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# Exam 3 Material MAC1105

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This page Study Guide was uploaded by Michaela Maynard on Saturday November 14, 2015. The Study Guide belongs to MAC1105 at Florida State University taught by Pennington LeNoir in Summer 2015. Since its upload, it has received 350 views. For similar materials see College Algebra in Calculus and Pre Calculus at Florida State University.

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Date Created: 11/14/15

Exam 5 Matemal 1033 Is Obj 17 Math Models or Constructing Functions Obj 17a Formula problems The surface area S of a right circular cylinder of height h and radius 7 is S 27rr2 27rrh If the height is g of the radius express the surface area S as a function of r or 4no in W 33 lt3arrh wees vacuova b n r 51h m 1 3ar 5 8 ra211r rgt 3 x 21Wquot 3rrrr WSW MuH ehoxce The surface area S of a right circular cylinder of height h and radius r is S 27TT2 27r39rh If the radius is g of the height express the surface area S as a function of h h D V W Sta farQnrh ched r6obon3hp bn rib r 7533 3 nh 31M 7 395 WW7 yamwt CW The volume V of a right circular cone with base radius r and height h is V l7rr2h If the radius 3 Wilt express the volume V as a function of h 4 3k t u Q V 3Tl f WOY V03 VC o h vlg r rsbh V nm h r V 393n Ebhaah R0 Obj 17 b Revenue problems Revenue 2 Price pfr Itch Cquahhda 1 The price p and quantity a sold of a certain product obey the price demand equation p x2050 for 0 S x S 4100 Express the revenue R as a function of 3 Enter only the right hand side of the expression Use correct varia e se proper or improper actions no mixed fractions or exact decimals Do not give decimal approximations C lwc to 496 divas113132 price 905th wNo c paresoolmeh Kw mcea muons bh n P 3 quot39 quot4W 8050 4 quot39 quotin 39f 5105014 1 The price p and quantity 3 sold of a certain product obey the pricedemand equation p 2c2050 for 0 g m lt 4100 Express the revenue R as a function of p Same online problem cautions as 0 Wm given above R5915 xWavvc Rev 03 tune a 3 Non foo r 0l39or5hvp 10 P516 at d EU 80 R PQ i iei Pquot quot 2 5D N For 4 a K HOO 3997P HOO 39 16 WIH taKC OHMfr typo3p 56 fl th 9 f W0 7 The price p and quantity 5 sold of a certain product obey the pricedemand equation 1 p7700 for 0 S p S 3300 Express the revenue R as a function of 3 Same online pro lem cautions as W given above MWOWB R as Pam o 1600 P K 9100 a 5 KPw 16 73 T7700 31 7P 23oo 3523Aoo 3 7P R x3937 3 3300 P quot774 3300 I K 7 330m 71 W m R Obj 170 Area of a Rectangle problems A farmer has 6500 feet of fencing available to enclose a rectangular area Let 1 represent the length if the side of the rectangle as define igfthe gure shown Express the area A of the rectangle a3 Einction of Enter only the righ an 81 e o t e expression Use correct variable Give fractions or exact decimals Do not give decimal approximations wam H as a fume of wCNO x Nfed re IahonShIP bn L 939 39 L VBwlA S CYUCWTf uslng a the mat rlal H Wj 3 92 20500 333 Vt For H 5035250 g APEC iKHDE WSDb L athfr wa A farmer has 8000 feet of fencing available to enclose a rectangular area and divide it into two plots One side will extend beyond a barn that is 100 feet long Fencing will not be needed along the barn Let 13 represent the length of the side of the rectangle as de ned in the gure shown Express the area A of the rectangle as a function of 13 Same online problem cautions as given above Wam H as Fume oi 16No 3 J ona k3 5L1Ild Shaunre uSm 2 material x 3L903 00gt 3000 3 23 8100 50 l f 64 33839I00 3x 454050 4 34 A farmer has 4800 feet of fencing available to enclose a rectangular area and divide it into two plots The rectangular area borders on a river and the side along the river does not require fencing Let 2 represent the length of the side of the rectangle as de ned in the gure shown Express the area A of the rectangle as a function of 11 Noam R 03 Puma of 160w a L Bwld MSm 0 W at en x I 3413 HB OO SOHe 90quot 35 3 jclIB OO39BJc luj Pr 16 01800quot 3 A H H3190 39 363 Obj 17d Distance problems w d my 7 3 Let P 2 be a oint on the a h of 21 3 Ex ress the distance d from P to the oint y p gr p 55 p p 1 4 as a function of an L4 7 W A swam dSt oA w quot 39OLH39 VOYlable I p a vkuuam d 03 645 0 3 4Y 2 n gt 5443 W C 0 55050 swam rs 4mm 254 1 lquot 16 2 NILHt Choice Let Pcy be a point on the graph of y 3x2 2 Express the distance d from P to the point 5 1 as a function of cc C gtP u 2n ant dist bh Kit 339 5 0 d x 5 rjq 7 6 5 4 3L 4 aquot WXM S 3x344I L2 D w x rwwmm Put hHe tfm toct N let lqb 7 11016 gb Mu Chowe Let Pcy be a point on the graph of y 5 22 Express the distance d from P to the point 1 3 asgfunction of 9 d quot39 We 39qu 63x33 79 JC 39quot 93 xC39x3 5 b 33 2 7 j 4 3 01 3 2 Q Rx 39 Vkawdr 4 38 139an Wvg 9 MMK Cmome m 5 5quot 30 pg Obj The Optimal Time problems A car rental agency rents 250 cars per day at a rate of 20 per day For each 1 increase in rate 5 fewer cars are rented Let x represent the number of 1 increases in rate I Express the income from car rentals 239 as a func ion 0 2 n e right a e expression Use correct pvaria 1e e p fractions no mixed fractions or exact decrma s Do no give incomecafs mb gt N wi FORDDY W one 4391 mt z39 450 5gtampOD 350 5J6gtRO gt Two quot39 C F39 1amp50 5395 3014 lime 4 mu 25lgt3933955 JorHHvl 9w 65 CCCPt Y PaLtDY K inc 1 239 11 PDVM or Malt out The owner of an orchard estimates that if 40 fruit trees are planted per acre then each mature tree will yield 60 pounds of fruit per season If 2 additional trees are planted per acre the average yield per tree is reduced by 3 pounds Let 1 represent the number of 2 tree increas planted per acre Express the yield per acre 3 as a function of 51 Same online p blem cautions as given above j Y665gtdgeld yer fe 39 Now 3 7 6390 90gtL3 J an 39 one Jtrce me 3 2 L10 aoo 3 M TWO 3 MC3 10220039 qOQQQaO c 1mm R ce MW 90222ao 33 X R39bvtc 1m oa 3 gt An orchard manager estimates that if the fruit is picked now it will bring 12 per pound with each tree yielding an average of 100 pounds per tree The average yield per tree increases 3 pounds each week he waits but the price drops 1 per pound per week Let 2 represent the number of weeks the manager waits before picking the fruit Express the income from the fruit trees 239 as a function of 51 Same online problem cautions as given above mcovvxc 5 WWC PU va d quot Pound per 66gt M 239 shower mew z39 03quot 06 wait a Wffy st 7 lR l D 39OOTB 0539 Walt M upx oo r3 Wall 0 46st5 Z 673 H30 T3Kgt RR Objective 18 Quadratic Functions 39 os m The simplest quadratic function is f x2 D vac00gt 0 0C 35quot 3 q a a c 43 e quot 39 6 w W 3mm wr j as O o 6 Dec 00 0 Inc 000 Vambom Veal0quot a Objective 18b Quadratic Functions in h k form Applying all of Obj 14 re ections and translations to the function 52le 12 T2055 bl G vertex h r If k Po tbl C S wl k a gt 0 parabola opens g 12 0551o M94 4lequot d rfhccmon quotlb113i a lt 0 parabola opens Own across 6 39am5 Objective 18a Quadratic Functions in Standard form fcacc2bmc A C What s the vertex We could CQm I hm Q 3 wand put it in 1219 form Goodnews OOVVb have no He39s been done For M fcax2brc vertex 390 a gt O parabola opens A B a lt 0 parabola opens My For either quadratic form To nd x intercepts let 9 O solve for c To nd yintercepts let 6 0 solve for 3 Sometimes we ask How many xintercepts are there Standard P39QYM 93 Rat gt0 7 0754 quotWquot or Obj 18a You can use the dlBCfmV6YH b2 0 a aceInc vaLlac 4 0 399 O L lnt For Obj 18b Just Qr Ph b f 0 I43 anm R7 Objective 180 Max Min of Quadratic Function 1 y YalUt A Mow MW 3 Z x4 x oak D RV M m 20 Ob 18a example The information included in this example would be asked in separate on line problems r M a 4 f93 12a92 42 1 mum Vf fx 1 Opens U 96 I M AD LL v eelil A Ircoordinate of vertex 2 756 EEC 9 23924 g o I 3 a 3 How many Isintercepts F c gt quot l 70 41 4 bg39 ac JDZ 40mm 5 WWW gt0 q39gtMiniS pl egt 307qvMgt 2V3gt a 31 3 bMin is atzvz 2 7 W 1 0 ags can we a hem vNot an at once 0 Find intercepts For on line problems Enter them in any order separated by a 2 3 3435 u m ght Hug bgogypaD y SCWHCOth C39W t 4 9O 7 397 ax 9 9 4 a 3K7quot9Kl Drum we 1 o CDZ39OG zK rb 0 0 39 Uquot92Ca 2 z loxquot2 0 2 40 Which of the tol wmg most closely resembles the graph of f x 12m2 411 c y y y n 39 x x x I x an 496an3 down m quot2 2 q s q a 430 W 103415 Ob 18b example The information included in this example would be asked in separate on line problems I H 2 fcax12 8agt0 gt Form 8 L l OpensDown Qgtvertex 391 398 BDHOW many xintercepts Gmh the 3 N l k H Q E l re a Lb mt 00K 56 down 5 HgtMaxis quotquot 39Zalg 5gtMaxis at a 39l 39 V6 4 9 Find all intercepts for f 13 2 222 12 8 For on line problems Enter them in any order separated by a comma wtlet xeo 57 gampo33 752gtX 510 xvlvrt t 370 gt CamelDRE Remn Ob Lug DP NOT mat out RQHYR lI KR f o xli2mz va 1 9 3 1 C 5 k s 3 Find all intercepts for f x 2251 12 10 For on line problems Enter them in any order separated by a comma Manx e39g ago 537 90 3aoa o 5 lo 393 xmt1 eh 7 o a lgta quot39IO 21 4 02 5 2 ZRn250 T QL if 0 quotI x l K quot I 0 Ob 180 example Studies have found that the relationship between advertising dollars a in thou sands and revenue R can be modeled by a quadratic function 11 If Ra 4a2 364a 25695 ow man thousands of advertisin dollars should be spent in order to maximize revenue h Oidfrcd Vans Q Kgt ad a want quotaquot value cl ucreuc Enter number answer integers or exact decimals mathematical operators are not allowed For example 152 must be entered as 75 Don t type any dollar signs commas or units The function given does not represent the results of an actual study 110 50 305 0156 Calculator Ra 74 8 a is 2 39r Arfvf adv W e 8 Ob 18c example A large swimming pool is treated regularly to control the growth of harmful bacteria If the concentration of bacteria 039 per cubic centimeter t days after treatment is given by C39 t 04t2 44t 301 What is the concentration of bacteria Same cautions as in previous example ve zc 6 P wan t C Ct o 4tquot HH r3cgt t c 56 0 60163 Cancenwa OW e c Hu 11 or boobEVIa Ra o X KR8m nb6r fo hfx t SS agtgt also C Ll55339 30 Wmda days Ci F tcxr vatm en b w M 214139 Covwhncratxon o5 baL thO be at 0 mm 397 5 da S Q Objective 19 Power Functions fa xquot where n is an integer n 2 2 The power functions are classi ed into 2 groups f x x where n is an even positive integer n 2 2 For example fr x2 34 fx x6 fr 958 eVCn WSSWHAA Duff 390 Dec 045 0 Ac 000 f as x where n is an odd positive integer n 2 3 For example f9 x3 x5 fa x7 fa 359 L 3 IL Jhgmer Power 3 8 Clef r to Vtgt53 bans near 00 3 59 I bl StefPer a quot 3 D1 00OOgt R6ijgt I WYJL oyj h ViC on P0a ltgtgt 71 buawn out For 0 A 059 500 0b 5 00 Objective 20 Solving Polynomial and Rational Ineggglities that mam that MP Introduction to Obj 20 Ifab 0 then it must be true that 0 0 or b 5 D However if a b gt 0 then it does whave to be true that a 7O or l0 7 Q In fact both number could be negative lt 0 It would require 2 cases to express this solution Case1QquotO and 1070 OrCase2 040 040 bin at are gt 0 bee are me I we consider c gt 0 thffeae numbers multiplied Witlf igbfesi being positi then it would require 4 cases to describe this solution and in only one case is it true that a gt 0 and b gt O and c gt O and We need a process that doesn t require determining and then analyzing cases The process is based on the Factor Theorem and the Intermediate Value Theorem Plan of Attack 1 Factor if needed Watch for Diff of Squares and Factoring y 39 O In the case a 2 Findl ggl 39tionin Points These are values that m che expression quot of a rationa inequa 39ty these are values that make the expression 7 0 from factors in the numerator or make the expression AtId Q n dlfrom factors in the denominator Set each factor equal to O to nd these Partitioning Points 3 Mark these on a number line must be in numberline order 4 Make a SignChart Select a value in each interval that s created by these partitioning points don t use one of endpoints Plug this value into each separate factor and record whether the result is or Consolidate the signs from all the factors The Factor Theorem guarantees that there are no other Partitioning Points beyond those found from the factorization The Intermediate Value Theorem guarantees that there can be no sign change within the intervals created by the Partitioning Points 32K 0 3 2 0 9 4332 SOlve39 1132 5305 Mane 64 076 3 393 Tag 39 3 3 32 40 x 2 L lt3 TESL 3 2K39Z 4 O W 0 C 395 0 9 moat use ioValufS Pd Vougrnowna mequahtv 35 Solve 332142x12lt0 3 55 05 aim62 lactor Pm red lla Rm 3 474 L 33 A O a L go 21x 2LO 6mm 390 X Vquot 2 kquot k r quot no 70ft 39 u 3391 1 1 439 g 3535 quotquot 139 v 39 39 J M aloe 39 39 a 39 9 W ft Cb lftw70 a 1545quot 10 a lL J to Solve 512 362a 14 x 40 Mfr co 39 r r my xx 39 x x eruDa o 9 I C 2 9 O 4 9 K 3w Ll K O 2 p lo I 6 Kai jlp X ct K 0 I L gt I rt 00 V0 F oOgtUL 1 x3x1 x 301 2c SOIVG MOKC Vt MM Solve 0 340 5 I 0 K5 K r 72 A c I Solve x5gt3w4 V X 3Lq gt0 3ampch o nomlal I 7O 4 Jr quot l f d x 33 c9 5 ED x o K73 Ice f lfb 3 Wk 16quot0 163 LVS 52 xncgt kCan t u5 an x 3P5 Par ohonivua PDlh cS town gt0 Solvex523x4 A 653 6920 L9 z 3 10 Q 3 S E Parhhonm POIIA lT5 WWW O l 25 O 3 A 5 Ufaco W Objective 200 How many partitioning points would be needed to solve Plan of Attack Obtain a single fraction on one side with 0 on the other That isz make the problem read to be solved b the si chart method P wt won t souf 5 NOTE You can t C 3905 M u l 39t or multiply by any expression that contains the variable because you won t know if the inequality sign should be reversed NOTE If the denominators are constants then you are allowed to crossmultiply multiply by LCM 3 2 a 4 6x1s2x 3 x2gtx 2 39 52 31 JC i gt O 046 W Wx 3wb Xv 02 xx NaraMm 9 0 Va 70 com ax33 If a blbOle39L grm i a O 7 Par 6 S anall 31933 39 f 3 part quot0l t a 40 Ian H 216 3 7A 35 W Lo 12 o Su 240 3 4 1 20 LC 2 7 gt625 2 3X1 LO one 1762 a l 7 1701M Objective 20d Find the domain when a signchart is needed Recall Obj 10c Give the domain for each fx 2xzo mas3 43 fltxgtm 3 21 20 RKZP3 Dquot0 o Ac 3 Dquot amp 3 Objective 20d Give the domain fx xa3c2 10x3 2 O x 3 3 on330 quot 1 1 quot 1 39 39 9 i Ice 0 Wk lee 5 69 on 165 I x33410X3gtZO mi 5 l X I 0700 5 0 XCan indude so 1 2 30 H55quot 90 EYOhly values 196 025939 5X 3quot 64W b0 0 6 392 5145 my W D I 9 75 U 02 39sz 7R rr f lquot 4gt f V35x 2c2 o 1399 quot 9 Mic MEL 0 3 20 re 39 35X392L2 icy 2 I 70 Numcrator 5 H r39 3 Kl u Domalm 39739 15gt 37450 Apr o kph cannot 0 So can t K 3 Lquota use ParUmOmmzd Pornt f3m fa535x 2a2 kocld mot L all 1 accept D 45 gt0 S xl7x WWW WK den40 D3 in Kf i zquot 15 33 I ND yam CWN G mama These are special cases you may see in your Practice or Hmwk Quiz problems but won t encounter in the Lab Quiz or Test problems We need to be aware that an evenroot radical can have Domain all Reals or even an empty domain fc a225 fz z2 25 at an pas aman 3 neg Doooampgt DUZ Copyright 2010present Annette Blackwelder all rights are reserved Reproduction or distribu tion of these notes is strictly forbidden 7R Objective 21b Example Select the graph of y f 1x 439 390 79 graph IVS over 6 25 fOY unversc I y K y R x39 Squot Lb quot2 L A f 39 unctlon can be 1138 own Inverse ConSIder 9 g i f a 72quot v 4 9539 J p Kl g ITfYQ 13 no LVV V5 Objective 21a Does every function have an inverse NDamast be one tv One 03 have an VEYSC y y k Fa v erb hw ten a g x 4 Noe a 39 RAWme Graph of a function must pass the hOYIZ h e 33k to be be the graph of a one to one function F x 395 to be a Ll f 5 3 3a UHC GOY WM 55 t0 96 one Cb ont 7R Objective 21b Example Select the graph of y f 391x 1 lot 6 29 IVS ovt r zz fair 1mver5c I quot 8 Lb quot x L A f unctlon can be 1138 own Inverse Con31der 9 5 C P z 32quot v 47 75 J v kl K WCYQ l3 no LAVEVSG Objective 21a Does every function have an inverse NDamusb be one w39om a have an mversc y y 39MB verb lune 9349 t 5 x quotquotNO 3 0 L RAWbow Graph of a function must pass the hOYIZ me ctS39b to be be the graph of a one to one function 39 22wa as to be a 6 7H3 5 U 729quot uMUOW mm 55 t0 3906 on b cnt Which are onetoone functions lt12gtlt13gtlt54gt 12gtlt32gtg5 u 5 0 9me at a not 39 e to DV quot lmquot one If a function is not oneto one restrict the domain in order to de ne an inverse function Recall intro to Obj 21 AL y gtltv Objective 21d Given a function nd the function rule for f 1 Plan of AttaclEE Write y for f to simplify the notation Solve for m For applied mathematicians when units are usually associated with the variables ou have the inverse function or our College Algebra course we will interchange a and 1 to write the inverse as a function of x Write f 132 for y to return to function notation Find the function rule for f 1 for each of the following fa3x5 at the quotGYM fx4 a57 w u 5 D 5 3x 5 j c 51 3 l 7 576 313 2x chaqu g i k j 3 5 67 I an 44 J34 via ig afgzm qr5x391 5 5 31 6quot 3 a 705 406 1 51 fltxgteampT 27 fxx1 133 D5 316 7 D g 345qu 7 f3 gay xa a 9397 1 w 93L13993f 2 5 4 C 391 c5 l 466 Kr9 Aquot 3L 4 163 3933 194quot at 2550 3i 7 quot39 l 5 4y553quot J6 j 310 35 x2a 3 DV39OQ quot L l yz 53 2 393 a wtj AKquot 3 x 6 9225 3 6 a 3 quot 7 2 O Jective 22 Exponential Functions f so a a gt 0 a 7E 1 K l 3 30615 l a was by xl 3nj J39b3 39X 73933 Does this de ne a function 4 0V1 M g Ya LS Cd to a PoW fY lVfS J on c a V WC Don t allow base to be negative because could be Yi ed for some 3 ie 539 539 1 X Don t allow base to be 1 because 4 c l quot graph would be linear not exponential What s the domain All reels If so we have to de ne what s meant by irrational exponents For example 4 5 or 4 r We haven t worked with irrational exponents Good News The limiting processes of calculus guarantee that irrational exponents are de ned and C line u as we want That means since 3 lt 7r lt 4 then a3 lt a r lt a4 That means since 1 lt lt 2 then a1 lt a 5 lt a2 52 The exponential functions are classi ed into 2 groups depending on the base x 1 x0lt lt1 fc aagt fc a a We will consider two speci c cases to develop the concept This is not an eGrade example you will not be making tables of values you will not be plotting points Consider 1192 iii Consider f x for an example of a gt 1 for an example of O lt a lt 1 y a y a y a l 3 0 400 000 k l 4 as w 1 s 39 q JD 3 10 f 100 V 6 394 W 100 1i q 0 9 1 IO I g 3 2 q 10 39 3 2 quot7 0 a bl l l 3 l 4 J 39 3 ll 3 4 3 q H 0 v 3 a 1 IO 39IO 10 q 9 2 1q up 10 4 Ll MOHAde q I 100 quotdo 100 000 l quotl 100 11 we nooo 390 w W w j 1 39r v v JPK39OMS 13 an 6131ijwa Objective 22a Properties and Graphs of Exponential Emctions fxa agt1 g0 faaquot 0ltalt1 y l Ly 39 lg exponent J39th39 39 l D 6 01 75 90 CLPOIA W45 K o 0 I 4 neve hlts R x rams x My oi BEL CWCWWO we w 35ij tote 5 j 0 hang R1 lljolB Objective 22b Graphing Exponential Functions with Re ections or Translations Don t nna e 0 laid Don t 4131b b Use Obj 14 W 05 values A 9 4 39e 0 F 6gt 7quot l Cgt Ell 7 rfPl across an ff I across 4615 Select the graph that best represents the graph of each of the following RHFI 05055 44 across fa r0105 MPG 7 mos 3 x I Mi lP mu other remcotton across x or grams un mama 1t Macaw1 ne tyuc w c a me Which function best descrlbes the graph sho n Whlch functlon best descn es the graph shown LQXAS y y 1quot yealt13 em I V f 5 a gt 1 0 4a 4 f25 f25 f x25quot 9325 f 04 21013 04 f0 04 f x 04 More Objective 22b Graphing Exponential Functions with Translations Don t 39 to l Don t 9 06 Point Use Obj 14 6 c quot0 Shvl t Milk 3 3 s Hz 0 2mg MP 3 Egg wot0319Q39C gt5lmh down 93966035h19t mam z39C 3 0 52 Select the graph that best represents the graph of each of the following m 3w swk down 93 3 S A 1 f I 7 a mc olta41 457686 ly y gthgt x I1 x xv 139 x Which function best describes the graph shown Which function best describes the graph shown 040 4 a 7 gt 1 C xv f 2 6 f 6quot 3 me 99543 0 6x3 z 2 x2 522 Objective 22c The exponential function is f1 ex because of so many areas of application 1 n e z 271828 6 umnm 1 Z Grewhface c t DEQWlll not need to Mwoo a 7 l 0 If PYDVC t V 13 7 m C Evaluate eat on a scienti c calculator the Mac calculator in lab class Strontium 90 is a radioactive material that decays over time The amount A in grams of Strontium 90 remaining in a certain sample can be approximated with the functionlA 22560037 where t is the number of years from now How many grams of Strontium 90 i be remaining in this Fr sample after 7 years gpw d F567 a e 037 Cblc 037EEEJ7ET ED g 131 aRSEV 9 17390 I a7525127 Banzai 751312 673 WJ 123 0371317 E10725 El 8000 is invested in a bond trust that earns 59 interest compounded continuously The account balance t years later can be found with the function A 800060 059t How much money will be in the account after 6 years 0 o 54 L03 0 03quot 0006 RA Objective 22d Solving exponential equations when we can obtain the same base Exponential functions are oneto one that means m n 2 I 0 39 a if and nly if m n Rewrite each side if needed in terms of a common base use the smallest base possible Be sure to replace equals 4 x 4 Solve equate BLPDV W Q 07K 1 23K ourl 9 07 gm Equatc F0n9nt5 9kl 0 79 gqgt3 3xgt 4 5 x QK zli 39hwras W W HL8O v IIL8 39 Solve 1 1 573 F5 i quotl X l 7 45 3 I X ca 5 t equate exponmt igazD 72 5 E cei 05 Aax37amp 5 quotCD 3 We 5 iqu 3 RF IllaI5 Objective 23 Logarithmic Functions 0 00 so Consider an exponential function y 095 QKPOH En is 40 rtsule 09 There is no algebraic operation to solve for as What s the inverse function eKPom chlahovw We must de ne a new function y log at p 0 OD a i v I 9 V Ulb 05 Objective 23a Evaluate Logarithmic Functions a 9 1 QKPOW Cntlatwh logz 8 3 i CK PM Twyws 439 R 3 1 quot P 391 lt3 1082557 39Tp w yg JP ab 5 25 3 5 5 1031162 L L meKquot Tl pz 2 46 Pe 2 iPTl I P i r 4 log22 0 LI 392 Ease rtsuie Match logz 1 O r twc I5 1 Which are de ned Be careful sometimes ask Which are unde ned 1081 21 10814 4 logizt 108129 aY HE quotDquot W x eww must be po mv comma we 0 a gamut RR Objective 23b Properties and Graphs of Logarithmic Functions f as log 2 b gt 0 b s 1 The logarithmic functions are classi ed into two groups comparable to the exponential functions Recall Oby 22a y 5 K a 1qu a General was 9 o quotgt ex ylogbbgt1 amt 0 g ylogbx0ltblt1 P K i 00 00 39 xlvwerse gm Moment vert avgo 0Q expOWEyi al xxw t l askum Mncmn 0 b Cons1der fc logy wmc ob Cons1der f x 10 30 m e for an example of b gt 1 5 for an example of 0 lt lt 1 L 9 5 z quot I quot I C a Pick the y values nd the xvalues Pick the y values nd the xvalues a y a y 3 frl w q U 2 1quotle 2 1 L 391 I q 1 20quot 4 1 W gt EVquotquotquot 4 sq 1 yq 1 2 17 Vim 2 27 Objective 23c Graphing Logarithmic Emotions with Re ections or Translations Don t Mal Q table Don t slot VODWtS Use Obj 14 39 W Kerchon grime r aehccuon 319063 across 26 6W5 a 3 acroas y amsi 5 lojapx Select the graph that best represents the graph of each of the following vch acvogtlXacl3 o ab fac loga 0 fwlog x lt gt 4 K lt gt agrm aw aromas Il 0404 Which function best describes the graph shown Which function best describes the graph shown it agt y 39 Graph Mc o y a ogtl no MH l P 5 1 6 0V 0404 s 1 x I re aCr x amo M wglm x QilQEW fx10g52fv f 710g52xquot x 1 g25 x it 10805 13 f a 10825 a ref am quotMzw m0 CWVWY rdll amass sz ablg xoan5 w deC it dec39 aim we VP More Ob39ective 23c Gra hin Lo arithmic Functions with Translations J p g g r9 ma a ma Don t maKe a table Don t aloe wne Use Obj 14 I v 39 W KeHeLuom 39 i lYOhSlamonS W o xNote 3 ny Compansom a 0 C u g 1 oja1cgtkc 3 C dOUJW lj f QOKgt C IhC Re t acre 39 b0g15 Rah across 3 quot 64 0 a 4ch 3g3ec 9 109a OptC 5252 9541 6 she Select the graph that best represents the graph of each of the following fx 10g3acj 2r 5nd fx 10311533 5 Q7 0404 Lav t 131m 4600 034mwa wow 1 Aly I I I 0h mal g I Which function best describes the graph shown Which function best describes the graph shown E Dec 4704014 I lvxc ogt 39 3W k lt5 I m bed nah K dew n I x X f0 10g5293 2 f0 10852 quot 2 an 10g525 3 2 53 10g5293 quot 2 33 z lquotES5 2 W 59 10g25x 2 it 10g255 3 quot 2 chub NO eh m am answer theme 520 Objective 23d Domain of Logarithmic Functions not by graphing Give the domain fa 10ng4 5x 70 q39Squot 70 1314 5 15167 4 k4 4 5 frv15logb 3avg o39nu 00 at UmBquot 70 39 3LgtO DO ogt 16quot0 log3t4 332 3 In 50b 20 439 70 J 042116 3ps o t LO 3X3quot 7 x 5 2 103415 Ob 18b example The information included in this example would be asked in separate on line problems I H 2 fcax12 8agt0 gt Form 8 L l OpensDown Qgtvertex 391 398 BDHOW many xintercepts Gmh the 3 N l k H Q E l re a Lb mt 00K 56 down 5 HgtMaxis quotquot 39Zalg 5gtMaxis at a 39l 39 V6 4 9 Find all intercepts for f 13 2 222 12 8 For on line problems Enter them in any order separated by a comma wtlet xeo 57 gampo33 752gtX 510 xvlvrt t 370 gt CamelDRE Remn Ob Lug DP NOT mat out RQHYR lI KR f o xli2mz va 1 9 3 1 C 5 k s 3 Find all intercepts for f x 2251 12 10 For on line problems Enter them in any order separated by a comma Manx e39g ago 537 90 3aoa o 5 lo 393 xmt1 eh 7 o a lgta quot39IO 21 4 02 5 2 ZRn250 T QL if 0 quotI x l K quot I 0 Ob 180 example Studies have found that the relationship between advertising dollars a in thou sands and revenue R can be modeled by a quadratic function 11 If Ra 4a2 364a 25695 ow man thousands of advertisin dollars should be spent in order to maximize revenue h Oidfrcd Vans Q Kgt ad a want quotaquot value cl ucreuc Enter number answer integers or exact decimals mathematical operators are not allowed For example 152 must be entered as 75 Don t type any dollar signs commas or units The function given does not represent the results of an actual study 110 50 305 0156 Calculator Ra 74 8 a is 2 39r Arfvf adv W e 8 Ob 18c example A large swimming pool is treated regularly to control the growth of harmful bacteria If the concentration of bacteria 039 per cubic centimeter t days after treatment is given by C39 t 04t2 44t 301 What is the concentration of bacteria Same cautions as in previous example ve zc 6 P wan t C Ct o 4tquot HH r3cgt t c 56 0 60163 Cancenwa OW e c Hu 11 or boobEVIa Ra o X KR8m nb6r fo hfx t SS agtgt also C Ll55339 30 Wmda days Ci F tcxr vatm en b w M 214139 Covwhncratxon o5 baL thO be at 0 mm 397 5 da S Q Objective 19 Power Functions fa xquot where n is an integer n 2 2 The power functions are classi ed into 2 groups f x x where n is an even positive integer n 2 2 For example fr x2 34 fx x6 fr 958 eVCn WSSWHAA Duff 390 Dec 045 0 Ac 000 f as x where n is an odd positive integer n 2 3 For example f9 x3 x5 fa x7 fa 359 L 3 IL Jhgmer Power 3 8 Clef r to Vtgt53 bans near 00 3 59 I bl StefPer a quot 3 D1 00OOgt R6ijgt I WYJL oyj h ViC on P0a ltgtgt 71 buawn out For 0 A 059 500 0b 5 00 Objective 20 Solving Polynomial and Rational Ineggglities that mam that MP Introduction to Obj 20 Ifab 0 then it must be true that 0 0 or b 5 D However if a b gt 0 then it does whave to be true that a 7O or l0 7 Q In fact both number could be negative lt 0 It would require 2 cases to express this solution Case1QquotO and 1070 OrCase2 040 040 bin at are gt 0 bee are me I we consider c gt 0 thffeae numbers multiplied Witlf igbfesi being positi then it would require 4 cases to describe this solution and in only one case is it true that a gt 0 and b gt O and c gt O and We need a process that doesn t require determining and then analyzing cases The process is based on the Factor Theorem and the Intermediate Value Theorem Plan of Attack 1 Factor if needed Watch for Diff of Squares and Factoring y 39 O In the case a 2 Findl ggl 39tionin Points These are values that m che expression quot of a rationa inequa 39ty these are values that make the expression 7 0 from factors in the numerator or make the expression AtId Q n dlfrom factors in the denominator Set each factor equal to O to nd these Partitioning Points 3 Mark these on a number line must be in numberline order 4 Make a SignChart Select a value in each interval that s created by these partitioning points don t use one of endpoints Plug this value into each separate factor and record whether the result is or Consolidate the signs from all the factors The Factor Theorem guarantees that there are no other Partitioning Points beyond those found from the factorization The Intermediate Value Theorem guarantees that there can be no sign change within the intervals created by the Partitioning Points 32K 0 3 2 0 9 4332 SOlve39 1132 5305 Mane 64 076 3 393 Tag 39 3 3 32 40 x 2 L lt3 TESL 3 2K39Z 4 O W 0 C 395 0 9 moat use ioValufS Pd

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