Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000


 Buddy Dalton
 3 years ago
 Views:
Transcription
1 hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails about th copyright on ths nots, plas s
2 Unit Mathmatics Contnts Vctors 8 Vctors and Scalars 8 Componnts 8 Magnitud 0 Equal Vctors Addition and Subtraction of Vctors Multiplication by a Scalar 7 Position Vctors 8 Basis Vctors 9 Collinarity 7 0 Dividing Lins in a Ratio 8 Th Scalar Product Th Angl Btwn Vctors Prpndicular Vctors 7 Proprtis of th Scalar Product 8 Furthr Calculus 9 Diffrntiating sin and cos 9 Intgrating sin and cos 0 Th Chain Rul Spcial Cass of th Chain Rul A Spcial Intgral Intgrating sin(a + b) and cos(a + b) 7 Eponntials and Logarithms 0 Eponntials 0 Logarithms Laws of Logarithms Eponntials and Logarithms to th Bas Eponntial and Logarithmic Equations Graphing with Logarithmic As 9 7 Graph Transformations 7 Wav Functions 7 Eprssing pcos + qsin in th form kcos( a) 7 Eprssing pcos + qsin in othr forms 7 Multipl Angls 77 Maimum and Minimum Valus 78 Solving Equations 79 Sktching Graphs of y pcos + qsin 8  ii  HSN000
3 Unit Vctors OUTCOME Vctors Vctors and Scalars A scalar is a quantity with magnitud (siz) only for ampl, an amount of mony or a lngth of tim Somtims siz alon is not nough to dscrib a quantity for ampl, dirctions to th narst shop For this w nd to know a magnitud (i how far), and a dirction Quantitis with both magnitud and dirction ar calld vctors A vctor is namd ithr by using th points at th nd of a dirctd lin sgmnt (g AB rprsnts th vctor starting at point A and nding at point B) or by using a bold lttr (g u) You will s bold lttrs usd in printd tt, but in handwriting you should just undrlin th lttr (g ) B Throughout ths nots, w will show vctors bold and undrlind (g u ) Componnts A A vctor may b rprsntd by its componnts, which w writ in a column For ampl, is a vctor in two dimnsions In this cas th first componnt is and this tlls us to mov units in th dirction Th scond componnt tlls us to mov units in th y dirction So if th vctor starts at th origin, it will look lik: y u O Pag 8 HSN000
4 Unit Vctors Not that w writ th componnts in a column to avoid confusing thm with coordinats Th following diagram also shows th vctor, but in this cas it dos not start at th origin y (, ) (, ) O Vctors in Thr Dimnsions In a vctor with thr componnts, th first two tll us how many units to mov in th  and ydirctions, as bfor Th third componnt spcifis how far to mov in th zdirction Whn looking at a pair of (, y ) as, th zais points out of th pag from th origin z A st of D as can b drawn on a pag as shown to th right z O y For ampl, is a vctor in thr dimnsions This vctor is shown in th diagram, starting from th origin z O y Zro Vctors Any vctor with all its componnts zro is calld a zro vctor and can b 0 writtn as 0, g Pag 9 HSN000
5 Unit Vctors Magnitud Th magnitud (or lngth) of a vctor u is writtn as u It can b calculatd as follows EXAMPLES Givn a If PQ b thn PQ a + b a If PQ b thn PQ a + b + c c u, find u u + ( ) 9 units Find th lngth of a a units Unit Vctors Any vctor with a magnitud of on is calld a unit vctor For ampl: if 0 u thn So u is a unit vctor u unit Pag 0 HSN000
6 Unit Vctors Distanc in Thr Dimnsions Th distanc btwn th points A and B is d AB AB units So givn AB, w find d AB In fact, thr is a thr dimnsional vrsion of th distanc formula Th distanc d btwn th points (, y, z ) and (,, ) EXAMPLE y z is units d + y y + z z Find th distanc btwn th points (,,) and ( 0,, 7) Th distanc is Equal Vctors ( ) + ( y y ) + ( z z ) ( 0 ( ) ) + ( ) + ( 7 ) + + ( 8) + + units Vctors with th sam magnitud and dirction ar said to b qual For ampl, all th vctors shown to th right ar qual If vctors ar qual to ach othr, thn all of thir componnts ar qual, i p r q s t a d if b thn a d, b and c f c f Convrsly, two vctors ar only qual if all of thir componnts ar qual Pag HSN000
7 Unit Vctors Addition and Subtraction of Vctors Considr th following vctors: a b c Addition W can construct a + b as follows: b a a + b a + b mans a followd by b Similarly, w can construct a + b + c as follows: a a + b + c mans a followd by b followd by c To add vctors, w position thm nostotail Thn th sum of th vctors is th vctor btwn th first tail and th last nos Subtraction b a + b + c c Now considr a b This can b writtn as a + ( b ), so if w first find b w can us vctor addition to obtain a b b b is just b but in th opposit dirction b b and b hav th sam magnitud, i b b Thrfor w can construct b a b as follows: a b a a b mans a followd by b Pag HSN000
8 Unit Vctors Using Componnts If w hav th componnts of vctors, thn things bcom much simplr Th following ruls can b usd for addition and subtraction a d a + d b + b + c f c + f add th componnts a d a d b b c f c f subtract th componnts EXAMPLES Givn u and u + v + 0 Givn 0 7 p and p q v, calculat u + v and 0 u v 0 u v q, calculat p q and q + p q + p Pag HSN000
9 Unit Vctors Multiplication by a Scalar A vctor u which is multiplid by a scalar k > 0 will giv th rsult ku This vctor will b k tims as long, i its magnitud will bk u Not that if k < 0 this mans that th vctor ku will b in th opposit dirction to u For ampl: u u u u a ka If u b thn ku kb c kc Each componnt is multiplid by th scalar EXAMPLES Givn v, find v v 9 Givn r, find r r Pag HSN000
10 Unit Vctors Ngativ Vctors Th ngativ of a vctor is th vctor multiplid by If w writ a vctor as a dirctd lin sgmnt AB, thn AB BA : AB A 7 Position Vctors B OA is calld th position vctor of point A rlativ to th origin O, and is writtn as a A B AB BA OB is calld th position vctor of point B, writtn b Givn P (, y, z ), th position vctor OP or p has componnts y z z O y P y O a A b B To mov from point A to point B w can mov back along th vctor a to th origin, and along vctor b to point B, i AB AO + OB OA + OB a + b b a For th vctor joining any two points P and Q, PQ q p Pag HSN000
11 Unit Vctors EXAMPLE R is th point (,, ) From th coordinats, RS s r 8 and S is th point (,, ) r and s Find RS Not You don t nd to writ this lin down in th am 8 Basis Vctors A vctor may also b dfind in trms of th basis vctors i, j and k Ths ar thr mutually prpndicular unit vctors (i thy ar prpndicular to ach othr) Ths basis vctors can b writtn in componnt form as i 0, 0 0 j and 0 0 k 0 Any vctor can b writtn in basis form using i, j and k For ampl: k i j 0 0 v i j + k 0 0 Thr is no nd for th working abov if th following is usd: a ai + bj + ck b c Pag HSN000
12 Unit Vctors 9 Collinarity In Straight Lins (Unit Outcom ), w larnd that points ar collinar if thy li on th sam straight lin Th points A, B and C in D spac ar collinar if AB is paralll to BC, with B a common point Not that w cannot find gradints in thr dimnsions instad w us th following Nonzro vctors ar paralll if thy ar scalar multipls of th sam vctor For ampl: u, v u So u and v ar paralll 0 p 9, q 8 So p and q ar paralll EXAMPLE A is th point (,, ), B( 8,, 9) and C(,,7 ) Show that A, B and C ar collinar AB b a BC c b BC AB, so AB and BC ar paralll, and sinc B is a common point, A, B and C ar collinar Pag 7 HSN000
13 Unit Vctors 0 Dividing Lins in a Ratio Thr is a simpl procss for finding th coordinats of a point which divids a lin sgmnt in a givn ratio EXAMPLE P is th point (,, ) and R is th point ( 8,, 9) Th point T divids PR in th ratio : Find th coordinats of T Stp Mak a sktch of th lin, showing th ratio in which th point divids th lin sgmnt Stp Using th sktch, quat th ratio of th two lin sgmnts with th givn ratio Stp Cross multiply, thn chang dirctd lin sgmnts to position vctors Stp Rarrang to giv th position vctor of th unknown point Stp From th position vctor, stat th coordinats of th unknown point R PT TR PT TR ( t p) ( r t ) T t p r t t + t r + p 8 t + 9 t t 0 t 7 So T is th point (,, 7 ) P Pag 8 HSN000
14 Unit Vctors Using th Sction Formula Th prvious mthod can b condnsd into a formula as shown blow If th point P divids th lin AB in th ratio m : n, thn na + mb p, n + m whr a, b and p ar th position vctors of A, B and P rspctivly This is rfrrd to as th sction formula It is not ncssary to know this, sinc th approach plaind abov will always work EXAMPLE P is th point (,, ) and R is th point ( 8,, 9) Th point T divids PR in th ratio : Find th coordinats of T Th ratio is :, so m and n Hnc: np + mr t n + m p + r ( ( ) + ( 8) ) ( ( ) + ( ) ) ( ( ) + ( 9) ) 7 So T is th point (,, 7 ) Not If you ar confidnt with arithmtic, this stp can b don mntally Pag 9 HSN000
15 Unit Vctors Furthr Eampls EXAMPLES Th cuboid OABCDEFG is shown in th diagram blow E F H D A G B O C Th point A has coordinats ( 0,0, ), C ( 8,0,0 ) and G( 8,,0 ) Th point H divids BF in th ratio : Find th coordinats of H From th diagram: OH OA + AB + BF OA + OC + CG ( g c) h a + c + a + c + g c a + c + g So H has coordinats 8 8,, Th points P(,, ), Q ( 8,,) and R ( 9,,) ar collinar Find th ratio in which Q divids PR Sinc th points ar collinar PQ kqr for som k Working with th first componnts: 8 k ( 9 8) k Thrfor PQ QR so Q divids PR in th ratio : Not BH, so BH BF BF Not Th ratio is : sinc PQ QR Pag 0 HSN000
16 Unit Vctors Th points A ( 7,, ), B(,, 7) and C ar collinar Givn that B divids AC in th ratio :, find th coordinats of C AB AC b a ( c a) b a c a c b a A c b a So C has coordinats ( 7,, 9) Not A sktch may hlp you to s this: B C Th Scalar Product So far w hav addd and subtractd vctors and multiplid a vctor by a scalar Now w will considr th scalar product, which is a form of vctor multiplication Th scalar product is dnotd by ab (somtims it is calld th dot product) and can b calculatd using th formula: ab a b cos θ, whr θ is th angl btwn th two vctors a and b This is givn in th am Pag HSN000
17 Unit Vctors Th dfinition abov assums that th vctors a and b ar positiond so that thy both point away from th angl, or both point into th angl a θ b a θ b Howvr, if on vctor is pointing away from th angl, whil th othr points into th angl, a θ b a θ b w find that ab a b cosθ EXAMPLES Two vctors, a and b hav magnituds 7 and units rspctivly and ar at an angl of 0 to ach othr as shown blow What is th valu of ab? ab a b cosθ 7 cos 0 b 0 Th vctor u has magnitud k and v is twic as long as u Th angl btwn u and v is 0, as shown blow v a 0 u Find an prssion for uv in trms of k uv u v cosθ k k cos0 k k Rmmbr Whn on vctor points in and on points out, uv u v cosθ Pag HSN000
18 Unit Vctors Th Componnt Form of th Scalar Product Th scalar product can also b calculatd as follows: ab a b + a b + a b whr This is givn in th am a a and a a b b b b EXAMPLES Find pq, givn that pq p q + p q + p q p and ( ) + ( ) + ( ) + 9 If A is th point ABAC C(,, ) B(,, ) A (,, 9) q,, 9, B(,, ) and C(,, ), calculat W nd to us th position vctors of th points: AB b a AC c a 9 ABAC (( ) ( ) ) + ( 0) + ( ) ( ) Pag HSN000
19 Unit Vctors Th Angl Btwn Vctors Th formula for th scalar product can b rarrangd to giv th following quations, both of which can b usd to calculat θ, th angl btwn two vctors cosθ ab a b or a b + a b + a b a b cos θ Look back to th formula for finding th scalar product, givn on th prvious pags Notic that th first quation is simply a rarrangd form of th on which can b usd to find th scalar product Also notic that th scond simply substituts ab for th componnt form of th scalar product Ths formula ar not givn in th am but can both b asily drivd from th formula on th prvious pags (which ar givn in th am) EXAMPLES Calculat th angl θ btwn vctors p i + j k and q i + j + k p and q p q + p q + p q cosθ p q ( ) + ( ) + (( ) ) + + ( ) θ cos 9 8 (to d p ) (or 98 radians (to d p )) Pag HSN000
20 Unit Vctors K is th point (, 7, ) Start with a sktch: L(,, ) θ, L(,, ) and M(,,) M,, Find ɵ KLM K (, 7, ) Now find th vctors pointing away from th angl: LK k l 7 0, LM m l Us th scalar product to find th angl: ɵ LKLM cosklm LK LM ( ) + ( 0 ) + ( ( ) ) + ( 0) + ( ) + + ( ) 0 8 ɵ KLM cos (to d p ) (or 8 radians (to d p )) Pag HSN000
21 Unit Vctors Th diagram blow shows th cub OPQRSTUV Th point R has coordinats (,0,0 ) (a) Writ down th coordinats of T and U (b) Find th componnts of RT and RU O (c) Calculat th siz of angl TRU (a) From th diagram, T( 0,, ) and U(,, ) z S P T V R U Q y (b) (c) 0 RT t r 0 0, 0 RU u r 0 0 RTRU cos TRU RT RU ( 0) + ( ) + ( ) ( ) TRU cos (to d p ) (or 0 radians (to d p )) Pag HSN000
22 Unit Vctors Prpndicular Vctors If a and b ar prpndicular thn ab 0 This is bcaus ab a b cosθ a b cos90 ( θ 90 sinc prpndicular) 0 (sinc cos90 0) Convrsly, if ab 0 thn a and b ar prpndicular EXAMPLES Two vctors ar dfind as a i + j k and b i + j + k Show that a and b ar prpndicular ab a b + a b + a b ( ) ( ) (( ) ) Sinc ab 0, a and b ar prpndicular PQ a 7 and RS a and RS whr a is a constant Givn that PQ Sinc PQ and RS ar prpndicular, PQRS 0 + ( a) + 7a 0 8 a + 7a a 0 a ar prpndicular, find th valu of a Pag 7 HSN000
23 Unit Vctors Proprtis of th Scalar Product Som usful proprtis of th scalar product ar as follows: ab ba a b + c ab + ac (Epanding brackts) aa a Not that ths ar not givn in th am, so you nd to rmmbr thm EXAMPLES In th diagram, p, r and q Calculat p ( q + r ) p q r p q + r pq + pr p q cosθ p r cosθ cos 0 + cos In th diagram blow, a c and b a c 0 b 0 a a b c Calculat ( + + ) ( + + ) a a b c aa + ab + ac a + a b cosθ a c cosθ cos0 cos0 Rmmbr a c a c cosθ sinc a points into θ and c points away Pag 8 HSN000
24 Unit Furthr Calculus OUTCOME Furthr Calculus Diffrntiating sin and cos In ordr to diffrntiat prssions involving trigonomtric functions, w us th following ruls: d d d d ( sin ) cos, ( cos ) sin Ths ruls only work whn is an angl masurd in radians A form of ths ruls is givn in th am EXAMPLES Diffrntiat y sin with rspct to dy cos d A function f is dfind by f ( ) sin cos for R Find f π f ( ) cos ( sin ) cos + sin f π cosπ + sinπ + + Find th quation of th tangnt to th curv y sin whn Whn, y sin π So th point is ( ) π W also nd th gradint at th point whr dy cos d π m π tangnt Whn, cos π, π : Rmmbr Th act valu triangl: π π π Pag 9 HSN000
25 Now w hav th point π, and th gradint y b m a ( π ) y y π y π + 0 Unit Furthr Calculus tangnt m, so: Intgrating sin and cos W know th drivativs of sin and cos, so it follows that th intgrals ar: cos d sin + c, sin d cos + c Again, ths rsults only hold if is masurd in radians EXAMPLES Find ( sin + cos ) d ( sin + cos ) d cos + sin + c d Find ( cos + sin ) π 0 0 π ( cos + sin ) [ sin cos ] d π ( sin π cos π ) ( sin0 cos0) (( ) ( )) Not It is good practic to rationalis th dnominator Find th valu of 0 0 sin d sin d cos 0 cos + cos 0 ( 0 + ) 0 87 (to dp) Rmmbr W must us radians whn intgrating or diffrntiating trig functions Pag 0 HSN000
26 Unit Furthr Calculus Th Chain Rul W will now look at how to diffrntiat composit functions, such as f g ( ) If th functions f and g ar dfind on suitabl domains, thn d d f ( g ( )) f g ( ) g Statd simply: diffrntiat th outr function, th brackt stays th sam, thn multiply by th drivativ of th brackt This is calld th chain rul You will nd to rmmbr it for th am EXAMPLE If y cos( π ) +, find dy ( π ) ( π ) ( π ) y cos + dy sin + d sin + d Not Th coms from d d( + π ) Spcial Cass of th Chain Rul W will now look at how th chain rul can b applid to particular typs of prssion Powrs of a Function [ ] n For prssions of th form f ( ), whr n is a constant, w can us a simplr vrsion of th chain rul: d d [ ] n n ( f ( )) n f ( ) f Statd simply: th powr ( n ) multiplis to th front, th brackt stays th sam, th powr lowrs by on (giving n ) and vrything is multiplid by th drivativ of th brackt ( f ( )) Pag HSN000
27 Unit Furthr Calculus EXAMPLES A function f is dfind on a suitabl domain by f + Find f ( ) + ( + ) f f + + ( ) ( ) ( )( ) Diffrntiat y sin with rspct to y sin ( sin ) dy ( sin ) cos d 8sin cos Powrs of a Linar Function Th rul for diffrntiating an prssion of th form ( a + b, whr a, b and n ar constants, is as follows: EXAMPLES d d n n a b an a b ( + ) ( + ) Diffrntiat y ( + ) with rspct to y ( + ) dy ( + ) d ( + ) ) n Pag HSN000
28 Unit Furthr Calculus If y ( + ), find dy d y ( + ) ( + ) dy ( + ) d ( + ) ( + ) A function f is dfind by f ( ) ( ) for R Find f ( ) f ( ) ( ) ( ) f ( ) ( ) Trigonomtric Functions Th following ruls can b usd to diffrntiat trigonomtric functions d d sin( a + b) a cos( a + b) d cos( a + b) a sin( a + b) Ths ar givn in th am EXAMPLE Diffrntiat y sin( 9 + π ) with rspct to dy 9cos( 9 + π) d d Pag HSN000
29 Unit Furthr Calculus A Spcial Intgral Th mthod for intgrating an prssion of th form ( a + b is: ( + ) n+ n a b ( a + b) d + c whr a 0 and n a ( n + ) Statd simply: rais th powr ( n ) by on, divid by th nw powr and also divid by th drivativ of th brackt ( a ( n + )), add c EXAMPLES d Find ( + ) 7 ( ) 7 ( + ) + d + c 8 8 ( + ) Find ( + ) d Find + c 8 ( + ) ( + ) d + c ( + ) + c 9 d + 8 whr 9 ) n d + 9 ( + 9) ( + 9 ) ( + 9) + c c 0 d d c Pag HSN000
30 Unit Furthr Calculus Evaluat + d whr d + d ( + ) 0 ( + ) ( ) ( 0 ) (or 8 8 to d p ) Not Changing powrs back into roots hr maks it asir to valuat th two brackts Rmmbr To valuat, it is asir to work out first Warning Mak sur you don t confus diffrntiation and intgration this could los you a lot of marks in th am Rmmbr th following ruls for diffrntiating and intgrating prssions of th form ( a + b) n : d d n ( a b) + an( a + b) ( + ) n+ n, n a b a + b d + c a( n + ) Ths ruls will not b givn in th am Pag HSN000
31 Unit Furthr Calculus Using Diffrntiation to Intgrat Rcall that intgration is just th procss of undoing diffrntiation So if w diffrntiat f ( ) to gt g ( ) thn w know that g ( ) d f ( ) + c EXAMPLES (a) Diffrntiat y ( ) (b) Hnc, or othrwis, find (a) y ( ) ( ) dy ( )( ) d 0 ( ) with rspct to ( ) d (b) From part (a) w know 0 d c ( ) ( ) + So: 0 d + c ( ) ( ) (a) Diffrntiat (b) Hnc, find (a) d ( ) 0 + c ( ) + c ( ) ( ) y ( ) ( ) ( ) y ( ) dy ( ) d d whr c with rspct to Not W could also hav usd th spcial intgral to obtain this answr is som constant Pag HSN000
32 Unit Furthr Calculus (b) From part (a) w know d + c So: ( ) ( ) d + c ( ) ( ) d + c + c ( ) ( ) ( ) whr c Not In this cas, th spcial intgral cannot b usd is som constant Intgrating sin(a + b) and cos(a + b) Sinc w know th drivativs of sin( a + b) and cos( a b) that thir intgrals ar: Ths ar givn in th am EXAMPLES Find sin( + ) d cos a + b d a sin a + b + c, sin a + b d a cos a + b + c sin + d cos + + c Find cos ( + π ) d π π cos + d sin + + c +, it follows Find th valu of cos ( ) d 0 0 cos( ) d sin( ) 0 (to dp) sin( ) sin( ) ( ) 0 Rmmbr W must us radians whn intgrating or diffrntiating trig functions Pag 7 HSN000
33 Unit Furthr Calculus Find th ara nclosd by th graph of y sin( π ) th lins 0 and π y +, th ais and y sin + π O π π 0 π ( + π ) d ( + π ) 0 ( cos( π ( ) + π )) ( cos( ( 0) + π )) (( ) ( )) + ( ) sin cos So th ara is squar units d Find cos( ) ( ) ( ) ( ) cos d sin + c sin + c ( ) d ( ) Find cos( ) + sin( ) cos + sin d sin cos + c Pag 8 HSN000
34 Unit Furthr Calculus 7 (a) Diffrntiat (b) Hnc find cos tan d cos with rspct to (a) ( cos ), and d ( cos ) ( cos ) sin cos d sin cos sin tan cos sin (b) cos cos cos sin From part (a) w know d + c cos cos Thrfor tan d + c cos cos Pag 9 HSN000
35 Unit Eponntials and Logarithms OUTCOME Eponntials and Logarithms Eponntials W hav alrady mt ponntial functions in Unit Outcom A function of th form f ( ) a whr a, R and a > 0 is known as an ponntial function to th bas a If a > thn th graph looks lik this: y y a, a > O (, a) This is somtims calld a growth function If 0 < a < thn th graph looks lik this: y y a, 0 < a < O (, a) This is somtims calld a dcay function Rmmbr that th graph of an ponntial function f ( ) a always passs through ( 0, ) and (, a ) sinc 0 f ( 0) a, f ( ) a a Pag 0 HSN000
36 Unit Eponntials and Logarithms EXAMPLES Th ottr population on an island incrass by % pr yar How many full yars will it tak for th population to doubl? Lt u 0 b th initial population u u (% as a dcimal) 0 u u u u 0 0 u u u u u 0 0 n n u 0 For th population to doubl aftr n yars, w rquir u u0 W want to know th smallst n which givs n a valu of or mor, sinc this will mak u n at last twic as big as u 0 Try valus of n until this is satisfid If n, < If n, < If n, 8 < If n, 0 > Thrfor aftr yars th population will doubl On a calculator: n ANS Th fficincy of a machin dcrass by % ach yar Whn th fficincy drops blow 7%, th machin nds to b srvicd Aftr how many yars will th machin nd srvicd? Lt u 0 b th initial fficincy u 0 9 u (9% as a dcimal) u u u u n 0 9 n 0 u u u u u u Whn th fficincy drops blow 0 7u 0 (7% of th initial valu) th machin must b srvicd So th machin nds srvicd aftr n yars if 0 9 n 0 7 Pag HSN000
37 Unit Eponntials and Logarithms Try valus of n until this is satisfid If n, > 0 7 If n, > 0 7 If n, > 0 7 If n, > 0 7 If n, < 0 7 Thrfor aftr yars, th machin will hav to b srvicd Logarithms Having prviously dfind what a logarithm is (s Unit Outcom ) w want to look in mor dtail at th proprtis of ths important functions Th rlationship btwn logarithms and ponntials is prssd as: y log a whr a, > 0 Hr, y is th powr of a which givs EXAMPLES Writ in logarithmic form log Evaluat log a Th powr of which givs is, so log Laws of Logarithms Thr ar thr laws of logarithms which you must know Rul y log + log y log y whr a,, y > 0 a a a If two logarithmic trms with th sam bas numbr (a abov) ar bing addd togthr, thn th trms can b combind by multiplying th argumnts ( and y abov) EXAMPLE Simplify log + log log + log log log 8 Pag HSN000
38 Unit Eponntials and Logarithms Rul ( y ) log log y log whr a,, y > 0 a a a If a logarithmic trm is bing subtractd from anothr logarithmic trm with th sam bas numbr (a abov), thn th trms can b combind by dividing th argumnts ( and y in this cas) Not that th argumnt which is bing takn away (y abov) appars on th bottom of th fraction whn th two trms ar combind EXAMPLE Evaluat log log Rul log log log log (sinc ) log a n nlog whr a, > 0 a Th powr of th argumnt (n abov) can com to th front of th trm as a multiplir, and vicvrsa EXAMPLE Eprss log7 in th form log7 a log 7 log7 log 9 7 Squash, Split and Fly You may find th following nams ar a simplr way to rmmbr th laws of logarithms log log y log ( y) + th argumnts ar squashd togthr by multiplying a a a loga loga y log a ( y ) log a th argumnts ar split into a fraction n nloga th powr of an argumnt can fly to th front of th log trm and vicvrsa Pag HSN000
39 Unit Eponntials and Logarithms Not Whn working with logarithms, you should rmmbr: EXAMPLE log 0 a sinc Evaluat log7 7 + log log 7 + log + 7 Combining svral log trms 0 a, log a a sinc a a Whn adding and subtracting svral log trms in th form log a b, thr is a simpl way to combin all th trms in on stp Multiply th argumnts of th positiv log trms in th numrator Multiply th argumnts of th ngativ log trms in th dnominator EXAMPLES Evaluat log0 + log log log 0 + log log 0 log log Evaluat log + log log + log log + log log + log 9 log 9 log log a (sinc ) argumnts of positiv log trms argumnts of ngativ log trms log OR log + log log + log log + log ( ) ( ) log + log log log + log0 + log log (sinc log ) Pag HSN000
40 Unit Eponntials and Logarithms Eponntials and Logarithms to th Bas Th constant is an important numbr in Mathmatics, and occurs frquntly in modls of rallif situations Its valu is roughly (to 9 dp), and is dfind as: ( n ) n + as n If you try larg valus of n on your calculator, you will gt clos to th valu of Lik π, is an irrational numbr Throughout this sction, w will us in prssions of th form:, which is calld an ponntial to th bas, log, which is calld a logarithm to th bas This is also known as th natural logarithm of, and is oftn writtn as ln (i ln log ) EXAMPLES Calculat th valu of log 8 log 8 08 (to dp) Solv log 9 log 9 so (to dp) Simplify log ( ) log ( ) prssing your answr in th form a + log b log c whr a, b and c ar whol numbrs log ( ) log ( ) log + log log log log + log + log log + log log + log log 7 On a calculator: ln 8 On a calculator: 9 OR log ( ) log ( ) log ( ) log ( ) ( ) log ( ) Rmmbr log 7 n n n ( ab) a b log 7 log + log log 7 + log log 7 Pag HSN000
41 Unit Eponntials and Logarithms Eponntial and Logarithmic Equations Many mathmatical modls of rallif situations us ponntials and logarithms It is important to bcom familiar with using th laws of logarithms to hlp solv quations EXAMPLES Solv log + log log 7 for > 0 a a a log + log log 7 a a a log log 7 a 7 (sinc log log y y) Solv ( ) ( ) a log + log for > ( ) ( ) log + log a log + + (sinc log y a y a ) + ( ) for p > Solv loga ( p ) loga ( p 0) loga ( p) loga ( p + ) + loga ( p 0) loga ( p) loga (( p + )( p 0) ) loga ( p) ( p + )( p 0) p p p + p p p + 0 p p ( p )( p ) p Sinc w rquir + 0 or p 0 p p >, p is th solution a Pag HSN000
42 Unit Eponntials and Logarithms Daling with Constants Somtims it may b ncssary to writ constants as logs, in ordr to solv quations EXAMPLE Solv log 7 log + for > 0 Writ in logarithmic form: log (sinc log ) log log 8 Us this in th quation: log 7 log + log 8 log 7 log OR log 7 log + log 7 log 7 log Convrting from log to ponntial form: Solving Equations with Unknown Eponnts If an unknown valu (g ) is th powr of a trm (g or 0 ), and its valu is to b calculatd, thn w must tak logs on both sids of th quation to allow it to b solvd Th sam solution will b rachd using any bas, but calculators can b usd for valuating logs ithr in bas or bas 0 EXAMPLES Solv 7 Taking log of both sids OR Taking log 0 of both sids log log 7 log log 7 ( log ) log 7 9 (to dp) 0 0 log log 7 log log log0 7 log 0 9 (to dp) Pag 7 HSN000
43 Unit Eponntials and Logarithms Solv log log 0 ( + ) log log 0 log 0 + log (to dp) Not log 0 could hav bn usd instad of log Eponntial Growth and Dcay Rcall from Sction that ponntial functions ar somtims known as growth or dcay functions Ths oftn occur in modls of rallif situations For instanc, radioactiv dcay can b modlld using an ponntial function An important masurmnt is th halflif of a radioactiv substanc, which is th tim takn for th mass of th radioactiv substanc to halv 7 Th mass G grams of a radioactiv sampl aftr tim t yars is givn by t th formula G 00 (a) What was th initial mass of radioactiv substanc in th sampl? (b) Find th halflif of th radioactiv substanc (a) Th initial mass was prsnt whn t 0 : G So th initial mass was 00 grams (b) Th halflif is th tim t at which G 0, so 0 00 t t 0 00 t log ( ) (convrting to log form) t 0 (to dp) So th halflif is 0 yars, roughly 0 8 days Pag 8 HSN000
44 Unit Eponntials and Logarithms 8 Th world population, in billions, t yars aftr 90 is givn by 0 078t P (a) What was th world population in 90? (b) Find, to th narst yar, th tim takn for th world population to doubl (a) For 90, t 0 : P So th world population in 90 was billion (b) For th population to doubl: t 0 078t 0 078t log (convrting to log form) t 8 9 (to dp) So th population doubld aftr 9 yars (to th narst yar) Graphing with Logarithmic As It is common in applications to find an ponntial rlationship btwn variabls; for instanc, th rlationship btwn th world population and tim in th prvious ampl Givn som data (g from an primnt) w would lik to find an plicit quation for th rlationship Rlationships of th form y ab Suppos w hav an ponntial graph y y ab, whr a, b > 0 y ab a O Pag 9 HSN000
45 Unit Eponntials and Logarithms Taking logarithms w find that log y log ( ab ) log a + log log a + log b W can scal th yais so that Y b ais Now our rlationship is of th form straight lin in th (, Y )plan Y log y ; th Yais is calld a logarithmic Y log b + log a, which is a ( log ) Y b + log a log a gradint is log b Sinc this is just a straight lin, w can us known points to find th gradint log b and th Yais intrcpt log a From ths w can asily find th valus of a and b, and hnc spcify th quation EXAMPLES y ab Th rlationship btwn two variabls, and y, is of th form y ab An primnt to tst this rlationship producd th data shown in th graph, whr log y is plottd against Find th valus of a and b W nd to obtain a straight lin quation: log y log ab (taking logs of both sids) y ab log y log a + log b log y log a + log b i Y log b + log a O log y From th graph, th Yais intrcpt is log a ; so O ( 7,) a Pag 70 HSN000
46 Unit Eponntials and Logarithms Using th gradint formula: log b b 7 Th rsults from an primnt wr notd as follows: Th rlationship btwn ths data can b writtn in th form y ab Find th valus of a and b, and stat th formula for y in trms of W nd to obtain a straight lin quation: log y log ab (taking logs of both sids) y ab log y log a + log b log y log a + log b i Y log b + log a W can find th gradint log b (and hnc b), using two points on th lin: using ( 0, 0 ) and ( 80, ), So log y log a log b So b (to dp) (to dp) Now w can work out log a (and hnc a) by substituting a point into this quation: using ( 0, 0 ), Thrfor y log y 0 78 log y 0 and 0 so log log a so a 09 (to dp) (to dp) a Not Dpnding on th points usd, slightly diffrnt valus for a and b may b obtaind Pag 7 HSN000
47 Unit Eponntials and Logarithms Equations in th form y a b Anothr common rlationship is y a, whr a, > 0 In this cas, th rlationship can b rprsntd by a straight lin if w chang both as to logarithmic ons EXAMPLE Th rsults from an primnt wr notd as follows: Th rlationship btwn ths data can b writtn in th form y a Find th valus of a and b, and stat th formula for y in trms of W nd to obtain a straight lin quation: y a log y log a (taking logs of both sids) 0 0 log y log a + log log y log a + b log i Y bx + log a b log0 log b 0 b W can find th gradint b using two points on th lin: using ( 70, ) and ( 8, 0 ), So log0 0 9 log0 log0 0 y + a b y b (to dp) Now w can work out a by substituting a point into this quation: using ( 70, ), log0 a log a a 0 0 (to dp) b Thrfor y 0 9 Pag 7 HSN000
48 Unit Eponntials and Logarithms 7 Graph Transformations Graph transformations wr covrd in Unit Outcom Functions and Graphs, but w will now look in mor dtail at applying transformations to graphs of ponntial and logarithmic functions EXAMPLES Shown blow is th graph of y f ( ) whr f ( ) log y y f ( ) ( 9,a) O (a) Stat th valu of a (b) Sktch th graph of y f ( + ) + (a) a log 9 (sinc 9) (b) Th graph shifts two units to th lft, and on unit upwards: y y f ( + ) + y ( 7,) O (,) Shown blow is part of th graph of y log y y f ( ) Sktch th graph of y log ( ) y y log ( ) y log ( ) log O log So rflct in th ais O (,) (, ) Pag 7 HSN000
49 Unit Eponntials and Logarithms Th diagram shows th graph of y y y (, ) O On sparat diagrams, sktch th graphs of: (a) y ; (b) y (a) Rflct in th yais: y (b) y (, ) So scal th graph from (a) by in th ydirction: (,8 ) y O y y O Pag 7 HSN000
50 Unit Wav Functions OUTCOME Wav Functions Eprssing pcos + qsin in th form kcos( a) An prssion of th form pcos + q sin can b writtn in th form k cos( a) whr k sina k p + q and tan a k cosa Th following ampl shows how to achiv this EXAMPLES Writ cos + sin in th form k cos( a ) whr 0 a 0 Stp Epand k cos( a) using th cos + sin compound angl formula k cos( a ) Stp Rarrang to compar with pcos + q sin Stp Compar th cofficints of cos and sin with pcos + q sin Stp Mark th quadrants on a CAST diagram, according to th signs of k cos a and k sin a Stp Find k and a using th formula abov (a lis in th quadrant markd twic in Stp ) Stp Stat pcos + q sin in th form k cos( a) using ths valus k cos cosa + k sin sin a k cos a cos + k sina sin k cosa k sin a 80 a a S A T C 80 + a 0 a k + 9 k sin a tana k cos a a tan 7 (to d p ) cos + sin cos 7 Pag 7 HSN000
51 Unit Wav Functions Writ cos sin in th form k cos( a) whr 0 a π cos sin k cos( a) k cosa k sina π a a S A T C π + a π a Hnc a is in th fourth quadrant k cos cosa + k sin sina k cosa cos + k sina sin k + ( ) Hnc cos sin cos( 7) Eprssing pcos + qsin in othr forms k sin a tan a k cos a First quadrant answr is: tan Not 0 0 (to d p ) Mak sur your calculator is in radian So a π 0 0 mod 7 (to d p ) An prssion in th form pcos + q sin can also b writtn in any of th following forms using a similar mthod: EXAMPLES k cos ( + a), k sin ( a), k sin ( + a) Writ cos + sin in th form k sin( + a ) whr 0 a 0 cos + sin k sin( + a ) k cosa k sina 80 a a S A T C 80 + a 0 a Hnc a is in th first quadrant k sin cos a + k cos sina k cos a sin + k sina cos k + Hnc cos + sin sin( + ) k sina tana k cosa So: a tan (to d p ) Pag 7 HSN000
52 Unit Wav Functions Writ cos sin in th form k cos( + a) whr 0 a π cos sin k cos( + a) k cosa k sin a π a a S A T C π + a π a Hnc a is in th first quadrant k cos cos a k sin sin a k cos a cos k sin a sin k + + Hnc cos sin cos( π ) Multipl Angls + k sin a tan a k cos a So: a tan π W can us th sam mthod with prssions involving th sam multipl angl, i pcos( n) + q sin( n), whr n is a constant EXAMPLE Writ cos + sin in th form k sin( + a ) whr 0 a 0 cos + sin k sin( + a ) k cosa k sina 80 a a S A T C 80 + a 0 a Hnc a is in th first quadrant k sin cos a + k cos sina k cosa sin + k sina cos k + 9 Hnc cos + sin sin( + ) k sina tana k cosa So: a tan (to d p ) Pag 77 HSN000
53 Unit Wav Functions Maimum and Minimum Valus To work out th maimum or minimum valus of pcos + q sin, w can rwrit it as a singl trigonomtric function, g k cos( a) Rcall that th maimum valu of th sin and cosin functions is, and thir minimum is y y sin y y cos ma ma O EXAMPLE Writ sin + cos in th form k cos( a) whr 0 a π and stat: (i) th maimum valu and th valu of 0 < π at which it occurs (ii) th minimum valu and th valu of 0 < π at which it occurs sin + cos k cos( a) k cosa k sin a π a π a S A T C π + a π a Hnc a is in th first quadrant k cos cosa + k sin sina k cosa cos + k sina sin k + 7 O Hnc sin + cos 7 cos( ) min π k sin a tan a k cosa So: a tan ( ) (to d p ) min Th maimum valu of 7 occurs whn: cos( ) cos 0 (to d p ) Th minimum valu of 7 occurs whn: cos( ) cos ( ) π 8 (to d p ) Pag 78 HSN000
54 Unit Wav Functions Solving Equations Th mthod of writing two trigonomtric trms as on can b usd to hlp solv quations involving both a sin( n ) and a cos( n ) trm EXAMPLES Solv cos + sin whr 0 0 First, w writ cos + sin in th form k cos( a ) : cos + sin k cos( a ) k cos a k sina 80 a a S A T C 80 + a 0 a Hnc a is in th first quadrant k cos cosa + k sin sina k cos a cos + k sina sin k + Hnc cos + sin cos( ) Now w us this to hlp solv th quation: cos + sin cos( ) cos( ) 9 or or 9 78 or 0 80 k sina tana k cosa So: a tan (to d p ) S A T C cos 9 (to d p ) Pag 79 HSN000
55 Unit Wav Functions Solv cos + sin whr 0 π First, w writ cos + sin in th form k cos( a ) : cos + sin k cos( a) k cosa k sina π a a S A T C π + a π a Hnc a is in th first quadrant k cos cos a + k sin sina k cosa cos + k sina sin k + ( ) + 9 Hnc cos + sin cos( 0 98) Now w us this to hlp solv th quation: k sina tana k cosa So: a tan 0 98 (to d p ) cos + sin π S A 0 < < π cos( 0 98) cos( 0 98) π + T C 0 < < π π 0 98 cos ( ) 90 (to d p ) or π 90 or π + 90 or π + π 90 or π + π or 99 or 7 7 or 7 7 or 97 or 8 or 9 7 or 988 or 78 or 0 Pag 80 HSN000
56 Unit Wav Functions Sktching Graphs of y pcos + qsin Eprssing pcos + q sin in th form k cos( a) nabls us to sktch th graph of y pcos + q sin EXAMPLES (a) Writ 7cos + sin in th form k cos( a ), 0 a 0 (b) Hnc sktch th graph of y 7cos + sin for 0 0 (a) First, w writ 7 cos + sin in th form k cos( a ) : 7 cos + sin k cos( a ) k cosa 7 k sina 80 a a S A T C 80 + a 0 a Hnc a is in th first quadrant k cos cosa + k sin sina k cosa cos + k sina sin k Hnc 7 cos + sin 8 cos( 0 ) k sina tana k cosa So: (b) Now w can sktch th graph of y 7cos + sin : y y 7 cos + sin 8 7 a tan 7 0 (to d p ) O Pag 8 HSN000
57 Unit Wav Functions Sktch th graph of y sin + cos for 0 0 First, w writ sin + cos in th form k cos( a ) : sin + cos k cos( a ) k cos a k sina 80 a a S A T C 80 + a 0 a Hnc a is in th first quadrant k cos cosa + k sin sin a k cosa cos + k sin a sin k + + Hnc sin + cos cos( 0 ) k sina tan a k cosa So: a tan 0 Now w can sktch th graph of y sin + cos : y y sin + cos O 0 0 Pag 8 HSN000
58 Unit Wav Functions (a) Writ sin cos in th form k sin( a ), 0 a 0 (b) Hnc sktch th graph of y sin cos +, 0 0 (a) sin cos k sin( a ) k cosa k sin a 80 a a S A T C 80 + a 0 a Hnc a is in th first quadrant k sin cosa + k cos sin a k cosa sin + k sin a cos k + + Hnc sin cos sin( ) (b) Now sktch th graph of y sin cos + sin( ) + : y 8 O y sin cos + 0 k sina tana k cosa So: ( ) a tan (to d p ) Pag 8 HSN000
Question 3: How do you find the relative extrema of a function?
ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating
More informationHigher. Exponentials and Logarithms 160
hsn uknt Highr Mthmtics UNIT UTCME Eponntils nd Logrithms Contnts Eponntils nd Logrithms 6 Eponntils 6 Logrithms 6 Lws of Logrithms 6 Eponntils nd Logrithms to th Bs 65 5 Eponntil nd Logrithmic Equtions
More informationAP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) 92.222  Linar Algbra II  Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial
More informationME 612 Metal Forming and Theory of Plasticity. 6. Strain
Mtal Forming and Thory of Plasticity mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.
More informationQUANTITATIVE METHODS CLASSES WEEK SEVEN
QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.
More informationNonHomogeneous Systems, Euler s Method, and Exponential Matrix
NonHomognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous firstordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach
More informationNew Basis Functions. Section 8. Complex Fourier Series
Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ralvalud Fourir sris is xplaind and formula ar givn for convrting
More information5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:
.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This
More informationThe Normal Distribution: A derivation from basic principles
Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn
More informationEcon 371: Answer Key for Problem Set 1 (Chapter 1213)
con 37: Answr Ky for Problm St (Chaptr 23) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc
More informationLecture 3: Diffusion: Fick s first law
Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th
More information5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power
Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim
More informationThe example is taken from Sect. 1.2 of Vol. 1 of the CPN book.
Rsourc Allocation Abstract This is a small toy xampl which is wllsuitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of Cnts. Hnc, it can b rad by popl
More informationby John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia
Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs
More informationSection 7.4: Exponential Growth and Decay
1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 117 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart
More informationTraffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. GangLn Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
More informationCloud and Big Data Summer School, Stockholm, Aug., 2015 Jeffrey D. Ullman
Cloud and Big Data Summr Scool, Stockolm, Aug., 2015 Jffry D. Ullman Givn a st of points, wit a notion of distanc btwn points, group t points into som numbr of clustrs, so tat mmbrs of a clustr ar clos
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationProjections  3D Viewing. Overview Lecture 4. Projection  3D viewing. Projections. Projections Parallel Perspective
Ovrviw Lctur 4 Projctions  3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl  Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog
More informationAdverse Selection and Moral Hazard in a Model With 2 States of the World
Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,
More informationFactorials! Stirling s formula
Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical
More information(Analytic Formula for the European Normal Black Scholes Formula)
(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually
More informationVersion 1.0. General Certificate of Education (Alevel) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.
Vrsion.0 Gnral Crtificat of Education (Alvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,
More informationLecture 20: Emitter Follower and Differential Amplifiers
Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.
More informationCPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions
CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:
More informationSPECIAL VOWEL SOUNDS
SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)
More informationParallel and Distributed Programming. Performance Metrics
Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:
More informationAP Calculus MultipleChoice Question Collection 1969 1998. connect to college success www.collegeboard.com
AP Calculus MultiplChoic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos
More informationPrinciples of Humidity Dalton s law
Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid
More informationIncomplete 2Port Vector Network Analyzer Calibration Methods
Incomplt Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar
More informationStatistical Machine Translation
Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm
More informationCurrent and Resistance
Chaptr 6 Currnt and Rsistanc 6.1 Elctric Currnt...66.1.1 Currnt Dnsity...66. Ohm s Law...64 6.3 Elctrical Enrgy and Powr...67 6.4 Summary...68 6.5 Solvd Problms...69 6.5.1 Rsistivity of a Cabl...69
More informationDeer: Predation or Starvation
: Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,
More informationA Note on Approximating. the Normal Distribution Function
Applid Mathmatical Scincs, Vol, 00, no 9, 4549 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and
More informationEFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS
25 Vol. 3 () JanuaryMarch, pp.375/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut
More informationBasis risk. When speaking about forward or futures contracts, basis risk is the market
Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also
More information[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lbinsec^2)
MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (voltssc/rad Motor torqu constant (lbin/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris
More informationVector Network Analyzer
Cours on Microwav Masurmnts Vctor Ntwork Analyzr Prof. Luca Prrgrini Dpt. of Elctrical, Computr and Biomdical Enginring Univrsity of Pavia mail: luca.prrgrini@unipv.it wb: microwav.unipv.it Microwav Masurmnts
More informationArchitecture of the proposed standard
Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th
More informationSharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means
Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s166001507411 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans WiMao Qian
More informationSection 55 Inverse of a Square Matrix
 Invrs of a Squar Matrix 9 (D) Rank th playrs from strongst to wakst. Explain th rasoning hind your ranking. 68. Dominan Rlation. Eah mmr of a hss tam plays on math with vry othr playr. Th rsults ar givn
More informationRemember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D
24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd
More informationa.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a
Bellwork a.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.
More information7 Timetable test 1 The Combing Chart
7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing
More informationUse a highlevel conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects
Chaptr 3: Entity Rlationship Modl Databas Dsign Procss Us a highlvl concptual data modl (ER Modl). Idntify objcts of intrst (ntitis) and rlationships btwn ths objcts Idntify constraints (conditions) End
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationFinancial Mathematics
Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution
More informationEntityRelationship Model
EntityRlationship Modl Kuanghua Chn Dpartmnt of Library and Information Scinc National Taiwan Univrsity A Company Databas Kps track of a company s mploys, dpartmnts and projcts Aftr th rquirmnts collction
More informationhttp://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force
ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd
More informationSUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* RostovonDon. Russia
SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* RostovonDon. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs
More informationC H A P T E R 1 Writing Reports with SAS
C H A P T E R 1 Writing Rports with SAS Prsnting information in a way that s undrstood by th audinc is fundamntally important to anyon s job. Onc you collct your data and undrstand its structur, you nd
More informationConstraintBased Analysis of Gene Deletion in a Metabolic Network
ConstraintBasd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFGRsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany
More informationHOMEWORK FOR UNIT 51: FORCE AND MOTION
Nam Dat Partnrs HOMEWORK FOR UNIT 51: FORCE AND MOTION 1. You ar givn tn idntial springs. Dsrib how you would dvlop a sal of for (i., a mans of produing rpatabl fors of a varity of sizs) using ths springs.
More informationCategory 7: Employee Commuting
7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil
More informationLong run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange
Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity
More informationMaking and Using the Hertzsprung  Russell Diagram
Making and Using th Hrtzsprung  Russll Diagram Nam In astronomy th HrtzsprungRussll Diagram is on of th main ways that w organiz data dscribing how stars volv, ags of star clustrs, masss of stars tc.
More informationNoble gas configuration. Atoms of other elements seek to attain a noble gas electron configuration. Electron configuration of ions
Valnc lctron configuration dtrmins th charactristics of lmnts in a group Nobl gas configuration Th nobl gass (last column in th priodic tabl) ar charactrizd by compltly filld s and p orbitals this is a
More informationForeign Exchange Markets and Exchange Rates
Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls
More informationCPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.
Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by
More informationInstallation Saving Spaceefficient Panel Enhanced Physical Durability Enhanced Performance Warranty The IRR Comparison
Contnts Tchnology Nwly Dvlopd Cllo Tchnology Cllo Tchnology : Improvd Absorption of Light Doublsidd Cll Structur Cllo Tchnology : Lss Powr Gnration Loss Extrmly Low LID Clls 3 3 4 4 4 Advantag Installation
More informationKeywords Cloud Computing, Service level agreement, cloud provider, business level policies, performance objectives.
Volum 3, Issu 6, Jun 2013 ISSN: 2277 128X Intrnational Journal of Advancd Rsarch in Computr Scinc and Softwar Enginring Rsarch Papr Availabl onlin at: wwwijarcsscom Dynamic Ranking and Slction of Cloud
More informationChapter 10 Function of a Matrix
EE448/58 Vrsion. John Stnsby Chatr Function of a atrix t f(z) b a comlxvalud function of a comlx variabl z. t A b an n n comlxvalud matrix. In this chatr, w giv a dfinition for th n n matrix f(a). Also,
More informationChapter 3: Capacitors, Inductors, and Complex Impedance
haptr 3: apacitors, Inductors, and omplx Impdanc In this chaptr w introduc th concpt of complx rsistanc, or impdanc, by studying two ractiv circuit lmnts, th capacitor and th inductor. W will study capacitors
More informationCHAPTER 4c. ROOTS OF EQUATIONS
CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03  Computation Mthod in Civil Enginring II Dpartmnt o Civil
More informationSPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM
RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs
More informationGold versus stock investment: An econometric analysis
Intrnational Journal of Dvlopmnt and Sustainability Onlin ISSN: 2688662 www.isdsnt.com/ijds Volum Numbr, Jun 202, Pag 7 ISDS Articl ID: IJDS20300 Gold vrsus stock invstmnt: An conomtric analysis Martin
More informationExamples. Epipoles. Epipolar geometry and the fundamental matrix
Epipoar gomtry and th fundamnta matrix Epipoar ins Lt b a point in P 3. Lt x and x b its mapping in two imags through th camra cntrs C and C. Th point, th camra cntrs C and C and th (3D points corrspon
More informationModern Portfolio Theory (MPT) Statistics
Modrn Portfolio Thory (MPT) Statistics Morningstar Mthodology Papr May 9, 009 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription
More informationMagic Message Maker Amaze your customers with this Gift of Caring communication piece
Magic Mssag Makr maz your customrs with this Gift of aring communication pic Girls larn th powr and impact of crativ markting with this attntion grabbing communication pic that will hlp thm o a World of
More informationSimulation of the electric field generated by a brown ghost knife fish
C H A P T R 2 7 Simulation of th lctric fild gnratd by a brown ghost knif fish lctric fild CONCPTS 27.1 Th fild modl 27.2 lctric fild diagrams 27.3 Suprposition of lctric filds 27.4 lctric filds and forcs
More informationVibrational Spectroscopy
Vibrational Spctroscopy armonic scillator Potntial Enrgy Slction Ruls V( ) = k = R R whr R quilibrium bond lngth Th dipol momnt of a molcul can b pandd as a function of = R R. µ ( ) =µ ( ) + + + + 6 3
More informationCIRCUITS AND ELECTRONICS. Basic Circuit Analysis Method (KVL and KCL method)
6. CIRCUITS AND ELECTRONICS Basic Circuit Analysis Mthod (KVL and KCL mthod) Cit as: Anant Agarwal and Jffry Lang, cours matrials for 6. Circuits and Elctronics, Spring 7. MIT 6. Fall Lctur Rviw Lumpd
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More information2312 test 2 Fall 2010 Form B
2312 test 2 Fall 2010 Form B 1. Write the slopeintercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function
More informationImportant Information Call Through... 8 Internet Telephony... 6 two PBX systems... 10 Internet Calls... 3 Internet Telephony... 2
Installation and Opration Intrnt Tlphony Adaptr Aurswald Box Indx C I R 884264 03 02/05 Call Duration, maximum...10 Call Through...7 Call Transportation...7 Calls Call Through...7 Intrnt Tlphony...3 two
More informationLecture notes: 160B revised 9/28/06 Lecture 1: Exchange Rates and the Foreign Exchange Market FT chapter 13
Lctur nots: 160B rvisd 9/28/06 Lctur 1: xchang Rats and th Forign xchang Markt FT chaptr 13 Topics: xchang Rats Forign xchang markt Asst approach to xchang rats Intrst Rat Parity Conditions 1) Dfinitions
More informationPerformance Evaluation
Prformanc Evaluation ( ) Contnts lists availabl at ScincDirct Prformanc Evaluation journal hompag: www.lsvir.com/locat/pva Modling Baylik rputation systms: Analysis, charactrization and insuranc mchanism
More informationProduction Costing (Chapter 8 of W&W)
Production Costing (Chaptr 8 of W&W).0 Introduction Production costs rfr to th oprational costs associatd with producing lctric nrgy. Th most significant componnt of production costs ar th ful costs ncssary
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is nonnegative
More informationJune 2012. Enprise Rent. Enprise 1.1.6. Author: Document Version: Product: Product Version: SAP Version: 8.81.100 8.8
Jun 22 Enpris Rnt Author: Documnt Vrsion: Product: Product Vrsion: SAP Vrsion: Enpris Enpris Rnt 88 88 Enpris Rnt 22 Enpris Solutions All rights rsrvd No parts of this work may b rproducd in any form or
More informationInternational Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)
Intrnational Association of Scintific Innovation and Rsarch (IASIR) (An Association Unifing th Scincs, Enginring, and Applid Rsarch) ISSN (Print): 79000 ISSN (Onlin): 79009 Intrnational Journal of Enginring,
More informationLG has introduced the NeON 2, with newly developed Cello Technology which improves performance and reliability. Up to 320W 300W
Cllo Tchnology LG has introducd th NON 2, with nwly dvlopd Cllo Tchnology which improvs prformanc and rliability. Up to 320W 300W Cllo Tchnology Cll Connction Elctrically Low Loss Low Strss Optical Absorption
More informationSPECIFIC HEAT AND HEAT OF FUSION
PURPOSE This laboratory consists of to sparat xprimnts. Th purpos of th first xprimnt is to masur th spcific hat of to solids, coppr and rock, ith a tchniqu knon as th mthod of mixturs. Th purpos of th
More information(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its
(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract 
More information811ISD Economic Considerations of Heat Transfer on Sheet Metal Duct
Air Handling Systms Enginring & chnical Bulltin 811ISD Economic Considrations of Hat ransfr on Sht Mtal Duct Othr bulltins hav dmonstratd th nd to add insulation to cooling/hating ducts in ordr to achiv
More informationNoise Power Ratio (NPR) A 65Year Old Telephone System Specification Finds New Life in Modern Wireless Applications.
TUTORIL ois Powr Ratio (PR) 65Yar Old Tlphon Systm Spcification Finds w Lif in Modrn Wirlss pplications ITRODUTIO by Walt Kstr Th concpt of ois Powr Ratio (PR) has bn around sinc th arly days of frquncy
More informationEpipolar Geometry and the Fundamental Matrix
9 Epipolar Gomtry and th Fundamntal Matrix Th pipolar gomtry is th intrinsic projctiv gomtry btwn two viws. It is indpndnt of scn structur, and only dpnds on th camras intrnal paramtrs and rlativ pos.
More informationThe Constrained SkiRental Problem and its Application to Online Cloud Cost Optimization
3 Procdings IEEE INFOCOM Th Constraind SkiRntal Problm and its Application to Onlin Cloud Cost Optimization Ali Khanafr, Murali Kodialam, and Krishna P. N. Puttaswam Coordinatd Scinc Laborator, Univrsit
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationFACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data
FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationBusiness rules FATCA V. 02/11/2015
Elmnt Attribut Siz InputTyp Rquirmnt BUSINESS RULES TYPE ERROR ACK Xpath I.Mssag Hadr FATCA_OECD Vrsion xsd: string = Validation WrongVrsion ftc:fatca_oecd/vrsion SndingCompanyIN Unlimit d xsd: string
More informationwww.sakshieducation.com
LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationFundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY
Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl
More informationIntroduction to Finite Element Modeling
Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation
More informationESCI 341 Atmospheric Thermodynamics Lesson 14 Humidity Dr. DeCaria
PARIAL PRESSURE ESCI 341 Atmoshric hrmoynamics Lsson 14 Humiity Dr. DCaria In a mixtur of gass, ach gas scis contributs to th total rssur. ο h rssur xrt by a singl gas scis is known as th artial rssur
More information