MATHEMATICS FOR MANAGEMENT BBMP1103
|
|
- George Stewart
- 5 years ago
- Views:
Transcription
1 Objctivs: TOPIC : EXPONENTIAL AND LOGARITHM FUNCTIONS. Idntif pnntils nd lgrithmic functins. Idntif th grph f n pnntil nd lgrithmic functins. Clcult qutins using prprtis f pnntils. Clcult qutins using prprtis f lgrithms 5. Slv pplictin prblms INTRODUCTION. Bth pnntil nd lgrithm functin r n-t-n functins.. Th invrs functin f n pnntil functin is clld th lgrithm functin.. PROPERTIES OF EXPONENTIALS. A functin is clld n pnntil functin if it hs frm whr th bs is psitiv, with nd tht its pnnt is n rl numbr. () b b () () () b () (7) () (5) (9) Empls : Find th vlus f () (b) (c) P g Prprd b Ezidin bin Nrmn Tutr
2 P g Prprd b Ezidin bin Nrmn Tutr (d) () (f) Slutins : () (b) (c) (d) 9 () (f) 9 9 Empls : Slv () (b) (c) (d) 9 Slutins : () (b)
3 (c) (d) 9, Ercis. Find th vlus f () (b) (c) 7 (d) () 5 (f) Ercis. () (b) (c) (d) () (f) P g Prprd b Ezidin bin Nrmn Tutr
4 . EQUATIONS AND GRAPHS SKETCHING. Thr r tw gnrl shps f pnntils grphs.. Th shp dpnd n th bs vlu f th pnntil functins. (i) whr > (ii) whr < < Th fllwing r th prprtis f th grph f pnntil functin f. (i) Th dmin fr pnntil functins is th ntir rl numbrs (ii) Its rng is ll psitiv numbrs (iii) Th intrcpt n th pnntil grph is,. (iv) Thr is n intrcpt (vi) If whr >, th grph is incrsing frm lft t right (vii) If whr < <, th grph is dcrsing frm lft t right P g Prprd b Ezidin bin Nrmn Tutr
5 Empl : Sktch grph f Slutin :. Stp (Cnstruct tbl cnsisting svrl vlus f nd.) Stp (Plt th pints n pln.) Stp (Drw smth curv thrugh ll th plttd pints.) P g 5 Prprd b Ezidin bin Nrmn Tutr
6 Empl : Sktch grph f. Slutin : Stp (Cnstruct tbl cnsisting svrl vlus f nd.) Stp (Plt th pints n pln.) Stp (Drw smth curv thrugh ll th plttd pints.) P g Prprd b Ezidin bin Nrmn Tutr
7 . LOGARITHM FUNCTIONS. A lgrithm functin with bs, is wittn s lg whr >,.. is th lgrithm fr with bs, dntd b lg.. Lgrithm with bs f is knwn s cmmn lgrithm nd is writtn s lg lg lg.. Lgrithm with bs, is clld nturl lgrithm, dntd b lg ln. lg Lgrithm Frm Epnntil Frm Empl : Cnvrt th fllwing qutins, frm lgrithm t pnntil frms. () lg 9 (b) lg (c) lg Slutin : () 9 (b) (c) Empl : Cnvrt th fllwing qutins, frm pnntil t lgrithm frms. () 5 (b) (c) 5 Slutin : () lg 5 (b) lg (c) lg 5 P g 7 Prprd b Ezidin bin Nrmn Tutr
8 . PROPERTIES OF LOGARITHMS () lg () lg m lg m () lg b m lg m (Lgrithm bs intrchngbl frmul lg () lg M lg N lg MN M (5) lg M lg N lg N () lg M lg N thn M N b Empls : Using th bv prprtis, find th vlu fr: () lg (b) ln (c) lg (d) lg () lg lg (f) lg 5 lg 9 Slutins: () lg lg lg (b) ln lg lg (c) lg lg lg (d) lg lg lg lg () lg lg lg lg lg (f) lg 5 lg 9 5 lg 9 lg P g Prprd b Ezidin bin Nrmn Tutr
9 Empls : Find th vlu f. () lg lg (b) lg (c) lg (d) lg () lg lg lg lg Slutin : (f) lg () lg lg 5 (c) lg 9 () lg lg lg lg (b) lg, (d) lg lg lg lg lg (f) lg lg lg P g 9 Prprd b Ezidin bin Nrmn Tutr
10 .5 EQUATIONS AND GRAPH SKETCHING. Thr r tw gnrl shps f lgrithm grphs. Th dpnd vr much n th bs vlu f lgrithm funfins. (i) lg whr > (ii) lg whr < < Th fllwing r th prprtis f th grph f pnntil functin f lg. (i) Th dmin fr lgrithm functins is ll psitiv numbrs (ii) Its rng is th ntir rl numbrs (iii) Th intrcpt n th lgrithm grph is,. (iv) Thr is n intrcpt (viii) If >, th grph is incrsing frm lft t right (i) If < <, th grph is dcrsing frm lft t right P g Prprd b Ezidin bin Nrmn Tutr
11 Empl : Sktch grph f lg. Slutin : Stp (Cnstruct th qutin, frm lgrithm t pnntil frm). lg Stp (Cnstruct tbl cnsisting svrl vlus f nd.) Stp (Plt th pints n pln.) Stp (Drw smth curv thrugh ll th plttd pints.) P g Prprd b Ezidin bin Nrmn Tutr
12 Empl : Sktch grph f lg. Slutin : Stp (Cnstruct th qutin, frm lgrithm t pnntil frm). lg Stp (Cnstruct tbl cnsisting svrl vlus f nd.) Stp (Plt th pints n pln.) Stp (Drw smth curv thrugh ll th plttd pints.) P g Prprd b Ezidin bin Nrmn Tutr
13 .5. APPLICATION ON GROWTH AND DECAY PROCESSES. Epnntil functin cn b pplid int grwth nd dc prcsss.. Th frmul fr ttl grwth is P P rt Whr P P r t = numbr f rsidnts ftr rs = numbr f riginl rsidnts = prcntg (rt) f grwth = tim prid Empl: Supps th ttl numbr f rsidnts in givn twn is, nd th rt f grwth f th rsidnts is 5% pr r. () Dtrmin th ttl numbr f rsidnts in this twn in th prid f rs frm nw. (b) Hw mn rs will it tk fr th numbr f rsidnts t dubl? Slutin: () Substitut ll th givn vlus int th frmul t find th vlu f P? P P rt P, r 5% t rt P P Th numbr f th rsidnts ftr si mr rs is 997. (b) Dubling th numbr f rsidnts implis P P. P g Prprd b Ezidin bin Nrmn Tutr
14 P P lg P P.5t lg rt.5t ln.5t.5t ln t.5 t. Th numbr f th rsidnts will dubl in but rs.. Th frmul fr dc prcss is P P rt Empl: Supps rdictiv lmnt is ging thrugh pwr dc ftr t ds bsd n pnntil t functin P.75. Hw much f th quntit is lft ftr ds? Slutin: Substitut ll th givn vlus int th frmul t find th vlu f P? P P rt INVESTMENT WITH COMPOUND INTEREST Th ttl munt f mn, dntd b S is th cmpund munt fr sum f mn P cmpunding ftr n th r, whr th intrst is pbl k tims t th rt f pr r % nnum, is givn b th frmul blw: S P r k nk Whr P g Prprd b Ezidin bin Nrmn Tutr
15 S P r k k = cmpund munt r th prspctiv vlu = initil invstmnt r th principl vlu = intrst rt pr nnum = numbr f intrst pid (cmpund) in r = numbr f r Empl : If RM is invstd t th rt f % pr nnum, cmpunding (pbl) vr qurtrl, wht wuld th ttl munt b in th ccunt ftr rs? Slutin: S?, P, r %, r, n S P. S r k S.5 S. S. nk Empl : Dtrmin th principl munt f ln, givn tht th prspctiv munt pbl ftr rs is RM,59. nd th cmpund rt f % pr nnum, cmpunding (pbl) n rl bsis. Slutin: S 59., P?, r %, r, n S P. 59. P 59. P. 59. P.59 P P r k nk P g 5 Prprd b Ezidin bin Nrmn Tutr
16 Ercis.. () Givn tht th pric n cr f lnd is incrsing t th rt f % pr r. Hw lng will it tk fr th pric t incrs t RM,, givn its currnt vlu is RM,. (b) Du t cnm dwnfll, th ttl numbr f rsidnts in twnship is rducing t th rt f % pr r. Initil ppultin ws, rsidnts. Wht is th ppultin ftr r?. Dtrmin th cmpund munt, givn th principl vlus, cmpund intrst rts nd tim prids: () RM55; % pr nnum cmpunding n mnthl bsis; mnths. (b) RM,; % pr nnum cmpunding rl; 5 rs (c) RM7; 7.% pr nnum cmpunding n qurtrl bsis; 5 rs mnths. (d) RM; 5.75% pr nnum cmpunding dil; 5 ds (ssum r = 5 ds).. Dtrmin th principl munt, givn th fllwing cmpund vlus, cmpund intrst rts nd tim prids: () RM,.; % pr nnum cmpunding n mnthl bsis; mnths (b) RM,97.; 5.% pr nnum cmpunding dil; 5 ds (ssum r = 5 ds) (c) RM57.;.% pr nnum cmpunding vr mnths; mnths (d) RM,.; 7.% pr nnum cmpunding vr mnths; 5 rs mnths P g Prprd b Ezidin bin Nrmn Tutr
17 P g 7 Prprd b Ezidin bin Nrmn Tutr
ELEG 413 Lecture #6. Mark Mirotznik, Ph.D. Professor The University of Delaware
LG 43 Lctur #6 Mrk Mirtnik, Ph.D. Prfssr Th Univrsity f Dlwr mil: mirtni@c.udl.du Wv Prpgtin nd Plritin TM: Trnsvrs lctrmgntic Wvs A md is prticulr fild cnfigurtin. Fr givn lctrmgntic bundry vlu prblm,
More informationIntegration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals
Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion
More informationALGEBRA 2/TRIGONMETRY TOPIC REVIEW QUARTER 3 LOGS
ALGEBRA /TRIGONMETRY TOPIC REVIEW QUARTER LOGS Cnverting frm Epnentil frm t Lgrithmic frm: E B N Lg BN E Americn Ben t French Lg Ben-n Lg Prperties: Lg Prperties lg (y) lg + lg y lg y lg lg y lg () lg
More information, between the vertical lines x a and x b. Given a demand curve, having price as a function of quantity, p f (x) at height k is the curve f ( x,
Clculus for Businss nd Socil Scincs - Prof D Yun Finl Em Rviw vrsion 5/9/7 Chck wbsit for ny postd typos nd updts Pls rport ny typos This rviw sht contins summris of nw topics only (This rviw sht dos hv
More informationPREPARATORY MATHEMATICS FOR ENGINEERS
CIVE 690 This qusti ppr csists f 6 pritd pgs, ch f which is idtifid by th Cd Numbr CIVE690 FORMULA SHEET ATTACHED UNIVERSITY OF LEEDS Jury 008 Emiti fr th dgr f BEg/ MEg Civil Egirig PREPARATORY MATHEMATICS
More informationSection 3: Antiderivatives of Formulas
Chptr Th Intgrl Appli Clculus 96 Sction : Antirivtivs of Formuls Now w cn put th is of rs n ntirivtivs togthr to gt wy of vluting finit intgrls tht is ct n oftn sy. To vlut finit intgrl f(t) t, w cn fin
More informationCh 1.2: Solutions of Some Differential Equations
Ch 1.2: Solutions of Som Diffrntil Equtions Rcll th fr fll nd owl/mic diffrntil qutions: v 9.8.2v, p.5 p 45 Ths qutions hv th gnrl form y' = y - b W cn us mthods of clculus to solv diffrntil qutions of
More information3. Classify the following Numbers (Counting (natural), Whole, Integers, Rational, Irrational)
After yu cmplete each cncept give yurself a rating 1. 15 5 2 (5 3) 2. 2 4-8 (2 5) 3. Classify the fllwing Numbers (Cunting (natural), Whle, Integers, Ratinal, Irratinal) a. 7 b. 2 3 c. 2 4. Are negative
More informationLecture 26: Quadrature (90º) Hybrid.
Whits, EE 48/58 Lctur 26 Pag f Lctur 26: Quadratur (9º) Hybrid. Back in Lctur 23, w bgan ur discussin f dividrs and cuplrs by cnsidring imprtant gnral prprtis f thrand fur-prt ntwrks. This was fllwd by
More information(i) b b. (ii) (iii) (vi) b. P a g e Exponential Functions 1. Properties of Exponents: Ex1. Solve the following equation
P g e 30 4.2 Eponentil Functions 1. Properties of Eponents: (i) (iii) (iv) (v) (vi) 1 If 1, 0 1, nd 1, then E1. Solve the following eqution 4 3. 1 2 89 8(2 ) 7 Definition: The eponentil function with se
More informationMore on FT. Lecture 10 4CT.5 3CT.3-5,7,8. BME 333 Biomedical Signals and Systems - J.Schesser
Mr n FT Lcur 4CT.5 3CT.3-5,7,8 BME 333 Bimdicl Signls nd Sysms - J.Schssr 43 Highr Ordr Diffrniin d y d x, m b Y b X N n M m N M n n n m m n m n d m d n m Y n d f n [ n ] F d M m bm m X N n n n n n m p
More informationSection 5.8 Notes Page Exponential Growth and Decay Models; Newton s Law
Sectin 5.8 Ntes Page 1 5.8 Expnential Grwth and Decay Mdels; Newtn s Law There are many applicatins t expnential functins that we will fcus n in this sectin. First let s lk at the expnential mdel. Expnential
More informationMore Foundations. Undirected Graphs. Degree. A Theorem. Graphs, Products, & Relations
Mr Funtins Grphs, Pruts, & Rltins Unirt Grphs An unirt grph is pir f 1. A st f ns 2. A st f gs (whr n g is st f tw ns*) Friy, Sptmr 2, 2011 Ring: Sipsr 0.2 ginning f 0.4; Stughtn 1.1.5 ({,,,,}, {{,}, {,},
More informationTopic 5: Discrete-Time Fourier Transform (DTFT)
ELEC36: Signals And Systms Tpic 5: Discrt-Tim Furir Transfrm (DTFT) Dr. Aishy Amr Cncrdia Univrsity Elctrical and Cmputr Enginring DT Furir Transfrm Ovrviw f Furir mthds DT Furir Transfrm f Pridic Signals
More informationELEC 372 LECTURE NOTES, WEEK 11 Dr. Amir G. Aghdam Concordia University
ELEC 37 LECTURE NOTES, WEE Dr Amir Aghdam Cncrdia Univrity Part f th nt ar adaptd frm th matrial in th fllwing rfrnc: Mdrn Cntrl Sytm by Richard C Drf and Rbrt H Bihp, Prntic Hall Fdback Cntrl f Dynamic
More informationTOPIC 5: INTEGRATION
TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 301 Signls & Systms Pf. Mk Fwl Discussin #1 Cmplx Numbs nd Cmplx-Vlud Functins Rding Assignmnt: Appndix A f Kmn nd Hck Cmplx Numbs Cmplx numbs is s ts f plynmils. Dfinitin f imginy # j nd sm sulting
More informationUNIT # 08 (PART - I)
. r. d[h d[h.5 7.5 mol L S d[o d[so UNIT # 8 (PRT - I CHEMICL INETICS EXERCISE # 6. d[ x [ x [ x. r [X[C ' [X [[B r '[ [B [C. r [NO [Cl. d[so d[h.5 5 mol L S d[nh d[nh. 5. 6. r [ [B r [x [y r' [x [y r'
More informationFind this material useful? You can help our team to keep this site up and bring you even more content consider donating via the link on our site.
Find this material useful? Yu can help ur team t keep this site up and bring yu even mre cntent cnsider dnating via the link n ur site. Still having truble understanding the material? Check ut ur Tutring
More informationINTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)
Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..
More informationLecture 2a. Crystal Growth (cont d) ECE723
Lctur 2a rystal Grwth (cnt d) 1 Distributin f Dpants As a crystal is pulld frm th mlt, th dping cncntratin incrpratd int th crystal (slid) is usually diffrnt frm th dping cncntratin f th mlt (liquid) at
More informationMinimum Spanning Trees
Minimum Spnning Trs Minimum Spnning Trs Problm A town hs st of houss nd st of rods A rod conncts nd only houss A rod conncting houss u nd v hs rpir cost w(u, v) Gol: Rpir nough (nd no mor) rods such tht:
More informationChapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1
Prctic qustions W now tht th prmtr p is dirctl rltd to th mplitud; thrfor, w cn find tht p. cos d [ sin ] sin sin Not: Evn though ou might not now how to find th prmtr in prt, it is lws dvisl to procd
More informationChapter 2 Linear Waveshaping: High-pass Circuits
Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Chaptr 2 Linar Wavshaping: High-pass Circuits. A ramp shwn in Fig.2p. is applid t a high-pass circuit. Draw t scal th utput wavfrm
More informationOVERVIEW Using Similarity and Proving Triangle Theorems G.SRT.4
OVRVIW Using Similrity nd Prving Tringle Therems G.SRT.4 G.SRT.4 Prve therems ut tringles. Therems include: line prllel t ne side f tringle divides the ther tw prprtinlly, nd cnversely; the Pythgren Therem
More informationCase Study VI Answers PHA 5127 Fall 2006
Qustion. A ptint is givn 250 mg immit-rls thophyllin tblt (Tblt A). A wk ltr, th sm ptint is givn 250 mg sustin-rls thophyllin tblt (Tblt B). Th tblts follow on-comprtmntl mol n hv first-orr bsorption
More informationME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören
ME 522 PINCIPLES OF OBOTICS FIST MIDTEM EXAMINATION April 9, 202 Nm Lst Nm M. Kml Özgörn 2 4 60 40 40 0 80 250 USEFUL FOMULAS cos( ) cos cos sin sin sin( ) sin cos cos sin sin y/ r, cos x/ r, r 0 tn 2(
More informationChapter 5: Diffusion (2)
Chapter 5: Diffusin () ISSUES TO ADDRESS... Nn-steady state diffusin and Fick s nd Law Hw des diffusin depend n structure? Chapter 5-1 Class Eercise (1) Put a sugar cube inside a cup f pure water, rughly
More information( ) Geometric Operations and Morphing. Geometric Transformation. Forward v.s. Inverse Mapping. I (x,y ) Image Processing - Lesson 4 IDC-CG 1
Img Procssing - Lsson 4 Gomtric Oprtions nd Morphing Gomtric Trnsformtion Oprtions dpnd on Pil s Coordints. Contt fr. Indpndnt of pil vlus. f f (, ) (, ) ( f (, ), f ( ) ) I(, ) I', (,) (, ) I(,) I (,
More informationChem 104A, Fall 2016, Midterm 1 Key
hm 104A, ll 2016, Mitrm 1 Ky 1) onstruct microstt tl for p 4 configurtion. Pls numrt th ms n ml for ch lctron in ch microstt in th tl. (Us th formt ml m s. Tht is spin -½ lctron in n s oritl woul writtn
More informationCONIC SECTIONS. MODULE-IV Co-ordinate Geometry OBJECTIVES. Conic Sections
Conic Sctions 16 MODULE-IV Co-ordint CONIC SECTIONS Whil cutting crrot ou might hv noticd diffrnt shps shown th dgs of th cut. Anlticll ou m cut it in thr diffrnt ws, nml (i) (ii) (iii) Cut is prlll to
More informationCONTINUITY AND DIFFERENTIABILITY
MCD CONTINUITY AND DIFFERENTIABILITY NCERT Solvd mpls upto th sction 5 (Introduction) nd 5 (Continuity) : Empl : Chck th continuity of th function f givn by f() = + t = Empl : Emin whthr th function f
More information3.1 EXPONENTIAL FUNCTIONS & THEIR GRAPHS
. EXPONENTIAL FUNCTIONS & THEIR GRAPHS EXPONENTIAL FUNCTIONS EXPONENTIAL nd LOGARITHMIC FUNCTIONS re non-lgebric. These functions re clled TRANSCENDENTAL FUNCTIONS. DEFINITION OF EXPONENTIAL FUNCTION The
More informationPart a: Writing the nodal equations and solving for v o gives the magnitude and phase response: tan ( 0.25 )
+ - Hmewrk 0 Slutin ) In the circuit belw: a. Find the magnitude and phase respnse. b. What kind f filter is it? c. At what frequency is the respnse 0.707 if the generatr has a ltage f? d. What is the
More informationLHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers
LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the
More informationFunctions. EXPLORE \g the Inverse of ao Exponential Function
ifeg Seepe3 Functins Essential questin: What are the characteristics f lgarithmic functins? Recall that if/(x) is a ne-t-ne functin, then the graphs f/(x) and its inverse,/'~\x}, are reflectins f each
More informationMath 153: Lecture Notes For Chapter 5
Mth 5: Lecture Notes For Chpter 5 Section 5.: Eponentil Function f()= Emple : grph f ) = ( if = f() 0 - - - - - - Emple : Grph ) f ( ) = b) g ( ) = c) h ( ) = ( ) f() g() h() 0 0 0 - - - - - - - - - -
More informationAccelerated Chemistry POGIL: Half-life
Name: Date: Perid: Accelerated Chemistry POGIL: Half-life Why? Every radiistpe has a characteristic rate f decay measured by its half-life. Half-lives can be as shrt as a fractin f a secnd r as lng as
More informationLecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture:
Lctur 11 Wvs in Priodic Potntils Tody: 1. Invrs lttic dfinition in 1D.. rphicl rprsnttion of priodic nd -priodic functions using th -xis nd invrs lttic vctors. 3. Sris solutions to th priodic potntil Hmiltonin
More informationEE 119 Homework 6 Solution
EE 9 Hmwrk 6 Slutin Prr: J Bkr TA: Xi Lu Slutin: (a) Th angular magniicatin a tlcp i m / th cal lngth th bjctiv ln i m 4 45 80cm (b) Th clar aprtur th xit pupil i 35 mm Th ditanc btwn th bjctiv ln and
More information5 Curl-free fields and electrostatic potential
5 Curl-fr filds and lctrstatic tntial Mathmaticall, w can gnrat a curl-fr vctr fild E(,, ) as E = ( V, V, V ), b taking th gradint f an scalar functin V (r) =V (,, ). Th gradint f V (,, ) is dfind t b
More information5.1 Properties of Inverse Trigonometric Functions.
Inverse Trignmetricl Functins The inverse f functin f( ) f ( ) f : A B eists if f is ne-ne nt ie, ijectin nd is given Cnsider the e functin with dmin R nd rnge [, ] Clerl this functin is nt ijectin nd
More informationExample. Determine the inverse of the given function (if it exists). f(x) = 3
Example. Determine the inverse of the given function (if it exists). f(x) = g(x) = p x + x We know want to look at two di erent types of functions, called logarithmic functions and exponential functions.
More information120~~60 o D 12~0 1500~30O, 15~30 150~30. ..,u 270,,,, ~"~"-4-~qno 240 2~o 300 v 240 ~70O 300
1 Find th plar crdinats that d nt dscrib th pint in th givn graph. (-2, 30 ) C (2,30 ) B (-2,210 ) D (-2,-150 ) Find th quatin rprsntd in th givn graph. F 0=3 H 0=2~ G r=3 J r=2 0 :.1 2 3 ~ 300 2"~ 2,
More informationHIGHER ORDER DIFFERENTIAL EQUATIONS
Prof Enriqu Mtus Nivs PhD in Mthmtis Edution IGER ORDER DIFFERENTIAL EQUATIONS omognous linr qutions with onstnt offiints of ordr two highr Appl rdution mthod to dtrmin solution of th nonhomognous qution
More informationExponential Functions, Growth and Decay
Name..Class. Date. Expnential Functins, Grwth and Decay Essential questin: What are the characteristics f an expnential junctin? In an expnential functin, the variable is an expnent. The parent functin
More informationThe Derivative of the Natural Logarithmic Function. Derivative of the Natural Exponential Function. Let u be a differentiable function of x.
Th Ntrl Logrithmic n Eponntil Fnctions: : Diffrntition n Intgrtion Objctiv: Fin rivtivs of fnctions involving th ntrl logrithmic fnction. Th Drivtiv of th Ntrl Logrithmic Fnction Lt b iffrntibl fnction
More informationPH2200 Practice Exam I Summer 2003
PH00 Prctice Exm I Summer 003 INSTRUCTIONS. Write yur nme nd student identifictin number n the nswer sheet.. Plese cver yur nswer sheet t ll times. 3. This is clsed bk exm. Yu my use the PH00 frmul sheet
More information3.1 Exponential Functions and Their Graphs
. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.
More informationy = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)
4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y
More informationIntroduction to Three-phase Circuits. Balanced 3-phase systems Unbalanced 3-phase systems
Intrductin t Three-hse Circuits Blnced 3-hse systems Unblnced 3-hse systems 1 Intrductin t 3-hse systems Single-hse tw-wire system: Single surce cnnected t ld using tw-wire system Single-hse three-wire
More informationLecture 35. Diffraction and Aperture Antennas
ctu 35 Dictin nd ptu ntnns In this lctu u will ln: Dictin f lctmgntic ditin Gin nd ditin pttn f ptu ntnns C 303 Fll 005 Fhn Rn Cnll Univsit Dictin nd ptu ntnns ptu ntnn usull fs t (mtllic) sht with hl
More informationMCR 3U Exam Review. 1. Determine which of the following equations represent functions. Explain. Include a graph. 2. y x
MCR U MCR U Em Review Introduction to Functions. Determine which of the following equtions represent functions. Eplin. Include grph. ) b) c) d) 0. Stte the domin nd rnge for ech reltion in question.. If
More informationMATHEMATICS SYLLABUS SECONDARY 5th YEAR
Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE
More informationthis is called an indeterninateformof-oior.fi?afleleitns derivatives can now differentiable and give 0 on on open interval containing I agree to.
hl sidd r L Hospitl s Rul 11/7/18 Pronouncd Loh mtims splld Non p t mtims w wnt vlut limit ii m itn ) but irst indtrnintmori?lltns indtrmint t inn gl in which cs th clld n i 9kt ti not ncssrily snsign
More informationLecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9
Lctur contnts Bloch thorm -vctor Brillouin zon Almost fr-lctron modl Bnds ffctiv mss Hols Trnsltionl symmtry: Bloch thorm On-lctron Schrödingr qution ch stt cn ccommo up to lctrons: If Vr is priodic function:
More information2.6 Logarithmic Functions. Inverse Functions. Question: What is the relationship between f(x) = x 2 and g(x) = x?
Inverse Functions Question: What is the relationship between f(x) = x 3 and g(x) = 3 x? Question: What is the relationship between f(x) = x 2 and g(x) = x? Definition (One-to-One Function) A function f
More informationChapter 3 Exponential and Logarithmic Functions Section 3.1
Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re non-lgebric functions. The re clled trnscendentl functions. The eponentil
More informationCOMP108 Algorithmic Foundations
Grdy mthods Prudn Wong http://www.s.liv..uk/~pwong/thing/omp108/01617 Coin Chng Prolm Suppos w hv 3 typs of oins 10p 0p 50p Minimum numr of oins to mk 0.8, 1.0, 1.? Grdy mthod Lrning outoms Undrstnd wht
More informationhttps://goo.gl/eaqvfo SUMMER REV: Half-Life DUE DATE: JULY 2 nd
NAME: DUE DATE: JULY 2 nd AP Chemistry SUMMER REV: Half-Life Why? Every radiistpe has a characteristic rate f decay measured by its half-life. Half-lives can be as shrt as a fractin f a secnd r as lng
More information( ) ( ) (a) w(x) = a v(x) + b. (b) w(x) = a v(x + b) w = the system IS linear. (1) output as the sum of the outputs from each signal individually
Hrk Sltis. Th fllig ipttpt rltis dscri ssts ith ipt d tpt. Which ssts r lir? Which r spc-irit? [6 pts.] i. tst lirit tstig lir sprpsiti lt thr t ipt sigls d tpt s th s f th tpts fr ch sigl idiidll ' tpt
More informationMath 656 Midterm Examination March 27, 2015 Prof. Victor Matveev
Math 656 Mdtrm Examnatn March 7, 05 Prf. Vctr Matvv ) (4pts) Fnd all vals f n plar r artsan frm, and plt thm as pnts n th cmplx plan: (a) Snc n-th rt has xactly n vals, thr wll b xactly =6 vals, lyng n
More informationCHEM 2400/2480. Lecture 19
Lecture 19 Metal In Indicatr - a cmpund whse clur changes when it binds t a metal in - t be useful, it must bind the metal less strngly than EDTA e.g. titratin f Mg 2+ with EDTA using erichrme black T
More informationPhysics 102. Final Examination. Spring Semester ( ) P M. Fundamental constants. n = 10P
ε µ0 N mp M G T Kuwit University hysics Deprtment hysics 0 Finl Exmintin Spring Semester (0-0) My, 0 Time: 5:00 M :00 M Nme.Student N Sectin N nstructrs: Drs. bdelkrim, frsheh, Dvis, Kkj, Ljk, Mrfi, ichler,
More information1 of 11. Adding Signed Numbers. MAT001 Chapter 9 Signed Numbers. Section 9.1. The Number Line. Ordering Numbers. CQ9-01. Replace? with < or >.
Sectin 9 Adding Signed Numbers The Number Line A number line is a line n which each pint is assciated with a number 0 Negative numbers Psitive numbers f The set f psitive numbers, negative numbers, and
More informationANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels
ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION Instructins: If asked t label the axes please use real wrld (cntextual) labels Multiple Chice Answers: 0 questins x 1.5 = 30 Pints ttal Questin Answer Number 1
More informationN J of oscillators in the three lowest quantum
. a) Calculat th fractinal numbr f scillatrs in th thr lwst quantum stats (j,,,) fr fr and Sl: ( ) ( ) ( ) ( ) ( ).6.98. fr usth sam apprach fr fr j fr frm q. b) .) a) Fr a systm f lcalizd distinguishabl
More informationConstruction 11: Book I, Proposition 42
Th Visul Construtions of Euli Constrution #11 73 Constrution 11: Book I, Proposition 42 To onstrut, in givn rtilinl ngl, prlllogrm qul to givn tringl. Not: Equl hr mns qul in r. 74 Constrution # 11 Th
More informationFind this material useful? You can help our team to keep this site up and bring you even more content consider donating via the link on our site.
Find this material useful? Yu can help ur team t keep this site up and bring yu even mre cntent cnsider dnating via the link n ur site. Still having truble understanding the material? Check ut ur Tutring
More informationInstructions for Section 1
Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks
More informationUniversity Chemistry Quiz /04/21 1. (10%) Consider the oxidation of ammonia:
University Chemistry Quiz 3 2015/04/21 1. (10%) Cnsider the xidatin f ammnia: 4NH 3 (g) + 3O 2 (g) 2N 2 (g) + 6H 2 O(l) (a) Calculate the ΔG fr the reactin. (b) If this reactin were used in a fuel cell,
More informationElectric Potential Energy
Electic Ptentil Enegy Ty Cnsevtive Fces n Enegy Cnsevtin Ttl enegy is cnstnt n is sum f kinetic n ptentil Electic Ptentil Enegy Electic Ptentil Cnsevtin f Enegy f pticle fm Phys 7 Kinetic Enegy (K) nn-eltivistic
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationThis Week. Computer Graphics. Introduction. Introduction. Graphics Maths by Example. Graphics Maths by Example
This Wk Computr Grphics Vctors nd Oprtions Vctor Arithmtic Gomtric Concpts Points, Lins nd Plns Eploiting Dot Products CSC 470 Computr Grphics 1 CSC 470 Computr Grphics 2 Introduction Introduction Wh do
More informationAnother Explanation of the Cosmological Redshift. April 6, 2010.
Anthr Explanatin f th Csmlgical Rdshift April 6, 010. Jsé Francisc García Juliá C/ Dr. Marc Mrncian, 65, 5. 4605 Valncia (Spain) E-mail: js.garcia@dival.s h lss f nrgy f th phtn with th tim by missin f
More informationTURFGRASS DISEASE RESEARCH REPORT J. M. Vargas, Jr. and R. Detweiler Department of Botany and Plant Pathology Michigan State University
I TURFGRASS DISEASE RESEARCH REPORT 9 J. M. Vrgs, Jr. n R. Dtwilr Dprtmnt f Btny n Plnt Pthlgy Mihign Stt Univrsity. Snw Ml Th 9 snw ml fungii vlutin trils wr nut t th Byn Highln Rsrt, Hrr Springs, Mihign
More informationVowel package manual
Vwl pckg mnl FUKUI R Grdt Schl f Hmnts nd Sclgy Unvrsty f Tky 28 ctbr 2001 1 Drwng vwl dgrms 1.1 Th vwl nvrnmnt Th gnrl frmt f th vwl nvrnmnt s s fllws. [ptn(,ptn,)] cmmnds fr npttng vwls ptns nd cmmnds
More informationCSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review
rmup CSE 7: AVL trs rmup: ht is n invrint? Mihl L Friy, Jn 9, 0 ht r th AVL tr invrints, xtly? Disuss with your nighor. AVL Trs: Invrints Intrlu: Exploring th ln invrint Cor i: xtr invrint to BSTs tht
More informationglo beau bid point full man branch last ior s all for ap Sav tree tree God length per down ev the fect your er Cm7 a a our
SING, MY TONGU, TH SAVIOR S GLORY mj7 Mlod Kbd fr nd S would tm flsh s D nd d tn s drw t crd S, Fth t So Th L lss m ful wn dd t, Fs4 F wd; v, snr, t; ngh, t: lod; t; tgu, now Chrst, h O d t bnd Sv God
More informationCHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India
CHAPTER 3 INEQUALITIES Cpyright -The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce
More informationAlgebra 2 Honors. Logs Test Review
Algebra 2 Honors Logs Test Review Name Date Let ( ) = ( ) = ( ) =. Perform the indicated operation and state the domain when necessary. 1. ( (6)) 2. ( ( 3)) 3. ( (6)) 4. ( ( )) 5. ( ( )) 6. ( ( )) 7. (
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More information1. Twelve less than five times a number is thirty three. What is the number
Alger 00 Midterm Review Nme: Dte: Directions: For the following prolems, on SEPARATE PIECE OF PAPER; Define the unknown vrile Set up n eqution (Include sketch/chrt if necessr) Solve nd show work Answer
More informationCalculus Placement Review. x x. =. Find each of the following. 9 = 4 ( )
Calculus Placement Review I. Finding dmain, intercepts, and asympttes f ratinal functins 9 Eample Cnsider the functin f ( ). Find each f the fllwing. (a) What is the dmain f f ( )? Write yur answer in
More informationMath 105: Review for Exam I - Solutions
1. Let f(x) = 3 + x + 5. Math 105: Review fr Exam I - Slutins (a) What is the natural dmain f f? [ 5, ), which means all reals greater than r equal t 5 (b) What is the range f f? [3, ), which means all
More informationEnd of Course Algebra I ~ Practice Test #2
End f Curse Algebra I ~ Practice Test #2 Name: Perid: Date: 1: Order the fllwing frm greatest t least., 3, 8.9, 8,, 9.3 A. 8, 8.9,, 9.3, 3 B., 3, 8, 8.9,, 9.3 C. 9.3, 3,,, 8.9, 8 D. 3, 9.3,,, 8.9, 8 2:
More informationPage 1
nswers: (997-9 HKMO Het vents) reted by: Mr. Frncis Hung Lst updted: 0 ecember 0 97-9 Individul 0 6 66 7 9 9 0 7 7 6 97-9 Grup 6 7 9 0 0 9 Individul vents I Given tht + + is divisible by ( ) nd ( ), where
More informationConservation of charge. Kirchhoff s current law. Current density. Conduction current Convection current Displacement current
5. TEADY ELECTRIC CURRENT Chrgs in mtin Currnt [ A ] Cnsrtin f chrg. Kirchhff s currnt lw. Currnt dnsity A/ m Thr typs f currnts Cnductin currnt Cnctin currnt Displcmnt currnt 5- Cnctin Currnt A currnt
More informationPre-Calculus Individual Test 2017 February Regional
The abbreviatin NOTA means Nne f the Abve answers and shuld be chsen if chices A, B, C and D are nt crrect. N calculatr is allwed n this test. Arcfunctins (such as y = Arcsin( ) ) have traditinal restricted
More informationFunctions and Graphs 1. (a) (b) (c) (f) (e) (d) 2. (a) (b) (c) (d)
Functions nd Grps. () () (c) - - - O - - - O - - - O - - - - (d) () (f) - - O - 7 6 - - O - -7-6 - - - - - O. () () (c) (d) - - - O - O - O - - O - -. () G() f() + f( ), G(-) f( ) + f(), G() G( ) nd G()
More informationAlgebra2/Trig: Trig Unit 2 Packet
Algebra2/Trig: Trig Unit 2 Packet In this unit, students will be able t: Learn and apply c-functin relatinships between trig functins Learn and apply the sum and difference identities Learn and apply the
More informationName: Date: Class: a. How many barium ions are there per formula unit (compound)? b. How many nitride ions are there per formula unit (compound)?
NOTES Name: Date: Class: Lessn 15 Part 2: Binary II Inic Bnding, Plyatmic Ins Bx 1: 1. Ba 3N 2 is the frmula fr. (name) a. Hw many barium ins are there per frmula unit (cmpund)? b. Hw many nitride ins
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationCIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7
CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - In som pplictions, it m mor dsirl to us n lmntl rprsnttion of th domin tht hs four sids, ithr rctngulr or qudriltrl in shp. Considr
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationLEARNING : At the end of the lesson, students should be able to: OUTCOMES a) state trigonometric ratios of sin,cos, tan, cosec, sec and cot
Mathematics DM 05 Tpic : Trignmetric Functins LECTURE OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :. Trignmetric Ratis and Identities LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES
More information² Ý ² ª ² Þ ² Þ Ң Þ ² Þ. ² à INTROIT. huc. per. xi, sti. su- sur. sum, cum. ia : ia, ia : am, num. VR Mi. est. lis. sci. ia, cta. ia.
str Dy Ps. 138 R 7 r r x, t huc t m m, l : p - í pr m m num m, l l : VR M rá s f ct st sc n -, l l -. Rpt nphn s fr s VR ftr ch vrs Ps. 1. D n, pr bá m, t c g ví m : c g ví ss s nm m m, t r r r c nm m
More informationComposition of Functions
Math 120 Intermediate Algebra Sec 9.1: Composite and Inverse Functions Composition of Functions The composite function f g, the composition of f and g, is defined as (f g)(x) = f(g(x)). Recall that a function
More informationM thematics. National 5 Practice Paper D. Paper 1. Duration 1 hour. Total marks 40
N5 M thematics Natinal 5 Practice Paper D Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain
More informationINF5820/INF9820 LANGUAGE TECHNOLOGICAL APPLICATIONS. Jan Tore Lønning, Lecture 4, 14 Sep
INF5820/INF9820 LANGUAGE TECHNOLOGICAL ALICATIONS Jn Tor Lønning Lctur 4 4 Sp. 206 tl@ii.uio.no Tody 2 Sttisticl chin trnsltion: Th noisy chnnl odl Word-bsd Trining IBM odl 3 SMT xpl 4 En kokk lgd n rtt
More information