Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 1 Review fr cumulative test O Mday, Jauary 7, Hrs Math 3 will have a curse-wide cumulative test cverig Chapters 1-. Yu ca expect the test t ctai -t-6 pe-respse prblems f a similar style ad frmat t thse the chapter tests. This test will cut as a rdiary test i yur Quarter 3 grade. Fllwig this tpic utlie, yu ll fid a set f review prblems fr each chapter, with aswers. Tpic utlie Chapter 1: Fuctis ad plymials Fittig plymial fucti frmulas t tables Lagrage Iterplati (p. 17) Sasha s alterate methd (p. exercise 5) addig a term t make a frmula fit a table at a additial pit (p. 5) Key therems ad their prfs Euclidea Prperty (p. 36) Remaider Therem (p. 37) Factr Therem (p. 1) ad its graphical iterpretati abut x-itercepts Crllaries abut degree plymials (pp. -3) Plymial graphs (frm haduts) Ed behavir based degree ad leadig cefficiet Multiplicity f factrs: just tuchig vs. passig thrugh x-axis Fittig frmulas t graphs, icludig fidig the leadig cefficiet Factrig methds Fid a rt, fid a factr (p. 59) Factrig quadratics (p. 51) icludig -mic quadratics (p. 5) Differece f squares, sum f cubes, differece f cubes (p. 55) Perfect squares (p. 55) ad perfect cubes (p. 60 exercise 1) Factrig by grupig (p. 57) Substitutis fr quadratic-like (p. 6) ad cubic-like (p. 63) plymials th degree with hidde differece-f-squares (p. 6) Factrig ver R (p. 65) Calculati skills Lg divisi (p. 35) ad sythetic divisi (Challege Prblem Set 1) Simplifyig, addig, ad subtractig ratial expressis (p. 69) Chapter : Sequeces ad series Wrkig with tables f umbers clums fr differeces (smetimes repeated:, 3, etc.) Σ clums shwig cumulative sums (smetimes repeated: ΣΣ, ΣΣΣ, etc.) Sequece ccept a sequece as a list f umbers a sequece as a fucti with dmai f egative itegers (r all itegers frm sme startig value) visual examples: figurate umbers
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page Series ccept defiite ad idefiite series writig i + tati r i Σ tati ways t visualize sums: staircases, pyramids Frmulas fr sequeces ad series clsed-frm frmulas fr the kth term f a sequece r fr the sum f a idefiite series recursive frmulas fr the kth term f a sequece r fr the sum f a idefiite series Evaluatig defiite ad idefiite series arithmetic series usig Gauss s Methd r its result gemetric series usig Euclid s Methd r its result Σk, Σk 3, Σk, Σk 5 usig Berulli Frmulas (these frmulas will be give if eeded) use f Σ idetities t break series it simpler pieces that ca be evaluated Limits limits f sequeces limits f series a limits f gemetric series: frmula 1 r visual examples: fractals (pp. 19-151 # 3,, 9); Achilles paradx (p. 151 #8) repeatig decimals usig limits f gemetric series Pascal s Triagle ad Bimial Therem patters, relatiships, ad symmetry i Pascal s Triagle summati prperties: hckey stick (pp. 160-161), Σ/ΣΣ/ΣΣΣ/etc. diagals (p. 163) eve/dd ad remaider patters (p. 16) Bimial Therem: use ad ifrmal justificati Chapter 3: Statistics Statistics fr a data set Fid the mea, mea abslute deviati, variace, ad stadard deviati. Prve ad use this alterate frmula fr variace: x (x). Prve ad use that meas ad variaces are additive. Repeated experimets Give the statistics (mea, variace, ad/r stadard deviati) fr a sigle experimet, fid the statistics fr the experimet repeated times. Fr repeated Berulli trials, fid the prbability that the ttal has a specific value (usig the frmula ivlvig a cmbiati umber). Fr repeated Berulli trials, fid the mea, variace, ad stadard deviati. Assessig the effectiveess f a treatmet Idetify apprpriate radmizati methds fr selectig the treatmet ad ctrl grups. Recgize utcmes that shw strg evidece, pssible evidece, r little r evidece that a treatmet is effective. Sample surveys Idetify apprpriate methds fr radm samplig. Give a sample prprti, use simulati results t assess what values are plausible fr the ppulati parameter. Estimate margis f errr with 95% r 99+% cfidece. Uderstad that crrelati des t imply causati.
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 3 Uderstad the differeces betwee sample surveys, bservatial studies, ad experimets. Prbability distributis Make prbability histgrams. Fr rmal distributis, apply the 68/95/99+% rule ad use the rmalpdf ad rmalcdf fuctis t aswer prbability questis. Apply the Cetral Limit Therem t idetify rmal distributis. Chapter : Trigmetry Uderstad ad apply the uit circle defiitis f the trigmetric fuctis Fid trigmetric fucti values usig the uit circle, usig special triagles (fr 30, 5, 60, ad agles related t these), ad i geeral usig a calculatr Prve ad apply the Pythagrea idetity (si α + cs α = 1) Fid trigmetric fucti values whe give ther values usig quadrat relatiships, the Pythagrea idetity, ad ther idetities Slve trigmetric equatis by had, usig iverses the calculatr, ad graphically the calculatr Graph the fuctis si x, cs x, ad ta x ad idetify the perids f the graphs Prve ad apply the agle sum idetities (frmulas fr si(α + β) ad cs (α + β), see p. 36) Prve ad apply the Law f Sies ad Law f Csies. Slve triagles (fid ukw sides ad agles) whe give SSS, SAS, ASA, r AAS. Slve triagles i the ptetially ambiguus case SSA. 1 Fid areas f triagles usig absi C ad (frm the Challege Prblem Set) usig Her s Frmula. Review prblems Chapter 1: Fuctis ad plymials 1. Fid a fucti frmula that matches the give table. D this usig tw differet methds (Lagrage iterplati 1.0 ad Sasha s agree-t-disagree methd 1.03). Iput, x Output, f(x) 1-6 3 8 5 750 7 856 9 7650. Suppse yu have a table that has six etries. What is the greatest pssible degree plymial that yu wuld eed t match the table? What is the least pssible degree f plymial that yu wuld eed? 3. a. Fid the qutiet ad remaider whe yu divide f (x) = x 3 + x + x 1 by x + 3. b. Fid f ( 3) withut pluggig i.. Suppse P(x) = x 3 x + 3x. Use the Remaider Therem t shw that P(x) P(5) is divisible by x 5.
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 5. Factr each expressi (ver Z except where it says therwise). a. x 19x 5 b. x 6 8 c. x +11x + 36 d. x 3 + x 8x 6, factrig ver R 6. Fid plymial fucti frmulas that culd fit each f these graphs. Remember that yu will eed t determie the value f the leadig cefficiet. a. b. Chapter : Sequeces ad series 7. Evaluate each sum belw usig Gauss methd r Euclid s methd. Shw all steps f each methd yu use! a. 3 + 8 + 13 + + 78 b. + 6 + 18 + + 86 8. Write each sum belw i sigma tati, the evaluate usig sum idetities. a. the sum f the eve itegers frm t 68, iclusive b. the sum f the first 8 terms f the gemetric sequece with iitial term 1 ad cmm rati ½ c. the sum f the first terms f the arithmetic sequece with first term 9 ad furth term 3 d. the sum f the first terms f the gemetric sequece 36, 1,, 9. Use the give table fr f() t aswer these questis. a. Fill i the Σ clum f cumulative sums. b. Write a fucti frmula fr f(). c. Express the series assciated with f usig tati. d. Fid a clsed-frm fucti frmula fr the series assciated with f. Shw hw yu get it. f() Σ 0 3 1 7 3 1 17
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 5 10. Fid a clsed-frm frmula that agrees with this recursively defied fucti: f () = 3 if = 0 f ( 1)+ + 7 5 if > 0 11. The Kch swflake is frmed as fllws. Begi with a equilateral triagle. O the middle third f each f the sides attach a equilateral triagle pitig utward, the remve the rigial middle third. Repeat this prcess frever. Stages 0 thrugh 3 are shw belw. a. Fid the rati betwee the area f a equilateral triagle with side legths 1 s ad the 3 area f a equilateral triagle with side legths s. b. If the area f the iitial triagle is 1 uit, fid the area f the Kch swflake i stage. The fid the area f a true Kch swflake, frmed by ifiitely may iteratis. 1. a. Draw the first eight rws f Pascal s triagle. b. Fid the sum 8 3 7 6 5 3 c. Use the Bimial Therem t expad ad simplify the expressi (x 3y) 5. Chapter 3: Statistics 13. Whe yu subtract a cstat c frm each elemet i a data set, what happes t the mea ad what happes t the stadard deviati? Prve yur claims. 1. Weather data frm a recet swstrm shws that the measured sw accumulati frm may lcatis i Lexigt had apprximately a rmal distributi with a mea f 8.5" ad a stadard deviati f 1.1". a. Yu ask tw f yur frieds wh live i Lexigt abut the sw depths i their yards. They say 7" ad 1". Discuss the likelihd f each f these claims. Iclude cmputatis f z-scres as part f yur aalysis. b. A ews stry the strm said Lexigt was blaketed by betwee 7 ad 10 iches f sw. What percet f lcatis received a amut f sw that was i this iterval? c. What is the prbability that a lcati i Lexigt gt less tha 6" f sw?.
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 6 15. A researcher ivestigates whether sw tires are mre effective tha regular tires at prevetig autmbile accidets. Suppse that a researcher cducts a well-desiged, apprpriately-radmized statistical study f this questi, ad tabulates hw may cars had a accidet ver a perid f three witers. Make tw cpies f this table the cmplete it as directed belw. accidet accidet ttal sw tires 1500 regular tires 1500 ttal 00 800 3000 a. Cmplete the table belw i such a way that there wuld be strg evidece that sw tires helped prevet accidets. b. Cmplete the table belw i such a way that it wuld be difficult t say whether r t the sw tires helped prevet accidets (that is, such that there s pssible evidece but t strg evidece). 16. A sw tubig area has 00 sw tubes that are used each day. Each tube has as 1% chace f eedig repair after each day f sw tubig. a. Calculate the prbability that there will t be ay sw tubes eedig repair after a day. b. Calculate the mea ad stadard deviati f the umber f sw tubes eedig repair each day. c. Calculate the prbability that the umber f sw tubes eedig repair will be 10 r mre. 17. The sw tubig area surveys radmly-selected custmers a particular day abut hw may times they have visited s far this seas. Number f sw tubig visits Hw may custmers 1 133 3 3 17 5 5 >5 0 a. Calculate the mea, variace, ad stadard deviati f umber f visits per custmer. b. Des this data have a rmal distributi? Explai. Chapter : Trigmetry 18. Slve the fllwig equatis ver the give dmai. Fid exact aswers where pssible. If yu use yur calculatr, slve fr degree values t the earest teth. a. cs x + = 3cs x fr geeral sluti(s) where x is i degrees b. si x si x = cs x fr 0 x 360
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 7 c. cs x = 3 cs x fr 0 x 360 d. cs x = 1 fr 180 x 180 e. si θ csθ csθ siθ = fr geeral sluti(s) where x is i degrees 19. Fid the exact value fr 0. Fid the area f quadrilateral ABCD. cs 15 tw ways by applyig tw differet idetities. D A 55 8 B 10 10 C 1. Slve ABC give a = 10, b = 1.93, ad C = 0.5. A ht air ball is see ver Bst simultaeusly by tw bservers at pits A ad B that are 1.75 miles apart. Their agles f elevati are 33 ad 37 respectively. Hw high abve the grud is the ball? Ht Air Ball A 33 1.75 miles 37 B 3. Use tw differet methds (uit circle ad agle sum idetities) t prve that cs 180 + θ = cs ( ) θ. a. Sketch a graph f y = ta x 0 x 360 b. Use yur graph t determie the umber f slutis t the equati 1 c. Use = ta 1 x t fid all values f x fr 0 x 360 1 ta x = t the earest degree.
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 8 Review prblem aswers (first draft; if ay aswers d t lk right, check with teacher) Chapter 1: Fuctis ad plymials f (x) = 1 6 (x 3)(x 5)(x 7)(x 9) 7 8 (x 1)(x 5)(x 7)(x 9) 1. Lagrage: + 375 3 (x 1)(x 3)(x 7)(x 9) 119 (x 1)(x 3)(x 5)(x 9) + 175 6 (x 1)(x 3)(x 5)(x 7) Sasha : f (x) = 6 + 5(x 1) + 7(x 1)(x 3) +18(x 1)(x 3)(x 5) + (x 1)(x 3)(x 5)(x 7) Bth equivalet t f (x) = x + x 3 x 5x.. Greatest degree 5; least degree 0. 3. a. qutiet x x + 7; remaider - b.. P(x) P(5) = x 3 x + 3x 90. Whe substitutig 5 fr x, the result is 0. S, P(x) P(5) is divisible by x 5. 5. a. (x +1)(x 5) b. (x )(x + x + ) c. (x x 6)(x + x 6) d. (x + 3)(x 1 3)(x 1+ 3) 6. Here are pssible aswers. It s OK if yur leadig cefficiets are slightly ff. a. f (x) = 1 (x )3 (x +1) usig (0, -) b. f (x) = 1 x(x ) (x + 3) 3 usig (-, 16) Chapter : Sequeces ad series 7. a. See the MIA p. 91 fr the steps required fr Gauss methd. S = 68. b. See the Example p. 9 fr the steps required fr Euclid s methd. S = 36. 3 3 3 3 3 8. a. ( + k) = + k = 1+ k = (33)+ (3)(33) =1188 7 b. 1 1 k 1 8 7 1 =1 k 1 =1, which simplifies t 3 9 1 3 1 c. Let d be the cmm differece. The 9 + 3d = 3, s d =. 1 (9 k) = 9 + k = 9 1 k = 9 1 1 1 1 d. Series is gemetric with a = 36, r = 1/3. ( 1)() = +11
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 9 9. a. 1 k 36 1 3 b. f () = 5 3 1 1 1 = 36 k 3 3 1 = 36 = 5 1 1 3 1 3 1 f() Σ 0 3 3 1 1 7 6 3 1 18 17 35 c. k= 0 f ( k) r (5k 3) d. (5k 3) = 5k + 3 = 5 k 3 1= 5()( +1) 3( +1) = 5 1 3 10. See the ustackig methd p. 119, ad the Fr Discussi prmpt p. 10. 3+ k + 7 k 5 1= 3+ k=1 k=1 k=1 ()( +1)( +1) + 7 6 ( +1) 5 = 3 3 + 9 7 6 + 3 11. a. The area f the equilateral triagle with sides 1 3 s is 1 the area f the triagle with sides s. 9 b. area = iitial area + area added stage 1 + area added stage + + area added stage area = 1+ 3 1 9 +1 1 9 + 8 1 3 9 +... + area added stage With the excepti f the first 1, the rest is a gemetric series with a = 1/3 ad r = /9. S ttal area at stage : 1+ 1 k 1 3 9 =1+ 1 1 k =1+ 1 1 9 =1 3 3 9 3 9 1 5 1 9. As appraches ifiity, area at stage appraches 1 3 ( 5 0 1 ), which simplifies t 8 5.
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 10 1. a. b. Use Pascal s triagle, where we kw is the etry i rw, etry k. Remember we k start by labelig the first rw = 0, ad the first etry i a rw k = 0. 8 3 7 c. By Bimial Therem, (x 3y) 5 = 5 0 6 5 3 = 56 + 6 + 1 + 15 + 10 = 108. x 5 ( 3y) 0 + 5 x ( 3y) 1 + 5 x 3 ( 3y) + 5 x ( 3y) 3 + 1 3 5 x 1 ( 3y) + 5 x 0 ( 3y) 5 5 Use the etries i rw = 5 f Pascal s triagle t fid values f 5 k, the simplify. Fial expasi: (x 3y) 5 = x 5 15x y + 90x 3 y 70x y 3 + 05x y 3y 5 Chapter 3: Statistics 13. The mea decreases by c. The variace stays the same. Prf (mea): Suppse x 1,, x have a mea f x. The x 1 c,, x c have a mea f ( x1 c) + L+ ( x c) x1 + L+ x c = = x c. Prf (std. dev.): Sice each data value decreases by c ad the mea als decreases by c, all f the deviatis are uchaged, s the squares f the deviatis are uchaged, s the variace ad the stadard deviati are uchaged. 1. a. The 7" amut is betwee 1 ad stadard deviatis belw the mea (z 1.36), s lwer tha average but well withi the iterval [6.3, 10.7] where 95% f the data wuld lie. Hwever, the 1" amut is mre tha 3 stadard deviatis abve the mea (z 3.18) where far less tha 1% f the data wuld fall. S the 7" amut was i the typical rage but the 1" was extrardiary.
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 11 b. rmalcdf(7,10,8.5,1.1) 8.73% c. rmalcdf(0,6,8.5,1.1) 1.15% 15. If there were assciati betwee the variables, the umber f accidets with sw tires 1 wuld have mea 100 ad a stadard deviati f 1500 1 15 15 9.66, r abut 10. a. T shw strg evidece, the umber f accidets with sw tires wuld eed t be sigificatly lwer tha 100. A apprpriate chice wuld be r mre stadard deviatis belw the mea, i.e., belw 80. a. T shw pssible evidece, the umber f accidets with sw tires wuld eed t be a bit lwer tha 100, but t eugh t shw strg evidece. Arud 90 wuld be a reasable chice. 16. a. (0.99) 00 0.018 r 1.8%. b. mea =, std. dev. = 00 0.01 0. 99 1.990. c. rmalcdf(10,00,,1.990) 0.0013 r 0.13%. 17. a. mea = 1.515, variace 0.780, std. dev. 0.883. b. N. The graph is t symmetrical abut the mea. Als, 7 ut f 00 custmers fall abut 3-t- stadard deviatis abve the mea, where i a rmal distributi it wuld be ucmm t have eve 1 custmer that high. Chapter : Trigmetry 18. a. x = 360 b. x = 90, 10, 330 c. x = 90, 150, 10, 70 d. x = ±60, ±10 e. x = 5 + 360, 135 + 360 3 + 19. cs 15 = cs 15 1 = 0. 108.3 square uits 1. c = 5 A =.5 B = 115. height = 0.6 miles cs15 3. Use cs( 180 + θ) = cs180 csθ si180siθ r use the diagram shw at the right.. a. Use yur calculatr t check yur graph. b. tw c. x = 7, 07 = cs 5 30 6 + r ( ) = 180+θ (-csθ,siθ) θ (csθ,siθ)