0.9 Radicals and Equations
|
|
- Shon Stewart
- 7 years ago
- Views:
Transcription
1 0.9 Radicals and Equations 0.9 Radicals and Equations In tis section we eview simlifying exessions and solving equations involving adicals. In addition to te oduct, quotient and owe ules stated in Teoem 0. in Section 0., we esent te following esult wic states tat n t oots and n t owes moe o less undo eac ote. Teoem 0.. Simlifying n t owes of n t oots: Suose n is a natual numbe, a is a eal numbe and n a is a eal numbe. Ten ( n a) n a if n is odd, n a n a; if n is even, n a n a. Since n a is defined so tat ( n a) n a, te fist claim in te teoem is just a e-woding of Definition 0.8. Te second at of te teoem beaks down along odd/even exonent lines due to ow exonents affect negatives. To see tis, conside te secific cases of ( ) and ( ). In te fist case, ( ) 8, so we ave an instance of wen n a n a. Te eason tat te cube oot undoes te tid owe in ( ) is because te negative is eseved wen aised to te tid (odd) owe. In ( ), te negative goes away wen aised to te fout (even) owe: ( ) 6. Accoding to Definition 0.8, te fout oot is defined to give only non-negative numbes, so 6. Hee we ave a case wee ( ), not. In geneal, we need te absolute values to simlify n a n only wen n is even because a negative to an even owe is always ositive. In aticula, x x, not just x (unless we know x 0.) We actice tese fomulas in te following examle. Examle Pefom te indicated oeations and simlify.. x +. t 0t x x + Solution. ( x ) (x) 6. 8x. q ( 8y 8y) +( 0 L 8 80). We told you back on age tat oots do not distibute acoss addition and since x + cannot be factoed ove te eal numbes, x + cannot be simlified. It may seem silly to stat wit tis examle but it is extemely imotant tat you undestand wat maneuves ae legal and wic ones ae not. See Section 5. fo a moe ecise undestanding of wat we mean ee. If tis discussion sounds familia, see te discussion following Definition 0.9 and te discussion following Extacting te Squae Root on age 8. You eally do need to undestand tis otewise oible evil will lague you futue studies in Mat. If you say someting totally wong like x +x + ten you may neve ass Calculus. PLEASE be caeful!
2 Peequisites. Again we note tat t 0t t 0t + 5, since adicals do not distibute acoss addition and subtaction. In tis case, oweve, we can facto te adicand and simlify as t 0t + 5 (t 5) t 5 Witout knowing moe about te value of t, we ave no idea if t so t 5 is ou final answe. 5 5 is ositive o negative. To simlify 8x, we need to look fo efect cubes in te adicand. Fo te cofficient, we ave To find te lagest efect cube facto in x, we divide (te exonent on x) by (since we ae looking fo a efect cube). We get wit a emainde of. Tis means +, so x x + x x (x ) x. Putting tis altogete gives: 8x 6 (x ) x (x ) 6x x 6x Facto out efect cubes Reaange factos, Poduct Rule of Radicals. In tis examle, we ae looking fo efect fout owes in te adicand. In te numeato is clealy a efect fout owe. Fo te denominato, we take te owe on te L, namely, and divide by to get. Tis means L 8 L (L ). We get L L (L ) L Quotient Rule of Radicals Poduct Rule of Radicals Simlify Witout moe infomation about, we cannot simlify any fute. Howeve, we can simlify L. Regadless of te coice of L, L 0. Actually, L > 0 because L is in te denominato wic means L 6 0. Hence, L L. Ou answe simlifies to: L L 5. Afte a quick cancellation (two of te s in te second tem) we need to obtain a common denominato. Since we can view te fist tem as aving a denominato of, te common denominato is ecisely te denominato of te second tem, namely ( x ). Wit Let t and see wat aens to t 0t + 5 vesus t 0t In geneal, t 5 6 t 5 and t 5 6 t + 5 so watc wat you e doing!
3 0.9 Radicals and Equations common denominatos, we oceed to add te two factions. Ou last ste is to facto te numeato to see if tee ae any cancellation ootunities wit te denominato. x x + ( (x) x x + x ) ( (x) Reduce x ) x x + x ( x ) Mutily (x x ) ( x ) ( x ) + x ( x ) Equivalent factions x( x ) x ( + x ) ( Multily x ) x(x ) ( x ) + x ( x ) Simlify x(x ) + x ( x ) Add x(x + ) ( x ) Facto x(x ) ( x ) We cannot educe tis any fute because x is ieducible ove te ational numbes. 6. We begin by woking inside eac set of aenteses, using te oduct ule fo adicals and combining like tems. q ( 8y 8y) +( q 0 80) ( 9 y y) +( 5 6 5) q ( 9 y y) +( 5 6 5) q ( y y) + ( 5 5) q ( y) +( 5) q y +( ) ( 5) y + 5 y + 0 To see if tis simlifies any fute, we facto te adicand: y + 0 (y + 0). Finding no efect squae factos, we ae done.
4 Peequisites Teoem 0. allows us to genealize te ocess of Extacting Squae Roots to Extacting n t oots wic in tun allows us to solve equations 6 of te fom X n c. Extacting n t oots: If c is a eal numbe and n is odd ten te eal numbe solution to X n c is X n c. If c 0 and n is even ten te eal numbe solutions to X n c ae X ± n c. Note: If c < 0 and n is even ten X n c as no eal numbe solutions. n Essentially, we solve X n c by taking te n t oot of bot sides: X n n c. Simlifying te left side gives us just X if n is odd o X if n is even. In te fist case, X n c, and in te second, X ± n c. Putting tis togete wit te ote at of Teoem 0., namely ( n a) n a, gives us a stategy fo solving equations wic involve n t and n t oots. Stategies fo Powe and Radical Equations If te equation involves an n t owe and te vaiable aeas in only one tem, isolate te tem wit te n t owe and extact n t oots. If te equation involves an n t oot and te vaiable aeas in tat n t oot, isolate te n t oot and aise bot sides of te equation to te n t owe. Note: Wen aising bot sides of an equation to an even owe, be sue to ceck fo extaneous solutions. Te note about extaneous solutions can be demonstated by te basic equation: x. Tis equation as no solution since, by definition, x 0 fo all eal numbes x. Howeve, if we squae bot sides of tis equation, we get ( x) ( ) o x. Howeve, x doesn t ceck in te oiginal equation, since, not. Once again, te oot 7 of all of ou oblems lies in te fact tat a negative numbe to an even owe esults in a ositive numbe. In ote wods, aising bot sides of an equation to an even owe does not oduce an equivalent equation, but ate, an equation wic may ossess moe solutions tan te oiginal. Hence te cautionay emak above about extaneous solutions. Examle Solve te following equations.. (5x + ) 6. (5 w) 7 9. t + t +6. y x + x 6. n ++n 0 Fo te emaining oblems, assume tat all of te vaiables eesent ositive eal numbes. 8 6 Well, not entiely. Te equation x 7 as seven answes: x and six comlex numbe solutions wic we ll find using tecniques in Section.7. 7 Pun intended! 8 Tat is, you needn t woy tat you e multilying o dividing by 0 o tat you e fogetting absolute value symbols.
5 0.9 Radicals and Equations 5 7. Solve fo : V (R ). 8. Solve fo M : M M 9. Solve fo v: m m 0. Assume all quantities eesent ositive eal numbes. v c Solution.. In ou fist equation, te quantity containing x is aleady isolated, so we extact fout oots. Since te exonent ee is even, wen te oots ae extacted we need bot te ositive and negative oots. (5x + ) 6 5x + ± 6 Extact fout oots 5x + ± 5x + o 5x + x 5 o x We leave it to te eade tat bot of tese solutions satisfy te oiginal equation.. In tis examle, we fist need to isolate te quantity containing te vaiable w. Hee, tid (cube) oots ae equied and since te exonent (index) is odd, we do not need te ±: (5 w) 7 9 (5 w) 7 8 Subtact (5 w) 56 Multily by 7 5 w 56 Extact cube oot 5 w ( 8)(7) 5 w 8 7 Poduct Rule 5 w 7 w 5 7 Subtact w w 5+ 7 Divide by Poeties of Negatives Te eade sould ceck te answe because it ovides a eaty eview of aitmetic.. To solve t + t + 6, we fist isolate te squae oot, ten oceed to squae bot sides of te equation. In doing so, we un te isk of intoducing extaneous solutions so cecking
6 6 Peequisites ou answes ee is a necessity. t + t + 6 t + 6 t Subtact t ( t + ) (6 t) Squae bot sides t + 6 t + t F.O.I.L. / Pefect Squae Tinomial 0 t t + Subtact t and 0 (t )(t ) Facto Fom te Zeo Poduct Poety, we know eite t 0 (wic gives t ) o t 0 (wic gives t ). Wen cecking ou answes, we find t satisfies te oiginal equation, but t does not. 9 So ou final answe is t only.. In ou next examle, we locate te vaiable (in tis case y) beneat a cube oot, so we fist isolate tat oot and cube bot sides. y + 0 y + Subtact y + Divide by y + Poeties of Negatives! ( y + ) Cube bot sides y + ( ) y + 7 y 7 y 7 Subtact 7 7 y 7 7 y 7 5 Common denominatos Subtact factions Divide by multily by Since we aised bot sides to an odd owe, we don t need to woy about extaneous solutions but we encouage te eade to ceck te solution just fo te fun of it. 9 It is wot noting tat wen t is substituted into te oiginal equation, we get If te + 5 wee 5, te solution would ceck. Once again, wen squaing bot sides of an equation, we lose tack of ±, wic is wat lets extaneous solutions in te doo.
7 0.9 Radicals and Equations 7 5. In te equation x + x, we ave not one but two squae oots. We begin by isolating one of te squae oots and squaing bot sides. x + x x x Subtact x fom bot sides ( x ) ( x) Squae bot sides x x + ( x) F.O.I.L. / Pefect Squae Tinomial x x + ( x) x x + 8x Distibute x 5 8x x Gate like tems At tis oint, we ave just one squae oot so we oceed to isolate it and squae bot sides a second time. 0 x 5 8x x x 6 x Subtact 5, add 8x (x 6) ( x) Squae bot sides x x + 6 6( x) x x x x x Subtact 6, add x (6x 8x + 5) 0 Facto (x )(8x 5) 0 Facto some moe Fom te Zeo Poduct Poety, we know eite x 0 o 8x 5 0. Te fome gives x 5 wile te latte gives us x 8. Since we squaed bot sides of te equation (twice!), we need to ceck fo extaneous solutions. We find x 5 8 to be extaneous, so ou only solution is x. 6. As usual, ou fist ste in solving n ++n 0 is to isolate te adical. We ten oceed to aise bot sides to te fout owe to eliminate te fout oot: n ++n 0 n + n Subtact n ( n + ) ( n) Raise bot sides to te t owe n + n Poeties of Negatives 0 n n Subtact n and 0 (n )(n + ) Facto - tis is a Quadatic in Disguise At tis oint, te Zeo Poduct Poety gives eite n 0 o n + 0. Fom n 0, we get n, so n ±. Fom n + 0, we get n, wic gives no eal solutions. 0 To avoid comlications wit factions, we ll foego dividing by te coefficient of x, namely. Tis is efectly fine so long as we don t foget to squae it wen we squae bot sides of te equation. Wy is tat again?
8 8 Peequisites Since we aised bot sides to an even (te fout) owe, we need to ceck fo extaneous solutions. We find tat n woks but n is extaneous. 7. In tis oblem, we ae asked to solve fo. Wile tee ae a lot of lettes in tis equation, aeas in only one tem:. Ou stategy is to isolate ten extact te cube oot. V (R ) V (R ) Multily by to clea factions V R Distibute V R Subtact R V R Divide by R V Poeties of Negatives R V Extact te cube oot Te ceck is, as always, left to te eade and igly encouaged. 8. Te equation we ae asked to solve in tis examle is fom te wold of Cemisty and is none ote tan Gaam s Law of effusion. As was mentioned in Examle 0.8., subscits in Matematics ae used to distinguis between vaiables and ave no aitmetic significance. In tis examle,,, M and M ae as diffeent as x, y, z and 7. Since we ae asked to solve fo M, we locate M and see it is in a denominato in a squae oot. We eliminate te squae oot by squaing bot sides and oceed fom tee. M M M M Squae bot sides M M M M Multily by M to clea factions, assume, M 60 M M Divide by, assume 60 As te eade may exect, cecking te answe amounts to a good execise in simlifying ational and adical exessions. Te fact tat we ae assuming all of te vaiables eesent ositive eal numbes comes in to lay, as well. including a Geek lette, no less!
9 0.9 Radicals and Equations 9 9. Ou last equation to solve comes fom Einstein s Secial Teoy of Relativity and elates te mass of an object to its velocity as it moves. We ae asked to solve fo v wic is located in just one tem, namely v, wic aens to lie in a faction undeneat a squae oot wic is itself a denominato. We ave quite a lot of wok aead of us! m m m m v m 0 v c c m 0 Multily by v to clea factions c! v c m 0 Squae bot sides v c m 0 Poeties of Exonents m m v c m 0 Distibute m v c m 0 m Subtact m m v c (m 0 m ) Multily by c (c 6 0) m v c m 0 + c m Distibute v c m c m 0 m Reaange tems, divide by m (m 6 0) c m c m 0 v m Extact Squae Roots, v > 0 so no ± c v (m m 0 ) Poeties of Radicals, facto m v c m m 0 m v c m m 0 m c > 0 and m > 0 so c c and m m Cecking te answe algebaically would ean te eade geat ono and esect on te Algeba battlefield so it is igly ecommended Rationalizing Denominatos and Numeatos In Section 0.7, tee wee a few instances wee we needed to ationalize a denominato - tat is, take a faction wit adical in te denominato and e-wite it as an equivalent faction witout See tis aticle on te Loentz Facto.
10 0 Peequisites a adical in te denominato. Tee ae vaious easons fo wanting to do tis, but te most essing eason is tat ationalizing denominatos - and numeatos as well - gives us an ootunity fo moe actice wit factions and adicals. To el efes you memoy, we ationalize a denominato and ten a numeato below: and In geneal, if te faction contains eite a single tem numeato o denominato wit an undesiable n t oot, we multily te numeato and denominato by wateve is equied to obtain a efect n t owe in te adicand tat we want to eliminate. If te faction contains two tems te situation is somewat moe comlicated. To see wy, conside te faction 5. Suose we wanted to id te denominato of te 5 tem. We could ty as above and multily numeato and denominato by 5 but tat just yields: 5 ( 5 5) We aven t emoved 5 fom te denominato - we ve just suffled it ove to te ote tem in te denominato. As you may ecall, te stategy ee is to multily bot numeato and denominato by wat s called te conjugate. Definition 0.7. Congugate of a Squae Root Exession: If a, b and c ae eal numbes wit c > 0 ten te quantities (a + b c) and (a b c) ae conjugates of one anote. a Conjugates multily accoding to te Diffeence of Squaes Fomula: (a + b c)(a b c)a (b c) a b c a As ae (b c a) and (b c + a): (b c a)(b c + a) b c a. Tat is, to get te conjugate of a two-tem exession involving a squae oot, you cange te to a +, o vice-vesa. Fo examle, te conjugate of 5 is + 5, and wen we multily tese two factos togete, we get ( 5)( + 5) ( 5) 6 5. Hence, to eliminate te 5 fom te denominato of ou oiginal faction, we multily bot te numeato and denominato by te conjugate of 5: 5 ( ( + 5) ( + 5) 5)( + 5) ( 5) ( + 5) Wat if we ad 5 instead of 5? We could ty multilying 5by+ 5 to get ( 5)( + 5) ( 5) 6 5, Befoe te advent of te andeld calculato, ationalizing denominatos made it easie to get decimal aoximations to factions containing adicals. Howeve, some (admittedly moe abstact) alications emain today one of wic we ll exloe in Section 0.0; one you ll see in Calculus.
11 0.9 Radicals and Equations wic leaves us wit a cube oot. Wat we need to undo te cube oot is a efect cube, wic means we look to te Diffeence of Cubes Fomula fo insiation: a b (a b)(a + ab + b ). If we take a and b 5, we multily ( 5)( + 5+( 5) ) ( 5) ( 5) So if we wee caged wit ationalizing te denominato of, we d ave: 5 5 ( ( + 5+( 5) ) 5)( + 5+( 5) ) Tis sot of ting extends to n t oots since (a b) is a facto of a n b n fo all natual numbes n, but in actice, we ll stick wit squae oots wit just a few cube oots town in fo a callenge. 5 Examle Rationalize te indicated numeato o denominato:. Rationalize te denominato: Solution. 5 x. Rationalize te numeato: 9+. We ae asked to ationalize te denominato, wic in tis case contains a fift oot. Tat means we need to wok to ceate fift owes of eac of te factos of te adicand. To do so, we fist facto te adicand: x 8 x x. To obtain fift owes, we need to multily by x inside te adical. 6 x 5 x 5 x 5 x 5 x 5 x 5 x x Equivalent Factions Poduct Rule 5 x x 5 5 x x 5 Poety of Exonents Poduct Rule 5 x x 5 8 x x 5 x x Poduct Rule Reduce Simlify 5 To see wat to do about fout oots, use long division to find (a b ) (a b), and aly tis to 5.
12 Peequisites. Hee, we ae asked to ationalize te numeato. Since it is a two tem numeato involving a squae oot, we multily bot numeato and denominato by te conjugate of 9+, namely Afte simlifying, we find an ootunity to educe te faction: 9+ ( 9+ )( 9+ + ) ( Equivalent Factions 9+ + ) ( 9+) ( 9+ + ) (9 + ) 9 ( 9+ + ) ( 9+ + ) ( 9+ + ) 9+ + Diffeence of Squaes Simlify Simlify Reduce We close tis section wit an awesome examle fom Calculus. Examle Simlify te comound faction ten ationalize te numeato of te esult. (x + )+ x + Solution. We stat by multilying te to and bottom of te big faction by x + + x +. (x + )+ x + x + + x + x + + x + x + + x + ((((((( x + + x + ((((((( x + + x + + x + x + + x + x + x + + x + + x + x + + x + x + Next, we multily te numeato and denominato by te conjugate of x + x + +,
13 0.9 Radicals and Equations namely x ++ x + +, simlify and educe: x + x + + x + + x + ( x + x + + )( x ++ x + + ) x + + x + ( x ++ x + + ) ( x + ) ( x + + ) x + + x + ( x ++ x + + ) (x + ) (x + + ) x + + x + ( x ++ x + + ) x + x x + + x + ( x ++ x + + ) x + + x + ( x ++ x + + ) x + + x + ( x ++ x + + ) Wile te denominato is quite a bit moe comlicated tan wat we stated wit, we ave done wat was asked of us. In te inteest of full disclosue, te eason we did all of tis was to cancel te oiginal fom te denominato. Tat s an awful lot of effot to get id of just one little, but you ll see te significance of tis in Calculus.
14 Peequisites 0.9. Execises In Execises -, efom te indicated oeations and simlify.. 9x. 8t. 50y 6. t +t + 5. w 6w x c v. z +z c 8. 5 L 9. x +. t +t x z In Execises - 5, find all eal solutions.! ( ). x + (x) q ( x s " 8 t x) + ( t) x. (x + ) ( y) t 7. x t + 9. x + x y + y + 0. t + 6 9t. x x +. w w. x + x 5 5. x ++ x In Execises 6-9, solve eac equation fo te indicated vaiable. Assume all quantities eesent ositive eal numbes.! () 6. Solve fo : I b L 8. Solve fo g: T g. 7. Solve fo a: I 0 5 a 6 9. Solve fo v: L L 0 v c. In Execises 0-5, ationalize te numeato o denominato, and simlify x 7. x x c c. x + + x +. x + x 7 5. x + x
Chapter 4: Matrix Norms
EE448/58 Vesion.0 John Stensby Chate 4: Matix Noms The analysis of matix-based algoithms often equies use of matix noms. These algoithms need a way to quantify the "size" of a matix o the "distance" between
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationConverting knowledge Into Practice
Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationAir Fuel Ratio It is expressed on a mass basis and defined as:
Cemical Reactions Wen analyzing eacting systems, we need to conside te cemical intenal enegy, wic is te enegy associated wit te destuction and omation o cemical bonds between te atoms. Temodynamic analysis
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationDefine What Type of Trader Are you?
Define What Type of Tade Ae you? Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 1 Disclaime and Risk Wanings Tading any financial maket involves isk. The content of this
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationInteger sequences from walks in graphs
otes on umbe Theoy and Discete Mathematics Vol. 9, 3, o. 3, 78 84 Intege seuences fom walks in gahs Enesto Estada, and José A. de la Peña Deatment of Mathematics and Statistics, Univesity of Stathclyde
More informationSymmetric polynomials and partitions Eugene Mukhin
Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation
More informationEquity compensation plans New Income Statement impact on guidance Earnings Per Share Questions and answers
Investos/Analysts Confeence: Accounting Wokshop Agenda Equity compensation plans New Income Statement impact on guidance Eanings Pe Shae Questions and answes IAC03 / a / 1 1 Equity compensation plans The
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationClassical Mechanics (CM):
Classical Mechanics (CM): We ought to have some backgound to aeciate that QM eally does just use CM and makes one slight modification that then changes the natue of the oblem we need to solve but much
More informationCLASS XI CHAPTER 3. Theorem 1 (sine formula) In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC
CLASS XI Anneue I CHAPTER.6. Poofs and Simple Applications of sine and cosine fomulae Let ABC be a tiangle. By angle A we mean te angle between te sides AB and AC wic lies between 0 and 80. Te angles B
More informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More informationLeft- and Right-Brain Preferences Profile
Left- and Right-Bain Pefeences Pofile God gave man a total bain, and He expects us to pesent both sides of ou bains back to Him so that He can use them unde the diection of His Holy Spiit as He so desies
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationConcept and Experiences on using a Wiki-based System for Software-related Seminar Papers
Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,
More informationMath 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
More informationDoes the Acquisition of Mines by Firms in Resource-importing Countries Decrease Resource Prices?
ET Discussion Pae Seies 3-E-073 Does te Acquisition o Mines by Fims in esouce-imoting Counties Decease esouce Pices? HGASHDA Keisaku Kwansei Gakuin Univesity MOTA Tamaki Yamanasi Peectual Univesity MANAG
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More informationCHAPTER 10 Aggregate Demand I
CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income
More informationPersonal Saving Rate (S Households /Y) SAVING AND INVESTMENT. Federal Surplus or Deficit (-) Total Private Saving Rate (S Private /Y) 12/18/2009
1 Pesonal Saving Rate (S Households /Y) 2 SAVING AND INVESTMENT 16.0 14.0 12.0 10.0 80 8.0 6.0 4.0 2.0 0.0-2.0-4.0 1959 1961 1967 1969 1975 1977 1983 1985 1991 1993 1999 2001 2007 2009 Pivate Saving Rate
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationDatabase Management Systems
Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012
More informationGauss Law. Physics 231 Lecture 2-1
Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationThank you for participating in Teach It First!
Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationVoltage ( = Electric Potential )
V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More informationQuantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w
1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationVoltage ( = Electric Potential )
V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationHow to create RAID 1 mirroring with a hard disk that already has data or an operating system on it
AnswesThatWok TM How to set up a RAID1 mio with a dive which aleady has Windows installed How to ceate RAID 1 mioing with a had disk that aleady has data o an opeating system on it Date Company PC / Seve
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More informationTrading Volume and Serial Correlation in Stock Returns in Pakistan. Abstract
Tading Volume and Seial Coelation in Stock Retuns in Pakistan Khalid Mustafa Assistant Pofesso Depatment of Economics, Univesity of Kaachi e-mail: khalidku@yahoo.com and Mohammed Nishat Pofesso and Chaiman,
More informationAMB111F Financial Maths Notes
AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More informationModel Question Paper Mathematics Class XII
Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat
More informationGraphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.
Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such
More informationMATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION
MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationHow Much Should a Firm Borrow. Effect of tax shields. Capital Structure Theory. Capital Structure & Corporate Taxes
How Much Should a Fim Boow Chapte 19 Capital Stuctue & Copoate Taxes Financial Risk - Risk to shaeholdes esulting fom the use of debt. Financial Leveage - Incease in the vaiability of shaeholde etuns that
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationVISCOSITY OF BIO-DIESEL FUELS
VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationWeek 3-4: Permutations and Combinations
Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationInstructions to help you complete your enrollment form for HPHC's Medicare Supplemental Plan
Instuctions to help you complete you enollment fom fo HPHC's Medicae Supplemental Plan Thank you fo applying fo membeship to HPHC s Medicae Supplement plan. Pio to submitting you enollment fom fo pocessing,
More informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
More informationChapter 10: Refrigeration Cycles
Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators
More informationProblem Set # 9 Solutions
Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease
More informationf(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
More informationSaturated and weakly saturated hypergraphs
Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 6-7 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B
More informationInstantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationHow to recover your Exchange 2003/2007 mailboxes and emails if all you have available are your PRIV1.EDB and PRIV1.STM Information Store database
AnswesThatWok TM Recoveing Emails and Mailboxes fom a PRIV1.EDB Exchange 2003 IS database How to ecove you Exchange 2003/2007 mailboxes and emails if all you have available ae you PRIV1.EDB and PRIV1.STM
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationUPS Virginia District Package Car Fleet Optimization
UPS Viginia Distit Pakage Ca Fleet Otimization Tavis Manning, Divaka Mehta, Stehen Sheae, Malloy Soldne, and Bian Togesen Abstat United Pael Sevie (UPS) is onstantly haged with ealigning its akage a fleet
More informationA r. (Can you see that this just gives the formula we had above?)
24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationA comparison result for perturbed radial p-laplacians
A comaison esult fo etubed adial -Lalacians Raul Manásevich and Guido Swees Diectoy Table of Contents Begin Aticle Coyight c 23 Last Revision Date: Ail 1, 23 Table of Contents 1. Intoduction and main esult
More informationOn Efficiently Updating Singular Value Decomposition Based Reduced Order Models
On Efficiently dating Singula alue Decoosition Based Reduced Ode Models Ralf Zieann GAMM oksho Alied and Nueical Linea Algeba with Secial Ehasis on Model Reduction Been Se..-3. he POD-based ROM aoach.
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More information30 H. N. CHIU 1. INTRODUCTION. Recherche opérationnelle/operations Research
RAIRO Rech. Opé. (vol. 33, n 1, 1999, pp. 29-45) A GOOD APPROXIMATION OF THE INVENTORY LEVEL IN A(Q ) PERISHABLE INVENTORY SYSTEM (*) by Huan Neng CHIU ( 1 ) Communicated by Shunji OSAKI Abstact. This
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationTop-Down versus Bottom-Up Approaches in Risk Management
To-Down vesus Bottom-U Aoaches in isk Management PETE GUNDKE 1 Univesity of Osnabück, Chai of Banking and Finance Kathainenstaße 7, 49069 Osnabück, Gemany hone: ++49 (0)541 969 4721 fax: ++49 (0)541 969
More informationMath Test Sections. The College Board: Expanding College Opportunity
Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt
More informationCONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS
CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D.
More informationModule Availability at Regent s School of Drama, Film and Media Autumn 2016 and Spring 2017 *subject to change*
Availability at Regent s School of Dama, Film and Media Autumn 2016 and Sping 2017 *subject to change* 1. Choose you modules caefully You must discuss the module options available with you academic adviso/
More informationDistributed Computing and Big Data: Hadoop and MapReduce
Distibuted Computing and Big Data: Hadoop and Map Bill Keenan, Diecto Tey Heinze, Achitect Thomson Reutes Reseach & Development Agenda R&D Oveview Hadoop and Map Oveview Use Case: Clusteing Legal Documents
More informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationYIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE
YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE Septembe 1999 Quoted Rate Teasuy Bills [Called Banke's Discount Rate] d = [ P 1 - P 1 P 0 ] * 360 [ N ] d = Bankes discount yield P 1 = face
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationThe impact of migration on the provision. of UK public services (SRG.10.039.4) Final Report. December 2011
The impact of migation on the povision of UK public sevices (SRG.10.039.4) Final Repot Decembe 2011 The obustness The obustness of the analysis of the is analysis the esponsibility is the esponsibility
More informationLecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3
Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each
More informationBasic Financial Mathematics
Financial Engineeing and Computations Basic Financial Mathematics Dai, Tian-Shy Outline Time Value of Money Annuities Amotization Yields Bonds Time Value of Money PV + n = FV (1 + FV: futue value = PV
More informationRecall from last time: Events are recorded by local observers with synchronized clocks. Event 1 (firecracker explodes) occurs at x=x =0 and t=t =0
1/27 Day 5: Questions? Time Dilation engt Contraction PH3 Modern Pysics P11 I sometimes ask myself ow it came about tat I was te one to deelop te teory of relatiity. Te reason, I tink, is tat a normal
More information6. Differentiating the exponential and logarithm functions
1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose
More informationHow to create a default user profile in Windows 7
AnswesThatWok TM How to ceate a default use pofile in Windows 7 (Win 7) How to ceate a default use pofile in Windows 7 When to use this document Use this document wheneve you want to ceate a default use
More informationTangent Lines and Rates of Change
Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe- and subsolution method with
More informationDynamic Ethic Risk In Business Process Outsourcing Contract: Prevention Mechanism & Simulation Analysis
Cen Yeua Qin Baoua Zang Guoing and Li Bo Dynamic Etic Risk In Business Pocess Outsoucing Contact: Pevention Mecanism & Simuation Anaysis Dynamic Etic Risk In Business Pocess Outsoucing Contact: Pevention
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently Fields-Hat
More information