XV Exponential and Logarithmic Functions

Size: px
Start display at page:

Download "XV Exponential and Logarithmic Functions"

Transcription

1 MATHEMATICS 0-0-RE Dirnial Calculus Marin Huard Winr 08 XV Eponnial and Logarihmic Funcions. Skch h graph o h givn uncions and sa h domain and rang. d) ) ) log. Whn Sarah was born, hr parns placd $000 or hr in an 8-ar criica ha pas 6% compoundd monhl. How much mon will h criica b worh on hr 8 h birhda?. Sam as bu h inrs is compoundd coninuousl a a ra on %?. Which invsmn will arn mor mon, a $0 000 invsmn or ars a 8% compoundd annuall, or a $0 000 invsmn or ars a 7%, compoundd quarrl? Wha is h balanc in ach cas? Th civ ra?. How much mon should a compan dposi in an accoun wih a 8% compoundd quarrl in ordr o hav $ or a crain pic o machinr ars rom now? 6. Eprss h quaion in ponnial orm. log8 log 9 log 7. Eprss h quaion in logarihmic orm Evalua acl (wihou h us o a calculaor). log log 0 log 9 log 6 d) log 8 ) ) log i) log g) log7 9 h) 9. Rwri ach prssion in a orm wih no logarihms o producs, quoins, or powrs. log6 0 d) z 0. Rwri h prssion as a singl logarihm. log log log ) 7 9

2 Mah 0 XV Eponnial and Logarihmic Funcions. Solv h quaion or. 7 ) g) 00 log d) 0 ) h) log log log 6 0 i) 0 j) 0 k) 0 l) 8. A dposi o $000 is mad in an accoun ha arns inrs a an annual ra o 8%. How long will i ak or h balanc o doubl i h inrs is compoundd annuall monhl dail d) coninuousl Th dmand uncion or a produc is p0 0.8 whr p is h pric pr uni and is h numbr o unis sold. Find h numbr o unis sold or a pric o $00 $. Dirnia ach uncion.. Find d. sin d) ) g log ) g) h log h) log i) an j) k) l) n) sin m) o) p) sin q) sin r) cos s) z ) g z z u) sc sc z z v) csc csc ) d) 6 csc cos an ) an Winr 08 Marin Huard

3 Mah 0 XV Eponnial and Logarihmic Funcions 6. Find g) h) cos sin sin sin cos i) j) sin cos k) sin l). sin cos 7. Find i. 8. Find a gnral ormula or coupl o drivaivs. n i. Hin: Tr o s a parn in h irs 9. Find all h poins o h graph o whr hr is a horizonal angn. 0. Find h quaion or h angn lin o h graph o a Th balanc A (in dollars) in a savings accoun is givn b A 000, whr is masurd in ars. Find h ra a which h balanc is changing whn ars 0 ars ar. Th valu o a compur ars ar purchas is 0. h compur s valu alling ar ars? v 000 dollars. A wha ra is. Th cos (in dollars) or producing unis o a commodi is givn b C whil h dmand is givn b p a producion lvl o 0 unis Find h marginal cos, rvnu and proi or 0 Winr 08 Marin Huard

4 Mah 0 XV Eponnial and Logarihmic Funcions ANSWERS. Domain: Domain: 0, Rang: 0, Rang: Domain: d) Domain: Rang: Rang:,, ) Domain: Rang: 0, ) Domain: Rang: / $ A 8% sinc a 8% $0. A 0000 $69.8 ( r 8% ) 07 whil a 7%: A $7.78 ( $ log 8 8. ) g) 6 6 r % ) log 9 log000 8 d) 99 ) h) i) log log 0. d) z ) log 8. log d) 999 ) i) ) g) h) j) k) 0 l), ars 8.69 ars ars d) ars unis unis Winr 08 Marin Huard

5 Mah 0 XV Eponnial and Logarihmic Funcions cos sin.. sin d) ) g ) log g) h log an i) sc j) 0 h) k) 60 l) m) n) sin cos 6sin cos 0 log Winr 08 Marin Huard sin o) 8 p) sin q) cos r) an z z z ) z s) z z z z u) z z z g z v) csc csc an sc an csc co csc co csc co d d) cos d sin d d ) an an sc d an an sc g) ) d d h) i) cos sin sin cos an co d 6 d sin sin sin sin cos j) co co an an d cos k) sin sin sin co d l) d cos sin ,0,, and $9.. Falling a a ra o $0.9 n n n, $0.9 $.96 pr ar. R 0 $. 80 $ C pr uni pr uni P uni 9 0 $.6 pr 6

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o

More information

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of Sampl Final 00 1. Suppos z = (, y), ( a, b ) = 0, y ( a, b ) = 0, ( a, b ) = 1, ( a, b ) = 1, and y ( a, b ) =. Thn (a, b) is h s is inconclusiv a saddl poin a rlaiv minimum a rlaiv maimum. * (Classiy

More information

5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t

5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =

More information

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35 MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h

More information

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall. Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin

More information

Elementary Differential Equations and Boundary Value Problems

Elementary Differential Equations and Boundary Value Problems Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ

More information

SOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz

SOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz STAT UIUC Pracic Problms #7 SOLUTIONS Spanov Dalpiaz Th following ar a numbr of pracic problms ha ma b hlpful for compling h homwor, and will lil b vr usful for suding for ams.. Considr wo coninuous random

More information

Chapter 2 The Derivative Business Calculus 99

Chapter 2 The Derivative Business Calculus 99 Chapr Th Drivaiv Businss Calculus 99 Scion 5: Drivaivs of Formulas In his scion, w ll g h rivaiv ruls ha will l us fin formulas for rivaivs whn our funcion coms o us as a formula. This is a vry algbraic

More information

EXERCISE - 01 CHECK YOUR GRASP

EXERCISE - 01 CHECK YOUR GRASP DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc

More information

Midterm Examination (100 pts)

Midterm Examination (100 pts) Econ 509 Spring 2012 S.L. Parn Midrm Examinaion (100 ps) Par I. 30 poins 1. Dfin h Law of Diminishing Rurns (5 ps.) Incrasing on inpu, call i inpu x, holding all ohr inpus fixd, on vnuall runs ino h siuaion

More information

Midterm exam 2, April 7, 2009 (solutions)

Midterm exam 2, April 7, 2009 (solutions) Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions

More information

Chap.3 Laplace Transform

Chap.3 Laplace Transform Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl

More information

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar

More information

MATH ANALYSIS HONORS UNIT 6 EXPONENTIAL FUNCTIONS TOTAL NAME DATE PERIOD DATE TOPIC ASSIGNMENT /19 10/22 10/23 10/24 10/25 10/26 10/29 10/30

MATH ANALYSIS HONORS UNIT 6 EXPONENTIAL FUNCTIONS TOTAL NAME DATE PERIOD DATE TOPIC ASSIGNMENT /19 10/22 10/23 10/24 10/25 10/26 10/29 10/30 NAME DATE PERIOD MATH ANALYSIS HONORS UNIT 6 EXPONENTIAL FUNCTIONS DATE TOPIC ASSIGNMENT 10 0 10/19 10/ 10/ 10/4 10/5 10/6 10/9 10/0 10/1 11/1 11/ TOTAL Mah Analysis Honors Workshee 1 Eponenial Funcions

More information

S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]

S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15] S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:

More information

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -

More information

The Natural Logarithmic Function: Differentiation. The Natural Logarithmic Function

The Natural Logarithmic Function: Differentiation. The Natural Logarithmic Function 60_00.q //0 :0 PM Pag CHAPTER Logarihmic, Eponnial, an Ohr Transcnnal Funcions Scion. Th Naural Logarihmic Funcion: Diffrniaion Dvlop an us propris of h naural logarihmic funcion. Unrsan h finiion of h

More information

Math 34A. Final Review

Math 34A. Final Review Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right

More information

Chapter 12 Introduction To The Laplace Transform

Chapter 12 Introduction To The Laplace Transform Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and

More information

46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r)

46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r) 98 Scion 7.. L w. Thn dw d, so d dw w dw. sin d (sin w)( wdw) w sin w dw L u w dv sin w dw du dw v cos w w sin w dw w cos w + cos w dw w cos w+ sin w+ sin d wsin wdw w cos w+ sin w+ cos + sin +. L w +

More information

Double Slits in Space and Time

Double Slits in Space and Time Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an

More information

Additional Math (4047) Paper 2 (100 marks) y x. 2 d. d d

Additional Math (4047) Paper 2 (100 marks) y x. 2 d. d d Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com C3 Eponnials and logarihms - Eponnial quaions. Rabbis wr inroducd ono an island. Th numbr of rabbis, P, yars afr hy wr inroducd is modlld by h quaion P = 3 0, 0 (a) Wri down h numbr of rabbis ha wr inroducd

More information

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions 4.0 rincipl of Macroconomic Fall 005 Quiz 3 Soluion Shor Quion (30/00 poin la a whhr h following amn ar TRUE or FALSE wih a hor xplanaion (3 or 4 lin. Each quion coun 5/00 poin.. An incra in ax oday alway

More information

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 3/28/2012. UW Madison

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 3/28/2012. UW Madison Economics 302 (Sc. 001) Inrmdia Macroconomic Thory and Policy (Spring 2011) 3/28/2012 Insrucor: Prof. Mnzi Chinn Insrucor: Prof. Mnzi Chinn UW Madison 16 1 Consumpion Th Vry Forsighd dconsumr A vry forsighd

More information

Partial Derivatives: Suppose that z = f(x, y) is a function of two variables.

Partial Derivatives: Suppose that z = f(x, y) is a function of two variables. Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv

More information

Chapter 6 Differential Equations and Mathematical Modeling

Chapter 6 Differential Equations and Mathematical Modeling 6 Scion 6. hapr 6 Diffrnial Equaions and Mahmaical Modling Scion 6. Slop Filds and Eulr s Mhod (pp. ) Eploraion Sing h Slops. Sinc rprsns a lin wih a slop of, w should d pc o s inrvals wih no chang in.

More information

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm

More information

Section 4.3 Logarithmic Functions

Section 4.3 Logarithmic Functions 48 Chapr 4 Sion 4.3 Logarihmi Funions populaion of 50 flis is pd o doul vry wk, lading o a funion of h form f ( ) 50(), whr rprsns h numr of wks ha hav passd. Whn will his populaion rah 500? Trying o solv

More information

Final Exam : Solutions

Final Exam : Solutions Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b

More information

DSP-First, 2/e. This Lecture: LECTURE #3 Complex Exponentials & Complex Numbers. Introduce more tools for manipulating complex numbers

DSP-First, 2/e. This Lecture: LECTURE #3 Complex Exponentials & Complex Numbers. Introduce more tools for manipulating complex numbers DSP-Fis, / LECTURE #3 Compl Eponnials & Compl umbs READIG ASSIGMETS This Lcu: Chap, Scs. -3 o -5 Appndi A: Compl umbs Appndi B: MATLAB Lcu: Compl Eponnials Aug 016 003-016, JH McClllan & RW Schaf 3 LECTURE

More information

Wave Equation (2 Week)

Wave Equation (2 Week) Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris

More information

First derivative analysis

First derivative analysis Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points

More information

Physics 160 Lecture 3. R. Johnson April 6, 2015

Physics 160 Lecture 3. R. Johnson April 6, 2015 Physics 6 Lcur 3 R. Johnson April 6, 5 RC Circui (Low-Pass Filr This is h sam RC circui w lookd a arlir h im doma, bu hr w ar rsd h frquncy rspons. So w pu a s wav sad of a sp funcion. whr R C RC Complx

More information

CHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano

CHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano Expcaions: Th Basic Prpard by: Frnando Quijano and Yvonn Quijano CHAPTER CHAPTER14 2006 Prnic Hall Businss Publishing Macroconomics, 4/ Olivir Blanchard 14-1 Today s Lcur Chapr 14:Expcaions: Th Basic Th

More information

CSE 245: Computer Aided Circuit Simulation and Verification

CSE 245: Computer Aided Circuit Simulation and Verification CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy

More information

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn

More information

Transfer function and the Laplace transformation

Transfer function and the Laplace transformation Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and

More information

MSLC Math 151 WI09 Exam 2 Review Solutions

MSLC Math 151 WI09 Exam 2 Review Solutions Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y

More information

( ) ( ) + = ( ) + ( )

( ) ( ) + = ( ) + ( ) Mah 0 Homwork S 6 Soluions 0 oins. ( ps I ll lav i o you vrify ha h omplimnary soluion is : y ( os( sin ( Th guss for h pariular soluion and is drivaivs ar, +. ( os( sin ( ( os( ( sin ( Y ( D 6B os( +

More information

1973 AP Calculus BC: Section I

1973 AP Calculus BC: Section I 97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f

More information

10. If p and q are the lengths of the perpendiculars from the origin on the tangent and the normal to the curve

10. If p and q are the lengths of the perpendiculars from the origin on the tangent and the normal to the curve 0. If p and q ar h lnghs of h prpndiculars from h origin on h angn and h normal o h curv + Mahmaics y = a, hn 4p + q = a a (C) a (D) 5a 6. Wha is h diffrnial quaion of h family of circls having hir cnrs

More information

5.1 - Logarithms and Their Properties

5.1 - 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 information

Logarithms Practice Exam - ANSWERS

Logarithms Practice Exam - ANSWERS Logarihms racice Eam - ANSWERS Answers. C. D 9. A 9. D. A. C. B. B. D. C. B. B. C NR.. C. B. B. B. B 6. D. C NR. 9. NR. NR... C 7. B. C. B. C 6. C 6. C NR.. 7. B 7. D 9. A. D. C Each muliple choice & numeric

More information

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields! Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr

More information

Parametrics and Vectors (BC Only)

Parametrics 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 information

Section 11.6: Directional Derivatives and the Gradient Vector

Section 11.6: Directional Derivatives and the Gradient Vector Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th

More information

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form:

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form: Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,

More information

are given in the table below. t (hours)

are given in the table below. t (hours) CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th

More information

2008 AP Calculus BC Multiple Choice Exam

2008 AP Calculus BC Multiple Choice Exam 008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl

More information

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +

More information

Calculus Revision A2 Level

Calculus Revision A2 Level alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ

More information

Institute of Actuaries of India

Institute of Actuaries of India Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6

More information

Integral Calculus What is integral calculus?

Integral Calculus What is integral calculus? Intgral Calulus What is intgral alulus? In diffrntial alulus w diffrntiat a funtion to obtain anothr funtion alld drivativ. Intgral alulus is onrnd with th opposit pross. Rvrsing th pross of diffrntiation

More information

+ f. e f. Ch. 8 Inflation, Interest Rates & FX Rates. Purchasing Power Parity. Purchasing Power Parity

+ f. e f. Ch. 8 Inflation, Interest Rates & FX Rates. Purchasing Power Parity. Purchasing Power Parity Ch. 8 Inlation, Intrst Rats & FX Rats Topics Purchasing Powr Parity Intrnational Fishr Ect Purchasing Powr Parity Purchasing Powr Parity (PPP: Th purchasing powr o a consumr will b similar whn purchasing

More information

Chapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.

Chapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7. Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation

More information

NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA

NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals

More information

Lecture #6: Continuous-Time Signals

Lecture #6: Continuous-Time Signals EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals Lecure #6: Coninuous-Time Signals. Inroducion In his lecure, we discussed he ollowing opics:. Mahemaical represenaion and ransormaions

More information

C From Faraday's Law, the induced voltage is, C The effect of electromagnetic induction in the coil itself is called selfinduction.

C From Faraday's Law, the induced voltage is, C The effect of electromagnetic induction in the coil itself is called selfinduction. Inducors and Inducanc C For inducors, v() is proporional o h ra of chang of i(). Inducanc (con d) C Th proporionaliy consan is h inducanc, L, wih unis of Hnris. 1 Hnry = 1 Wb / A or 1 V sc / A. C L dpnds

More information

AP Calculus BC Chapter 10 Part 1 AP Exam Problems

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

Math 221: Mathematical Notation

Math 221: Mathematical Notation Mah 221: Mahemaical Noaion Purpose: One goal in any course is o properly use he language o ha subjec. These noaions summarize some o he major conceps and more diicul opics o he uni. Typing hem helps you

More information

DIFFERENTIAL EQUATION

DIFFERENTIAL EQUATION MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous

More information

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 ) AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc

More information

Strongly Connected Components

Strongly Connected Components Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts

More information

Chapter 6 Test December 9, 2008 Name

Chapter 6 Test December 9, 2008 Name Chapr 6 Ts Dcmbr 9, 8 Nam. Evalua - ÄÄ ÄÄ - + - ÄÄ ÄÄ ÄÄ - + H - L u - and du u - du - - - A - u - D - - - ÄÄÄÄ 6 6. Evalua i j + z i j + z i 7 j ÄÄÄ + z 9 + ÄÄÄ ÄÄÄ 9 ÄÄÄ + C 6 ÄÄÄ + ÄÄÄ 9 ÄÄÄ + C 9.

More information

Poisson process Markov process

Poisson process Markov process E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc

More information

2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes

2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion

More information

Homework: Introduction to Motion

Homework: Introduction to Motion Homwork: Inroducon o Moon Dsanc vs. Tm Graphs Nam Prod Drcons: Answr h foowng qusons n h spacs provdd. 1. Wha do you do o cra a horzona n on a dsancm graph? 2. How do you wak o cra a sragh n ha sops up?

More information

Exponential and Logarithmic Functions -- ANSWERS -- Logarithms Practice Diploma ANSWERS 1

Exponential and Logarithmic Functions -- ANSWERS -- Logarithms Practice Diploma ANSWERS 1 Eponenial and Logarihmic Funcions -- ANSWERS -- Logarihms racice Diploma ANSWERS www.puremah.com Logarihms Diploma Syle racice Eam Answers. C. D 9. A 7. C. A. C. B 8. D. D. C NR. 8 9. C 4. C NR. NR 6.

More information

Control System Engineering (EE301T) Assignment: 2

Control System Engineering (EE301T) Assignment: 2 Conrol Sysm Enginring (EE0T) Assignmn: PART-A (Tim Domain Analysis: Transin Rspons Analysis). Oain h rspons of a uniy fdack sysm whos opn-loop ransfr funcion is (s) s ( s 4) for a uni sp inpu and also

More information

1973 AP Calculus AB: Section I

1973 AP Calculus AB: Section I 97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=

More information

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches. Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M

More information

Math 111 Midterm I, Lecture A, version 1 -- Solutions January 30 th, 2007

Math 111 Midterm I, Lecture A, version 1 -- Solutions January 30 th, 2007 NAME: Suden ID #: QUIZ SECTION: Mah 111 Miderm I, Lecure A, version 1 -- Soluions January 30 h, 2007 Problem 1 4 Problem 2 6 Problem 3 20 Problem 4 20 Toal: 50 You are allowed o use a calculaor, a ruler,

More information

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim. MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function

More information

Math 105 Second Midterm March 16, 2017

Math 105 Second Midterm March 16, 2017 Mah 105 Second Miderm March 16, 2017 UMID: Insrucor: Iniials: Secion: 1. Do no open his exam unil you are old o do so. 2. Do no wrie your name anywhere on his exam. 3. This exam has 9 pages including his

More information

a 1and x is any real number.

a 1and x is any real number. Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars

More information

Southern Taiwan University

Southern Taiwan University Chaptr Ordinar Diffrntial Equations of th First Ordr and First Dgr Gnral form:., d +, d 0.a. f,.b I. Sparabl Diffrntial quations Form: d + d 0 C d d E 9 + 4 0 Solution: 9d + 4d 0 9 + 4 C E + d Solution:

More information

1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.

1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h. NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim

More information

On the Speed of Heat Wave. Mihály Makai

On the Speed of Heat Wave. Mihály Makai On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.

More information

priority queue ADT heaps 1

priority queue ADT heaps 1 COMP 250 Lctur 23 priority quu ADT haps 1 Nov. 1/2, 2017 1 Priority Quu Li a quu, but now w hav a mor gnral dinition o which lmnt to rmov nxt, namly th on with highst priority..g. hospital mrgncy room

More information

a dt a dt a dt dt If 1, then the poles in the transfer function are complex conjugates. Let s look at f t H t f s / s. So, for a 2 nd order system:

a dt a dt a dt dt If 1, then the poles in the transfer function are complex conjugates. Let s look at f t H t f s / s. So, for a 2 nd order system: Undrdamd Sysms Undrdamd Sysms nd Ordr Sysms Ouu modld wih a nd ordr ODE: d y dy a a1 a0 y b f If a 0 0, hn: whr: a d y a1 dy b d y dy y f y f a a a 0 0 0 is h naural riod of oscillaion. is h daming facor.

More information

Multiple Choice Solutions 1. E (2003 AB25) () xt t t t 2. A (2008 AB21/BC21) 3. B (2008 AB7) Using Fundamental Theorem of Calculus: 1

Multiple Choice Solutions 1. E (2003 AB25) () xt t t t 2. A (2008 AB21/BC21) 3. B (2008 AB7) Using Fundamental Theorem of Calculus: 1 Paricle Moion Soluions We have inenionally included more maerial han can be covered in mos Suden Sudy Sessions o accoun for groups ha are able o answer he quesions a a faser rae. Use your own judgmen,

More information

INTEGRATION BY PARTS

INTEGRATION BY PARTS Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd

More information

Math 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2

Math 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2 Mah 0 Homwork S 6 Soluions 0 oins. ( ps) I ll lav i o you o vrify ha y os sin = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add s ono h las wo rms sin hos ar xaly h omplimnary

More information

(π 3)k. f(t) = 1 π 3 sin(t)

(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 information

Mundell-Fleming I: Setup

Mundell-Fleming I: Setup Mundll-Flming I: Sup In ISLM, w had: E ( ) T I( i π G T C Y ) To his, w now add n xpors, which is a funcion of h xchang ra: ε E P* P ( T ) I( i π ) G T NX ( ) C Y Whr NX is assumd (Marshall Lrnr condiion)

More information

Discussion 06 Solutions

Discussion 06 Solutions STAT Discussion Soluions Spring 8. Th wigh of fish in La Paradis follows a normal disribuion wih man of 8. lbs and sandard dviaion of. lbs. a) Wha proporion of fish ar bwn 9 lbs and lbs? æ 9-8. - 8. P

More information

Microscopic Flow Characteristics Time Headway - Distribution

Microscopic Flow Characteristics Time Headway - Distribution CE57: Traffic Flow Thory Spring 20 Wk 2 Modling Hadway Disribuion Microscopic Flow Characrisics Tim Hadway - Disribuion Tim Hadway Dfiniion Tim Hadway vrsus Gap Ahmd Abdl-Rahim Civil Enginring Dparmn,

More information

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion

More information

UNIT #4 TEST REVIEW EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT #4 TEST REVIEW EXPONENTIAL AND LOGARITHMIC FUNCTIONS Name: Par I Quesions UNIT #4 TEST REVIEW EXPONENTIAL AND LOGARITHMIC FUNCTIONS Dae: 1. The epression 1 is equivalen o 1 () () 6. The eponenial funcion y 16 could e rewrien as y () y 4 () y y. The epression

More information

Topics covered. ECON6021 Microeconomic Analysis. Price Effects. Price effect I

Topics covered. ECON6021 Microeconomic Analysis. Price Effects. Price effect I Topics covrd ECON6 Microconomic nalsis Consumption Thor. Pric Chang. Pric Elasticitis 3. ncom Elasticitis 4. Markt Dmand P P P Pric ffct ' P Pric consumption curv (PCC) Or Pric pansion path (PEP) ( P,

More information

2. The Laplace Transform

2. The Laplace Transform Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin

More information

DEPARTMENT OF ECONOMICS /11. dy =, for each of the following, use the chain rule to find dt

DEPARTMENT OF ECONOMICS /11. dy =, for each of the following, use the chain rule to find dt SCHOO OF ORIENTA AND AFRICAN STUDIES UNIVERSITY OF ONDON DEPARTMENT OF ECONOMICS 14 15 1/11-15 16 MSc Economics PREIMINARY MATHEMATICS EXERCISE 4 (Skech answer) Course websie: hp://mercur.soas.ac.uk/users/sm97/eaching_msc_premah.hm

More information

Mor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration

More information

Be able to sketch a function defined parametrically. (by hand and by calculator)

Be 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 information

where: u: input y: output x: state vector A, B, C, D are const matrices

where: u: input y: output x: state vector A, B, C, D are const matrices Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D H D I " $ " & $ ' " & $ ' " &

More information

Ma/CS 6a Class 15: Flows and Bipartite Graphs

Ma/CS 6a Class 15: Flows and Bipartite Graphs //206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d

More information

Differential Equations

Differential Equations UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs

More information

3.6 Derivatives as Rates of Change

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