0.2 ABSOLUTE VALUE AND DISTANCE ON THE REAL NUMBER LINE


 Loraine Jackson
 1 years ago
 Views:
Transcription
1 360040_0002.q 1/3/05 11:17 AM Pge 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. Fin the istnce etween two numers on the rel numer line. Define intervls on the rel numer line. Fin the mipoint of n intervl n use intervls to moel n solve rellife prolems. Asolute Vlue of Rel Numer TEHNOLOGY Asolute vlue epressions cn e evlute on grphing utility. When n epression such s 3 8 is evlute, prentheses shoul surroun the epression, s in s 3 8. Definition of Asolute Vlue The solute vlue of rel numer is,, if 0 if < 0. At first glnce, it my pper from this efinition tht the solute vlue of rel numer cn e negtive, ut this is not possile. For emple, let 3. Then, ecuse 3 < 0, you hve The following properties re useful for working with solute vlues. Properties of Asolute Vlue 1. Multipliction: 2. Division:, 3. Power: n n 4. Squre root: 2 0 Be sure you unerstn the fourth property in this list. A common error in lger is to imgine tht y squring numer n then tking the squre root, you come ck to the originl numer. But this is true only if the originl numer is nonnegtive. For instnce, if 2, then ut if 2, then The reson for this is tht (y efinition) the squre root symol the nonnegtive root. enotes only
2 360040_0002.q 1/3/05 11:17 AM Pge 09 SETION 0.2 Asolute Vlue n Distnce on the Rel Numer Line 09 Distnce on the Rel Numer Line onsier two istinct points on the rel numer line, s shown in Figure The irecte istnce from to is. 2. The irecte istnce from to is. 3. The istnce etween n is or. In Figure 0.9, note tht ecuse is to the right of, the irecte istnce from to (moving to the right) is positive. Moreover, ecuse is to the left of, the irecte istnce from to (moving to the left) is negtive. The istnce etween two points on the rel numer line cn never e negtive. Distnce Between Two Points on the Rel Numer Line The istnce etween points 1 n 2 on the rel numer line is given y Directe istnce from to : Directe istnce from to : Distnce etween n : FIGURE 0.9 or Note tht the orer of sutrction with n oes not mtter ecuse n EXAMPLE 1 Fining Distnce on the Rel Numer Line Determine the istnce etween 3 n 4 on the rel numer line. Wht is the irecte istnce from 3 to 4? Wht is the irecte istnce from 4 to 3? SOLUTION The istnce etween 3 n 4 is given y s shown in Figure Distnce = 7 or or FIGURE 0.10 The irecte istnce from 3 to 4 is The irecte istnce from 4 to 3 is TRY IT 1 Determine the istnce etween 2 n 6 on the rel numer line. Wht is the irecte istnce from 2 to 6? Wht is the irecte istnce from 6 to 2?
3 360040_0002.q 1/3/05 11:17 AM Pge HAPTER 0 A Preclculus Review Intervls Define y Asolute Vlue EXAMPLE 2 Defining n Intervl on the Rel Numer Line Fin the intervl on the rel numer line tht contins ll numers tht lie no more thn two units from 3. SOLUTION Let e ny point in this intervl. You nee to fin ll such tht the istnce etween n 3 is less thn or equl to 2. This implies tht 3 2. Requiring the solute vlue of 3 to e less thn or equl to 2 mens tht 3 must lie etween 2 n 2. So, you cn write units 2 units Solving this pir of inequlities, you hve Solution set FIGURE 0.11 So, the intervl is 1, 5, s shown in Figure TRY IT 2 Fin the intervl on the rel numer line tht contins ll numers tht lie no more thn four units from 6. Two Bsic Types of Inequlities Involving Asolute Vlue Let n e rel numers, where > 0. if n only if. Inequlity if n only if or. Interprettion Grph ALGEBRA REVIEW Be sure you see tht inequlities of the form hve solution sets consisting of two intervls. To escrie the two intervls without using solute vlues, you must use two seprte inequlities, connecte y n or to inicte union. All numers whose istnce from is less thn or equl to. All numers whose istnce from is greter thn or equl to. + +
4 360040_0002.q 1/3/05 11:17 AM Pge 011 SETION 0.2 Asolute Vlue n Distnce on the Rel Numer Line 011 Appliction EXAMPLE 3 Qulity ontrol A lrge mnufcturer hire qulity control firm to etermine the reliility of prouct. Using sttisticl methos, the firm etermine tht the mnufcturer coul epect 0.35% ± 0.17% of the units to e efective. If the mnufcturer offers moneyck gurntee on this prouct, how much shoul e ugete to cover the refuns on 100,000 units? (Assume tht the retil price is $8.95.) SOLUTION Let r represent the percent of efective units (written in eciml form). You know tht r will iffer from y t most r Figure 0.12() Now, letting e the numer of efective units out of 100,000, it follows tht 100,000r n you hve Figure 0.12() Finlly, letting e the cost of refuns, you hve So, the totl cost of refuns for 100,000 units shoul fll within the intervl given y $ r , ,000r , $ Figure 0.12(c) r () Percent of efective units () Numer of efective units (c) ost of refuns FIGURE 0.12 TRY IT 3 Use the informtion in Emple 3 to etermine how much shoul e ugete to cover refuns on 250,000 units. In Emple 3, the mnufcturer shoul epect to spen etween $1611 n $4654 for refuns. Of course, the sfer uget figure for refuns woul e the higher of these estimtes. However, from sttisticl point of view, the most representtive estimte woul e the verge of these two etremes. Grphiclly, the verge of two numers is the mipoint of the intervl with the two numers s enpoints, s shown in Figure Mipoint = 2 = FIGURE 0.13 Mipoint of n Intervl The mipoint of the intervl with enpoints n is foun y tking the verge of the enpoints. Mipoint 2
5 360040_0002.q 1/3/05 11:17 AM Pge HAPTER 0 A Preclculus Review EXERISES 0.2 In Eercises 1 6, fin () the irecte istnce from to,() the irecte istnce from to, n (c) the istnce etween n , , , , , 61 5, In Eercises 7 18, use solute vlues to escrie the given intervl (or pir of intervls) on the rel numer line. 7. 2, , 3 9., 2 2, 10., 3 3, 11. 2, , 1 13., 0 4, 14., 20 24, 15. All numers less thn two units from All numers more thn si units from y is t most two units from. 18. y is less thn h units from c. In Eercises 19 34, solve the inequlity n sketch the grph of the solution on the rel numer line > 10 2 < 5 2 < 6 > < < 5 10 > < < 1, > 0 2 > 0 3 4, < 2, 5 2 >, In Eercises 35 40, fin the mipoint of the given intervl , , , , , 3 4 > 0 > 0 5 6, hemistry opper hs melting point M within 0.2 of Use solute vlues to write the rnge s n inequlity. 42. Stock Price A stock mrket nlyst preicts tht over the net yer the price p of stock will not chnge from its current price of $ y more thn $2. Use solute vlues to write this preiction s n inequlity. 43. Sttistics The heights h of twothirs of the memers of popultion stisfy the inequlity h where h is mesure in inches. Determine the intervl on the rel numer line in which these heights lie. 44. Biology The Americn Kennel lu hs evelope guielines for juging the fetures of vrious rees of ogs. For collies, the guielines specify tht the weights for mles stisfy the inequlity w where w is mesure in pouns. Determine the intervl on the rel numer line in which these weights lie. 45. Prouction The estimte ily prouction t refinery is given y 200,000 25,000 where is mesure in rrels of oil. Determine the high n low prouction levels. 46. Mnufcturing The cceptle weights for 20ounce cerel o re given y where is mesure in ounces. Determine the high n low weights for the cerel o. Buget Vrince In Eercises 47 50, () use solute vlue nottion to represent the two intervls in which epenses must lie if they re to e within $500 n within 5% of the specifie uget mount n () using the more stringent constrint, etermine whether the given epense is t vrince with the uget restriction. Item Buget Epense 47. Utilities $ $ Insurnce $15, $14, Mintennce $20, $22, Tes $ $
1. Area under a curve region bounded by the given function, vertical lines and the x axis.
Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given
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 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 information1 of 7 9/14/15, 10:27 AM
Stuent: Dte: Instructor: Doug Ensle Course: MAT117 1 Applie Sttistics  Ensle Assignment: Online 6  Section 3.2 1 of 7 9/14/15, 1:27 AM 1. 2 of 7 9/14/15, 1:27 AM The t on the right shows the percent
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 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 information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationMATH PLACEMENT REVIEW GUIDE
MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your
More informationGeometry Notes SIMILAR TRIANGLES
Similr Tringles Pge 1 of 6 SIMILAR TRIANGLES Objectives: After completing this section, you shoul be ble to o the following: Clculte the lengths of sies of similr tringles. Solve wor problems involving
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 informationExponential 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 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 informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More 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 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 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 informationKnow the sum of angles at a point, on a straight line and in a triangle
2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke
More informationMaximum area of polygon
Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is
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 information1 Fractions from an advanced point of view
1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning
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 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 informationLecture 3 Basic Probability and Statistics
Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The
More informationASYMPTOTES HORIZONTAL ASYMPTOTES VERTICAL ASYMPTOTES. An asymptote is a line which a function gets closer and closer to but never quite reaches.
UNFAMILIAR FUNCTIONS (Chpter 19) 547 B ASYMPTOTES An smptote is line whih funtion gets loser n loser to but never quite rehes. In this ourse we onsier smptotes whih re horizontl or vertil. HORIZONTAL ASYMPTOTES
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 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 informationTests 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 informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More 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 informationMath 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 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 informationExtrema of a Function
60_00.qd //0 : PM Pge 6 6 CHAPTER Applictions o Dierentition Section. Etrem on n Intervl Understnd the deinition o etrem o unction on n intervl. Understnd the deinition o reltive etrem o unction on n open
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 informationAngles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example
2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel
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 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 informationPerfect competition model (PCM)
18/9/21 Consumers: Benefits, WT, nd Demnd roducers: Costs nd Supply Aggregting individul curves erfect competition model (CM) Key ehviourl ssumption Economic gents, whether they e consumers or producers,
More informationThe AVL Tree Rotations Tutorial
The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr222007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk
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 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 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 informationExam 1 Study Guide. Differentiation and Antidifferentiation Rules from Calculus I
Exm Stuy Guie Mth 2020  Clculus II, Winter 204 The following is list of importnt concepts from ech section tht will be teste on exm. This is not complete list of the mteril tht you shoul know for the
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 information6.7. In this section. Ratios Proportions
6.7 Applictions of Rtios n Proportions (64) 4 In this section Rtios Proportions E X A M P L E 6.7 APPLICATIONS OF RATIOS AND PROPORTIONS In this section we will use the ies of rtionl expressions in rtio
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 informationC h a p t e r 1. Functions, Graphs, and Lines. 1.1 Functions
C h p t e r Functions, Grphs, nd Lines Trying to do clculus without using functions would e one of the most pointless things you could do. If clculus hd n ingredients list, functions would e first on it,
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 informationFunctions A B C D E F G H I J K L. Contents:
Funtions Contents: A reltion is n set of points whih onnet two vriles. A funtion, sometimes lled mpping, is reltion in whih no two different ordered pirs hve the sme oordinte or first omponent. Algeri
More informationMaths 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 informationAREA 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 informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
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 informationLecture 3.1 Scalars and Vectors, Kinematics in Two and Three Dimensions
1. Sclrs n Vectors Lecture 3.1 Sclrs n Vectors, Kinemtics in Two n Three Dimensions Phsics is quntittive science, where everthing cn be escribe in mthemticl terms. As soon s the sstem of units hs been
More informationChapter L  Problems
Chpter L  Problems Blinn College  Physics 46  Terry Honn Problem L.1 Young's ouble slit experiment is performe by shooting HeNe lser bem (l 63.8 nm) through two slits seprte by 0.15 mm onto screen
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationVectors. 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 informationSequences 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 informationAddition and subtraction of rational expressions
Lecture 5. Addition nd subtrction of rtionl expressions Two rtionl expressions in generl hve different denomintors, therefore if you wnt to dd or subtrct them you need to equte the denomintors first. The
More informationPHYS1231 Higher Physics 1B Solutions Tutorial 2
PHYS3 Higher Physics Solutions Tutoril sic info: lthough the term voltge is use every y, in physics it is mesure of firly bstrct quntity clle Electric Potentil. It s importnt to istinguish electric potentil
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 informationSection 4.3. By the Mean Value Theorem, for every i = 1, 2, 3,..., n, there exists a point c i in the interval [x i 1, x i ] such that
Difference Equtions to Differentil Equtions Section 4.3 The Fundmentl Theorem of Clculus We re now redy to mke the longpromised connection between differentition nd integrtion, between res nd tngent lines.
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More information50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS
0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS BASIC KNOWLEDGE () RATIOS: Rtios re use to ompre two or more numers For n two numers n ( 0), the rtio is written s : = / Emple : If 4 stuents in
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More 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 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 informationBasics of Logic Design: Boolean Algebra, Logic Gates. Administrative
Bsics of Logic Design: Boolen Alger, Logic Gtes Computer Science 104 Administrtive Homework #3 Due Sundy Midterm I Mondy in clss, closed ook, closed notes Ø Will provide IA32 instruction set hndout Ø Lst
More informationArea Between Curves: We know that a definite integral
Are Between Curves: We know tht definite integrl fx) dx cn be used to find the signed re of the region bounded by the function f nd the x xis between nd b. Often we wnt to find the bsolute re of region
More informationRotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationN Mean SD Mean SD Shelf # Shelf # Shelf #
NOV xercises smple of 0 different types of cerels ws tken from ech of three grocery store shelves (1,, nd, counting from the floor). summry of the sugr content (grms per serving) nd dietry fiber (grms
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 information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More 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 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 informationVolumes by Cylindrical Shells: the Shell Method
olumes Clinril Shells: the Shell Metho Another metho of fin the volumes of solis of revolution is the shell metho. It n usull fin volumes tht re otherwise iffiult to evlute using the Dis / Wsher metho.
More informationContent Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem
Pythgors Theorem S Topic 1 Level: Key Stge 3 Dimension: Mesures, Shpe nd Spce Module: Lerning Geometry through Deductive Approch Unit: Pythgors Theorem Student ility: Averge Content Ojectives: After completing
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 informationBoğaziçi University Department of Economics Spring 2016 EC 102 PRINCIPLES of MACROECONOMICS Problem Set 5 Answer Key
Boğziçi University Deprtment of Eonomis Spring 2016 EC 102 PRINCIPLES of MACROECONOMICS Prolem Set 5 Answer Key 1. One yer ountry hs negtive net exports. The next yer it still hs negtive net exports n
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 informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationStrokeBased Performance Metrics for Handwritten Mathematical Expressions
StrokeBse Performnce Metrics for Hnwritten Mthemticl Expressions Richr Znii rlz@cs.rit.eu Amit Pilly p2731@rit.eu eprtment of Computer Science Rochester Institute of Technology, NY, SA Hrol Mouchère hrol.mouchere@univnntes.fr
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 informationDATABASDESIGN FÖR INGENJÖRER  1056F
DATABASDESIGN FÖR INGENJÖRER  06F Sommr 00 En introuktionskurs i tssystem http://user.it.uu.se/~ul/tsommr0/ lt. http://www.it.uu.se/eu/course/homepge/esign/st0/ Kjell Orsorn (Rusln Fomkin) Uppsl Dtse
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 informationLesson 2.1 Inductive Reasoning
Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,
More informationAP 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 informationHelicopter 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 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 informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More information10.3 Systems of Linear Equations: Determinants
758 CHAPTER 10 Systems of Equtions nd Inequlities 10.3 Systems of Liner Equtions: Determinnts OBJECTIVES 1 Evlute 2 y 2 Determinnts 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles
More informationMathematics. Circles. hsn.uk.net. Higher. Contents. Circles 119 HSN22400
hsn.uk.net Higher Mathematics UNIT OUTCOME 4 Circles Contents Circles 119 1 Representing a Circle 119 Testing a Point 10 3 The General Equation of a Circle 10 4 Intersection of a Line an a Circle 1 5 Tangents
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 informationVariable Dry Run (for Python)
Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 2050 minutes
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 informationAP Calculus AB Cram Sheet
ABCrmSheet.n AP Clcls AB Crm Sheet Definition of the Derivtive Fnction: f ' (x) = lim hø0 f Hx+hL  Å h ÅÅÅ Definition of Derivtive t Point: f H+hL f HL f ' () = lim hø0 (note: the first efinition reslts
More informationBasically, logarithmic transformations ask, a number, to what power equals another number?
Wht i logrithm? To nwer thi, firt try to nwer the following: wht i x in thi eqution? 9 = 3 x wht i x in thi eqution? 8 = 2 x Biclly, logrithmic trnformtion k, number, to wht power equl nother number? In
More information