coin To etermine whether coin is fir (hs n equl proility of lning tils up or hes up), you toss the coin 1 times n recor the numer of hes. The coin is eclre unfir if For wht vlues of will the coin e eclre unfir? 6. Dily Prouction The estimte ily oil prouction p t refinery is p,, < 1, where p is mesure in rrels. Determine the high n low prouction levels. In Eercises 63 n 64, etermine which of the two rel numers is greter. 63. () or () 11 or 73 () or () or 6. Approimtion Powers of 1 Light trvels t the spee of meters per secon. Which est estimtes the istnce in meters tht light trvels in yer? () (c) () () Writing The ccurcy of n pproimtion to numer is relte to how mny significnt igits there re in the pproimtion. Write efinition for significnt igits n illustrte the concept with emples. True or Flse? In Eercises 67 7, etermine whether the sttement is true or flse. If it is flse, eplin why or give n emple tht shows it is flse. 67. The reciprocl of nonzero integer is n integer. 68. The reciprocl of nonzero rtionl numer is rtionl numer. 69. Ech rel numer is either rtionl or irrtionl. 7. The solute vlue of ech rel numer is positive. 71. If <, then. 7. If n re ny two istinct rel numers, then < or >. In Eercises 73 8, prove the property Hint: 1 7., n n, n 1,, 3, k, if n only if k k, 8. if n only if k or k, k k >. 81. Fin n emple for which n n emple for which Then prove tht. for ll,. 8. Show tht the mimum of two numers n is given y the formul m, k >. >, Derive similr formul for min,.

### x ) dx dx x sec x over the interval (, ).

Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### set is not closed under matrix [ multiplication, ] and does not form a group.

Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

### Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

### Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

### Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

### Math 211A Homework. Edward Burkard. = tan (2x + z)

Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =

Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

### INTEGRALS. Chapter Introduction

INTEGRALS 87 hpter 7 INTEGRALS Just s mountineer clims mountin ecuse it is there, so goo mthemtics stuent stuies new mteril ecuse it is there. JAMES B. BRISTOL 7. Introuction Differentil lculus is centre

### 2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

### Chapter 9 Definite Integrals

Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

### Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

### FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

### Section 4: Integration ECO4112F 2011

Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

### Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

### B Veitch. Calculus I Study Guide

Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### Chapter 6 Continuous Random Variables and Distributions

Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete

### MATH 573 FINAL EXAM. May 30, 2007

MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.

### The Fundamental Theorem of Algebra

The Fundmentl Theorem of Alger Jeremy J. Fries In prtil fulfillment of the requirements for the Mster of Arts in Teching with Speciliztion in the Teching of Middle Level Mthemtics in the Deprtment of Mthemtics.

### Interpreting Integrals and the Fundamental Theorem

Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

### Sturm-Liouville Theory

LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

### How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

### Chapter Five - Eigenvalues, Eigenfunctions, and All That

Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl

### The Trapezoidal Rule

_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

### Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

### STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

### This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

### Algebra 2 Semester 1 Practice Final

Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers

### What Is Calculus? 42 CHAPTER 1 Limits and Their Properties

60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

### Homework Solution - Set 5 Due: Friday 10/03/08

CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

### CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS

CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS LEARNING OBJECTIVES After stuying this chpter, you will be ble to: Unerstn the bsics

### 8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

### Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

### MPE Review Section I: Algebra

MPE Review Section I: lger t Colordo Stte Universit, the College lger sequence etensivel uses the grphing fetures of the Tes Instruments TI-8 or TI-8 grphing clcultor. Whenever possile, the questions on

### 2.4 Linear Inequalities and Problem Solving

Section.4 Liner Inequlities nd Problem Solving 77.4 Liner Inequlities nd Problem Solving S 1 Use Intervl Nottion. Solve Liner Inequlities Using the Addition Property of Inequlity. 3 Solve Liner Inequlities

### Quadratic Equations. Brahmagupta gave. Solving of quadratic equations in general form is often credited to ancient Indian mathematicians.

9 Qudrtic Equtions Qudrtic epression nd qudrtic eqution Pure nd dfected qudrtic equtions Solution of qudrtic eqution y * Fctoristion method * Completing the squre method * Formul method * Grphicl method

### Math 017. Materials With Exercises

Mth 07 Mterils With Eercises Jul 0 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions....... Lesson

### Math 259 Winter Solutions to Homework #9

Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier

### 38 Riemann sums and existence of the definite integral.

38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

### 5: The Definite Integral

5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

### CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)

### 4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### 10. AREAS BETWEEN CURVES

. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### STRAND B: NUMBER THEORY

Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,

### LINEAR ALGEBRA APPLIED

5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### P 1 (x 1, y 1 ) is given by,.

MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

### Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

### The Trapezoidal Rule

SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion Approimte

### Improper Integrals with Infinite Limits of Integration

6_88.qd // : PM Pge 578 578 CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls Section 8.8 f() = d The unounded region hs n re of. Figure 8.7 Improper Integrls Evlute n improper integrl

### REVIEW Chapter 1 The Real Number System

Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole

### Best Approximation in the 2-norm

Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

### Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

### Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1

Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show

### Matrix & Vector Basic Linear Algebra & Calculus

Mtrix & Vector Bsic Liner lgebr & lculus Wht is mtrix? rectngulr rry of numbers (we will concentrte on rel numbers). nxm mtrix hs n rows n m columns M x4 M M M M M M M M M M M M 4 4 4 First row Secon row

### Math Calculus with Analytic Geometry II

orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

### 2. VECTORS AND MATRICES IN 3 DIMENSIONS

2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

### MTH 505: Number Theory Spring 2017

MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of \$ nd \$ s two denomintions of coins nd \$c

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

### Math 113 Exam 2 Practice

Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

### 3.1 Exponential Functions and Their Graphs

. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

### BRIEF NOTES ADDITIONAL MATHEMATICS FORM

BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht

### The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

### Section 7.1 Area of a Region Between Two Curves

Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

### Exponents and Polynomials

C H A P T E R 5 Eponents nd Polynomils ne sttistic tht cn be used to mesure the generl helth of ntion or group within ntion is life epectncy. This dt is considered more ccurte thn mny other sttistics becuse

### Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

### CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

### CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018

CS 301 Lecture 04 Regulr Expressions Stephen Checkowy Jnury 29, 2018 1 / 35 Review from lst time NFA N = (Q, Σ, δ, q 0, F ) where δ Q Σ P (Q) mps stte nd n lphet symol (or ) to set of sttes We run n NFA

### 5.1 Estimating with Finite Sums Calculus

5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during

### Name Ima Sample ASU ID

Nme Im Smple ASU ID 2468024680 CSE 355 Test 1, Fll 2016 30 Septemer 2016, 8:35-9:25.m., LSA 191 Regrding of Midterms If you elieve tht your grde hs not een dded up correctly, return the entire pper to

### Year 11 Matrices. A row of seats goes across an auditorium So Rows are horizontal. The columns of the Parthenon stand upright and Columns are vertical

Yer 11 Mtrices Terminology: A single MATRIX (singulr) or Mny MATRICES (plurl) Chpter 3A Intro to Mtrices A mtrix is escribe s n orgnise rry of t. We escribe the ORDER of Mtrix (it's size) by noting how

### Continuity. Recall the following properties of limits. Theorem. Suppose that lim. f(x) =L and lim. lim. [f(x)g(x)] = LM, lim

Recll the following properties of limits. Theorem. Suppose tht lim f() =L nd lim g() =M. Then lim [f() ± g()] = L + M, lim [f()g()] = LM, if M = 0, lim f() g() = L M. Furthermore, if f() g() for ll, then

### Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.

TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s

### Convex Sets and Functions

B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line

### We know that if f is a continuous nonnegative function on the interval [a, b], then b

1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

### ONLINE PAGE PROOFS. Anti-differentiation and introduction to integral calculus

Anti-differentition nd introduction to integrl clculus. Kick off with CAS. Anti-derivtives. Anti-derivtive functions nd grphs. Applictions of nti-differentition.5 The definite integrl.6 Review . Kick off

### PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

### u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + \$

### The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

### Precalculus Spring 2017

Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

### Math 4200: Homework Problems

Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,

### CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

### Proportions: A ratio is the quotient of two numbers. For example, 2 3

Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)

### 63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

### Math 360: A primitive integral and elementary functions

Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

### Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

### Chapter 2. Random Variables and Probability Distributions

Rndom Vriles nd Proilit Distriutions- 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts

### Lecture 2 : Propositions DRAFT

CS/Mth 240: Introduction to Discrete Mthemtics 1/20/2010 Lecture 2 : Propositions Instructor: Dieter vn Melkeeek Scrie: Dlior Zelený DRAFT Lst time we nlyzed vrious mze solving lgorithms in order to illustrte

### Review Factoring Polynomials:

Chpter 4 Mth 0 Review Fctoring Polynomils:. GCF e. A) 5 5 A) 4 + 9. Difference of Squres b = ( + b)( b) e. A) 9 6 B) C) 98y. Trinomils e. A) + 5 4 B) + C) + 5 + Solving Polynomils:. A) ( 5)( ) = 0 B) 4

### Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!

Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite: