Net Change and Displacement


 Merry Lamb
 1 years ago
 Views:
Transcription
1 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 region below the xxis contributes negtivelysigned re to this net clcultion To find the totl re enclosed by f on [, b], one needs to evlute the definite integrl of the bsolute vlue of f (x): f (x) dx We cn pply this ide to other contexts In Clculus I we interpreted the first nd second derivtives s velocity nd ccelertion in the context of motion In prticulr, l s(t) position t time t s (t) v(t) velocity t time t s (t) v (t) (t) ccelertion t time t Reversing our point of view nd using ntiderivtives, since position s(t) is n ntiderivtive of veloctty v(t), by the FTC (II) we hve v(t) dt s(t) b s(b) s() But s(b) s() represents the net distnce trveled or displcement of the object during the time intervl becuse it is the difference in loctions t the end nd strt of the time period If we wnted the totl distnce trveled rther thn the net distnce trveled, just s with the re problem, we would compute the integrl of v(t) insted Tht is, totl distnce trveled This cn be summrized s follows v(t) dt DEFINITION 611 (Displcement versus Distnce) Assume tht the position of n object moving long stright line t time t is denoted by s(t) reltive to the origing nd tht its velocity is denoted by v(t) Then 1 The displcement or net chnge in position of the object between times t nd t b > is given by s(b) s() v(t) dt The totl distnce trveled by the object between times t nd t b > is given by v(t) dt EXAMPLE 61 Suppose n object moves with velocity v(t) t 1t + 16 km/hr long stright rod (1) Determine the displcement of the object on the time intervl [1, ] nd [, ] nd interpret your nswer () Determine the distnce trvelled on the time intervl [, ] Solution NetChngetex Version: Mitchell15/9/7::5
2 mth 11, pplictions motion: velocity nd net chnge (1) The displcement is esy to clculte: For the intervl [, ], On [, ], Displcement on [, ] s() s() On [1, ], s() s(1) 1 t 1t + 16 dt t 6t + 16t t 1t + 16 dt t 6t + 16t ( ) ( ) () The distnce trvelled is hrder to determine since we need to integrte v(t) We must first determine where v(t) is positive nd negtive t 1t + 16 (t 6t + 8) (t )(t ) t or t The number line to the right shows tht t 1t + 16 only [, ] We cn now find the distnce trvelled (totl re) by splitting the intervl into two pieces [, ] nd [, ], chnging the sign of v(t) on the second piece to obtin the bsolute vlue of v(t) Dist Trv Totl Are t 1t + 16 dt [ t t 1t + 16 dt t 1t + 16 dt ] [ t ] 6t + 16t 6t + 16t EXAMPLE 61 Suppose n object moves with velocity t 5t + t m/s (1) Determine the displcement of the object on the time intervl [, 6] nd interpret your nswer () Determine the distnce trvelled on [, 6] t 1t Figure 61: The distnce trvelled on [, ] is the re under the bsolute vlue of the velocity curve Solution (1) For displcement on [, 6], s(6) s() 6 t 5t + t dt t 5t + t 6 ( 6 + 7) 6 () For the distnce trvelled we must first determine where v(t) is positive nd negtive t 5t + t t(t 5t + ) t(t 1)(t ) t, 1, t 5t + t The number line to the right shows tht t 5t + t only [1, ] We cn now find the distnce trvelled (totl re) by splitting the intervl into three pieces [, 1], [1, ] nd [, 6], chnging the sign of v(t) on the second piece to NetChngetex Version: Mitchell15/9/7::5
3 mth 11, pplictions motion: velocity nd net chnge obtin the bsolute vlue of v(t) 6 Dist Trv [ t t 1t + 16 dt 1 t 5t + t dt 1 ] [ 1 t 5t + t t 5t + t dt + ] 5t + t 1 + t 5t + t dt [ t ] 5t + 6 t 5 1 Future Positions Suppose tht we know the velocity of n object moving long stright line is v(x) nd we know its position s() t some time t [Note: Often we know the initil position s()] We cn determine the future position t time generl time t using the displcement eqution Since s(x) is n ntiderivtive of v(x), gin FTC (II) tells us tht v(x) dx s(x) s(t) s() Solving for the position s(t), we see THEOREM 61 (Position from Velocity) Position t time t s(t) s() + v(x) dx In similr wy, if (t) represents ccelertion of the object, velocity t time t v(t) v() + (x) dx t EXAMPLE 615 Suppose tht the ccelertion of n object is given by (x) cos x for x with Find s(t) v() 1 s() Solution First find v(t) using Theorem 61 v(t) v() + (x) dx 1 + cos x dt 1 + [x sin x] t 1 + (t sin t) () 1 + t sin t Now solve for s(t) by using Theorem 61 s(t) s() + v(x) dx x sin x dx + (x + x + cos x) t + (t + t + cos t) ( + + 1) + t + t + cos t EXAMPLE 616 If ccelertion is given by (t) 1 + t t, find the exct position function, if s() 1 nd s() 11 Solution This time we don t hve velocity t time So let v() v be some unknown constnt We will see if we cn work it out lter Then v(t) v() (x) dx v x x dx v + 1x + x x t v + 1t + t t NetChngetex Version: Mitchell15/9/7::5
4 mth 11, pplictions motion: velocity nd net chnge Now s(t) s() + v(x) dx 1 + v + 1x + x x + c dx 1 + (v x + 5x + 1 x 1 x ) t 1 + v t + 5t + 1 t 1 t So s(t) 1 + v t + 5t + 1 t 1 t To solve for v, evlute t s(t) t t s() 1 + v so v 1 v 5 Thus, s(t) 1 5t + 5t + 1 t 1 t Constnt Accelertion: Grvity In mny motion problems the ccelertion is constnt This hppens when n object is thrown or dropped nd the only ccelertion is due to grvity In such sitution we hve (t), constnt ccelertion with initil velocity v() v nd initil position s() s Then using Theorem 61 So Next, v(t) v() + (x) dx v + s(t) s() + Therefore v(x) dx s + ds v + x t v + (t ) t + v v(t) t + v s + v ds s + ( 1 x + v x) t s + ( 1 t + v t ) 1 t + v t + s s(t) 1 t + v t + s EXAMPLE 617 Suppose bll is thrown with initil velocity 96 ft/s from roof top feet high The ccelertion due to grvity is constnt (t) ft/s Find v(t) nd s(t) Then find the mximum height of the bll nd the time when the bll hits the ground Solution Recognizing tht v 96 nd s nd tht the ccelertion is constnt, we my use the generl formuls we just developed v(t) t + v t + 96 nd s(t) 1 t + v t + s 16t + 96t + The mx height occurs when the velocity is (when the bll stops rising): v(t) t + 96 t s() ft The bll hits the ground when s(t) s(t) 16t + 96t + 16(t 6t 7) 16(t 9)(t + ) So t 9 only (since t does not mke sense) NetChngetex Version: Mitchell15/9/7::5
5 mth 11, pplictions motion: velocity nd net chnge 5 EXAMPLE 618 A person drops stone from bridge Wht is the height (in feet) of the bridge if the person hers the splsh 5 seconds fter dropping it? Solution Here s wht we know v (dropped) nd s(5) (hits wter) And we know ccelertion is constnt, ft/s We wnt to find the height of the bridge, which is just s Use our constnt ccelertion motion formuls to solve for v(t) t + v t nd s(t) 1 t + v t + s 16t + s Now we use the position we know: s(5) s(5) 16(5) + s s ft Notice tht we did not need to use the velocity function YOU TRY IT 61 (Extr Credit) In the previous problem we did not tke into ccount tht sound does not trvel instntneously in your clcultion bove Assume tht sound trvels t 11 ft/s Wht is the height (in feet) of the bridge if the person hers the splsh 5 seconds fter dropping it? Check on your nswer: Should the bridge be higher or lower thn in the preceding exmple? Why? EXAMPLE 619 Here s vrition This time we will use metric units Suppose bll is thrown with unknown initil velocity v m/s from roof top 9 meters high nd the position of the bll t time t is s() The ccelertion due to grvity is constnt (t) 98 m/s Find v(t) nd s(t) Solution This time v is unknown but s 9 nd s() Agin the ccelertion is constnt so we my use the generl formuls for this sitution v(t) t + v 98t + v nd But we know tht which mens s(t) 1 t + v t + s 9t + v t + 9 s() 9() + v + 9 v 9(9) 9(1) 9 v 9/ So nd Interpret v 9/ v(t) 98t 9 s(t) 9t 9 t + 9 More on Net Chnge nd Future Vlues We ve interpreted net re s displcement nd totl re s totl distnce when working with velocity function But we cn pply these sme ides nytime we hve rte of chnge function Exmples include when f (t) is flow rte of liquid (in which cse we cn compute net chnge in volume), or f (t) is popultion growth rte (in which cse we cn compute future popultion estimtes), or p(t) is NetChngetex Version: Mitchell15/9/7::5
6 mth 11, pplictions motion: velocity nd net chnge 6 the growth of finncil ccount (in which cse we might compute the net chnge in vlue of the ccount) In generl, suppose we know the rte of chnge in some quntity is given by Q (x) on the intervl [, t] Then by integrting using FTC II nd the fct tht Q(x) is n ntiderivtive of Q (x), we get Q (x) dx Q(x) t Q(t) Q() Agin, s in Theorem 61, we cn rerrnge terms to get THEOREM 611 (Net Chnge nd Future Vlue) Assume tht quntity Q chnges over time t know rte Q Then the net chnge in Q between t nd t b > is Q(b) Q() Q (t) dt Further, if Q() is the initil vlue, then the future vlue of Q t time t is Q(t) Q() + Q (x) dx EXAMPLE 6111 (Popultion Growth) Suppose rfter of wild turkeys in Genev hs n initil vlue of P() nd the community grows t rte of P (t) t, where time is mesured in yers Determine the popultion in yers Then find the generl formul for the popultion, P(t) A group of turkeys is clled rfter A group of turtledoves is clled pitying Solution By the second prt of Theorem 611, 6 P() P() + P (t) dt + 15 t dt ( ) + t 6t/ More generlly, the popultion t time t is P(t) P() + P (x) dx + 15 x dx ( ) + x 6x/ + (8 16) () 87 t + t 6t/ EXAMPLE 611 (Svings) Grndprents of newborn deposit $1, in college svings ccount tht hs growth rte of 555e 55t How much will be vilble in the ccount when the child is 18 nd strting college? Solution By the second prt of Theorem 611, Q(18) Q() + Q (t) dt 1, + 555e 555t dt 1, e555t 18 1, + 1, e 555t 18 Not bd, but not worth semester t HWS 1, + (1, e 999 1, ) 7, NetChngetex Version: Mitchell15/9/7::5
7 mth 11, pplictions motion: velocity nd net chnge 7 EXAMPLE 611 (Economics) The mrginl cost of product is dditionl the cost incurred by producing one more item of the product Typiclly, s production increses, the mrginl cost decreses, t lest up to point This is wht people men by the term "economy of scle" Mrginl cost is pproximted by the derivtive of the cost function C (x) tht depends on the number of units x being produced Suppose tht the mrginl cost is given by the function C (x) + 1x 1x Find the dditionl cost incurred when production rises from 5 to 55 units Then find the cost incurred when production rises from 55 to 6 units Solution By the first prt of Theorem 611, C(55) C(5) For the second prt C(6) C(55) C (x) dx C (x) dx x 1x dx x + 5x 1x 55 5 ((55) + 5(55) 1(55) 19, x 1x dx x + 5x 1x 6 55 ((6) + 5(6) 1(6) 17, 8 ) ) Notice tht the cost mking 5 more units hs decresed, illustrting the ide tht mrginl cost decreses s production increses ) ((5) + 5(5) 1(5) ) ((55) + 5(55) 1(55) NetChngetex Version: Mitchell15/9/7::5
8 mth 11, pplictions motion: velocity nd net chnge 8 For Fun: Additionl Motion Problems with Constnt Accelertion The following problems ll involve motion with constnt ccelertion When ccelertion is constnt, we cn use the equtions we developed few pges erlier But remember, when ccelertion is not constnt, be sure to use Theorem 61 EXAMPLE 611 Mo Green is ttempting to run the 1m dsh in the Genev Invittionl Trck Meet in 98 seconds He wnts to run in wy tht his ccelertion is constnt,, over the entire rce Determine his velocity function ( will still pper s n unknown constnt) Determine his position function There should be no unknown constnts in your eqution t this point Wht is his velocity t the end of the rce? Do you think this is relistic? Solution We hve: constnt ccelertion m/s ; v m/s; s m So v(t) t + v t nd s(t) 1 t + v t + s 1 t But s(98) 1 (98) 1, so (98) 85 m/s So s(t) 85t Mo s velocity t the end of the rce is v(98) 98 85(98) 1 m/s not relistic EXAMPLE 6115 A stone dropped off cliff hits the ground with speed of 1 ft/s Wht ws the height of the cliff? Solution Notice tht v (dropped!) nd s is unknown but is equl to the cliff height, nd tht the ccelertion is constnt ft/ Use the generl formuls for motion with constnt ccelertion: v(t) t + v t + t Now we use the velocity function nd the one velocity vlue we know: v 1 when it hits the ground So the time when it hits the ground is given by v(t) t 1 t 1/ 15/ when it hits the ground Now remember when it hits the ground the height is So s(15/) But we know s(t) 1 t + v t + s 16t + t + s 16t + s Now substitute in t 15/ nd solve for s The cliff height is 5 feet s(15/) 16(15/) + s s 15 5 EXAMPLE 6116 A cr is trveling t 9 km/h when the driver sees deer 75 m hed nd slms on the brkes Wht constnt decelertion is required to void hitting Bmbi? [Note: First convert 9 km/h to m/s] Solution Let s list ll tht we know v 9 km/h or m/s nd s Let time t represent the time it tkes to stop Then s(t ) 75 m Now the cr is stopped t time t, so we know v(t ) Finlly we know tht ccelertion is n unknown constnt,, which is wht we wnt to find NetChngetex Version: Mitchell15/9/7::5
9 mth 11, pplictions motion: velocity nd net chnge 9 Now we use our constnt ccelertion motion formuls to solve for v(t) t + v t + 5 nd s(t) 1 t + v t + s 1 t + 5t Now use the other velocity nd position we know: v(t ) nd s(t ) 75 when the cr stops So v(t ) t + 5 t 5/ nd s(t ) 1 (t ) + 5t 1 ( 5/) + 5( 5/) 75 Simplify to get so m/s (Why is ccelertion negtive?) EXAMPLE 6117 One cr intends to pss nother on bck rod Wht constnt ccelertion is required to increse the speed of cr from mph ( ft/s) to 5 mph ( ft/s) in 5 seconds? Solution Given: (t) constnt v ft/s s And v(5) ft/s Find But v(t) t + v t + So Thus v(5) YOU TRY IT 6 A toy bumper cr is moving bck nd forth long stright trck Its ccelertion is (t) cos t + sin t Find the prticulr velocity nd position functions given tht v(π/) nd s(π) 1 + c c Thus, v(t) sin t cos t Now s(t) v(t) dt sin t cos t dt cos t sin t + c Since s(π) ( 1) + c 1 c So s(t) cos t sin t nswer to you try it 6 v(t) (t) dt cos t + sin t dt sin t cos t + c So v(π/) webwork: Click to try Problems 71 through 7 Use guest login, if not in my course NetChngetex Version: Mitchell15/9/7::5
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 informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More 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 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 informationDerivatives and Rates of Change
Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te
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 informationVersion 001 Summer Review #03 tubman (IBII20142015) 1
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This printout should he 35 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
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 informationThe Definite Integral
Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationAnswer, Key Homework 4 David McIntyre Mar 25,
Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his printout should hve 18 questions. Multiplechoice questions my continue on the next column or pe find ll choices before mkin your selection.
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 information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationMathematics 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 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 informationDouble 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 informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More 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 informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
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 information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
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 information6 Energy Methods And The Energy of Waves MATH 22C
6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this
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 information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
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 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 informationVolumes of solids of revolution
Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the xxis. There is strightforwrd technique which enbles this to be done, using
More information10.6 Applications of Quadratic Equations
10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,
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 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 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 informationCypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:
Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More 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 informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
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 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 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 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 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 informationSOLUTIONS TO CONCEPTS CHAPTER 5
1. m k S 10m Let, ccelertion, Initil velocity u 0. S ut + 1/ t 10 ½ ( ) 10 5 m/s orce: m 5 10N (ns) 40000. u 40 km/hr 11.11 m/s. 3600 m 000 k ; v 0 ; s 4m v u ccelertion s SOLUIONS O CONCEPS CHPE 5 0 11.11
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
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 informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
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 informationNewton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt
Newton s Three Lws For couple centuries before Einstein, Newton s Lws were the bsic principles of Physics. These lws re still vlid nd they re the bsis for much engineering nlysis tody. Forml sttements
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 informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More 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 informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More 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 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 informationwww.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 information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
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 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 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 informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
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 informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More informationModule 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 informationMATLAB Workshop 13  Linear Systems of Equations
MATLAB: Workshop  Liner Systems of Equtions pge MATLAB Workshop  Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB
More informationWeek 11  Inductance
Week  Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n
More informationChapter 6. The Integral. 6.1 Measuring Work. Human Work
Chpter 6 The Integrl There re mny contexts work, energy, re, volume, distnce trvelled, nd profit nd loss re just few where the quntity in which we re interested is product of known quntities. For exmple,
More informationPHY 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 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 informationAAPT UNITED STATES PHYSICS TEAM AIP 2010
2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS  75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD
More information19. The FermatEuler Prime Number Theorem
19. The FermtEuler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout
More informationv T R x m Version PREVIEW Practice 7 carroll (11108) 1
Version PEVIEW Prctice 7 crroll (08) his printout should he 5 questions. Multiplechoice questions y continue on the next colun or pge find ll choices before nswering. Atwood Mchine 05 00 0.0 points A
More informationFirm Objectives. The Theory of the Firm II. Cost Minimization Mathematical Approach. First order conditions. Cost Minimization Graphical Approach
Pro. Jy Bhttchry Spring 200 The Theory o the Firm II st lecture we covered: production unctions Tody: Cost minimiztion Firm s supply under cost minimiztion Short vs. long run cost curves Firm Ojectives
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationPROJECTILE 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 informationSlow roll inflation. 1 What is inflation? 2 Equations of motions for a homogeneous scalar field in an FRW metric
Slow roll infltion Pscl udrevnge pscl@vudrevnge.com October 6, 00 Wht is infltion? Infltion is period of ccelerted expnsion of the universe. Historiclly, it ws invented to solve severl problems: Homogeneity:
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More information3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
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 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 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 informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationThe Fundamental Theorem of Calculus
Section 5.4 Te Funmentl Teorem of Clculus Kiryl Tsiscnk Te Funmentl Teorem of Clculus EXAMPLE: If f is function wose grp is sown below n g() = f(t)t, fin te vlues of g(), g(), g(), g(3), g(4), n g(5).
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 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 informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions
More informationChapter 5: Elasticity. measures how strongly people respond to changes in prices and changes in income.
Chpter 5: Elsticity Elsticity responsiveness mesures how strongly people respond to chnges in prices nd chnges in income. Exmples of questions tht elsticity helps nswer Wht hppens to ttendnce t your museum
More informationDIFFERENTIATING UNDER THE INTEGRAL SIGN
DIFFEENTIATING UNDE THE INTEGAL SIGN KEITH CONAD I hd lerned to do integrls by vrious methods shown in book tht my high school physics techer Mr. Bder hd given me. [It] showed how to differentite prmeters
More informationMechanics Cycle 1 Chapter 5. Chapter 5
Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies
More informationUnderstanding Life Cycle Costs How a Northern Pump Saves You Money
Understnding Life Cycle Costs How Nrn Pump Sves You Money Reference: Hydrulic Institute (www.s.g) Introduction Wht Life Cycle Cost (LCC) Clculting Totl LCC LCC Components Wht Life Cycle Cost Life Cycle
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More information