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


 Sharlene Wheeler
 11 months ago
 Views:
Transcription
1 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 Negtive integers Positive integers RATIONAL NUMBERS All numers tht cn e written in the form >, where nd re integers nd Z 0 IRRATIONAL NUMBERS Rel numers tht cnnot e written s the quotient of two integers ut cn e represented on the numer line REAL NUMBERS Include ll numers tht cn e represented on the numer line, tht is, ll rtionl nd irrtionl numers Irrtionl Numers 5VN 3, VN, p, etc Rel Numers Zero 4_ Rtionl Numers 3, 4, 1, 0, 06, 1, etc Integers p 3,, 1, 0, 1,, 3, p Whole Numers 0, 1,, 3, p 5 Importnt Properties PROPERTIES OF ADDITION Identity Property of Zero: Inverse Property: Commuttive Property: Associtive Property: PROPERTIES OF MULTIPLICATION Property of Zero: Identity Property of One: # 1 =, when Z 0 Inverse Property: # 1, when = 1 Commuttive Property: Associtive Property: PROPERTIES OF DIVISION 0 Property of Zero:, when Z 0 = 0 Property of One:, when Z 0 = 1 Identity Property of One: Asolute Vlue # 0 = = + 1 = 0 + = c = c # = # Z 0 # 1 # c = 1 # # c 1 = # 1 The solute vlue of numer is lwys 0 If 7 0, ƒ ƒ = If 6 0,  ƒ = For exmple, ƒ 5 ƒ = 5 nd ƒ 5 ƒ = 5 In ech cse, the nswer is positive Nturl Numers 1,, 3, p PRIME NUMBERS A prime numer is numer greter thn 1 tht hs only itself nd 1 s fctors Some exmples:, 3, nd 7 re prime numers COMPOSITE NUMBERS A composite numer is numer tht is not prime For exmple, 8 is composite numer since 8 = # # = 3 ISBN13: ISBN10:
2 Key Words nd Symols The following words nd symols re used for the opertions listed Addition Sum, totl, increse, plus ddend ddend = sum Sutrction Difference, decrese, minus minuend sutrhend = difference Multipliction Product, of, times *, #, 11, fctor fctor = product Division Quotient, per, divided y Order of Opertions dividend divisor = quotient 1 st : Prentheses Simplify ny expressions inside prentheses nd : Exponents Work out ny exponents 3 rd : Multipliction nd Division Solve ll multipliction nd division, working from left to right 4 th : Addition nd Sutrction These re done lst, from left to right For exmple, Integers > 15  # , 3 = 15  # 3 + 7, 9 = = 1 ADDING AND SUBTRACTING WITH NEGATIVES   = = = + Some exmples: = = = 419 = 15 Integers MULTIPLYING AND DIVIDING WITH NEGATIVES Some exmples: 3 # 5 = = 4 14>18 = or Frctions  # =   #  =   = , =  LEAST COMMON MULTIPLE The LCM of set of numers is the smllest numer tht is multiple of ll the given numers For exmple, the LCM of 5 nd 6 is 30, since 5 nd 6 hve no fctors in common GREATEST COMMON FACTOR The GCF of set of numers is the lrgest numer tht cn e evenly divided into ech of the given numers For exmple, the GCF of 4 nd 7 is 3, since oth 4 nd 7 re divisile y 3, ut they re not oth divisile y ny numers lrger thn 3 FRACTIONS Frctions re nother wy to express division The top numer of frction is clled the numertor, nd the ottom numer is clled the denomintor ADDING AND SUBTRACTING FRACTIONS Frctions must hve the sme denomintor efore they cn e dded or sutrcted, with d Z 0 d + d = + d, with d Z 0 d  d =  d If the frctions hve different denomintors, rewrite them s equivlent frctions with common denomintor Then dd or sutrct the numertors, keeping the denomintors the sme For exmple, = = 11 1
3 Frctions Equivlent frctions re found y multiplying the numertor nd denomintor of the frction y the sme numer In the previous exmple, 1 nd 4 = 1 # 3 4 # = MULTIPLYING AND DIVIDING FRACTIONS When multiplying nd dividing frctions, common denomintor is not needed To multiply, tke the product of the numertors nd the product of the denomintors: To divide frctions, invert the second frction nd then multiply the numertors nd denomintors:, c d = # d c = d c Some exmples: 3 = # 4 3 # 4 = 8 1 # c d = # c # = c d d 3 5 # 7 = , 1 = 5 1 # 1 = 10 1 = 5 6 REDUCING FRACTIONS To reduce frction, divide oth the numertor nd denomintor y common fctors In the lst exmple, 10 1 = 10, 1, = 5 6 MIXED NUMBERS A mixed numer hs two prts: whole numer prt nd frctionl prt An exmple of mixed numer is This relly represents 5 + 3, 8 which cn e written s = 43 8 Similrly, n improper frction cn e written s mixed numer For exmple, 0 cn e written s 6 3, 3 since 0 divided y 3 equls 6 with reminder of Rtes, Rtios, Proportions, nd Percents RATES AND RATIOS A rte is comprison of two quntities with different units For exmple, cr tht trvels 110 miles in hours is moving t rte of 110 miles/ hours or 55 mph A rtio is comprison of two quntities with the sme units For exmple, clss with 3 students hs 3 student techer rtio of 3:1 or PROPORTIONS A proportion is sttement in which two rtios or rtes re equl An exmple of proportion is the following sttement: 30 dollrs is to 5 hours s 60 dollrs is to 10 hours This is written $30 5 hr = $60 10 hr A typicl proportion prolem will hve one unknown quntity, such s 1 mile 0 min = x miles 60 min We cn solve this eqution y cross multiplying s shown: 0x = 60 # 1 x = 60 0 = 3 So, it tkes 60 minutes to wlk 3 miles PERCENTS A percent is the numer of prts out of 100 To write percent s frction, divide y 100 nd drop the percent sign For exmple, 57% = To write frction s percent, first check to see if the denomintor is 100 If it is not, write the frction s n equivlent frction with 100 in the denomintor Then the numertor ecomes the percent For exmple, 4 5 = = 80% To find percent of quntity, multiply the percent y the quntity For exmple, 30% of 5 is 30 # 5 = = 3 1 3
4 Bsic Mth Review Deciml Numers The numers fter the deciml point represent frctions with denomintors tht re powers of 10 The deciml point seprtes the whole numer prt from the frctionl prt 9 For exmple, 09 represents Plce Vlue Chrt illions ADDING AND SUBTRACTING DECIMAL NUMBERS To dd or sutrct deciml numers, line up the numers so tht the deciml points re ligned Then dd or sutrct s usul, keeping the deciml point in the sme plce For exmple, = 300 MULTIPLYING AND DIVIDING DECIMAL NUMBERS To multiply deciml numers, multiply them s though they were whole numers The numer of deciml plces in the product is the sum of the numer of deciml plces in the fctors For exmple, 37 * 45 is hundred thousnds hundred millions ten millions millions ten thousnds thousnds hundreds tens ones tenths Whole numers hundredths thousndths ten thousndths Decimls deciml plces 1 deciml plce 3 deciml plces millionths hundred thousndths To divide deciml numers, first mke sure the divisor is whole numer If it is not, move the deciml plce to the right (multiply y 10, 100, nd so on) to mke it whole numer Then move the deciml point the sme numer of plces in the dividend For exmple, 04, 1 = 4, The deciml point in the nswer is plced directly ove the new deciml point in the dividend Percents to Decimls nd Decimls to Percents To chnge numer from percent to deciml, divide y 100 nd drop the percent sign: 58% = 58/100 = 058 To chnge numer from deciml to percent, multiply y 100 nd dd the percent sign: 073 = 73 * 100 = 73% Simple Interest Given the principl (mount of money to e orrowed or invested), interest rte, nd length of time, the mount of interest cn e found using the formul where I = interest 1dollr mount p = principl r = percentge rte of interest t = time period For exmple, find the mount of simple interest on $3800 lon t n nnul rte of 55% for 5 yers: p = $3800 r = 55% = 0055 t = 5 yers I = = 1045 The mount of interest is $1045 Scientific Nottion I = p # r # t Scientific nottion is convenient wy to express very lrge or very smll numers A numer in this form is written s * 10 n, where 1 ƒ ƒ 6 10 nd n is n integer For exmple, 36 * 10 5 nd 1 * 104 re expressed in scientific nottion To chnge numer from scientific nottion to numer without exponents, look t the power of ten If tht numer is positive, move the deciml point to the right If it is negtive, move the deciml point to the left The numer tells you how mny plces to move the deciml point For exmple, 397 * 10 3 = 3970 To chnge numer to scientific nottion, move the deciml point so it is to the right of the first nonzero digit If the deciml point is moved n plces to the left nd this mkes the numer smller, n is positive; otherwise, n is negtive If the deciml point is not moved, n is 0 For exmple, = 16 *
5 Scientific Nottion MULTIPLYING AND DIVIDING IN SCIENTIFIC NOTATION To multiply or divide numers in scientific nottion, we cn chnge the order nd grouping, so tht we multiply or divide first the deciml prts nd then the powers of 10 For exmple, Sttistics There re severl wys to study list of dt Men, or verge, is the sum of ll the dt vlues divided y the numer of vlues Medin is the numer tht seprtes the list of dt into two equl prts To find the medin, list the dt in order from smllest to lrgest If the numer of dt is odd, the medin is the middle numer If the numer of dt is even, the medin is the verge of the two middle numers Mode is the numer in the list tht occurs the most frequently There cn e more thn one mode For exmple, consider the following list of test scores: {87, 56, 69, 87, 93, 8} To find the men, first dd: = 474 Then divide y 6: = 79 The men score is 79 To find the medin, first list the dt in order: 56, 69, 8, 87, 87, 93 Since there is n even numer of dt, we tke the verge of 8 nd 87: = 169 = 845 The medin score is 845 The mode is 87, since this numer ppers twice nd ech of the other numers ppers only once Distnce Formul 137 * 103 # 15 * 10 8 = 137 * 5 # * 10 8 = 95 * 10 5 Given the rte t which you re trveling nd the length of time you will e trveling, the distnce cn e found y using the formul d = r # t where d = distnce r = rte t = time US Mesurement Units in = inch ft = foot min = minute sec = second gl = gllon yd = yrd pt = pint Mesurements Metric Units mm = millimeter cm = centimeter km = kilometer m = meter ml = milliliter cl = centiliter L = liter kl = kiloliter mg = milligrm cg = centigrm g = grm kg = kilogrm US AND METRIC CONVERSIONS US oz = ounce c = cup mi = mile hr = hour l = pound qt = qurt T = ton 1 in = 1 ft 3 ft = 1 yd 1760 yd = 1 mi 580 ft = 1 mi c = 1 pt 1 c = 8 oz 4 qt = 1 gl pt = 1 qt 000 l = 1 T 16 oz = 1 l Metric 1000 mm = 1 m 100 cm = 1 m 1000 m = 1 km 100 cl = 1 L 1000 ml = 1 L 100 cg = 1 g 1000 mg = 1 g 1000 g = 1 kg 0001 m = 1 mm 001 m = 1 cm 0001 g = 1 mg 001 g = 1 cg 0001 L = 1 ml 001 L = 1 cl 5
6 Geometry The perimeter of geometric figure is the distnce round it or the sum of the lengths of its sides The perimeter of rectngle is times the length plus times the width: W P = L + W The perimeter of squre is 4 times the length of side: P = 4s Are is lwys expressed in squre units, since it is twodimensionl The formul for re of rectngle is A = L # W The formul for re of squre is A = s # s or A = s s The re of tringle is onehlf the product of the height nd se: L s Geometry PYTHAGOREAN THEOREM In ny right tringle, if nd re the lengths of the legs nd c is the length of the hypotenuse, then + = c CIRCLES Are: A = p # r Circumference: C = p # d = # p # r where d is the dimeter, r is the rdius, or hlf the dimeter, nd p is pproximtely 314 or A circle hs n ngle of 360 degrees A stright line hs n ngle of 180 degrees c d 7 r The sum of ll three ngles in ny tringle lwys equls 180 degrees x A = 1 # h y x + y + z = 180 A right tringle is tringle with 90 (right) ngle The hypotenuse of right tringle is the side opposite the right ngle h z hypotenuse Algeric Terms Vrile: A vrile is letter tht represents numer ecuse the numer is unknown or ecuse it cn chnge For exmple, the numer of dys until your vction chnges every dy, so it could e represented y vrile, x Constnt: A constnt is term tht does not chnge For exmple, the numer of dys in the week, 7, does not chnge, so it is constnt Expression: An lgeric expression consists of constnts, vriles, numerls nd t lest one opertion For exmple, x + 7 is n expression Eqution: An eqution is siclly mthemticl sentence indicting tht two expressions re equl For exmple, x + 7 = 18 is n eqution Solution: A numer tht mkes n eqution true is solution to tht eqution For exmple, in using the ove eqution, x + 7 = 18, we know tht the sttement is true if x =
Addition 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 informationTHE 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 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 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 informationMultiplication 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 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 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 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 informationQuadratic 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 informationPrealgebra 7* In your group consider the following problems:
Prelger * Group Activit # Group Memers: In our group consider the following prolems: 1) If ever person in the room, including the techer, were to shke hnds with ever other person ectl one time, how mn
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 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 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 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 informationP.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 informationThe area of the larger square is: IF it s a right triangle, THEN + =
8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationGRADE 7 ADAPTED NJDOE ASSESSMENT. Assessed Standards: 7.NS.1 7.NS.2 7.NS.3. (To be administered after NPS Grade 7 Scope and Sequence Units 1&2)
ADAPTED NJDOE ASSESSMENT GRADE 7 (To e dministered fter NPS Grde 7 Scope nd Sequence Units &2) Assessed Stndrds: 7.NS. 7.NS.2 7.NS.3 The Newrk Pulic Schools  Office of Mthemtics 203 Nme Period Dte Grde
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 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 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 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 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 information5.6 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 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 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 informationChapter. 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 informationAlgebra 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 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 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 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 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 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 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 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 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 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 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 informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
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 informationThe 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 informationSection 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 informationExponentiation: 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 informationALGEBRAIC 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 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 informationPROBLEMS 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 informationGeometry 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 information1. 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 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 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 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 information4 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 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 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 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 informationLet 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 informationD e c i m a l s DECIMALS.
D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationMA 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 informationChapter 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 informationSCHOOL 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 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 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 informationDimensional Analysis is a simple method for changing from one unit of measure to another. How many yards are in 49 ft?
HFCC Math Lab NAT 05 Dimensional Analysis Dimensional Analysis is a simple method for changing from one unit of measure to another. Can you answer these questions? How many feet are in 3.5 yards? Revised
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 informationThree squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.
1 The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor SelfChek Quiz Three squres with
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 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 informationHomework 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}.
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 informationLinear Equation In Two Variables
. Liner Eqution In Two Vriles o Liner Eqution : An eqution in which the highest power of the vrile is one, tht eqution clled liner eqution An eqution hving only one vrile (unknown) nd the highest power
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 information9.1 PYTHAGOREAN THEOREM (right triangles)
Simplifying Rdicls: ) 1 b) 60 c) 11 d) 3 e) 7 Solve: ) x 4 9 b) 16 80 c) 9 16 9.1 PYTHAGOREAN THEOREM (right tringles) c If tringle is right tringle then b, b re the legs * c is clled the hypotenuse (side
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 informationCONIC 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 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 informationRational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and
Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions
More informationPure 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 informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
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 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 informationRadius of the Earth  Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth  dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
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 informationGraphs 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 informationSection 54 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 information5.2 The Definite Integral
5.2 THE DEFINITE INTEGRAL 5.2 The Definite Integrl In the previous section, we sw how to pproximte totl chnge given the rte of chnge. In this section we see how to mke the pproximtion more ccurte. Suppose
More informationUnit 1, Concept 1 Number Sense, Fractions, and Algebraic Thinking Instructional Resources: Carnegie Learning: Bridge to Algebra
Unit 1, 1 Number Sense, Fractions, and Algebraic Thinking 7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers
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 informationCOMPLEX FRACTIONS. section. Simplifying Complex Fractions
58 (66) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours. 00 600 6 FIGURE FOR EXERCISE 9 95. Selling. George sells one gzine suscription every 0 inutes, wheres Theres
More informationConverting Units of Measure Measurement
Converting Units of Measure Measurement Outcome (lesson objective) Given a unit of measurement, students will be able to convert it to other units of measurement and will be able to use it to solve contextual
More informationExercise Worksheets. Copyright. 2002 Susan D. Phillips
Exercise Worksheets Copyright 00 Susan D. Phillips Contents WHOLE NUMBERS. Adding. Subtracting. Multiplying. Dividing. Order of Operations FRACTIONS. Mixed Numbers. Prime Factorization. Least Common Multiple.
More information8th Grade Unit of Study Exponents
DRAFT 8th Grde Unit of Study Exponents Grde: 8 Topic: Exponent opertions nd rules Length of Unit: 6 dys Focus of Lerning Common Core Stte Stndrds: Expressions nd Equtions 8.EE Work with rdicls nd integer
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More information11. PYTHAGORAS THEOREM
11. PYTHAGORAS THEOREM 111 Along the Nile 2 112 Proofs of Pythgors theorem 3 113 Finding sides nd ngles 5 114 Semiirles 7 115 Surds 8 116 Chlking hndll ourt 9 117 Pythgors prolems 10 118 Designing
More informationNumerical Solutions of Linear Systems of Equations
EE 6 Clss Notes Numericl Solutions of Liner Systems of Equtions Liner Dependence nd Independence An eqution in set of equtions is linerly independent if it cnnot e generted y ny liner comintion of the
More informationMEASUREMENTS. U.S. CUSTOMARY SYSTEM OF MEASUREMENT LENGTH The standard U.S. Customary System units of length are inch, foot, yard, and mile.
MEASUREMENTS A measurement includes a number and a unit. 3 feet 7 minutes 12 gallons Standard units of measurement have been established to simplify trade and commerce. TIME Equivalences between units
More informationRatio and Proportion
Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty
More informationReview 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 informationWorksheet 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 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 information