Unit 6: Exponents and Radicals

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Unit 6: Exponents and Radicals"

Transcription

1 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): - positive nd negtive whole numers. {,,,,,, 0,,,,,, } Rtionl Numers (Q): - numers tht cn e turned into frction, where, I, nd 0. - include ll Terminting or Repeting Decimls. - include ll Nturl Numers, Whole Numers nd Integers. - include n perfect roots (rdicls). ) Terminting Decimls: decimls tht stops ) Repeting Decimls: decimls tht repets in pttern nd goes on. 0.. c) Perfect Roots: - rdicls when evluted will result in either Terminting or repeting decimls, or 0. ± 0. frctions, where, I, nd ± 0... ± ± Irrtionl Numers (Q ): - numers tht CANNOT e turned into frction, where, I, nd 0. - include ll non-terminting, non-repeting decimls. - include n non-perfect roots (rdicls). ) Non-terminting, Non-repeting Decimls: - decimls tht do not repet ut go on nd on. π. 0. ) Non-Perfect Roots: rdicls when evluted will result in Non-Terminting, Non-Repeting decimls. ± ± ± Rel Numers (R): - n numers tht cn e put on numer line. - include ll nturl numers, whole numers, integers, rtionl nd irrtionl numers. Asolute Vlue : - the positive vlue of. () + Pge. Coprighted Griel Tng, B.Ed., BSc.

2 Pure Mth 0 Notes Eponents nd Rdicls In generl, we cn displ the reltionships etween ll tpes of rel numers in digrm. Rel Numers (R) Q Q I W N ) All Nturl Numers elong to the set of Whole Numer. N W ) All Nturl nd Whole Numers elong to the set of Integers. N nd W I c) All Nturl, Whole Numers nd Integers elong to the set of Rtionl Numers. N, W nd I Q d) Rtionl Numers nd Irrtionl Numers do NOT elong to ech other. (You cn hve oth tpes t the sme time). Q Q e) All Nturl, Whole Numers, Integers, Rtionl nd Irrtionl Numers elong to the set of Rel Numers. N, W, I, Q nd Q R Emple : Clssif the following numers... c.. d.. I, Q nd R N, W, I, Q nd R Q nd R Q nd R e. f.. g. h. Q nd R Q nd R Q nd R Q nd R i) 0. j) k) l) Q nd R N, W, I, Q nd R N, W, I, Q nd R Q nd R Coprighted Griel Tng, B.Ed., B.Sc. Pge.

3 Eponents nd Rdicls Pure Mth 0 Notes Inequlities Smols > Greter thn < Less thn lower upper lower < < upper lower nd upper Greter thn or equl to Less thn or equl to NOT Equl to Menings is etween the lower nd upper oundries (inclusive). is etween the lower nd upper oundries (eclusive). is less thn the lower oundr nd is greter thn the upper oundr (inclusive). < lower nd > upper is less thn the lower oundr nd is greter thn the upper oundr (eclusive). Emple : Grph the following inequlities. ) n > ) 0 0 c) <. d) m π. π 0 0 e) r f) < w 0 0 g) > nd 0 h) t 0 0 Pge. Coprighted Griel Tng, B.Ed., BSc.

4 Pure Mth 0 Notes Eponents nd Rdicls (AP) Emple : Grph the following inequlities. ) <. ) r. Sme s. Sme s r < nd r > 0 0 c) > nd 0 Sme s 0 < r 0 Emple : Convert the following decimls to frctions lgericll. ) 0. ). Let 0. (To cncel out the repeting Let. 0. decimls, we hve to move 00. the deciml plce to the right, which mens 0) (Move the deciml plces to the right will line up the repeting decimls) c) 0. d). Let 0. First, ignore the Let negtive sign. 0. (Move the deciml 000. plces to the right will line up the repeting 000. decimls) Put the negtive sign ck! 0 0 Put the negtive sign ck! First, ignore the negtive sign. (Move the deciml plce to the right will mke the repeting decimls pper right fter the deciml point.) Homework Assignments Regulr: pg. to # to,,,,, nd AP: pg. to # to, to, to, nd Coprighted Griel Tng, B.Ed., B.Sc. Pge.

5 Eponents nd Rdicls Pure Mth 0 Notes -: Evluting Irrtionl Numers Rdicls: - the result of numer fter root opertion. Rdicl Sign: - the mthemticl smol. Rdicnd: - the numer inside rdicl sign. inde n rdicl sign rdicnd Inde: - the smll numer to the left of the rdicl sign indicting how mn times numer (nswer to the rdicl) hs to multipl itself to equl to the rdicnd. squre root cue root fourth root fifth root To cll up the cue root or Choose Option for cue root higher root functions, press MATH Choose Option for higher root. But e sure to enter the numer for the inde first! Emple : Evlute the followings... c. d. ± ± ()() () ()() () ()()() ()()()() () ()()()() () ()()()()() A rdicl with n even inde lws hs two nswers (±), nd cn onl hve rdicnd greter thn or equl to 0 inside rdicl sign. A rdicl with n odd inde lws hs one nswer onl nd cn hve negtive rdicnd inside the rdicl sign. Emple : A formul v f v i + d cn e used to find the finl velocit (speed) of n ccelerted oject, where v f finl velocit, v i initil velocit, ccelertion, nd d distnce trvelled. An pple is thrown from the tll uilding 00 m high with n initil velocit of m/s. The ccelertion due to grvit is. m/s. Wht is the finl velocit of the pple s it reches the ground? Solve for v f : f Pge 00. v f? v i m/s d 00 m. m/s v v f f vi + d v ( ) + (.)( 00) v i + d v v f f + v f. m/s Coprighted Griel Tng, B.Ed., BSc.

6 Pure Mth 0 Notes Eponents nd Rdicls Estimting Squre Roots. Estimting Squre Roots GREATER thn :. Group the rdicnd two digits strting directl to the LEFT of the deciml plce. The digit 0 m e dded to the eginning of the rdicnd if there re n odd numer of digits.. Estimte ech group of two digits finding the squre root of the nerest lower squre numer.. Estimting Squre Roots LESS thn :. Group the rdicnd two digits strting directl to the RIGHT of the deciml plce. The digit 0 m e dded to the end of the rdicnd if there re n odd numer of digits.. Estimte ech group of two digits finding the squre root of the nerest lower squre numer. Emple : Estimte. Then, find the pproimted vlue to the fifth deciml plce using clcultor with onl positive roots.... c. 0. d Actul.0 Actul.00 Actul 0. Actul 0.0 Emple : Evlute estimting, then, find the pproimted vlue to the fifth deciml plce using clcultor with onl positive roots.. ( )( 0). 0 ( ) c. 0 0 () 0 (). Actul.0 Actul.0 Actul. Coprighted Griel Tng, B.Ed., B.Sc. Pge 0.

7 Eponents nd Rdicls Pure Mth 0 Notes Emple : Evlute the followings using onl positive roots... c. d Emple : Evlute the followings using onl positive roots. Verif using clcultor.. +. () + () () 0 + Emple : Evlute the followings using onl positive roots. Verif using clcultor ( ) ( ) ( 0.0 ) ( 0.) 0. - Homework Assignments Regulr: pg. to # to (odd), to (no estimtes), to, to,, nd AP: pg. to # to (even), to (no estimtes), to, to, nd Pge 0. Coprighted Griel Tng, B.Ed., BSc.

8 Pure Mth 0 Notes Eponents nd Rdicls -: Simplifing Rdicls where 0 nd 0 where 0 nd > 0 Entire Rdicls: - rdicls tht hve no coefficient in front of them. Emples: nd Mied Rdicls: - rdicls tht hve coefficients in front of them. - the coefficient is the squre root of the perfect squre fctor of the rdicnd. Emples: nd To convert n entire rdicl to mied rdicl, find the lrgest perfect squre fctor of the rdicnd nd write its root s coefficient of the remining rdicnd fctor. Emple : Simplif the followings. (Convert them to mied rdicls) c. d. To convert mied rdicl to n entire rdicl, squre the coefficient nd multipl it to the rdicnd. Emple : Write the followings s entire rdicls... c Coprighted Griel Tng, B.Ed., B.Sc. Pge 0.

9 Eponents nd Rdicls Pure Mth 0 Notes Emple : Order,, nd from lest to gretest. < < 0 0 < < Emple : Simplif... ( )( ) Rtionliztion: - turning rdicl denomintor into nturl numer denomintor multipling frction of the rdicl over itself. Emple : Simplif Pge 0. Coprighted Griel Tng, B.Ed., BSc. c. OR 0 0 0

10 Pure Mth 0 Notes Emple : Simplif... 0 c. Need to find perfect cue fctor of the rdicnd. We cn hve negtive perfect cue fctor. Emple : Write the followings s entire rdicls... c. 0 We need to cue the ( ) coefficient nd multipl it into the rdicnd Eponents nd Rdicls d. Need to find perfect fourth fctor of the rdicnd. d. We need to tke the coefficient to the fourth power nd multipl it into the rdicnd. 0 (AP) Emple : Simplif.. We hve to multipl the cue root of the squre of the rdicnd to form perfect cue.. (AP) Emple : Solve for... ( ) Coprighted Griel Tng, B.Ed., B.Sc. Pge 0. 0 c. - Homework Assignments Regulr: pg. to 0 # to (odd), to, to (even), to, AP: pg. to 0 # to (even), to, to (odd), to

11 Eponents nd Rdicls Pure Mth 0 Notes -: Opertions with Rdicls Adding nd Sutrcting Rdicls: Convert n entire rdicls into mied rdicls first. Then, comine like terms (rdicls with the sme rdicnd) dding or sutrcting their coefficients. Emple : Simplif the followings ( ) ( ) ( ) Multipling Rdicls: When multipling two mied rdicls, multipl the coefficients first, nd then multipl the rdicnds. Simplif ech term fterwrds if necessr. Emple : Simplif the followings.. ( + ). ( ) c. ( + )( ) ( + ) ( ) ( + )( ) ( ) ( ) ( ) ( ) 0 () 0 d. ( )( + ) e. ( ) ( )( + ) + 0 ( ) + 0() + ( ) ( )( ) () + ( ) + Pge 0. Coprighted Griel Tng, B.Ed., BSc. + +

12 Pure Mth 0 Notes Eponents nd Rdicls Conjugtes: - inomils tht hve the ect sme terms opposite signs in etween. + c d nd c Emple: ( + ) nd ( ) ( ) ( d ) Multipling Conjugte Rdicls: The result of multipling conjugte rdicls is ALWAYS Rtionl Numer (the rdicl terms will lws cncel out). Emple : Simplif the followings.. ( + )( ). ( )( + ) ( + )( ) ( )( + ) ( ) ( ) ( ) 0 Notice the middle two terms lws cncel out! Rtionlizing Binomil Rdicl Denomintors: Multipl the rdicl epressions frction consist of the conjugte of the denomintor over itself Emple : Simplif the followings.. + ( + ) ( ) ( ) ( ) ( + ) ( ) ( + ) ( ) ( ) ( ) ( ) ( ) Emple : A rectngle hs perimeter of 0 + nd its width is 0. Wht is the length of this rectngle? ( 0 + ) l ( 0 ) P ( l + w) Sutrcting l ( 0 + ) ( 0 ) w 0 P 0 + P rcket! l + w ( 0 ) ( 0 + ) l Switch signs in the second rcket! P w l length 0 + Coprighted Griel Tng, B.Ed., B.Sc. Pge 0.

13 Eponents nd Rdicls Pure Mth 0 Notes Emple : Find the volume of clinder tht hs rdius of, nd its height is r h ( ) V V V V V V V V πr π π π π π π π h ( ) ( ) ( )( )( ) ( ( ( ) () ) ( )( ) ( ( 0( ) ( )( 0 ) ( 0 0) Volume π 0π 0 Emple : A prllelogrm hs n re of +. Clculte the mesure of its height if the se is h A ( ). ( + ) A h h h h A ( + ) ( ) ( + ) ( + ) ( ) ( + ) Height () + Emple : Simplif (AP) c. ( + )( ) ( ) + 0( ) 0( ) ( ) ( ) + ( ) - Homework Assignments Regulr: pg. to # to (odd), to 0, to, 0 AP: pg. to # to 0 (even), to 0, to 0 Pge 0. Coprighted Griel Tng, B.Ed., BSc. + +

14 Pure Mth 0 Notes Eponents nd Rdicls -: Reviewing the Eponent Lws power m eponent se Eponentil Lws m ( ) ( m )( n ) m + n n ( ) n n mn ( m ) n mn 0 () m m m m m m Emple : Simplif. Epress ll nswers in positive eponents onl.. (c d )(c d ). c + c d d c. ( ) ()( )( ) d. ( ) ( ) ( ) 0 ( )( ) ( m n ) ( ) ( m n ) e. (m n ) (m n ) f. m m m () n n n n 0 0 When reciprocting n entire rcket, do NOT mess with its content. n p q p q p p p q p q q q p q p q ( ) ( ) p q Coprighted Griel Tng, B.Ed., B.Sc. Pge 0.

15 Eponents nd Rdicls Pure Mth 0 Notes g. ( ) ( ) + ( ) h. ( h k ) ( h k ) ( h k ) ( h k ) ( h k ) ( h k ) h ( h k )( h k ) h 0 0 ( ) ( ) ( ) k k h k Emple : In stronom, one light er is the distnce light cn trvel in one er. Light hs constnt speed of 0 km/s in the vcuum of spce.. Clculte the distnce of one light er.. The closest str to the Sun, Alph Centuri, is. 0 km. How mn light ers is it to our sun?. One Light Yer ( 0 km/s)( ds/r)( hr/d)(0 min/hr)(0 s/min) EE, Emple : Solve for.. ( )( ) ± 0 (AP) Emple : Simplif.. ( m ) () m +. m ( ) ( ) m+ ( m+ ) ( ) m+ ( ) m+.0 0 km/r + ( ) + (AP) c. Pge 0. Coprighted Griel Tng, B.Ed., BSc... 0 km.0 0 km / r ( ) + + ( + ) ( + ) ( ) ( + ) ± light ers - Homework Assignments Regulr: pg. # to (even), 0 to,,,, AP: pg. # to (odd), 0 to

16 Pure Mth 0 Notes Eponents nd Rdicls -: Rtionl Eponents m n n m The inde of the rdicl is the denomintor of the frctionl eponent. Emple : Evlute... ( ) e. ( ) f. Emple : Evlute using clcultor.. ( ). ( ). c. ( ) 0 d. ( ) 0 ( ) g. c.. ( ) Emple : Write the followings using eponents... c. ( ) ( ) d. ( ) ( ) ( ) Coprighted Griel Tng, B.Ed., B.Sc. Pge.

17 Eponents nd Rdicls Pure Mth 0 Notes Pge. Coprighted Griel Tng, B.Ed., BSc. Emple : Evlute, if possile.. ( )( ). c. ( ) Emple : Write the following epressions using eponents... c. ( )( ) d. ( ) (AP) Emple : Solve for... + c. - Homework Assignments Regulr: pg. to # to (odd), to, to (no estimtes), 0, AP: pg. to # to (even), to, to (no estimtes), 0 to + ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) + + +

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

More information

Chapter 9: Quadratic Equations

Chapter 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 information

Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }

Basic 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 information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

Operations with Polynomials

Operations 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 information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 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 information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

Quadratic Equations - 1

Quadratic Equations - 1 Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL 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 information

THE RATIONAL NUMBERS CHAPTER

THE RATIONAL NUMBERS CHAPTER CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix 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 information

Factoring Polynomials

Factoring 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 information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values) www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or 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 information

Square Roots Teacher Notes

Square 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 information

4.0 5-Minute Review: Rational Functions

4.0 5-Minute Review: Rational Functions mth 130 dy 4: working with limits 1 40 5-Minute 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 information

Chapter 6 Solving equations

Chapter 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 information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial 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 information

Sect 8.3 Triangles and Hexagons

Sect 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 two-dimensionl geometric figure consisting of t lest three line segments for its

More information

Square & Square Roots

Square & 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 information

Binary Representation of Numbers Autar Kaw

Binary 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 information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. 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 information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

More information

The Quadratic Formula and the Discriminant

The Quadratic Formula and the Discriminant 9-9 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 information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 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 information

Solutions to Section 1

Solutions 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 information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

More information

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929  Math Learning Center Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

More information

Section A-4 Rational Expressions: Basic Operations

Section A-4 Rational Expressions: Basic Operations A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

0.1 Basic Set Theory and Interval Notation

0.1 Basic Set Theory and Interval Notation 0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we

More information

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

More information

10.5 Graphing Quadratic Functions

10.5 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 information

Reasoning to Solve Equations and Inequalities

Reasoning 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 information

Pythagoras theorem and trigonometry (2)

Pythagoras theorem and trigonometry (2) HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These

More information

Answers (Anticipation Guide and Lesson 7-1)

Answers (Anticipation Guide and Lesson 7-1) Answers (Anticiption Guide nd Lesson 7-) NAME DATE PERID 7 Anticiption Guide Rdicl Equtions STEP Chpter 7 Glencoe Algebr Answers Chpter Resources Before ou begin Chpter 7 Red ech sttement. Decide whether

More information

not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions

not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the x-coordintes of the points where the grph of y = p(x) intersects the x-xis.

More information

Geometry and Measure. 12am 1am 2am 3am 4am 5am 6am 7am 8am 9am 10am 11am 12pm

Geometry and Measure. 12am 1am 2am 3am 4am 5am 6am 7am 8am 9am 10am 11am 12pm Reding Scles There re two things to do when reding scle. 1. Mke sure you know wht ech division on the scle represents. 2. Mke sure you red in the right direction. Mesure Length metres (m), kilometres (km),

More information

1+(dy/dx) 2 dx. We get dy dx = 3x1/2 = 3 x, = 9x. Hence 1 +

1+(dy/dx) 2 dx. We get dy dx = 3x1/2 = 3 x, = 9x. Hence 1 + Mth.9 Em Solutions NAME: #.) / #.) / #.) /5 #.) / #5.) / #6.) /5 #7.) / Totl: / Instructions: There re 5 pges nd totl of points on the em. You must show ll necessr work to get credit. You m not use our

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Math 135 Circles and Completing the Square Examples

Math 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 information

Solving Linear Equations - Formulas

Solving 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 information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

Net Change and Displacement

Net 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 information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 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 information

Sequences and Series

Sequences and Series Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic

More information

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

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

More information

Tallahassee Community College. Simplifying Radicals

Tallahassee Community College. Simplifying Radicals Tllhssee Communit College Simplifing Rdils The squre root of n positive numer is the numer tht n e squred to get the numer whose squre root we re seeking. For emple, 1 euse if we squre we get 1, whih is

More information

Answer, Key Homework 8 David McIntyre 1

Answer, Key Homework 8 David McIntyre 1 Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your

More information

ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS

ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS CHAPTER ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS Although people tody re mking greter use of deciml frctions s they work with clcultors, computers, nd the metric system, common

More information

0.2 ABSOLUTE VALUE AND DISTANCE ON THE REAL NUMBER LINE

0.2 ABSOLUTE VALUE AND DISTANCE ON THE REAL NUMBER LINE 360040_0002.q 1/3/05 11:17 AM Pge 0-8 0-8 HAPTER 0 A Preclculus Review 0.2 ABSOLUTE VALUE AND DISTANE ON THE REAL NUMBER LINE Fin the solute vlues of rel numers n unerstn the properties of solute vlue.

More information

Review Problems for the Final of Math 121, Fall 2014

Review Problems for the Final of Math 121, Fall 2014 Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since

More information

Lecture 15 - Curve Fitting Techniques

Lecture 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 information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use 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 information

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,

More information

FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation

FUNCTIONS 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 information

5.6 POSITIVE INTEGRAL EXPONENTS

5.6 POSITIVE INTEGRAL EXPONENTS 54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section

More information

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Math 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 information

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

More information

AP QUIZ #2 GRAPHING MOTION 1) POSITION TIME GRAPHS DISPLACEMENT Each graph below shows the position of an object as a function of time.

AP QUIZ #2 GRAPHING MOTION 1) POSITION TIME GRAPHS DISPLACEMENT Each graph below shows the position of an object as a function of time. AP QUIZ # GRAPHING MOTION ) POSITION TIME GRAPHS DISPLAEMENT Ech grph below shows the position of n object s function of time. A, B,, D, Rnk these grphs on the gretest mgnitude displcement during the time

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

The remaining two sides of the right triangle are called the legs of the right triangle.

The remaining two sides of the right triangle are called the legs of the right triangle. 10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivatives/Indefinite Integrals of Basic Functions Anti-derivtives/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 information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com C Integrtion Volumes PhsicsAndMthsTutor.com. Using the sustitution cos u, or otherwise, find the ect vlue of d 7 The digrm ove shows sketch of prt of the curve with eqution, <

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

NQF Level: 2 US No: 7480

NQF Level: 2 US No: 7480 NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................

More information

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent! MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

More information

Tests for One Poisson Mean

Tests for One Poisson Mean Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution

More information

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

More information

11. PYTHAGORAS THEOREM

11. PYTHAGORAS THEOREM 11. PYTHAGORAS THEOREM 11-1 Along the Nile 2 11-2 Proofs of Pythgors theorem 3 11-3 Finding sides nd ngles 5 11-4 Semiirles 7 11-5 Surds 8 11-6 Chlking hndll ourt 9 11-7 Pythgors prolems 10 11-8 Designing

More information

Let us recall some facts you have learnt in previous grades under the topic Area.

Let us recall some facts you have learnt in previous grades under the topic Area. 6 Are By studying this lesson you will be ble to find the res of sectors of circles, solve problems relted to the res of compound plne figures contining sectors of circles. Ares of plne figures Let us

More information

Chapter. Contents: A Constructing decimal numbers

Chapter. Contents: A Constructing decimal numbers Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting

More information

Maths Assessment Year 4: Number and Place Value

Maths Assessment Year 4: Number and Place Value Nme: Mths Assessment Yer 4: Numer nd Plce Vlue 1. Count in multiples of 6, 7, 9, 25 nd 1 000; find 1 000 more or less thn given numer. 2. Find 1,000 more or less thn given numer. 3. Count ckwrds through

More information

Sample Problems. Practice Problems

Sample 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 information

Surface Area and Volume

Surface Area and Volume Surfce Are nd Volume Student Book - Series J- Mthletics Instnt Workooks Copyright Surfce re nd volume Student Book - Series J Contents Topics Topic - Surfce re of right prism Topic 2 - Surfce re of right

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Mean and the Pythagorean Theorem Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

More information

1 Numerical Solution to Quadratic Equations

1 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 information

Singapore Mathematical Olympiad Training Handbook - Sec 1

Singapore Mathematical Olympiad Training Handbook - Sec 1 Develop The Mths Genius in You Singpore Mthemticl Olympid Trining Hndbook - Sec Includes Questions from Includes Questions from other Olympids Der young students of mthemtics Mthemtics is wonderful subject

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS 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 point-direction nd twopoint

More information

Integration. 148 Chapter 7 Integration

Integration. 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 information

Quadratic Equations. Math 99 N1 Chapter 8

Quadratic 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 information

Chapter 2 Decimals. (A reminder) In the whole number chapter, we looked at ones, tens, hundreds, thousands and larger numbers. = 1

Chapter 2 Decimals. (A reminder) In the whole number chapter, we looked at ones, tens, hundreds, thousands and larger numbers. = 1 Chpter 2 Decimls Wht is Deciml? (A reminder) In the whole numer chpter, we looked t ones, tens, hundreds, thousnds nd lrger numers. When single unit is divided into 10 (or 100) its, we hve deciml frctions

More information

Assuming 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;

Assuming 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; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

Exponents base exponent power exponentiation

Exponents 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 information

r 2 F ds W = r 1 qe ds = q

r 2 F ds W = r 1 qe ds = q Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study

More information

Strong acids and bases

Strong acids and bases Monoprotic Acid-Bse Equiliri (CH ) ϒ Chpter monoprotic cids A monoprotic cid cn donte one proton. This chpter includes uffers; wy to fi the ph. ϒ Chpter 11 polyprotic cids A polyprotic cid cn donte multiple

More information

to the area of the region bounded by the graph of the function y = f(x), the x-axis y = 0 and two vertical lines x = a and x = b.

to the area of the region bounded by the graph of the function y = f(x), the x-axis y = 0 and two vertical lines x = a and x = b. 5.9 Are in rectngulr coordintes If f() on the intervl [; ], then the definite integrl f()d equls to the re of the region ounded the grph of the function = f(), the -is = nd two verticl lines = nd =. =

More information

Right Triangles and Trigonometry

Right Triangles and Trigonometry 9 Right Tringles nd Trigonometry 9.1 The Pythgoren Theorem 9. Specil Right Tringles 9.3 Similr Right Tringles 9.4 The Tngent Rtio 9.5 The Sine nd osine Rtios 9.6 Solving Right Tringles 9.7 Lw of Sines

More information

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to

More information

Algebra Review. How well do you remember your algebra?

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 information

Math I EB127. Arab Academy For Science & Technology. [Basic and Applied Science Dept.]

Math I EB127. Arab Academy For Science & Technology. [Basic and Applied Science Dept.] Ar Acdem For Science & Technolog [Bsic nd Applied Science Dept] Mth [EB] Anltic Geometr Determinnts Mtrices Sstem of Liner Equtions Curve Fitting Liner Progrmming Mth I EB Sllus for Mthemtics I Course

More information

Worksheet 24: Optimization

Worksheet 24: Optimization Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I

More information

Double Integrals over General Regions

Double Integrals over General Regions Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

More information