10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e
|
|
- Milton McLaughlin
- 5 years ago
- Views:
Transcription
1 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device o produce he graphs for he cases a,,.5,.,,.5,, and shown in Figure 7. Noice ha all of hese curves (ecep he case a ) have wo branches, and boh branches approach he verical asmpoe a as approaches a from he lef or righ. a=_ a=_ a=_.5 a=_. a= a=.5 a= a= FIGURE 7 Members of he famil =a+cos, =a an +sin, all graphed in he viewing recangle _, b _, When a, boh branches are smooh; bu when a reaches, he righ branch acquires a sharp poin, called a cusp. For a beween and he cusp urns ino a loop, which becomes larger as a approaches. When a, boh branches come ogeher and form a circle (see Eample ). For a beween and, he lef branch has a loop, which shrinks o become a cusp when a. For a, he branches become smooh again, and as a increases furher, he become less curved. Noice ha he curves wih a posiive are reflecions abou he -ais of he corresponding curves wih a negaive. These curves are called conchoids of Nicomedes afer he ancien Greek scholar Nicomedes. He called hem conchoids because he shape of heir ouer branches resembles ha of a conch shell or mussel shell. M. EXERCISES Skech he curve b using he parameric equaions o plo poins. Indicae wih an arrow he direcion in which he curve is raced as increases.. s,,. cos, cos,. 5 sin,,. e, e, 5 (a) Skech he curve b using he parameric equaions o plo poins. Indicae wih an arrow he direcion in which he curve is raced as increases. Eliminae he parameer o find a Caresian equaion of he curve. 5. 5, 6., 5, 5 7., 5, 8., 9. s,., 8 (a) Eliminae he parameer o find a Caresian equaion of he curve. Skech he curve and indicae wih an arrow he direcion in which he curve is raced as he parameer increases.. sin, cos,. cos, 5 sin,. sin, csc,. e, 5. e, e 6. ln, s, 7. sinh, cosh
2 SECTIN. CURVES DEFINED BY PARAMETRIC EQUATINS cosh, 5 sinh 9 Describe he moion of a paricle wih posiion, as varies in he given inerval. 9. cos, sin,. sin, cos,. 5 sin, cos,. sin, cos, Use he graphs of f and o skech he parameric curve f,. Indicae wih arrows he direcion in which he curve is raced as increases. 5. _ 6.. Suppose a curve is given b he parameric equaions f,, where he range of f is, and he range of is,. Wha can ou sa abou he curve?. Mach he graphs of he parameric equaions f and in (a) (d) wih he parameric curves labeled I IV. Give reasons for our choices. (a) I 7. II 8. Mach he parameric equaions wih he graphs labeled I-VI. Give reasons for our choices. (Do no use a graphing device.) (a),, s (c) sin, sin sin (d) cos 5, sin (e) sin, cos sin cos (f), I II III (c) III IV V VI (d) IV ; 9. Graph he curve 5. ;. Graph he curves 5 and and find heir poins of inersecion correc o one decimal place.
3 68 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES. (a) Show ha he parameric equaions where, describe he line segmen ha joins he poins P, and P,. Find parameric equaions o represen he line segmen from, 7 o,. ;. Use a graphing device and he resul of Eercise (a) o draw he riangle wih verices A,, B,, and C, 5.. Find parameric equaions for he pah of a paricle ha moves along he circle in he manner described. (a) nce around clockwise, saring a, Three imes around counerclockwise, saring a, (c) Halfwa around counerclockwise, saring a, ;. (a) Find parameric equaions for he ellipse a b. [Hin: Modif he equaions of he circle in Eample.] Use hese parameric equaions o graph he ellipse when a and b,,, and 8. (c) How does he shape of he ellipse change as b varies? ; 5 6 Use a graphing calculaor or compuer o reproduce he picure If a and b are fied numbers, find parameric equaions for he curve ha consiss of all possible posiions of he poin P in he figure, using he angle as he parameer. Then eliminae he parameer and idenif he curve. a b P. If a and b are fied numbers, find parameric equaions for he curve ha consiss of all possible posiions of he poin P in he figure, using he angle as he parameer. The line segmen AB is angen o he larger circle. A a b P B Compare he curves represened b he parameric equaions. How do he differ? 7. (a), 6, (c) e, e 8. (a), cos, (c) e, e 9. Derive Equaions for he case.. Le P be a poin a a disance d from he cener of a circle of radius r. The curve raced ou b P as he circle rolls along a sraigh line is called a rochoid. (Think of he moion of a poin on a spoke of a biccle wheel.) The ccloid is he special case of a rochoid wih d r. Using he same parameer as for he ccloid and, assuming he line is he -ais and when P is a one of is lowes poins, show ha parameric equaions of he rochoid are r d sin r d cos Skech he rochoid for he cases d r and d r. sec. A curve, called a wich of Maria Agnesi, consiss of all possible posiions of he poin P in he figure. Show ha parameric equaions for his curve can be wrien as a co a sin Skech he curve. =a a. (a) Find parameric equaions for he se of all poins P as shown in he figure such ha P AB. (This curve is called he cissoid of Diocles afer he Greek scholar Diocles, who inroduced he cissoid as a graphical mehod for consrucing he edge of a cube whose volume is wice ha of a given cube.) A C P
4 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES abou he -ais. Therefore, from Formula 7, we ge S r sin sr sin r cos d r sin sr sin cos d r sin r d r sin d r cos ] r M. EXERCISES Find dd.. sin,., s e 9. cos,. cos, sin sin 6 Find an equaion of he angen o he curve a he poin corresponding o he given value of he parameer.., ;., ; 5. e s, ln ; 6. cos sin, sin cos ; 7 8 Find an equaion of he angen o he curve a he given poin b wo mehods: (a) wihou eliminaing he parameer and b firs eliminaing he parameer. 7. ln, ;, 8. an, sec ; (, s) ; 9 Find an equaion of he angen(s) o he curve a he given poin. Then graph he curve and he angen(s) sin, ;. cos cos, sin sin ; 6 Find dd and d d. For which values of is he curve concave upward?.,.,. e, e. ln, ln 5. sin, cos, 6. cos, cos, 7 Find he poins on he curve where he angen is horizonal or verical. If ou have a graphing device, graph he curve o check our work. 7.,, 8.,, ;. Use a graph o esimae he coordinaes of he righmos poin on he curve 6, e. Then use calculus o find he eac coordinaes. ;. Use a graph o esimae he coordinaes of he lowes poin and he lefmos poin on he curve,. Then find he eac coordinaes. ; Graph he curve in a viewing recangle ha displas all he imporan aspecs of he curve..,. 8, 5. Show ha he curve cos, sin cos has wo angens a, and find heir equaions. Skech he curve. ; 6. Graph he curve cos cos, sin sin o discover where i crosses iself. Then find equaions of boh angens a ha poin. 7. (a) Find he slope of he angen line o he rochoid r d sin, r d cos in erms of. (See Eercise in Secion..) Show ha if d r, hen he rochoid does no have a verical angen. 8. (a) Find he slope of he angen o he asroid, a sin a cos in erms of. (Asroids are eplored in he Laboraor Projec on page 69.) A wha poins is he angen horizonal or verical? (c) A wha poins does he angen have slope or? 9. A wha poins on he curve, does he angen line have slope?. Find equaions of he angens o he curve, ha pass hrough he poin,.. Use he parameric equaions of an ellipse, a cos, b sin,, o find he area ha i encloses.
5 SECTIN. CALCULUS WITH PARAMETRIC CURVES 67. Find he area enclosed b he curve, s and he -ais.. Find he area enclosed b he -ais and he curve e,.. Find he area of he region enclosed b he asroid a cos, a sin. (Asroids are eplored in he Laboraor Projec on page 69.) a _a a 9. Use Simpson s Rule wih n 6 o esimae he lengh of he curve e, e, In Eercise in Secion. ou were asked o derive he parameric equaions a co, a sin for he curve called he wich of Maria Agnesi. Use Simpson s Rule wih n o esimae he lengh of he arc of his curve given b. 5 5 Find he disance raveled b a paricle wih posiion, as varies in he given ime inerval. Compare wih he lengh of he curve. 5. sin, cos, 5. cos, cos, 5. Find he area under one arch of he rochoid of Eercise in Secion. for he case d r. 6. Le be he region enclosed b he loop of he curve in Eample. (a) Find he area of. If is roaed abou he -ais, find he volume of he resuling solid. (c) Find he cenroid of. 7 Se up an inegral ha represens he lengh of he curve. Then use our calculaor o find he lengh correc o four decimal places. 7.,, 8. e,, 9. cos, sin,. ln, s, Find he eac lengh of he curve..,,. e e, 5,., ln,. cos cos, sin sin, ; 5 7 Graph he curve and find is lengh. 5. e cos, e sin, 6. cos ln(an ), sin, 7. e, e, _a Find he lengh of he loop of he curve,. CAS CAS 5. Show ha he oal lengh of he ellipse a sin, b cos, a b, is L a where e is he eccenrici of he ellipse (e ca, where c sa b ). 5. Find he oal lengh of he asroid a cos, a sin, where a. 55. (a) Graph he epirochoid wih equaions cos cos sin sin Wha parameer inerval gives he complee curve? Use our CAS o find he approimae lengh of his curve. 56. A curve called Cornu s spiral is defined b he parameric equaions C cosu du S sinu du where C and S are he Fresnel funcions ha were inroduced in Chaper 5. (a) Graph his curve. Wha happens as l and as l? Find he lengh of Cornu s spiral from he origin o he poin wih parameer value Se up an inegral ha represens he area of he surface obained b roaing he given curve abou he -ais. Then use our calculaor o find he surface area correc o four decimal places. 57. e, e, s e sin 58. sin, sin, d
6 APPENDIX I ANSWERS T DD-NUMBERED EXERCISES A (a) Populaion sabilizes a 5. (i) W, R : Zero populaions (ii) W, R 5: In he absence of wolves, he rabbi populaion is alwas 5. (iii) W 6, R : Boh populaions are sable. (c) The populaions sabilize a rabbis and 6 wolves. (d) Species R Species CHAPTER 9 REVIEW N PAGE 65 True-False Quiz. True. False 5. True 7. True Eercises. (a) c ; 6,, (iv) (iii) (ii) (i) = = = =, 5 W R = W (a) P ; 56 ln e. 7. (a) L L L 5 e. L Le k 9. 5 das. k ln h h RV C. (a) Sabilizes a, (i), : Zero populaions (ii),, : In he absence of birds, he insec populaion is alwas,. (iii) 5,, 75: Boh populaions are sable. (c) The populaions sabilize a 5, insecs and 75 birds. (d) (insecs) 5, 5, 5, 5, 5, 5. (a) k cosh k a k or k cosh k k cosh kb h PRBLEMS PLUS N PAGE 68. f e C 9. f L L ln n (c) No L L. (a) 9.8 h ; 68 f h (c) 5. h. 6 5 CHAPTER EXERCISES. N PAGE 66.. =5 {+œ 5, 5} insecs birds (birds),9, f k sinh kb = (a)..8 = (, ) = (, ) 5 = (, ) 5 _ (c) and ; here is a local maimum or minimum 5. ( C)e sin 7. sln C 9. r 5e.. C ln 5. (a) (_5, ) = (_8, _) =_ (_, ) = (, 5) =
7 A APPENDIX I ANSWERS T DD-NUMBERED EXERCISES 7. (a) 5, (_, 5) = 9. (a), (, ) = 5, = (, ) = (, _) = (7, ) =_ (, _) =. 5, 7 8,. (a) cos, sin, cos, sin, 6 (c) cos, sin, 7. The curve is generaed in (a). In, onl he porion wih is generaed, and in (c) we ge onl he porion wih.. a cos, b sin ; a b, ellipse. a 5. (a) Two poins of inersecion. (a),. (a), (, ) (, _) 5. (a) ln 7. (a), (, ) ne collision poin a, when (c) There are sill wo inersecion poins, bu no collision poin. 7. For c, here is a cusp; for c, here is a loop whose size increases as c increases. _ Moves counerclockwise along he circle from, o,. Moves imes clockwise around he ellipse 5, saring and ending a,. I is conained in he recangle described b and (, ) = = _ 9. As n increases, he number of oscillaions increases; a and b deermine he widh and heigh. EXERCISES. N PAGE 66. cos sin. 5. e _ (_, ) = (, _) =_ = _,, e, e e, an, sec, 6, 6 7. Horizonal a, verical a, 9. Horizonal a (s, ) (four poins), verical a,..6, ; (5 6 65, e 6 5 ) _
8 APPENDIX I ANSWERS T DD-NUMBERED EXERCISES A , (c) 8.5 _, 7. (a) d sin r d cos 9. 7, 9 9 ),,. ab. e 5. r d 7. s d s s sin cos d.67. s ln( s) s ln( s) 5. s e 8 ( 6. (a) (c),,, 5 (, ),, _ (, s) _ 7. e e (a) s, s, (s, s) 5. (a) (i) (s, 7) (ii) (s, ) (i), (ii), r= _, r= =_ = 6 r=. = r= r= e se d (7s 6) 6. 5a (99s6 ) EXERCISES. N PAGE 67. (a) 5,, _ _. s 5. Circle, cener, radius 7. Circle, cener (, ), radius 9. Horizonal line, uni above he -ais. r sec. r co csc 5. r c cos 7. (a) = 5, =_ 6, 7,,, 5,,
AP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More information10.6 Parametric Equations
0_006.qd /8/05 9:05 AM Page 77 Secion 0.6 77 Parameric Equaions 0.6 Parameric Equaions Wha ou should learn Evaluae ses of parameric equaions for given values of he parameer. Skech curves ha are represened
More informationChapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationBe able to sketch a function defined parametrically. (by hand and by calculator)
Pre Calculus Uni : Parameric and Polar Equaions (7) Te References: Pre Calculus wih Limis; Larson, Hoseler, Edwards. B he end of he uni, ou should be able o complee he problems below. The eacher ma provide
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
More informationCheck in: 1 If m = 2(x + 1) and n = find y when. b y = 2m n 2
7 Parameric equaions This chaer will show ou how o skech curves using heir arameric equaions conver arameric equaions o Caresian equaions find oins of inersecion of curves and lines using arameric equaions
More informationAnswers to 1 Homework
Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:
More informationMotion along a Straight Line
chaper 2 Moion along a Sraigh Line verage speed and average velociy (Secion 2.2) 1. Velociy versus speed Cone in he ebook: fer Eample 2. Insananeous velociy and insananeous acceleraion (Secions 2.3, 2.4)
More information15. Bicycle Wheel. Graph of height y (cm) above the axle against time t (s) over a 6-second interval. 15 bike wheel
15. Biccle Wheel The graph We moun a biccle wheel so ha i is free o roae in a verical plane. In fac, wha works easil is o pu an exension on one of he axles, and ge a suden o sand on one side and hold he
More information1. Kinematics I: Position and Velocity
1. Kinemaics I: Posiion and Velociy Inroducion The purpose of his eperimen is o undersand and describe moion. We describe he moion of an objec by specifying is posiion, velociy, and acceleraion. In his
More informationChapter 11. Parametric, Vector, and Polar Functions. aπ for any integer n. Section 11.1 Parametric Functions (pp ) cot
Secion. 6 Chaper Parameric, Vecor, an Polar Funcions. an sec sec + an + Secion. Parameric Funcions (pp. 9) Eploraion Invesigaing Cclois 6. csc + co co +. 7. cos cos cos [, ] b [, 8]. na for an ineger n..
More informationMath 116 Practice for Exam 2
Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem
More informationParametrics and Vectors (BC Only)
Paramerics and Vecors (BC Only) The following relaionships should be learned and memorized. The paricle s posiion vecor is r() x(), y(). The velociy vecor is v(),. The speed is he magniude of he velociy
More information3, so θ = arccos
Mahemaics 210 Professor Alan H Sein Monday, Ocober 1, 2007 SOLUTIONS This problem se is worh 50 poins 1 Find he angle beween he vecors (2, 7, 3) and (5, 2, 4) Soluion: Le θ be he angle (2, 7, 3) (5, 2,
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationIB Physics Kinematics Worksheet
IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?
More informationPhys 221 Fall Chapter 2. Motion in One Dimension. 2014, 2005 A. Dzyubenko Brooks/Cole
Phys 221 Fall 2014 Chaper 2 Moion in One Dimension 2014, 2005 A. Dzyubenko 2004 Brooks/Cole 1 Kinemaics Kinemaics, a par of classical mechanics: Describes moion in erms of space and ime Ignores he agen
More informationDisplacement ( x) x x x
Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh
More informationPhysics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.
Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More informationPhysics 20 Lesson 5 Graphical Analysis Acceleration
Physics 2 Lesson 5 Graphical Analysis Acceleraion I. Insananeous Velociy From our previous work wih consan speed and consan velociy, we know ha he slope of a posiion-ime graph is equal o he velociy of
More informationTeaching parametric equations using graphing technology
Teaching parameric equaions using graphing echnology The session will sar by looking a problems which help sudens o see ha parameric equaions are no here o make life difficul bu are imporan and give rise
More information3 at MAC 1140 TEST 3 NOTES. 5.1 and 5.2. Exponential Functions. Form I: P is the y-intercept. (0, P) When a > 1: a = growth factor = 1 + growth rate
1 5.1 and 5. Eponenial Funcions Form I: Y Pa, a 1, a > 0 P is he y-inercep. (0, P) When a > 1: a = growh facor = 1 + growh rae The equaion can be wrien as The larger a is, he seeper he graph is. Y P( 1
More informationMidterm Exam Review Questions Free Response Non Calculator
Name: Dae: Block: Miderm Eam Review Quesions Free Response Non Calculaor Direcions: Solve each of he following problems. Choose he BEST answer choice from hose given. A calculaor may no be used. Do no
More informationa. Show that these lines intersect by finding the point of intersection. b. Find an equation for the plane containing these lines.
Mah A Final Eam Problems for onsideraion. Show all work for credi. Be sure o show wha you know. Given poins A(,,, B(,,, (,, 4 and (,,, find he volume of he parallelepiped wih adjacen edges AB, A, and A.
More informationSolution: b All the terms must have the dimension of acceleration. We see that, indeed, each term has the units of acceleration
PHYS 54 Tes Pracice Soluions Spring 8 Q: [4] Knowing ha in he ne epression a is acceleraion, v is speed, is posiion and is ime, from a dimensional v poin of view, he equaion a is a) incorrec b) correc
More information72 Calculus and Structures
72 Calculus and Srucures CHAPTER 5 DISTANCE AND ACCUMULATED CHANGE Calculus and Srucures 73 Copyrigh Chaper 5 DISTANCE AND ACCUMULATED CHANGE 5. DISTANCE a. Consan velociy Le s ake anoher look a Mary s
More informationThe average rate of change between two points on a function is d t
SM Dae: Secion: Objecive: The average rae of change beween wo poins on a funcion is d. For example, if he funcion ( ) represens he disance in miles ha a car has raveled afer hours, hen finding he slope
More informationName: Total Points: Multiple choice questions [120 points]
Name: Toal Poins: (Las) (Firs) Muliple choice quesions [1 poins] Answer all of he following quesions. Read each quesion carefully. Fill he correc bubble on your scanron shee. Each correc answer is worh
More informationAP Calculus BC - Parametric equations and vectors Chapter 9- AP Exam Problems solutions
AP Calculus BC - Parameric equaions and vecors Chaper 9- AP Exam Problems soluions. A 5 and 5. B A, 4 + 8. C A, 4 + 4 8 ; he poin a is (,). y + ( x ) x + 4 4. e + e D A, slope.5 6 e e e 5. A d hus d d
More information3.6 Derivatives as Rates of Change
3.6 Derivaives as Raes of Change Problem 1 John is walking along a sraigh pah. His posiion a he ime >0 is given by s = f(). He sars a =0from his house (f(0) = 0) and he graph of f is given below. (a) Describe
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationACCUMULATION. Section 7.5 Calculus AP/Dual, Revised /26/2018 7:27 PM 7.5A: Accumulation 1
ACCUMULATION Secion 7.5 Calculus AP/Dual, Revised 2019 vie.dang@humbleisd.ne 12/26/2018 7:27 PM 7.5A: Accumulaion 1 APPLICATION PROBLEMS A. Undersand he quesion. I is ofen no necessary o as much compuaion
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationAP CALCULUS BC 2016 SCORING GUIDELINES
6 SCORING GUIDELINES Quesion A ime, he posiion of a paricle moving in he xy-plane is given by he parameric funcions ( x ( ), y ( )), where = + sin ( ). The graph of y, consising of hree line segmens, is
More informationa 10.0 (m/s 2 ) 5.0 Name: Date: 1. The graph below describes the motion of a fly that starts out going right V(m/s)
Name: Dae: Kinemaics Review (Honors. Physics) Complee he following on a separae shee of paper o be urned in on he day of he es. ALL WORK MUST BE SHOWN TO RECEIVE CREDIT. 1. The graph below describes he
More informationNon-uniform circular motion *
OpenSax-CNX module: m14020 1 Non-uniform circular moion * Sunil Kumar Singh This work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License 2.0 Wha do we mean by non-uniform
More information4.3 Trigonometry Extended: The Circular Functions
8 CHAPTER Trigonomeric Funcions. Trigonomer Eended: The Circular Funcions Wha ou ll learn abou Trigonomeric Funcions of An Angle Trigonomeric Funcions of Real Numbers Periodic Funcions The 6-Poin Uni Circle...
More information2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance
Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion
More informationSPH3U: Projectiles. Recorder: Manager: Speaker:
SPH3U: Projeciles Now i s ime o use our new skills o analyze he moion of a golf ball ha was ossed hrough he air. Le s find ou wha is special abou he moion of a projecile. Recorder: Manager: Speaker: 0
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More informationUniversity Physics with Modern Physics 14th Edition Young TEST BANK
Universi Phsics wih Modern Phsics 14h Ediion Young SOLUTIONS MANUAL Full clear download (no formaing errors) a: hps://esbankreal.com/download/universi-phsics-modern-phsics- 14h-ediion-oung-soluions-manual-/
More informationIn this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion
More informationx i v x t a dx dt t x
Physics 3A: Basic Physics I Shoup - Miderm Useful Equaions A y A sin A A A y an A y A A = A i + A y j + A z k A * B = A B cos(θ) A B = A B sin(θ) A * B = A B + A y B y + A z B z A B = (A y B z A z B y
More informationPracticing Problem Solving and Graphing
Pracicing Problem Solving and Graphing Tes 1: Jan 30, 7pm, Ming Hsieh G20 The Bes Ways To Pracice for Tes Bes If need more, ry suggesed problems from each new opic: Suden Response Examples A pas opic ha
More informationKinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.
Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages
More informationMath 116 Second Midterm March 21, 2016
Mah 6 Second Miderm March, 06 UMID: EXAM SOLUTIONS Iniials: Insrucor: Secion:. Do no open his exam unil you are old o do so.. Do no wrie your name anywhere on his exam. 3. This exam has pages including
More informationTHE ESSENTIALS OF CALCULUS ANSWERS TO SELECTED EXERCISES
Assignmen - page. m.. f 7 7.. 7..8 7..77 7. 87. THE ESSENTIALS OF CALCULUS ANSWERS TO SELECTED EXERCISES m.... no collinear 8...,,.,.8 or.,..78,.7 or.7,.8., 8.87 or., 8.88.,,, 7..7 Assignmen - page 7.
More informationPHYSICS 220 Lecture 02 Motion, Forces, and Newton s Laws Textbook Sections
PHYSICS 220 Lecure 02 Moion, Forces, and Newon s Laws Texbook Secions 2.2-2.4 Lecure 2 Purdue Universiy, Physics 220 1 Overview Las Lecure Unis Scienific Noaion Significan Figures Moion Displacemen: Δx
More informationSolutions for homework 12
y Soluions for homework Secion Nonlinear sysems: The linearizaion of a nonlinear sysem Consider he sysem y y y y y (i) Skech he nullclines Use a disincive marking for each nullcline so hey can be disinguished
More informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
SCORING GUIDELINES (Form B) Quesion A blood vessel is 6 millimeers (mm) long Disance wih circular cross secions of varying diameer. x (mm) 6 8 4 6 Diameer The able above gives he measuremens of he B(x)
More informationLesson 3.1 Recursive Sequences
Lesson 3.1 Recursive Sequences 1) 1. Evaluae he epression 2(3 for each value of. a. 9 b. 2 c. 1 d. 1 2. Consider he sequence of figures made from riangles. Figure 1 Figure 2 Figure 3 Figure a. Complee
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationTopics covered in tutorial 01: 1. Review of definite integrals 2. Physical Application 3. Area between curves. 1. Review of definite integrals
MATH4 Calculus II (8 Spring) MATH 4 Tuorial Noes Tuorial Noes (Phyllis LIANG) IA: Phyllis LIANG Email: masliang@us.hk Homepage: hps://masliang.people.us.hk Office: Room 3 (Lif/Lif 3) Phone number: 3587453
More informationExponential and Logarithmic Functions
Chaper 5 Eponenial and Logarihmic Funcions Chaper 5 Prerequisie Skills Chaper 5 Prerequisie Skills Quesion 1 Page 50 a) b) c) Answers may vary. For eample: The equaion of he inverse is y = log since log
More informationPROBLEMS FOR MATH 162 If a problem is starred, all subproblems are due. If only subproblems are starred, only those are due. SLOPES OF TANGENT LINES
PROBLEMS FOR MATH 6 If a problem is sarred, all subproblems are due. If onl subproblems are sarred, onl hose are due. 00. Shor answer quesions. SLOPES OF TANGENT LINES (a) A ball is hrown ino he air. Is
More informationMEI Mechanics 1 General motion. Section 1: Using calculus
Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a 6 4 4 When =, v 4 a 6 4 6. (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy
More informationReview Exercises for Chapter 3
60_00R.qd //0 :9 M age CHATER Applicaions of Differeniaion Review Eercises for Chaper. Give he definiion of a criical number, and graph a funcion f showing he differen pes of criical numbers.. Consider
More informationPARAMETRIC EQUATIONS AND POLAR COORDINATES
PARAMETRIC EQUATINS AND PLAR CRDINATES Parametric equations and polar coordinates enable us to describe a great variet of new curves some practical, some beautiful, some fanciful, some strange. So far
More informationPhysics for Scientists and Engineers I
Physics for Scieniss and Engineers I PHY 48, Secion 4 Dr. Beariz Roldán Cuenya Universiy of Cenral Florida, Physics Deparmen, Orlando, FL Chaper - Inroducion I. General II. Inernaional Sysem of Unis III.
More informationAP Calculus BC 2004 Free-Response Questions Form B
AP Calculus BC 200 Free-Response Quesions Form B The maerials included in hese files are inended for noncommercial use by AP eachers for course and exam preparaion; permission for any oher use mus be sough
More informationd = ½(v o + v f) t distance = ½ (initial velocity + final velocity) time
BULLSEYE Lab Name: ANSWER KEY Dae: Pre-AP Physics Lab Projecile Moion Weigh = 1 DIRECTIONS: Follow he insrucions below, build he ramp, ake your measuremens, and use your measuremens o make he calculaions
More informationCALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version):
CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS 6 cos Secon Funamenal Theorem of Calculus: f a 4 a f 6 cos Secon Funamenal Theorem of Calculus (Chain Rule Version): g f a E. Use he Secon
More informationChapter 3 Kinematics in Two Dimensions
Chaper 3 KINEMATICS IN TWO DIMENSIONS PREVIEW Two-dimensional moion includes objecs which are moing in wo direcions a he same ime, such as a projecile, which has boh horizonal and erical moion. These wo
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationReview - Quiz # 1. 1 g(y) dy = f(x) dx. y x. = u, so that y = xu and dy. dx (Sometimes you may want to use the substitution x y
Review - Quiz # 1 (1) Solving Special Tpes of Firs Order Equaions I. Separable Equaions (SE). d = f() g() Mehod of Soluion : 1 g() d = f() (The soluions ma be given implicil b he above formula. Remember,
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More information5.2. The Natural Logarithm. Solution
5.2 The Naural Logarihm The number e is an irraional number, similar in naure o π. Is non-erminaing, non-repeaing value is e 2.718 281 828 59. Like π, e also occurs frequenly in naural phenomena. In fac,
More informationand v y . The changes occur, respectively, because of the acceleration components a x and a y
Week 3 Reciaion: Chaper3 : Problems: 1, 16, 9, 37, 41, 71. 1. A spacecraf is raveling wih a veloci of v0 = 5480 m/s along he + direcion. Two engines are urned on for a ime of 84 s. One engine gives he
More informationProblem Set 7-7. dv V ln V = kt + C. 20. Assume that df/dt still equals = F RF. df dr = =
20. Assume ha df/d sill equals = F + 0.02RF. df dr df/ d F+ 0. 02RF = = 2 dr/ d R 0. 04RF 0. 01R 10 df 11. 2 R= 70 and F = 1 = = 0. 362K dr 31 21. 0 F (70, 30) (70, 1) R 100 Noe ha he slope a (70, 1) is
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationDecimal moved after first digit = 4.6 x Decimal moves five places left SCIENTIFIC > POSITIONAL. a) g) 5.31 x b) 0.
PHYSICS 20 UNIT 1 SCIENCE MATH WORKSHEET NAME: A. Sandard Noaion Very large and very small numbers are easily wrien using scienific (or sandard) noaion, raher han decimal (or posiional) noaion. Sandard
More informationWeek 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)
Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F =
More informationAP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1. 1 : estimate = = 120 liters/hr
AP CALCULUS AB/CALCULUS BC 16 SCORING GUIDELINES Quesion 1 (hours) R ( ) (liers / hour) 1 3 6 8 134 119 95 74 7 Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for 8, where is measured
More informationUCLA: Math 3B Problem set 3 (solutions) Fall, 2018
UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 This problem se concenraes on pracice wih aniderivaives. You will ge los of pracice finding simple aniderivaives as well as finding aniderivaives graphically
More informationConceptual Physics Review (Chapters 2 & 3)
Concepual Physics Review (Chapers 2 & 3) Soluions Sample Calculaions 1. My friend and I decide o race down a sraigh srech of road. We boh ge in our cars and sar from res. I hold he seering wheel seady,
More informationStarting from a familiar curve
In[]:= NoebookDirecory Ou[]= C:\Dropbox\Work\myweb\Courses\Mah_pages\Mah_5\ You can evaluae he enire noebook by using he keyboard shorcu Al+v o, or he menu iem Evaluaion Evaluae Noebook. Saring from a
More informationQ2.1 This is the x t graph of the motion of a particle. Of the four points P, Q, R, and S, the velocity v x is greatest (most positive) at
Q2.1 This is he x graph of he moion of a paricle. Of he four poins P, Q, R, and S, he velociy is greaes (mos posiive) a A. poin P. B. poin Q. C. poin R. D. poin S. E. no enough informaion in he graph o
More informationLimits at Infinity. Limit at negative infinity. Limit at positive infinity. Definition of Limits at Infinity Let L be a real number.
0_005.qd //0 : PM Page 98 98 CHAPTER Applicaions of Differeniaion f() as Secion.5 f() = + f() as The i of f as approaches or is. Figure. Limis a Infini Deermine (finie) is a infini. Deermine he horizonal
More informationPrinciple of Least Action
The Based on par of Chaper 19, Volume II of The Feynman Lecures on Physics Addison-Wesley, 1964: pages 19-1 hru 19-3 & 19-8 hru 19-9. Edwin F. Taylor July. The Acion Sofware The se of exercises on Acion
More informationChapter 2 Trigonometric Functions
Chaper Trigonomeric Funcions Secion.. 90 7 80 6. 90 70 89 60 70 9 80 79 60 70 70 09. 90 6 89 9 60 6 6 80 6 79 9 60 6 6 7. 9.. 0. 60 0 + 60 α is a quadran III angle coerminal wih an angle of measure 0..
More informationWelcome Back to Physics 215!
Welcome Back o Physics 215! (General Physics I) Thurs. Jan 19 h, 2017 Lecure01-2 1 Las ime: Syllabus Unis and dimensional analysis Today: Displacemen, velociy, acceleraion graphs Nex ime: More acceleraion
More informationMEI STRUCTURED MATHEMATICS 4758
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Cerificae of Educaion Advanced General Cerificae of Educaion MEI STRUCTURED MATHEMATICS 4758 Differenial Equaions Thursday 5 JUNE 006 Afernoon
More information1.6. Slopes of Tangents and Instantaneous Rate of Change
1.6 Slopes of Tangens and Insananeous Rae of Change When you hi or kick a ball, he heigh, h, in meres, of he ball can be modelled by he equaion h() 4.9 2 v c. In his equaion, is he ime, in seconds; c represens
More informationAdditional Exercises for Chapter What is the slope-intercept form of the equation of the line given by 3x + 5y + 2 = 0?
ddiional Eercises for Caper 5 bou Lines, Slopes, and Tangen Lines 39. Find an equaion for e line roug e wo poins (, 7) and (5, ). 4. Wa is e slope-inercep form of e equaion of e line given by 3 + 5y +
More informationEchocardiography Project and Finite Fourier Series
Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every
More information( ) 2. Review Exercise 2. cos θ 2 3 = = 2 tan. cos. 2 x = = x a. Since π π, = 2. sin = = 2+ = = cotx. 2 sin θ 2+
Review Eercise sin 5 cos sin an cos 5 5 an 5 9 co 0 a sinθ 6 + 4 6 + sin θ 4 6+ + 6 + 4 cos θ sin θ + 4 4 sin θ + an θ cos θ ( ) + + + + Since π π, < θ < anθ should be negaive. anθ ( + ) Pearson Educaion
More information1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
More information(π 3)k. f(t) = 1 π 3 sin(t)
Mah 6 Fall 6 Dr. Lil Yen Tes Show all our work Name: Score: /6 No Calculaor permied in his par. Read he quesions carefull. Show all our work and clearl indicae our final answer. Use proper noaion. Problem
More informationUnit 1 Test Review Physics Basics, Movement, and Vectors Chapters 1-3
A.P. Physics B Uni 1 Tes Reiew Physics Basics, Moemen, and Vecors Chapers 1-3 * In sudying for your es, make sure o sudy his reiew shee along wih your quizzes and homework assignmens. Muliple Choice Reiew:
More informationBiol. 356 Lab 8. Mortality, Recruitment, and Migration Rates
Biol. 356 Lab 8. Moraliy, Recruimen, and Migraion Raes (modified from Cox, 00, General Ecology Lab Manual, McGraw Hill) Las week we esimaed populaion size hrough several mehods. One assumpion of all hese
More informationPosition, Velocity, and Acceleration
rev 06/2017 Posiion, Velociy, and Acceleraion Equipmen Qy Equipmen Par Number 1 Dynamic Track ME-9493 1 Car ME-9454 1 Fan Accessory ME-9491 1 Moion Sensor II CI-6742A 1 Track Barrier Purpose The purpose
More informationApplications of the Basic Equations Chapter 3. Paul A. Ullrich
Applicaions of he Basic Equaions Chaper 3 Paul A. Ullrich paullrich@ucdavis.edu Par 1: Naural Coordinaes Naural Coordinaes Quesion: Why do we need anoher coordinae sysem? Our goal is o simplify he equaions
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationBrock University Physics 1P21/1P91 Fall 2013 Dr. D Agostino. Solutions for Tutorial 3: Chapter 2, Motion in One Dimension
Brock Uniersiy Physics 1P21/1P91 Fall 2013 Dr. D Agosino Soluions for Tuorial 3: Chaper 2, Moion in One Dimension The goals of his uorial are: undersand posiion-ime graphs, elociy-ime graphs, and heir
More informationSPH3U1 Lesson 03 Kinematics
SPH3U1 Lesson 03 Kinemaics GRAPHICAL ANALYSIS LEARNING GOALS Sudens will Learn how o read values, find slopes and calculae areas on graphs. Learn wha hese values mean on boh posiion-ime and velociy-ime
More information