1. (16 points) Answer the following derivative-related questions. dx tan sec x. dx tan u = du d. dx du tan u. du tan u d v.
|
|
- Gyles Merritt
- 6 years ago
- Views:
Transcription
1 Exam #2 Soluions. (6 poins) Answer he following eriaie-relae quesions. (a) (8 poins) If y an sec x, fin. This is an applicaion of he chain rule in wo sages, in which we shall le u sec x, an sec x: an sec x an u u u an u ( ) ( u an u ) u an u (sec x an x) 2 sec2 u sec x an x 2 sec x sec2 sec x (b) (4 poins) Fin ((ex + 2x) arcsin x). This is an applicaion of he prouc rule: ((ex + 2x) arcsin x) (ex + 2x) arcsin x + (e x + 2x) arcsin x (e x + 2) arcsin x+)e x + 2x) x 2 (c) (4 poins) If g() ln 3, fin g (). This is an applicaion of he quoien rule: g () ln 3 3 ( ln ) ln /3 ( 3 ) 2 3 ln ( 3 ) 2 3 2/3 This problem coul alernaiely be sole as a prouc-rule inocaion on g () ln /3. 2. (2 poins) Calculae cos(x2 e x ). Page of Karika 6
2 Exam #2 Soluions Le u x 2 e x ; hen his is a sraighforwar chain rule inocaion, alhough wih a prouc rule embee wihin i: cos(x2 e x ) cos(u) u u cos(u) ( x 2 e x) ( sin u) ( 2xe x + x 2 e x) ( sin u) ( 2xe x + x 2 e x) ( sin(x 2 e x ) ) 3. (0 poins) Fin an equaion of he angen line o he cure y e x (x 2 3x + ) a (0, ). Using he prouc rule, ex (x 2 3x + ) + e x (2x 3) an specifically when x 0, e0 + e 0 ( 3) 2, so he esire line has slope 2, an hus equaion y 2x + b. Since i passes hrough he poin (0, ) we may specifically eermine ha b, an hus ha b, for an equaion y 2x + of he angen line. 4. (5 poins) The cissoi of Diocles is a cure saisfying he equaion x(x 2 + y 2 ) 4y 2. (a) (2 poins) Fin a formula for on his cure. Taking he eriaie of each sie, we ge ( x(x 2 + y 2 ) ) ( ) 4y 2 (x 2 + y 2 ) + x (x2 + y 2 ) ( ) 4y 2 (x 2 + y 2 ) + 2x 2 + x y2 8y 3x 2 + y 2 + x y2 8y 3x 2 + y 2 + 2xy 8y 2xy 8y 3x2 y 2 (2xy 8y) 3x2 y 2 3x2 y 2 2xy 8y 3x2 + y 2 8y 2xy (b) (3 poins) Fin he equaion of he angen line o he cure a (2, 2). Page 2 of Karika 6
3 Exam #2 Soluions A he specific alues x 2, y 2, we see from he aboe ha ( 2) 2 8( 2) 2 2( 2) Thus we hae a line of he form y 2x + b; plugging in (2, 2) we can sole for b: so y 2x + 2 is he angen line b 2 b 5. (9 poins) Fin he absolue maxima an minima of he funcion f(x) x 3 2x 2 4x + 3 on he ineral [0, 3]. We know ha f (x) 3x 2 4x 4 We hen seek ou criical poins. f (x) is a polynomial, so poins where f (x) oes no exis are no uner consieraion. We also look for alues where f (x) 0. This occurs when 3x 2 4x 4 0, which, eiher by facorizaion or by he quaraic formula can be foun o be he case when x 2 or x 2. x 2 is ousie our ineral, so he only criical poin 3 3 wihin he ineral is x 2. Thus, he only caniaes for minimum an maximum are he criical poin x 2 an he enpoins x 0 an x 3. f(0) , f(2) , an f(3) Of hese, we may hus say wih cerainy ha he highes alue (he absolue maximum) is a x 0, an he lowes alue (he absolue minimum) a x (6 poins) Amy is saning moionless 50 meers eas of a norh-souh roa wih a raar gun, while Bob, who is 20 meers o he norh, is riing souh. The raar gun repors how quickly he isance beween Amy an Bob is changing (which may no be Bob s acual spee). Our seup for his problem inoles a righ riangle beween Bob s car, he place on he roa parallel o Amy, an Amy s posiion well off he roa. The eas-wes leg beween Amy an he roa is a consan lengh of 50, while he lengh of he norh-souh leg an he lengh of he hypoenuse are changing. We may enoe hese respeciely by y an s, an noe ha, by he Pyhagorean Theorem, hey hae he relaionship y s 2. We also know ha y is currenly 20, an can compue in aion ha s is currenly y (a) (2 poins) If Bob is riing souh a 30 meers per secon, wha will he raar repor as he rae of change of he isance beween Bob an Amy? The informaion gien o us here is ha 30 (since y is ecreasing a a rae of 30 meers per secon), an ha we wish o fin s. Using he relaionship escribe in he Page 3 of Karika 6
4 Exam #2 Soluions genral problem, we iffereniae boh sies wih respec o : y s 2 ( y ) s2 ( y ) s s 2y 2s y s s s s2 an hus s (b) (4 poins) Conersely, if he raar repore a change-rae of 25 meers per secon, wha woul Bob s acual spee be? In his case we are gien s ±25, an aske o fin. We may sar wih he secon-o las line of he eriaion seen in he preious secion: an sole for s s ±25 30 y 20 ± 325 s 2y 2s 7. (9 poins) Esimae he following alues using appropriae linear approximaions. 2. (a) (4 poins) (.993) 4. We consier he funcion f(x) x 4, whose eriaie is f (x) 4x 3. For x close o 2 (as.993 is), we can use he linear approximaion: f(x) f( 2) + (x + 2)f ( 2) Since f( 2) 6 an f ( 2) 32, i follows ha f(.993) ( 32) For purposes of comparison, he acual alue of (.993) 4 is aroun (b) (5 poins) We consier he funcion f(x) x, whose eriaie is 2. For x close o 25 (as x is), we can use he linear approximaion: f(x) f(25) + (x 25)f (25) Since f(25) 5 an f (25) 2, i follows ha 25 0 f(25.07) 5 + (0.07) For purposes of comparison, he acual alue of is aroun Page 4 of Karika 6
5 Exam #2 Soluions 8. (3 poins) If f(x) arcan 3x x 4 +2, hen fin f (x). We shall nee he quoien rule immeiaely, an will eenually nee recourse o he chain rule; preempiely we shall efine u 3x an x 4 + 2, which we will nee laer. Then: f (x) arcan u ( arcan u) arcan u ( ) 2 ( u arcan u) arcan u u ( 3x) arcan u ( +u 2 x4 + 2 ) 2 x (3x) 2 arcan(3x) 4x 3 2 x 4 +2 x x x 2 4x3 arcan(3x) 2 x 4 +2 x The las line of he aboe calculaion is cleanup an is opional. 9. (6 poin bonus) Currenly Yee is 0 miles norh of he Library of Babel, walking souh a 3mph, while Zachary is mile eas of he Library, walking eas a 5mph. How soon will i be he case ha he isance beween hem is (if only momenarily) unchanging? This can be sole eiher purely as a eriaie calculaion, or as a relae raes problem. The key obseraion is ha we are inerese in arious properies a imes oher han he presen, so i woul be erroneous o work uner he presumpion ha Yee an Zachary s isances from he library are specifically 0 an mile respeciely; insea, in hours, heir isances will be 0 3 an + 5 respeciely. To sole his purely as a eriaie, one coul hen compue he isance beween hem a ime o be s (0 3) 2 + ( + 5) an hen using he chain rule, s an since we wan o know when s is zero, we se his expression o zero an sole for, geing Alernaiely, we coul le Yee s an Zachary s isances from he library be gien by he ariables y 0 3 an z + 5, an by he furher known eriaies 3 an Page 5 of Karika 6
6 Exam #2 Soluions z 5. We know y2 + z 2 s 2, so iffereniaing boh sies wih respec o : ( y 2 + z 2) s2 z z2 s y2 + z s s2 z s 2y + 2z 2s 2 ( 3)(0 3) + 2 5( + 5) s 2s an since we are inerese in when s wih soluion , his equaion can be simplifie o , Page 6 of Karika 6
Section 5: Chain Rule
Chaper The Derivaive Applie Calculus 11 Secion 5: Chain Rule There is one more ype of complicae funcion ha we will wan o know how o iffereniae: composiion. The Chain Rule will le us fin he erivaive of
More information( ) ( ) ( ) ( u) ( u) = are shown in Figure =, it is reasonable to speculate that. = cos u ) and the inside function ( ( t) du
Porlan Communiy College MTH 51 Lab Manual The Chain Rule Aciviy 38 The funcions f ( = sin ( an k( sin( 3 38.1. Since f ( cos( k ( = cos( 3. Bu his woul imply ha k ( f ( = are shown in Figure =, i is reasonable
More informationLinear Motion I Physics
Linear Moion I Physics Objecives Describe he ifference beween isplacemen an isance Unersan he relaionship beween isance, velociy, an ime Describe he ifference beween velociy an spee Be able o inerpre a
More informationChapter 2 The Derivative Applied Calculus 107. We ll need a rule for finding the derivative of a product so we don t have to multiply everything out.
Chaper The Derivaive Applie Calculus 107 Secion 4: Prouc an Quoien Rules The basic rules will le us ackle simple funcions. Bu wha happens if we nee he erivaive of a combinaion of hese funcions? Eample
More informationPhysics 218 Exam 1. with Solutions Fall 2010, Sections Part 1 (15) Part 2 (20) Part 3 (20) Part 4 (20) Bonus (5)
Physics 18 Exam 1 wih Soluions Fall 1, Secions 51-54 Fill ou he informaion below bu o no open he exam unil insruce o o so! Name Signaure Suen ID E-mail Secion # ules of he exam: 1. You have he full class
More informationAdvanced Integration Techniques: Integration by Parts We may differentiate the product of two functions by using the product rule:
Avance Inegraion Techniques: Inegraion by Pars We may iffereniae he prouc of wo funcions by using he prouc rule: x f(x)g(x) = f (x)g(x) + f(x)g (x). Unforunaely, fining an anierivaive of a prouc is no
More informationThe Natural Logarithm
The Naural Logarihm 5-4-007 The Power Rule says n = n + n+ + C provie ha n. The formula oes no apply o. An anierivaive F( of woul have o saisfy F( =. Bu he Funamenal Theorem implies ha if > 0, hen Thus,
More informationThe Fundamental Theorems of Calculus
FunamenalTheorems.nb 1 The Funamenal Theorems of Calculus You have now been inrouce o he wo main branches of calculus: ifferenial calculus (which we inrouce wih he angen line problem) an inegral calculus
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 informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationINSTANTANEOUS VELOCITY
INSTANTANEOUS VELOCITY I claim ha ha if acceleraion is consan, hen he elociy is a linear funcion of ime and he posiion a quadraic funcion of ime. We wan o inesigae hose claims, and a he same ime, work
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 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 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 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 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 informationHomework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
More informationChapter Three Systems of Linear Differential Equations
Chaper Three Sysems of Linear Differenial Equaions In his chaper we are going o consier sysems of firs orer orinary ifferenial equaions. These are sysems of he form x a x a x a n x n x a x a x a n x n
More informationPractice Problems - Week #4 Higher-Order DEs, Applications Solutions
Pracice Probles - Wee #4 Higher-Orer DEs, Applicaions Soluions 1. Solve he iniial value proble where y y = 0, y0 = 0, y 0 = 1, an y 0 =. r r = rr 1 = rr 1r + 1, so he general soluion is C 1 + C e x + C
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 informationtotal distance cov ered time int erval v = average speed (m/s)
Physics Suy Noes Lesson 4 Linear Moion 1 Change an Moion a. A propery common o eeryhing in he unierse is change. b. Change is so imporan ha he funamenal concep of ime woul be meaningless wihou i. c. Since
More informationPhysics Notes - Ch. 2 Motion in One Dimension
Physics Noes - Ch. Moion in One Dimension I. The naure o physical quaniies: scalars and ecors A. Scalar quaniy ha describes only magniude (how much), NOT including direcion; e. mass, emperaure, ime, olume,
More informationChapter 2: The Derivative
Chaper The Derivaive Business Calculus 1 Chaper : The Derivaive Precalculus Iea: Slope an Rae of Change The slope of a line measures how fas a line rises or falls as we move from lef o righ along he line.
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationLAB # 2 - Equilibrium (static)
AB # - Equilibrium (saic) Inroducion Isaac Newon's conribuion o physics was o recognize ha despie he seeming compleiy of he Unierse, he moion of is pars is guided by surprisingly simple aws. Newon's inspiraion
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
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 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 information(c) Several sets of data points can be used to calculate the velocity. One example is: distance speed = time 4.0 m = 1.0 s speed = 4.
Inquiry an Communicaion 8. (a) ensiy eermine by Group A is he mos reasonable. (b) When roune off o wo significan igis, Group B has he same alue as Group A. Howeer, saing an experimenal measuremen o six
More informationInstructor: Barry McQuarrie Page 1 of 5
Procedure for Solving radical equaions 1. Algebraically isolae one radical by iself on one side of equal sign. 2. Raise each side of he equaion o an appropriae power o remove he radical. 3. Simplify. 4.
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 informationMath 4600: Homework 11 Solutions
Mah 46: Homework Soluions Gregory Handy [.] One of he well-known phenomenological (capuring he phenomena, bu no necessarily he mechanisms) models of cancer is represened by Gomperz equaion dn d = bn ln(n/k)
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 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 informationEquations of motion for constant acceleration
Lecure 3 Chaper 2 Physics I 01.29.2014 Equaions of moion for consan acceleraion Course websie: hp://faculy.uml.edu/andriy_danylo/teaching/physicsi Lecure Capure: hp://echo360.uml.edu/danylo2013/physics1spring.hml
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationSection 2.6 Derivatives of products and quotients
Secion 2.6 Derivaives of proucs an quoiens (3/19/08) Overview: In his secion, we erive formulas for erivaives of funcions ha are consruce by aking proucs an quoiens of oher funcions, an we use hese formulas
More informationSections 2.2 & 2.3 Limit of a Function and Limit Laws
Mah 80 www.imeodare.com Secions. &. Limi of a Funcion and Limi Laws In secion. we saw how is arise when we wan o find he angen o a curve or he velociy of an objec. Now we urn our aenion o is in general
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 informationAP 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 informationMath 115 Final Exam December 14, 2017
On my honor, as a suden, I have neiher given nor received unauhorized aid on his academic work. Your Iniials Only: Iniials: Do no wrie in his area Mah 5 Final Exam December, 07 Your U-M ID # (no uniqname):
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationSection 4.4 Logarithmic Properties
Secion. Logarihmic Properies 59 Secion. Logarihmic Properies In he previous secion, we derived wo imporan properies of arihms, which allowed us o solve some asic eponenial and arihmic equaions. Properies
More informationHOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.
HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =
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 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 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 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 informationMath 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11.
1 Mah 334 Tes 1 KEY Spring 21 Secion: 1 Insrucor: Sco Glasgow Daes: Ma 1 and 11. Do NOT wrie on his problem saemen bookle, excep for our indicaion of following he honor code jus below. No credi will be
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 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 informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More informationCircuit Variables. AP 1.1 Use a product of ratios to convert two-thirds the speed of light from meters per second to miles per second: 1 ft 12 in
Circui Variables 1 Assessmen Problems AP 1.1 Use a produc of raios o conver wo-hirds he speed of ligh from meers per second o miles per second: ( ) 2 3 1 8 m 3 1 s 1 cm 1 m 1 in 2.54 cm 1 f 12 in 1 mile
More informationNEWTON S SECOND LAW OF MOTION
Course and Secion Dae Names NEWTON S SECOND LAW OF MOTION The acceleraion of an objec is defined as he rae of change of elociy. If he elociy changes by an amoun in a ime, hen he aerage acceleraion during
More informationModule II, Part C. More Insight into Fiber Dispersion
Moule II Par C More Insigh ino Fiber Dispersion . Polariaion Moe Dispersion Fiber Birefringence: Imperfec cylinrical symmery leas o wha is known as birefringence. Recall he HE moe an is E x componen which
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 information2.4 Cuk converter example
2.4 Cuk converer example C 1 Cuk converer, wih ideal swich i 1 i v 1 2 1 2 C 2 v 2 Cuk converer: pracical realizaion using MOSFET and diode C 1 i 1 i v 1 2 Q 1 D 1 C 2 v 2 28 Analysis sraegy This converer
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 informationVelocity is a relative quantity
Veloci is a relaie quani Disenangling Coordinaes PHY2053, Fall 2013, Lecure 6 Newon s Laws 2 PHY2053, Fall 2013, Lecure 6 Newon s Laws 3 R. Field 9/6/2012 Uniersi of Florida PHY 2053 Page 8 Reference Frames
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 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 informationPhysics 101: Lecture 03 Kinematics Today s lecture will cover Textbook Sections (and some Ch. 4)
Physics 101: Lecure 03 Kinemaics Today s lecure will coer Texbook Secions 3.1-3.3 (and some Ch. 4) Physics 101: Lecure 3, Pg 1 A Refresher: Deermine he force exered by he hand o suspend he 45 kg mass as
More information1. The graph below shows the variation with time t of the acceleration a of an object from t = 0 to t = T. a
Kinemaics Paper 1 1. The graph below shows he ariaion wih ime of he acceleraion a of an objec from = o = T. a T The shaded area under he graph represens change in A. displacemen. B. elociy. C. momenum.
More informationExam 1 Solutions. 1 Question 1. February 10, Part (A) 1.2 Part (B) To find equilibrium solutions, set P (t) = C = dp
Exam Soluions Februar 0, 05 Quesion. Par (A) To find equilibrium soluions, se P () = C = = 0. This implies: = P ( P ) P = P P P = P P = P ( + P ) = 0 The equilibrium soluion are hus P () = 0 and P () =..
More informationChapter 12: Velocity, acceleration, and forces
To Feel a Force Chaper Spring, Chaper : A. Saes of moion For moion on or near he surface of he earh, i is naural o measure moion wih respec o objecs fixed o he earh. The 4 hr. roaion of he earh has a measurable
More informationApplication 5.4 Defective Eigenvalues and Generalized Eigenvectors
Applicaion 5.4 Defecive Eigenvalues and Generalized Eigenvecors The goal of his applicaion is he soluion of he linear sysems like where he coefficien marix is he exoic 5-by-5 marix x = Ax, (1) 9 11 21
More informationLab #2: Kinematics in 1-Dimension
Reading Assignmen: Chaper 2, Secions 2-1 hrough 2-8 Lab #2: Kinemaics in 1-Dimension Inroducion: The sudy of moion is broken ino wo main areas of sudy kinemaics and dynamics. Kinemaics is he descripion
More informationSolutionbank Edexcel AS and A Level Modular Mathematics
Page of 4 Soluionbank Edexcel AS and A Level Modular Mahemaics Exercise A, Quesion Quesion: Skech he graphs of (a) y = e x + (b) y = 4e x (c) y = e x 3 (d) y = 4 e x (e) y = 6 + 0e x (f) y = 00e x + 0
More informationChapter 2. Motion in One-Dimension I
Chaper 2. Moion in One-Dimension I Level : AP Physics Insrucor : Kim 1. Average Rae of Change and Insananeous Velociy To find he average velociy(v ) of a paricle, we need o find he paricle s displacemen
More information1 Solutions to selected problems
1 Soluions o seleced problems 1. Le A B R n. Show ha in A in B bu in general bd A bd B. Soluion. Le x in A. Then here is ɛ > 0 such ha B ɛ (x) A B. This shows x in B. If A = [0, 1] and B = [0, 2], hen
More informationKinematics. introduction to kinematics 15A
15 15A Inroducion o kinemaics 15B Velociy ime graphs and acceleraion ime graphs 15C Consan acceleraion formulas 15D Insananeous raes of change Kinemaics AreAS of STuDy Diagrammaic and graphical represenaion
More informationPhysics for Scientists and Engineers. Chapter 2 Kinematics in One Dimension
Physics for Scieniss and Engineers Chaper Kinemaics in One Dimension Spring, 8 Ho Jung Paik Kinemaics Describes moion while ignoring he agens (forces) ha caused he moion For now, will consider moion in
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
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 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 informationMATH 128A, SUMMER 2009, FINAL EXAM SOLUTION
MATH 28A, SUMME 2009, FINAL EXAM SOLUTION BENJAMIN JOHNSON () (8 poins) [Lagrange Inerpolaion] (a) (4 poins) Le f be a funcion defined a some real numbers x 0,..., x n. Give a defining equaion for he Lagrange
More informationCOMPETITIVE GROWTH MODEL
COMPETITIVE GROWTH MODEL I Assumpions We are going o now solve he compeiive version of he opimal growh moel. Alhough he allocaions are he same as in he social planning problem, i will be useful o compare
More informationTwo Dimensional Dynamics
Physics 11: Lecure 6 Two Dimensional Dynamics Today s lecure will coer Chaper 4 Exam I Physics 11: Lecure 6, Pg 1 Brie Reiew Thus Far Newon s Laws o moion: SF=ma Kinemaics: x = x + + ½ a Dynamics Today
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationOne-Dimensional Kinematics
One-Dimensional Kinemaics One dimensional kinemaics refers o moion along a sraigh line. Een hough we lie in a 3-dimension world, moion can ofen be absraced o a single dimension. We can also describe moion
More informationMathcad Lecture #8 In-class Worksheet Curve Fitting and Interpolation
Mahcad Lecure #8 In-class Workshee Curve Fiing and Inerpolaion A he end of his lecure, you will be able o: explain he difference beween curve fiing and inerpolaion decide wheher curve fiing or inerpolaion
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 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 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 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 informationTwo Dimensional Dynamics
Physics 11: Lecure 6 Two Dimensional Dynamics Today s lecure will coer Chaper 4 Saring Wed Sep 15, W-F oice hours will be in 3 Loomis. Exam I M oice hours will coninue in 36 Loomis Physics 11: Lecure 6,
More informationDerivatives of Inverse Trig Functions
Derivaives of Inverse Trig Fncions Ne we will look a he erivaives of he inverse rig fncions. The formlas may look complicae, b I hink yo will fin ha hey are no oo har o se. Yo will js have o be carefl
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationSeminar 5 Sustainability
Seminar 5 Susainabiliy Soluions Quesion : Hyperbolic Discouning -. Suppose a faher inheris a family forune of 0 million NOK an he wans o use some of i for himself (o be precise, he share ) bu also o beques
More informationSpiral Symmetry on the TI-92 Paul Beem Indiana University South Bend South Bend, IN
Proceeings of he Thir Inernaional DERIVE/TI-92 Conference Spiral Symmery on he TI-92 Paul Beem Iniana Universiy Souh Ben Souh Ben, IN pbeem@iusb.eu We all hink an reason in conex. In a linear algebra class,
More informationSolutions to Odd Number Exercises in Chapter 6
1 Soluions o Odd Number Exercises in 6.1 R y eˆ 1.7151 y 6.3 From eˆ ( T K) ˆ R 1 1 SST SST SST (1 R ) 55.36(1.7911) we have, ˆ 6.414 T K ( ) 6.5 y ye ye y e 1 1 Consider he erms e and xe b b x e y e b
More informationRole of Material Functions in Rheological Analysis. Role of Material Functions in Rheological Analysis
Role of Maerial Funcions in Rheological Analysis QUALIY CONROL compare wih oher in-house daa on qualiaie basis unknown maerial QUALIAIVE ANALYSIS compare daa wih lieraure repors on arious fluids conclude
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 informationVS203B Lecture Notes Spring, 2011 Topic: Diffraction
Diffracion Diffracion escribes he enency for ligh o ben aroun corners. Huygens principle All poins on a wavefron can be consiere as poin sources for he proucion of seconary waveles, an a a laer ime he
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More information