MATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)

Size: px
Start display at page:

Download "MATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)"

Transcription

1 MATH 14 AND 15 FINAL EXAM REVIEW PACKET (Revised spring 8) The following quesions cn be used s review for Mh 14/ 15 These quesions re no cul smples of quesions h will pper on he finl em, bu hey will provide ddiionl prcice for he meril h will be covered on he finl em When solving hese problems keep he following in mind: Full credi for correc nswers will only be wrded if ll work is shown Ec vlues mus be given unless n pproimion is required Credi will no be given for n pproimion when n ec vlue cn be found by echniques covered in he course The nswers, long wih commens, re posed s sepre file on hp://mhrizonedu/~clc 1 A funcion f () is coninuous nd differenible, nd hs vlues given in he ble below f () Fill in he ble wih pproime vlues for he funcion f () f () Arrnge he following numbers from smlles (1) o lrges (5) using he grph of f shown below: f( + h) f() lim h h The slope of f = 1 f (16) The verge re of chnge of f from = 1 o = 4 dy d = 8 y Suppose g () = 3 nd g () = 1 Find g( ) nd g ( ) ssuming ) g ( ) is n even funcion b) g ( ) is n odd funcion 4 For priculr pin medicion, he size of he dose, D, depends on he weigh of he pien, W We cn wrie D = fw ( ) where D is mesured in milligrms nd W is mesured in pounds ) Inerpre f (15) = 15 nd f (15) = 3in erms of his pin medicion b) Use he informion in pr ) o esime f (155)

2 5 Use he grph of f ( ) given below o skech grph of f ( ) y 6 Deermine if he semen is rue (T) or flse (F) No need o mke correcions ) If g ( ) is coninuous =, hen g ( ) mus be differenible = b) If r ( ) is posiive hen r ( ) mus be incresing c) If ( ) is concve down, hen ( ) mus be negive d) If h ( ) hs locl mimum or minimum = hen h ( ) mus be zero 7 Skech grph of f ( ) h sisfies ll of he following condiions: i) f ( ) is coninuous nd differenible everywhere ii) he only soluions of f( ) = re =,, nd 4 iii) he only soluions of f ( ) = re = 1 nd 3 iv) he only soluion of f ( ) = is = Find he following limis for f( ) = e ) lim f ( ) b) lim f ( ) c) lim f ( ) d) 1 + lim f ( ) e) lim f ( ) e e 9 Find lim h h nd some vlue (3 + h) (3) by recognizing he limi s he definiion of f ( ) for some funcion f 1 A priculr cr ws purchsed for \$5, in 4 Suppose i loses 15% of is vlue ech yer Le V () represen he vlue of he cr s funcion of he yers since i ws purchsed Find V () nd use i o find he ec vlue of V (3) 11 Use he grph of f ( ) he righ o find he vlue(s) of so h ) f( ) = b) f ( ) = c) f ( ) = f ( )

3 1 Use he grph of f ( ) he righ o find inervls where ) f ( ) is decresing b) f ( ) is concve down f ( ) Le be posiive consn Find dy for ech of he following: d 3 ) y= rcn( +) b) y = c) y = cos ( ) d) y= + e) y = sinh 14 Le f ( ) be coninuous funcion wih f (4) = 3 nd f (4) = 5 ) Find he equion of he ngen line o h ( ) = f( ) + 7 = 4 b) Is g ( ) = incresing or decresing = 4? f ( ) c) Find k () where k ( ) = f( ) ( ) d) Find m (4) where m ( ) = e f 3 15 If g ( ) = nd g ( ) = 3, find 16 Find he indiced derivives: ) dm m m = o 1 v c dv for ( ) b) g ( ) for g ( ) = < < 17 Le f( ) = 4 = 4 4 > ) Is f ( ) coninuous = 1? Differenible = 1? b) Is f ( ) coninuous =? Differenible =? c) Find f ( ) Epress your nswer s piecewise funcion

4 18 Torricelli s Theorem ses h if here is hole in coniner of liquid h fee below he surfce of he liquid, hen he liquid will flow ou re given by R( h ) = gh where g = 3f sec Find liner funcion h cn be used o pproime his re for holes h re close o 5 fee below he surfce of he wer 19 For wh vlue(s) of k will 3 f ( ) k k k = + + hve n inflecion poin = 5? The funcion ( ) is defined implicily by he equion y cos( π ) y = ln y ) Find he vlue of he derivive of y wih respec o he poin (1, 1) b) Find he equion of he ngen line o he curve (1, 1) 3 1 A cble is mde of n insuling meril in he shpe of long, hin cylinder of rdius R I hs elecricl chrge disribued evenly hroughou i The elecricl field, E, disnce r from he cener of he cble is given below k is posiive consn kr r R E = kr r > R r ) Is E coninuous r = R? b) Is E differenible r = R? c) Skech E s funcion of r d) Find de dr 1 Le f() = + for Find 3 ) he criicl poin(s) nd deermine if i is locl mimum or minimum b) he inflecion poin(s) c) he globl mimum nd minimum on he given inervl 3 Le f ( ) 3 > 1 ) he coordines of he locl mim nd he locl minim b) he coordines of he inflecion poin(s) 3 4 = + wih consn Find (nswers will be in erms of ) 4 Find he ec vlue of he following limis: π sin( θ ) ) lim b) lim c) lim rcn π sin θ sin(7 θ )

5 B 5 Consider he fmily of funcions f() = Find he vlues of A nd B so h f () hs 1 + A criicl poin (4,1) 6 Consider he fmily of funcions y () = ln for > ) Find he -inercep Your nswer will be in erms of b) Find he criicl poin nd deermine if i is locl mimum or minimum (or neiher) 7 Find he vlues of, b, nd k so h he prmeric equions given below rce ou circle of rdius 3 cenered (,4) = + kcos, y= b+ ksin, π = Consider he lines prmeerized by y = 4 9 nd = 5+ 6 y = c+ 8 ) For wh vlue of c, if ny, will hese wo lines be prllel? b) For wh vlue of c, if ny, will hese wo lines inersec (5, 3)? 9 Suppose n objec moves in he y plne long ph given by prmeric equions 3 = 3 +1, y = 4 1, ) Deermine he ime when he objec sops Where will i sop? b) Deermine he ime when he objec his he -is 3 Wire wih ol lengh of L inches will be used o consruc he edges of recngulr bo nd hus provide frmework for he bo The boom of he bo mus be squre Find he mimum volume h such bo cn hve 31 Wh re he dimensions of he lrges recngle h cn be inscribed under he grph of y = 5 so h one side is on he -is? 3 A closed recngulr bo wih squre boom hs fied volume V I mus be consruced from hree differen ypes of merils The meril used for he four sides coss \$18 per squre foo; he meril for he boom coss \$339 per squre foo, nd he meril for he op coss \$161 per squre foo Find he minimum cos for such bo in erms of V w c 33 The speed of wve rveling in deep wer is given by V( w) = k c + w where w is he wvelengh of he wve Assume c nd k re posiive consns Find he wvelengh h minimizes he speed of he wve

6 34 The grph of he funcion f ( ) nd is derivive f ( ) re given he righ ) Deermine which grph is f ( ) nd which grph is f ( ) b) Use he grphs o find he vlues of h mimize nd minimize he funcion g ( ) = f( e ) The grph below on he lef shows he number of gllons, G, of gsoline used on rip of M miles The grph below on he righ shows disnce rveled, M, s funcion of ime, in hours since he sr of he rip You cn ssume he segmens of he grphs re liner G (gllons) (7, 8) (1, 46) M (miles) ( 1, 7 ) (, 1 ) M (miles) (hours) ) Wh is he gs consumpion in miles per gllon during he firs 7 miles of he rip? During he ne 3 miles? b) If G= f( M) nd M = h (), wh does k () = f( h ()) represen? Find k(5) c) Find k (5) nd k (15) Wh do hese quniies ell us? 36 A cmer is focused on rin s he rin moves long rck owrds sion s shown he righ The rin rvels consn speed of 1 km hr How fs is he cmer roing (in rdins/min ) when he rin is km from he cmer? 37 Snd is poured ino pile from bove I forms righ circulr cone wih bse rdius h is lwys 3 imes he heigh of he cone If he snd is being poured re of 15 f 3 per minue, how fs is he heigh of he pile growing when he pile is 1 f high? h r

7 38 A volge, V vols, pplied o resisor of R ohms produces n elecricl curren of I mps where V = I R As he curren flows, he resisor hes up nd is resisnce flls If 1 vols is pplied o resisor of 1 ohms, he curren is iniilly 1 mps bu increses by 1 mps per minue A wh re is he resisnce chnging if he volge remins consn? 39 A funcion f () is coninuous nd differenible, nd hs vlues given in he ble below The vlues in he ble re represenive of he properies of he funcion f () ) Find upper nd lower esimes for b) Find 16 f () d 1 18 f () dusing 4 1 n = 4 Severl objecs re moving in srigh line from ime = o ime = 1 seconds The following re grphs of he velociies of hese objecs (in cm/sec ) ) Which objec(s) is frhes from he originl posiion he end of 1 seconds? b) Which objec(s) is closes o is originl posiion he end of 1 seconds? c) Which objec(s) hs rveled he grees ol disnce during hese 1 seconds? d) Which objec(s) hs rveled he les disnce during hese 1 seconds? Velociy of Objec A Velociy of Objec B Velociy of Objec C Velociy of Objec D

8 41 Illusre he following on he grph of f ( ) given below Assume F ( ) = f( ) ) f () b f() b) f ( b) f( ) b fhl fhl b b c) Fb () F () d) Fb ( ) F ( ) b fhl fhl b b 4 A funcion g () is posiive nd decresing everywhere Arrnge he following numbers from smlles (1) o lrges (3) 1 g ( ) Δ k = 1 k 9 g ( k ) Δ k = n lim g ( k ) Δ n k = 1 43 Le b be posiive consn Evlue he following: b+ ) ( b + 1) d b) d c) d d) b+ 1+ d ( b) 44 Find he res of he regions Include skech of he regions ) The region bounded beween y= (4 ) nd he -is b) The region bounded beween y= + nd = 3 + y 45 I is prediced h he populion of priculr ciy will grow he re of p () = 3 + (mesured in hundreds of people per yer) How mny people will be dded o he ciy in he firs four yers ccording o his model?

9 46 A ime = wer is pumped ino nk consn re of 75 gllons per hour Afer hours, he re decreses unil he flow of wer is zero ccording o r ( ) = 3( ) + 75, gllons per hour Find he ol gllons of wer pumped ino he nk 47 Use he grph of g ( ) given he righ o skech grph of g ( ) so h g () = 3 g'() A cr going 8 f sec brkes o sop in five seconds Assume he decelerion is consn ) Find n equion for v (), he velociy funcion Skech he grph of v () b) Find he ol disnce rveled from he ime he brkes were pplied unil he cr cme o sop Illusre his quniy on he grph of v () in pr ) c) Find n equion for s (), he posiion funcion Skech he grph of s () 49 Consider he funcion F ( ) = e d ) Find F () b) Find F ( ) c) Is F ( ) incresing or decresing for? d) Is F ( ) concve up or concve down for? 5 The verge vlue of f from o b is defined s 3 π f( ) = over he inervl cos 4 1 b ( ) b f d Find he verge vlue of ( ) 51 According o book of mhemicl bles, ln 5 + 4cos d = π ln π b) Find ln ( 5 4cos( )) π + π ) Find ln ( cos ) d π d

10 5 Use he grph of f ( ) below o nswer he following Circle True or Flse ) b) 1 f ( d ) f( d ) True Flse 4 5 f ( d ) f( d ) True Flse 1 1 f d f d True Flse c) ( ) ( ( )) 1 d) f( ) d True Flse e) 1 f ( ) d 1 True Flse ) If 6 f( ) d= 17, find b) If g ( ) is n odd funcion nd c) If ( ) is n even funcion nd 5 f ( d ) h ( h ) 3 3 gd ( ) =, find gd ( ) ( ) 3 d= 5, find hd ( ) 54 Use he grph of g ( ) he righ o deermine which sign is pproprie: g ( ) ) gc ( ) < = > gd ( ) b) g ( B) < = > g ( C) A B C D c) g ( A) < = > g ( B)

Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl

More information

PHYSICS 1210 Exam 1 University of Wyoming 14 February points

PHYSICS 1210 Em 1 Uniersiy of Wyoming 14 Februry 2013 150 poins This es is open-noe nd closed-book. Clculors re permied bu compuers re no. No collborion, consulion, or communicion wih oher people (oher

More information

MTH 146 Class 11 Notes

8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he

More information

ENGR 1990 Engineering Mathematics The Integral of a Function as a Function

ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under

More information

Physics 2A HW #3 Solutions

Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen

More information

1. Consider a PSA initially at rest in the beginning of the left-hand end of a long ISS corridor. Assume xo = 0 on the left end of the ISS corridor.

In Eercise 1, use sndrd recngulr Cresin coordine sysem. Le ime be represened long he horizonl is. Assume ll ccelerions nd decelerions re consn. 1. Consider PSA iniilly res in he beginning of he lef-hnd

More information

MATH 122B AND 125 FINAL EXAM REVIEW PACKET (Fall 2014)

MATH B AND FINAL EXAM REVIEW PACKET (Fll 4) The following questions cn be used s review for Mth B nd. These questions re not ctul smples of questions tht will pper on the finl em, but they will provide

More information

Average & instantaneous velocity and acceleration Motion with constant acceleration

Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission

More information

f t f a f x dx By Lin McMullin f x dx= f b f a. 2

Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes

More information

FM Applications of Integration 1.Centroid of Area

FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is

More information

September 20 Homework Solutions

College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum

More information

MATH 122B AND 125 FINAL EXAM REVIEW PACKET ANSWERS (Fall 2016) t f () t 1/2 3/4 5/4 7/4 2

MATH B AND FINAL EXAM REVIEW PACKET ANSWERS (Fall 6).....6.8 f () / / / 7/ f( + h) f(). lim h h The slope of f a = f (6) The average rae of change of f from = o = dy = 8. a) f ( a) b) f ( a) + f( a). a)

More information

Solutions to Problems from Chapter 2

Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5

More information

0 for t < 0 1 for t > 0

8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside

More information

4.8 Improper Integrals

4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls

More information

REAL ANALYSIS I HOMEWORK 3. Chapter 1

REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs

More information

First Semester Review Calculus BC

First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

More information

Magnetostatics Bar Magnet. Magnetostatics Oersted s Experiment

Mgneosics Br Mgne As fr bck s 4500 yers go, he Chinese discovered h cerin ypes of iron ore could rc ech oher nd cerin mels. Iron filings "mp" of br mgne s field Crefully suspended slivers of his mel were

More information

Version 001 test-1 swinney (57010) 1. is constant at m/s.

Version 001 es-1 swinne (57010) 1 This prin-ou should hve 20 quesions. Muliple-choice quesions m coninue on he nex column or pge find ll choices before nswering. CubeUniVec1x76 001 10.0 poins Acubeis1.4fee

More information

3 Motion with constant acceleration: Linear and projectile motion

3 Moion wih consn ccelerion: Liner nd projecile moion cons, In he precedin Lecure we he considered moion wih consn ccelerion lon he is: Noe h,, cn be posiie nd neie h leds o rie of behiors. Clerl similr

More information

( ) ( ) ( ) ( ) ( ) ( y )

8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll

More information

Phys 110. Answers to even numbered problems on Midterm Map

Phys Answers o een numbered problems on Miderm Mp. REASONING The word per indices rio, so.35 mm per dy mens.35 mm/d, which is o be epressed s re in f/cenury. These unis differ from he gien unis in boh

More information

EXERCISE - 01 CHECK YOUR GRASP

UNIT # 09 PARABOLA, ELLIPSE & HYPERBOLA PARABOLA EXERCISE - 0 CHECK YOUR GRASP. Hin : Disnce beween direcri nd focus is 5. Given (, be one end of focl chord hen oher end be, lengh of focl chord 6. Focus

More information

5.1-The Initial-Value Problems For Ordinary Differential Equations

5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil

More information

Chapter 2. Motion along a straight line. 9/9/2015 Physics 218

Chper Moion long srigh line 9/9/05 Physics 8 Gols for Chper How o describe srigh line moion in erms of displcemen nd erge elociy. The mening of insnneous elociy nd speed. Aerge elociy/insnneous elociy

More information

PARABOLA. moves such that PM. = e (constant > 0) (eccentricity) then locus of P is called a conic. or conic section.

wwwskshieducioncom PARABOLA Le S be given fixed poin (focus) nd le l be given fixed line (Direcrix) Le SP nd PM be he disnce of vrible poin P o he focus nd direcrix respecively nd P SP moves such h PM

More information

Physics Worksheet Lesson 4: Linear Motion Section: Name:

Physics Workshee Lesson 4: Liner Moion Secion: Nme: 1. Relie Moion:. All moion is. b. is n rbirry coorine sysem wih reference o which he posiion or moion of somehing is escribe or physicl lws re formule.

More information

(b) 10 yr. (b) 13 m. 1.6 m s, m s m s (c) 13.1 s. 32. (a) 20.0 s (b) No, the minimum distance to stop = 1.00 km. 1.

Answers o Een Numbered Problems Chper. () 7 m s, 6 m s (b) 8 5 yr 4.. m ih 6. () 5. m s (b).5 m s (c).5 m s (d) 3.33 m s (e) 8. ().3 min (b) 64 mi..3 h. ().3 s (b) 3 m 4..8 mi wes of he flgpole 6. (b)

More information

Contraction Mapping Principle Approach to Differential Equations

epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of

More information

CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES

CHAPTER PARAMETRIC EQUATIONS AND POLAR COORDINATES. PARAMETRIZATIONS OF PLANE CURVES., 9, _ _ Ê.,, Ê or, Ÿ. 5, 7, _ _.,, Ÿ Ÿ Ê Ê 5 Ê ( 5) Ê ˆ Ê 6 Ê ( 5) 7 Ê Ê, Ÿ Ÿ \$ 5. cos, sin, Ÿ Ÿ 6. cos ( ), sin (

More information

Physics 101 Lecture 4 Motion in 2D and 3D

Phsics 11 Lecure 4 Moion in D nd 3D Dr. Ali ÖVGÜN EMU Phsics Deprmen www.ogun.com Vecor nd is componens The componens re he legs of he righ ringle whose hpoenuse is A A A A A n ( θ ) A Acos( θ) A A A nd

More information

Chapter Direct Method of Interpolation

Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o

More information

Question Details Int Vocab 1 [ ] Question Details Int Vocab 2 [ ]

/3/5 Assignmen Previewer 3 Bsic: Definie Inegrls (67795) Due: Wed Apr 5 5 9: AM MDT Quesion 3 5 6 7 8 9 3 5 6 7 8 9 3 5 6 Insrucions Red ody's Noes nd Lerning Gols. Quesion Deils In Vocb [37897] The chnge

More information

CBSE 2014 ANNUAL EXAMINATION ALL INDIA

CBSE ANNUAL EXAMINATION ALL INDIA SET Wih Complee Eplnions M Mrks : SECTION A Q If R = {(, y) : + y = 8} is relion on N, wrie he rnge of R Sol Since + y = 8 h implies, y = (8 ) R = {(, ), (, ), (6, )}

More information

!!"#"\$%&#'()!"#&'(*%)+,&',-)./0)1-*23)

"#"\$%&#'()"#&'(*%)+,&',-)./)1-*) #\$%&'()*+,&',-.%,/)*+,-&1*#\$)()5*6\$+\$%*,7&*-'-&1*(,-&*6&,7.\$%\$+*&%'(*8\$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1\$&\$.\$%&'()*1\$\$.,'&',-9*(&,%)?%*,('&5

More information

The solution is often represented as a vector: 2xI + 4X2 + 2X3 + 4X4 + 2X5 = 4 2xI + 4X2 + 3X3 + 3X4 + 3X5 = 4. 3xI + 6X2 + 6X3 + 3X4 + 6X5 = 6.

[~ o o :- o o ill] i 1. Mrices, Vecors, nd Guss-Jordn Eliminion 1 x y = = - z= The soluion is ofen represened s vecor: n his exmple, he process of eliminion works very smoohly. We cn elimine ll enries

More information

e t dt e t dt = lim e t dt T (1 e T ) = 1

Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie

More information

Minimum Squared Error

Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > for ll smples y i solve sysem of liner inequliies MSE procedure y i = i for ll smples

More information

Name: Per: L o s A l t o s H i g h S c h o o l. Physics Unit 1 Workbook. 1D Kinematics. Mr. Randall Room 705

Nme: Per: L o s A l o s H i g h S c h o o l Physics Uni 1 Workbook 1D Kinemics Mr. Rndll Room 705 Adm.Rndll@ml.ne www.laphysics.com Uni 1 - Objecies Te: Physics 6 h Ediion Cunel & Johnson The objecies

More information

Minimum Squared Error

Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > 0 for ll smples y i solve sysem of liner inequliies MSE procedure y i i for ll smples

More information

A 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m

PHYS : Soluions o Chper 3 Home Work. SSM REASONING The displcemen is ecor drwn from he iniil posiion o he finl posiion. The mgniude of he displcemen is he shores disnce beween he posiions. Noe h i is onl

More information

2D Motion WS. A horizontally launched projectile s initial vertical velocity is zero. Solve the following problems with this information.

Nme D Moion WS The equions of moion h rele o projeciles were discussed in he Projecile Moion Anlsis Acii. ou found h projecile moes wih consn eloci in he horizonl direcion nd consn ccelerion in he ericl

More information

P441 Analytical Mechanics - I. Coupled Oscillators. c Alex R. Dzierba

Lecure 3 Mondy - Deceber 5, 005 Wrien or ls upded: Deceber 3, 005 P44 Anlyicl Mechnics - I oupled Oscillors c Alex R. Dzierb oupled oscillors - rix echnique In Figure we show n exple of wo coupled oscillors,

More information

Motion in a Straight Line

Moion in Srigh Line. Preei reched he mero sion nd found h he esclor ws no working. She wlked up he sionry esclor in ime. On oher dys, if she remins sionry on he moing esclor, hen he esclor kes her up in

More information

ECE Microwave Engineering. Fall Prof. David R. Jackson Dept. of ECE. Notes 10. Waveguides Part 7: Transverse Equivalent Network (TEN)

EE 537-635 Microwve Engineering Fll 7 Prof. Dvid R. Jcson Dep. of EE Noes Wveguides Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model Our gol is o come up wih rnsmission line model for

More information

Properties of Logarithms. Solving Exponential and Logarithmic Equations. Properties of Logarithms. Properties of Logarithms. ( x)

Properies of Logrihms Solving Eponenil nd Logrihmic Equions Properies of Logrihms Produc Rule ( ) log mn = log m + log n ( ) log = log + log Properies of Logrihms Quoien Rule log m = logm logn n log7 =

More information

3.6 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 information

1.0 Electrical Systems

. Elecricl Sysems The ypes of dynmicl sysems we will e sudying cn e modeled in erms of lgeric equions, differenil equions, or inegrl equions. We will egin y looking fmilir mhemicl models of idel resisors,

More information

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

KINEMATICS 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 information

Physic 231 Lecture 4. Mi it ftd l t. Main points of today s lecture: Example: addition of velocities Trajectories of objects in 2 = =

Mi i fd l Phsic 3 Lecure 4 Min poins of od s lecure: Emple: ddiion of elociies Trjecories of objecs in dimensions: dimensions: g 9.8m/s downwrds ( ) g o g g Emple: A foobll pler runs he pern gien in he

More information

SOME USEFUL MATHEMATICS

SOME USEFU MAHEMAICS SOME USEFU MAHEMAICS I is esy o mesure n preic he behvior of n elecricl circui h conins only c volges n currens. However, mos useful elecricl signls h crry informion vry wih ime. Since

More information

ECE Microwave Engineering

EE 537-635 Microwve Engineering Adped from noes y Prof. Jeffery T. Willims Fll 8 Prof. Dvid R. Jcson Dep. of EE Noes Wveguiding Srucures Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model

More information

Released Assessment Questions, 2017 QUESTIONS

Relese Assessmen Quesions, 17 QUESTIONS Gre 9 Assessmen of Mhemis Aemi Re he insruions elow. Along wih his ookle, mke sure ou hve he Answer Bookle n he Formul Shee. You m use n spe in his ook for rough

More information

Lecture 3: 1-D Kinematics. This Week s Announcements: Class Webpage: visit regularly

Lecure 3: 1-D Kinemics This Week s Announcemens: Clss Webpge: hp://kesrel.nm.edu/~dmeier/phys121/phys121.hml isi regulrly Our TA is Lorrine Bowmn Week 2 Reding: Chper 2 - Gincoli Week 2 Assignmens: Due:

More information

PROBLEMS 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 information

PHY2048 Exam 1 Formula Sheet Vectors. Motion. v ave (3 dim) ( (1 dim) dt. ( (3 dim) Equations of Motion (Constant Acceleration)

Insrucors: Field/Mche PHYSICS DEPATMENT PHY 48 Em Ferur, 5 Nme prin, ls firs: Signure: On m honor, I he neiher gien nor receied unuhoried id on his eminion. YOU TEST NUMBE IS THE 5-DIGIT NUMBE AT THE TOP

More information

AP 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 information

A Kalman filtering simulation

A Klmn filering simulion The performnce of Klmn filering hs been esed on he bsis of wo differen dynmicl models, ssuming eiher moion wih consn elociy or wih consn ccelerion. The former is epeced o beer

More information

Midterm 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 information

S Radio transmission and network access Exercise 1-2

S-7.330 Rdio rnsmission nd nework ccess Exercise 1 - P1 In four-symbol digil sysem wih eqully probble symbols he pulses in he figure re used in rnsmission over AWGN-chnnel. s () s () s () s () 1 3 4 )

More information

Sph3u Practice Unit Test: Kinematics (Solutions) LoRusso

Sph3u Prcice Uni Te: Kinemic (Soluion) LoRuo Nme: Tuey, Ocober 3, 07 Ku: /45 pp: /0 T&I: / Com: Thi i copy of uni e from 008. Thi will be imilr o he uni e you will be wriing nex Mony. you cn ee here re

More information

CHAPTER 2 KINEMATICS IN ONE DIMENSION ANSWERS TO FOCUS ON CONCEPTS QUESTIONS

Physics h Ediion Cunell Johnson Young Sdler Soluions Mnul Soluions Mnul, Answer keys, Insrucor's Resource Mnul for ll chpers re included. Compleed downlod links: hps://esbnkre.com/downlod/physics-h-ediion-soluions-mnulcunell-johnson-young-sdler/

More information

6. Gas dynamics. Ideal gases Speed of infinitesimal disturbances in still gas

6. Gs dynmics Dr. Gergely Krisóf De. of Fluid echnics, BE Februry, 009. Seed of infiniesiml disurbnces in sill gs dv d, dv d, Coninuiy: ( dv)( ) dv omenum r r heorem: ( ( dv) ) d 3443 4 q m dv d dv llievi

More information

Forms of Energy. Mass = Energy. Page 1. SPH4U: Introduction to Work. Work & Energy. Particle Physics:

SPH4U: Inroducion o ork ork & Energy ork & Energy Discussion Definiion Do Produc ork of consn force ork/kineic energy heore ork of uliple consn forces Coens One of he os iporn conceps in physics Alernive

More information

15. 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 information

Mathematics 805 Final Examination Answers

. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se

More information

A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES

A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR j IAN KNOWLES 1. Inroducion Consider he forml differenil operor T defined by el, (1) where he funcion q{) is rel-vlued nd loclly

More information

INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).

INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely

More information

Probability, Estimators, and Stationarity

Chper Probbiliy, Esimors, nd Sionriy Consider signl genered by dynmicl process, R, R. Considering s funcion of ime, we re opering in he ime domin. A fundmenl wy o chrcerize he dynmics using he ime domin

More information

Math 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 information

A new model for limit order book dynamics

Anewmodelforlimiorderbookdynmics JeffreyR.Russell UniversiyofChicgo,GrdueSchoolofBusiness TejinKim UniversiyofChicgo,DeprmenofSisics Absrc:Thispperproposesnewmodelforlimiorderbookdynmics.Thelimiorderbookconsiss

More information

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 9: The High Beta Tokamak

.65, MHD Theory of Fusion Sysems Prof. Freidberg Lecure 9: The High e Tokmk Summry of he Properies of n Ohmic Tokmk. Advnges:. good euilibrium (smll shif) b. good sbiliy ( ) c. good confinemen ( τ nr )

More information

K The slowest step in a mechanism has this

CM 6 Generl Chemisry II Nme SLUTINS Exm, Spring 009 Dr. Seel. (0 pins) Selec he nswer frm he clumn n he righ h bes mches ech descripin frm he clumn n he lef. Ech nswer cn be used, ms, nly nce. E G This

More information

Math 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 information

The order of reaction is defined as the number of atoms or molecules whose concentration change during the chemical reaction.

www.hechemisryguru.com Re Lw Expression Order of Recion The order of recion is defined s he number of oms or molecules whose concenrion chnge during he chemicl recion. Or The ol number of molecules or

More information

, where P is the number of bears at time t in years. dt (a) If 0 100, lim Pt. Is the solution curve increasing or decreasing?

CALCULUS BC WORKSHEET 1 ON LOGISTIC GROWTH Work he following on noebook paper. Use your calculaor on 4(b) and 4(c) only. 1. Suppose he populaion of bears in a naional park grows according o he logisic

More information

SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)

QUESTION BANK 6 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddhrh Ngr, Nrynvnm Rod 5758 QUESTION BANK (DESCRIPTIVE) Subjec wih Code :Engineering Mhemic-I (6HS6) Coure & Brnch: B.Tech Com o ll Yer & Sem:

More information

Solutions 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 information

Motion in One Dimension 2

The curren bsolue lnd speed record holder is he Briish designed ThrusSSC, win urbofn-powered cr which chieved 763 miles per hour (1,8 km/h) for he mile (1.6 km), breking he sound brrier. The cr ws driven

More information

Think of the Relationship Between Time and Space Again

Repor nd Opinion, 1(3),009 hp://wwwsciencepubne sciencepub@gmilcom Think of he Relionship Beween Time nd Spce Agin Yng F-cheng Compny of Ruid Cenre in Xinjing 15 Hongxing Sree, Klmyi, Xingjing 834000,

More information

1. 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

ME 391 Mechanical Engineering Analysis

Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of

More information

Collision Detection and Bouncing

Collision Deecion nd Bouncing Collisions re Hndled in Two Prs. Deecing he collision Mike Biley mj@cs.oregonse.edu. Hndling he physics of he collision collision-ouncing.ppx If You re Lucky, You Cn Deec

More information

IB 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 information

Introduction to LoggerPro

Inroducion o LoggerPro Sr/Sop collecion Define zero Se d collecion prmeers Auoscle D Browser Open file Sensor seup window To sr d collecion, click he green Collec buon on he ool br. There is dely of second

More information

AP 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 information

WEEK-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 information

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 2

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER Seion Eerise -: Coninuiy of he uiliy funion Le λ ( ) be he monooni uiliy funion defined in he proof of eisene of uiliy funion If his funion is oninuous y hen

More information

2.1: 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 information

AP 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 information

Chapter 2 PROBLEM SOLUTIONS

Chper PROBLEM SOLUTIONS. We ssume h you re pproximely m ll nd h he nere impulse rels uniform speed. The elpsed ime is hen Δ x m Δ = m s s. s.3 Disnces reled beween pirs of ciies re ( ) Δx = Δ = 8. km h.5

More information

UCLA: 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

x(m) t(sec ) Homework #2. Ph 231 Introductory Physics, Sp-03 Page 1 of 4

Homework #2. Ph 231 Inroducory Physics, Sp-03 Page 1 of 4 2-1A. A person walks 2 miles Eas (E) in 40 minues and hen back 1 mile Wes (W) in 20 minues. Wha are her average speed and average velociy (in ha

More information

Math 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 information

KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow

1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering

More information

An integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples.

Improper Inegrls To his poin we hve only considered inegrls f(x) wih he is of inegrion nd b finie nd he inegrnd f(x) bounded (nd in fc coninuous excep possibly for finiely mny jump disconinuiies) An inegrl

More information

Logistic growth rate. Fencing a pen. Notes. Notes. Notes. Optimization: finding the biggest/smallest/highest/lowest, etc.

Opimizaion: finding he bigges/smalles/highes/lowes, ec. Los of non-sandard problems! Logisic growh rae 7.1 Simple biological opimizaion problems Small populaions AND large populaions grow slowly N: densiy

More information

Physics 101 Fall 2006: Exam #1- PROBLEM #1

Physics 101 Fall 2006: Exam #1- PROBLEM #1 1. Problem 1. (+20 ps) (a) (+10 ps) i. +5 ps graph for x of he rain vs. ime. The graph needs o be parabolic and concave upward. ii. +3 ps graph for x of he person

More information

1. 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 information