Analytical Techniques Chromatography

Size: px
Start display at page:

Download "Analytical Techniques Chromatography"

Transcription

1 Anlytil Tehniques hromtogrphy hromtogrphy is n nlytil tehnique tht seprtes omponents in mixture etween moile phse nd sttionry phse. Types of hromtogrphy inlude: thin-lyer hromtogrphy (TL) plte is oted with solid nd solvent moves up the plte olumn hromtogrphy () olumn is pked with solid nd solvent moves down the olumn gs hromtogrphy (G) olumn is pked with solid or with solid oted y liquid, nd gs is pssed through the olumn under pressure t high temperture. The moile phse my e liquid or gs. The sttionry phse my e solid (s in thin-lyer hromtogrphy, TL) or either liquid or solid on solid support (s in gs hromtogrphy, G) In gs-liquid hromtogrphy G the moile phse is inert gs suh s nitrogen, helium, rgon. The Sttionry phse is liquid on n inert solid. Seprtion y olumn hromtogrphy depends on the lne etween soluility in the moving phse nd retention in the sttionry phse. A solid sttionry phse seprtes y dsorption, A liquid sttionry phse seprtes y reltive soluility If the sttionry phse ws polr nd the moving phse ws non- polr e.g. exne. Then non-polr ompounds would pss through the olumn more quikly thn polr ompounds s they would hve greter soluility in the non-polr moving phse. (Think out intermoleulr fores) Retention times nd Rf vlues re used to identify different sustnes. Method: Thin-lyer hromtogrphy ) Wering gloves, drw penil line m ove the ottom of TL plte nd mrk spots for eh smple, eqully sped long line. ) Use pillry tue to dd tiny drop of eh solution to different spot nd llow the plte to ir dry. ) Add solvent to hmer or lrge eker with lid so tht is no more thn m in depth d) Ple the TL plte into the hmer, mking sure tht the level of the solvent is elow the penil line. Reple the lid to get tight sel. e) When the level of the solvent rehes out m from the top of the plte, remove the plte nd mrk the solvent level with penil. Allow the plte to dry in the fume upord. f) Ple the plte under UV lmp or if using mino ids spry with ninhydrin in order to see the spots. Drw round them lightly in penil. g) lulte the Rf vlues of the oserved spots. Wer plsti gloves to prevent ontmintion from the hnds to the plte penil line will not dissolve in the solvent tiny drop too ig drop will use different spots to merge Depth of solvent if the solvent is too deep it will dissolve the smple spots from the plte lid to prevent evportion of toxi solvent Will get more urte results if the solvent is llowed to rise to ner the top of the plte ut the Rf vlue n e lulted if the solvent front does not reh the top of the plte dry in fume upord s the solvent is toxi UV lmp used if the spots re olourless nd not visile R f vlue = distne moved y mino id distne moved y the solvent N Goly hemrevise.org

2 olumn hromtogrphy () Simple olumn hromtogrphy A glss tue is filled with the sttionry phse usully sili or lumin in powder form to inrese the surfe re. A filter or plug is used to retin the solid in the tue. Solvent is dded to over ll the powder. The mixture to e nlysed is dissolved in minimum of solvent nd dded to the olumn. A solvent or mixture of solvents is then run through the olumn. The time for eh omponent in the mixture to reh the end of the olumn is reorded (retention time) solvent glss tue filled with sli omponents PL stnds for high performne liquid hromtogrphy nd it type of olumn hromtogrphy ommonly used in industry. PL: sttionry phse is solid sili PL: moile phse liquid filter Gs-Liquid hromtogrphy Gs-liquid hromtogrphy n e used to seprte mixtures of voltile liquids. The time tken for prtiulr ompound to trvel from the injetion of the smple to where it leves the olumn to the detetor is known s its retention time. This n e used to identify sustne. Flow ontrol In gs-liquid hromtogrphy, the moile phse is gs suh s helium nd the sttionry phse is high oiling point liquid sored onto solid. Smple in oven Some ompounds hve similr retention times so will not e distinguished. disply Bsi gs-liquid hromtogrphy will tell us how mny omponents there re in the mixture y the numer of peks. It will lso tell us the undne of eh sustne. The re under eh pek will e proportionl to the undne of tht omponent. rrier gs olumn detetor Wste outlet It is lso possile for gs-liquid hromtogrphy mhine to e onneted to mss spetrometer, IR or NMR mhine, enling ll the omponents in mixture to e identified. G-MS is used in nlysis, in forensis, environmentl nlysis, irport seurity nd spe proes. Most ommonly mss spetrometer is omined with G to generte mss spetr whih n e nlysed or ompred with spetrl dtse y omputer for positive identifition of eh omponent in the mixture. N Goly hemrevise.org

3 Infrred spetrosopy ertin onds in moleule sor infr-red rdition t hrteristi frequenies using the ovlent onds to virte ABVE 500 m - Funtionl group identifition omplited spetr n e otined thn provide informtion out the types of onds present in moleule BELW 500 m - Fingerprinting omplited nd ontins mny signls piking out funtionl group signls diffiult. This prt of the spetrum is unique for every ompound, nd so n e used s "fingerprint". e.g. = m - - (id) m - Use the IR sorption tle provided in the dt ook to dedue presene or sene of prtiulr onds or funtionl groups Bond Wvenumer A omputer will ompre the IR spetr ginst dtse of known pure ompounds to identify the ompound use spetr to identify prtiulr funtionl groups limited to dt presented in wvenumer form e.g. n lohol from n sorption pek of the ond, = roxyli ids Very rod N Aohols, phenols rod Spetr for utnl = Asorption or trough in etween m - rnge indites presene of = ond Alwys quote the wve numer rnge from the dt sheet - sorptions tend to e rod Asorption or trough in etween m - rnge indites presene of - ond in n id Spetr for ethnoi id = N Goly hemrevise.org

4 Mss spetrometry Mesuring the M r of n orgni moleule If moleule is put through mss spetrometer it will often rek up nd give series of peks used y the frgments. The pek with the lrgest m/z, however, will e due to the omplete moleule nd will e equl to the M r of the moleule. This M pek is lled the moleulr ion Spetr for 4 0 Mss spetrum for utne 4 9 Moleulr ion M+ pek A smll pek will often our in mss spetrum next to the moleulr ion pek. This is due to the presene of the isotope in the moleule. The reltive height of this M+ pek ompred to the M pek n e used to lulte the numer of rons in moleule Use the following eqution to lulte the numer (n) of rons in moleule n = 00 x height of M+ pek. x height of M pek Lern this eqution Exmple A hydroron showed rtio of heights of the M : M+ peks of 9. : 0.5. lulte the numer of ron toms present in one moleule of this hydroron n = 00 x height of M+ pek. x height of M pek n = 00 x 0.5. x 9. =4.89 Round to 5 rons M+ pek If ompound ontins hlorine or romine tom then M+ pek will our due to the two nturlly ourring isotopes of hlorine or romine. hlorine exists s l 5 (75%) nd l 7 (5%) Bromine exists s Br 79 (50%) nd Br 8 (50%) l will hve m/z vlue of M of 50 l 5 nd M+ of 5 l 7 The rtio of heights M:M+ will e : Br will hve m/z vlue of M of 94 Br 79 nd M+ of 96 Br 8 The rtio of heights M:M+ will e : If ompound ontins two hlorine or romine toms then M+ nd M+4 pek will our 4 l will hve m/z vlue of M of 98 4 l 5 l 5, M+ of 00 4 l 5 l 7 nd M+4 of 0 4 l 5 l 7 The rtio of heights M:M+: M+4 will e 9:6: 4 Br will hve m/z vlue of M of 86 4 Br 79 Br 79, M+ of 88 4 Br 79 Br 8 nd M+4 of 90 4 Br 8 Br 8 The rtio of heights M:M+: M+4 will e :: Reltive undnes l 5 l 5 = 0.75 x0.75 = =>9 l 5 l 7 = 0.75 x0.5x= 0.75 =>6 l 7 l 5 l 7 l 7 = 0.5 x0.5 =0.065 => smllest to get whole numer rtio l will hve m/z vlue of M of 4 l 5 l 5 l 5, M+ of 4 4 l 5 l 5 l 7, M+4 of 6 4 l 5 l 7 l 7 nd M+6 of 8 4 l 7 l 7 l 7 The rtio of heights M:M+:M+4:M+6 will e 7:7:9: Reltive undnes l 5 l 5 l 5 = 0.75 x0.75x0.75 = 0.49 =>7 l 5 l 5 l 7 = 0.75 x0.75x0.5x= 0.49 =>7 l 5 l 7 l 7 = 0.75 x0.5x0.5x= =>9 l 7 l 7 l 7 = 0.5 x0.5x0.5 =0.056 => smllest to get whole numer rtio N Goly hemrevise.org 4

5 igh Resolution Mss spetrosopy igh resolution mss spetrometry n e used to determine the moleulr formul of ompound from the urte mss of the moleulr ion For exmple, the following moleulr formuls ll hve rough M r of 60, ut more preise M r n give the moleulr formul. e.g. M r = 60.0 moleulr formul = 4 M r M r = moleulr formul = 8 = moleulr formul = 4 N igh resolution mss spetrosopy n mesure the mss to 5 d.p. This n help differentite etween ompounds tht pper to hve similr Mr (to the nerest whole numer) Aurte msses of toms: =.0078 =.0000 = N = 4.00 Exmple A ompound is found to hve n urte reltive formul mss of It is thought to e either or N N. lulte the M r of eh ompound to 4 deiml ples to work out whih one it is. = (.0000 x ) + ( x ) + (.0078 x6) = N N. = (.0000 x ) + (4.00 x ) + (.0078 x6) = Frgmenttion When orgni moleules re pssed through mss spetrometer, it detets oth the whole moleule nd frgments of the moleule. Moleulr ion formed: M [M] +. + e The moleule loses n eletron nd eomes oth n ion nd free rdil Severl peks in the mss spetrum our due to frgmenttion. The Moleulr ion frgments due to ovlent onds reking: [M] +. X + + Y. Reltively stle ions suh s rotions R + suh s + nd ylium ions [R-=] + re ommon. The more stle the ion, the greter the pek intensity. This proess produes n ion nd free rdil. The ion is responsile for the pek The pek with the highest mss/hrge rtio will e normlly due to the originl moleule tht hsn t frgmented (lled the moleulr ion). As the hrge of the ion is + the mss/ hrge rtio is equl to Mr. Mss spetrum for utne = 58 Eqution for formtion moleulr ion 4 0 [ 4 0 ] +. + e m/z 58 Equtions for formtion of frgment ions from moleulr ions [ 4 0 ] +. [ ] + +. m/z 4 [ 4 0 ] +. [ ] + +. m/z 9 Mss spetrum for utnone The high pek t 4 4 due to stility [ ] + of yl group Eqution for formtion moleulr ion [ ] +. + e m/z 7 Equtions for formtion of frgment ions from moleulr ions [ ] +. [ ] + +. m/z 57 9 [ ] + [ ] + 57 [ ] +. 7 [ ] +. [ ] + +. m/z 4 [ ] +. [ ] + +. m/z 9 N Goly hemrevise.org 5

6 NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the low-energy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht used in mgneti resonne imging (MRI) to otin dignosti informtion out internl strutures in ody snners There re two min types of NMR. NMR. (proton) NMR There is only round % in orgni moleules ut modern NMR mhines re sensitive enough to give full spetr for The spetr is simpler spetrum thn the NMR The rdio wves used in proton nmr use the hydrogen nuleus to hnge its spin stte. Equivlent ron toms. In NMR spetrum, there is one signl (pek) for eh set of equivlent toms. d d 4 peks peks 4 peks, dinitroenzene, dinitroenzene,4 dinitroenzene N l peks peks N 4 peks peks d peks d e N 5 peks N Goly hemrevise.org 6

7 Equivlent ydrogen toms. In n NMR spetrum, there is one signl for eh set of equivlent toms. In ddition the intensity (integrtion vlue) of eh signl is proportionl to the numer of equivlent toms it represents. Ethnol hs groups of different hydrogen toms d d 4sets of equivlent s: rtio 6::: signl d sets of equivlent s: rtio ::9 sets of equivlent s: rtio :: sets of equivlent s: rtio :: Br d 4 sets of equivlent s: rtio ::: Solvents Smples re dissolved in solvents without ny toms, e.g. l 4, Dl. This mens tht in the NMR the solvent will not give ny peks The sme solvent is used in NMR nd in this se there will e one pek due to the solvent tht will pper on the spetrum. owever, it is known where this pek is so it n e ignored. In the exm it is likely this pek will not our on the spetr. lirtion nd shift A smll mount of TMS (tetrmethylsilne) is dded to the smple to lirte the spetrum TMS is used euse: its signl is wy from ll the others it only gives one signl it is non-toxi it is inert it hs low oiling point nd so n e removed from smple esily Si tetrmethylsilne The sme lirtion ompound is used for oth nd NMR The spetr re reorded on sle known s the hemil shift (δ), whih is how muh the field hs shifted wy from the field for TMS.. The δ is mesure in prts per million (ppm) is reltive sle of how fr the frequeny of the proton signl hs shifted wy from tht for TMS d hemil shift (ppm) 0 N Goly hemrevise.org 7

8 NMR shift The δ depends on wht other toms/groups re ner the more eletronegtive groups gives greter shift. δ ppm Proton exhnge using D δ ppm If D is dded to smple then proess of proton exhnge hppens with the in ny - nd N- onds. This hs the effet of removing the peks from the - NMR spetr. This n help with the identifition of nd N peks on the spetr. Addition of D to the smple of Propnoi id would mke the pek t δ =.7 (ppm) in the ove spetrum dispper N Goly hemrevise.org 8

9 NMR shift It will not e possile to identify the ext ron orresponding to eh pek if severl rons re in the sme rnge ppm It is not possile to distinguish etween similr shifts for eh ron in enzene ring. In this exmple it should e possile to work out there re four different rons in the enzene ring nd these orrespond to the four peks etween peks etween 0-60 N Goly hemrevise.org 9

10 Spin-Spin oupling in NMR In high resolution NMR eh signl in the spetrum n e split into further lines due to inequivlent s on neighouring toms. Splitting of pek = numer of inequivlent s on neighouring toms + signl singlet doulet triplet qurtet quintet pperne Split numer of peks numer of neighouring inequivlent toms reltive size : :: ::: :4:6:4: Nulei in identil hemil environments do not show oupling mongst themselves! The pek due to group will e triplet s it is next to ( ron with s) The pek due to group will e qurtet s it is next to ( ron with s) The pek due to group will e singlet s it is next to ron with no s) For 6 split peks use the term hextet or multiplet The pek due to group will e triplet s it is next to ron with s Shift Integrtion tre ppm The pek due to group will e singlet s it is next to ron with 0 s Shift.-.6 Integrtion tre The pek due to group will e qurtet s it is next to ron with s Shift.7-4. Integrtion tre ydrogens onded to Nitrogen or xygen usully do not ouple with other protons nd pper s singlets on the NMR spetr The pek due to group will e triplet s it is next to ron with s Shift Integrtion tre The pek due to group will e qurtet s it is next to ron with s Shift.7-4. Integrtion tre The pek due to group will e singlet s the ydrogen is onded to n oxygen nd this does not split Shift Integrtion tre Ethnol You will not e sked to interpret splitting ptterns for the protons tthed to enzene ring N Goly hemrevise.org 0

11 Bringing it ll together. Work out empiril formul Elementl nlysis 66.6%.8%.9%. Using moleulr ion pek m/z vlue from mss spetrum lulte Moleulr formul moleulr ion pek m/z vlue= /.8/.9/6 =5.555 =.8 = =4 =8 = Mr empiril formul 4 8 = 7 If Mr moleulr formul 44 then ompound is 8 6. Use IR spetr to identify min onds/funtionl group 8 6 ould e n ester, roxyli id or omintion of lohol nd ronyl. Look for IR spetr for = nd - onds There is = ut no - sorptions, so must e n ester. - = 4. Use NMR spetr to give detils of ron hin 4 peks only 4 different environments. singlet of re 9 At δ =0.9 Mens groups 9 Pek t δ 4 shows Are suggests Qurtet mens next to Pek t δ. shows = Are suggests Singlet mens djent to with no hydrogens Pek t δ. shows R- Are mens Triplet mens next to 5 4 δ ppm Put ll together to give finl struture N Goly hemrevise.org

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3.

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3. . Spetrosopy Mss spetrosopy igh resolution mss spetrometry n e used to determine the moleulr formul of ompound from the urte mss of the moleulr ion For exmple, the following moleulr formuls ll hve rough

More information

22.Analytical Techniques Chromatography

22.Analytical Techniques Chromatography .Anlytil Tehniques hromtogrphy hromtogrphy is n nlytil tehnique tht seprtes omponents in mixture etween moile phse nd sttionry phse. Types of hromtogrphy inlude: thin-lyer hromtogrphy (TL) plte is oted

More information

3.15 NMR spectroscopy Different types of NMR There are two main types of NMR 1. C 13 NMR 2. H (proton) NMR

3.15 NMR spectroscopy Different types of NMR There are two main types of NMR 1. C 13 NMR 2. H (proton) NMR .5 NMR spetrosopy Different types of NMR There re two min types of NMR. NMR. (proton) NMR There is only round % in orgni moleules ut modern NMR mhines re sensitive enough to give full spetr for The spetr

More information

1 This question is about mean bond enthalpies and their use in the calculation of enthalpy changes.

1 This question is about mean bond enthalpies and their use in the calculation of enthalpy changes. 1 This question is out men ond enthlpies nd their use in the lultion of enthlpy hnges. Define men ond enthlpy s pplied to hlorine. Explin why the enthlpy of tomistion of hlorine is extly hlf the men ond

More information

Chapter 4rth LIQUIDS AND SOLIDS MCQs

Chapter 4rth LIQUIDS AND SOLIDS MCQs Chpter 4rth LIQUIDS AND SOLIDS MCQs Q.1 Ioni solis re hrterize y () low melting points () goo onutivity in soli stte () high vpour pressure () soluility in polr solvents Q.2 Amorphous solis. () hve shrp

More information

Trigonometry Revision Sheet Q5 of Paper 2

Trigonometry Revision Sheet Q5 of Paper 2 Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd

More information

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R /10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?

More information

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light.

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light. 1 This igrm represents the energy hnge tht ours when eletron in trnsition metl ion is exite y visile light. Give the eqution tht reltes the energy hnge ΔE to the Plnk onstnt, h, n the frequeny, v, of the

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

Generalization of 2-Corner Frequency Source Models Used in SMSIM

Generalization of 2-Corner Frequency Source Models Used in SMSIM Generliztion o 2-Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville

More information

First compression (0-6.3 GPa) First decompression ( GPa) Second compression ( GPa) Second decompression (35.

First compression (0-6.3 GPa) First decompression ( GPa) Second compression ( GPa) Second decompression (35. 0.9 First ompression (0-6.3 GP) First deompression (6.3-2.7 GP) Seond ompression (2.7-35.5 GP) Seond deompression (35.5-0 GP) V/V 0 0.7 0.5 0 5 10 15 20 25 30 35 P (GP) Supplementry Figure 1 Compression

More information

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL: PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles

More information

Logarithms LOGARITHMS.

Logarithms LOGARITHMS. Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +

More information

CALCULATING REACTING QUANTITIES

CALCULATING REACTING QUANTITIES MODULE 2 14 WORKSHEET WORKSHEET For multiple-hoie questions 1 5 irle the letter orresponding to the most orret nswer. 1 The lned eqution for the urning of utnol (C 4 H 9 OH) is given elow: C 4 H 9 OH(l)

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

Spacetime and the Quantum World Questions Fall 2010

Spacetime and the Quantum World Questions Fall 2010 Spetime nd the Quntum World Questions Fll 2010 1. Cliker Questions from Clss: (1) In toss of two die, wht is the proility tht the sum of the outomes is 6? () P (x 1 + x 2 = 6) = 1 36 - out 3% () P (x 1

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

GM1 Consolidation Worksheet

GM1 Consolidation Worksheet Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM Chem 44 - Homework due ondy, pr. 8, 4, P.. . Put this in eq 8.4 terms: E m = m h /m e L for L=d The degenery in the ring system nd the inresed sping per level (4x bigger) mkes the sping between the HOO

More information

Comparing the Pre-image and Image of a Dilation

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

More information

Review Topic 14: Relationships between two numerical variables

Review Topic 14: Relationships between two numerical variables Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph

More information

1.3 SCALARS AND VECTORS

1.3 SCALARS AND VECTORS Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

4-cyanopentanoic acid dithiobenzoate (CPADB) was synthesized as reported by Y.

4-cyanopentanoic acid dithiobenzoate (CPADB) was synthesized as reported by Y. Eletroni upplementry Mteril (EI) for Journl of Mterils Chemistry B This journl is The Royl oiety of Chemistry 2012 ynthesis of 4-ynopentnoi id dithioenzote (CPADB). 4-ynopentnoi id dithioenzote (CPADB)

More information

CHENG Chun Chor Litwin The Hong Kong Institute of Education

CHENG Chun Chor Litwin The Hong Kong Institute of Education PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using

More information

8 THREE PHASE A.C. CIRCUITS

8 THREE PHASE A.C. CIRCUITS 8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),

More information

Appendix C Partial discharges. 1. Relationship Between Measured and Actual Discharge Quantities

Appendix C Partial discharges. 1. Relationship Between Measured and Actual Discharge Quantities Appendi Prtil dishrges. Reltionship Between Mesured nd Atul Dishrge Quntities A dishrging smple my e simply represented y the euilent iruit in Figure. The pplied lternting oltge V is inresed until the

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

CEM143 MWF 8:00 8:50 am. October 5, 2018

CEM143 MWF 8:00 8:50 am. October 5, 2018 CEM43, Fll 208 st Miterm CEM43 MWF 8:00 8:50 m st Miterm toer 5, 208 Nme: Setion: PID: TA: This is lose ook n note exmintion. This exm hs 35 questions. Answer ll questions on the seprte nswer sheet (ule

More information

CEM143 MWF 8:00 8:50 am. October 5, 2018

CEM143 MWF 8:00 8:50 am. October 5, 2018 CEM43, Fll 208 st Miterm CEM43 MWF 8:00 8:50 m st Miterm toer 5, 208 Nme: Setion: PID: TA: This is lose ook n note exmintion. This exm hs 35 questions. Answer ll questions on the seprte nswer sheet (ule

More information

Non Right Angled Triangles

Non Right Angled Triangles Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

More information

Nondeterministic Automata vs Deterministic Automata

Nondeterministic Automata vs Deterministic Automata Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n

More information

NON-DETERMINISTIC FSA

NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

More information

Lecture Notes No. 10

Lecture Notes No. 10 2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite

More information

Instructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting.

Instructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting. ID: B CSE 2021 Computer Orgniztion Midterm Test (Fll 2009) Instrutions This is losed ook, 80 minutes exm. The MIPS referene sheet my e used s n id for this test. An 8.5 x 11 Chet Sheet my lso e used s

More information

21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae

21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Contents STRAND G: Alger Unit 1 Formule Student Tet Contents Setion 1.1 Using Formule 1. Construt nd Use Simple Formule 1.3 Revision of Negtive Numers 1.4

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their

More information

Pythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245.

Pythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245. Pythgors Theorem Pythgors Theorem Curriulum Redy ACMMG:, 45 www.mthletis.om Fill in these spes with ny other interesting fts you n find out Pythgors. In the world of Mthemtis, Pythgors is legend. He lived

More information

Discrete Structures Lecture 11

Discrete Structures Lecture 11 Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

Iowa Training Systems Trial Snus Hill Winery Madrid, IA

Iowa Training Systems Trial Snus Hill Winery Madrid, IA Iow Trining Systems Tril Snus Hill Winery Mdrid, IA Din R. Cohrn nd Gil R. Nonneke Deprtment of Hortiulture, Iow Stte University Bkground nd Rtionle: Over the lst severl yers, five sttes hve een evluting

More information

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points. Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You

More information

Factorising FACTORISING.

Factorising FACTORISING. Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will

More information

THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL

THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL P3.1 Kot Iwmur*, Hiroto Kitgw Jpn Meteorologil Ageny 1. INTRODUCTION Jpn Meteorologil Ageny

More information

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

CS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014

CS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014 S 224 DIGITAL LOGI & STATE MAHINE DESIGN SPRING 214 DUE : Mrh 27, 214 HOMEWORK III READ : Relte portions of hpters VII n VIII ASSIGNMENT : There re three questions. Solve ll homework n exm prolems s shown

More information

Algorithms & Data Structures Homework 8 HS 18 Exercise Class (Room & TA): Submitted by: Peer Feedback by: Points:

Algorithms & Data Structures Homework 8 HS 18 Exercise Class (Room & TA): Submitted by: Peer Feedback by: Points: Eidgenössishe Tehnishe Hohshule Zürih Eole polytehnique fédérle de Zurih Politenio federle di Zurigo Federl Institute of Tehnology t Zurih Deprtement of Computer Siene. Novemer 0 Mrkus Püshel, Dvid Steurer

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

Mathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers

Mathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers Mthemtis SKE: STRAND F UNIT F1 Formule: Tet STRAND F: Alger F1 Formule Tet Contents Setion F1.1 Using Formule F1. Construt nd Use Simple Formule F1.3 Revision of Negtive Numers F1.4 Sustitution into Formule

More information

Chapter 8 Roots and Radicals

Chapter 8 Roots and Radicals Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi

More information

Scientific notation is a way of expressing really big numbers or really small numbers.

Scientific notation is a way of expressing really big numbers or really small numbers. Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific

More information

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of: 22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)

More information

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6 CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

Finite State Automata and Determinisation

Finite State Automata and Determinisation Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions

More information

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}

More information

12.4 Similarity in Right Triangles

12.4 Similarity in Right Triangles Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right

More information

Chemistry Practice Exam

Chemistry Practice Exam Chemistry Prtie Exm 1 2 3 Whih of the following is n element? A rop of merury. A splinter of woo. A rystl of sugr. A rop of wter. Whih of the following nswers ontins only elements? Soium, soium hlorie,

More information

Lecture 6: Coding theory

Lecture 6: Coding theory Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists. AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

System Validation (IN4387) November 2, 2012, 14:00-17:00

System Validation (IN4387) November 2, 2012, 14:00-17:00 System Vlidtion (IN4387) Novemer 2, 2012, 14:00-17:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

Table of Content. c 1 / 5

Table of Content. c 1 / 5 Tehnil Informtion - t nd t Temperture for Controlger 03-2018 en Tble of Content Introdution....................................................................... 2 Definitions for t nd t..............................................................

More information

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t

More information

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y Nernst-Plnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly

More information

Part 4. Integration (with Proofs)

Part 4. Integration (with Proofs) Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1

More information

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

Supporting Information

Supporting Information tom-thik Interlyer Mde of VD-Grown Grphene Film on Seprtor for dvned thium-sulfur tteries Zhenzhen Du 1, hengkun Guo 2, njun Wng 3, jun Hu 1, Song Jin 1, Timing Zhng 1, Honghng Jin 1, Zhiki Qi 1, Sen Xin

More information

CS 573 Automata Theory and Formal Languages

CS 573 Automata Theory and Formal Languages Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple

More information

Exam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B

Exam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B PHY 249, Fll 216 Exm 1 Solutions nswer 1 is correct for ll problems. 1. Two uniformly chrged spheres, nd B, re plced t lrge distnce from ech other, with their centers on the x xis. The chrge on sphere

More information

18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106

18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106 8. Problem Set Due Wenesy, Ot., t : p.m. in - Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the one-element, linerly epenent sets forme from these. (b) Wht re the two-element, linerly

More information

Physics 505 Homework No. 11 Solutions S11-1

Physics 505 Homework No. 11 Solutions S11-1 Physis 55 Homework No 11 s S11-1 1 This problem is from the My, 24 Prelims Hydrogen moleule Consider the neutrl hydrogen moleule, H 2 Write down the Hmiltonin keeping only the kineti energy terms nd the

More information

Eigenvectors and Eigenvalues

Eigenvectors and Eigenvalues MTB 050 1 ORIGIN 1 Eigenvets n Eigenvlues This wksheet esries the lger use to lulte "prinipl" "hrteristi" iretions lle Eigenvets n the "prinipl" "hrteristi" vlues lle Eigenvlues ssoite with these iretions.

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret

More information

Part I: Study the theorem statement.

Part I: Study the theorem statement. Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

Reading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5.

Reading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5. PHY1 Electricity Topic 5 (Lectures 7 & 8) pcitors nd Dielectrics In this topic, we will cover: 1) pcitors nd pcitnce ) omintions of pcitors Series nd Prllel 3) The energy stored in cpcitor 4) Dielectrics

More information

3-Way Mixing and Sequencing Globe Valves, Flared (5/8 in. O.D.) with Electric, Hydraulic, and Pneumatic Actuators

3-Way Mixing and Sequencing Globe Valves, Flared (5/8 in. O.D.) with Electric, Hydraulic, and Pneumatic Actuators lectric, Hydrulic, nd Pneumtic ctutors TL 1. Select Vlve ody including P ode (Vlve Size, v Rting, Port ode) or select Vlve ssemly correct (refer to Tle 3 nd Tle 3 lso) less ctutor ode (XXX) including the

More information

CHAPTER 20: Second Law of Thermodynamics

CHAPTER 20: Second Law of Thermodynamics CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het

More information

1 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum?

1 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum? Which of the following summrises the chnge in wve chrcteristics on going from infr-red to ultrviolet in the electromgnetic spectrum? frequency speed (in vcuum) decreses decreses decreses remins constnt

More information

(h+ ) = 0, (3.1) s = s 0, (3.2)

(h+ ) = 0, (3.1) s = s 0, (3.2) Chpter 3 Nozzle Flow Qusistedy idel gs flow in pipes For the lrge vlues of the Reynolds number typilly found in nozzles, the flow is idel. For stedy opertion with negligible body fores the energy nd momentum

More information

= state, a = reading and q j

= state, a = reading and q j 4 Finite Automt CHAPTER 2 Finite Automt (FA) (i) Derterministi Finite Automt (DFA) A DFA, M Q, q,, F, Where, Q = set of sttes (finite) q Q = the strt/initil stte = input lphet (finite) (use only those

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

HS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem

HS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem HS Pre-Alger Notes Unit 9: Roots, Rel Numers nd The Pythgoren Theorem Roots nd Cue Roots Syllus Ojetive 5.4: The student will find or pproximte squre roots of numers to 4. CCSS 8.EE.-: Evlute squre roots

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

MAT 403 NOTES 4. f + f =

MAT 403 NOTES 4. f + f = MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn

More information