Portland Community College MTH 251 Laboratory Manual

Transcription

1 Porlan Communiy College MTH 5 Laboraory Manual

2 Porlan Communiy College MTH 5 Lab Manual Table of Conens Raes of Cange (Aciviies -) Lab Aciviies Pages 6 Supplemenary Eercises (E) Pages A A (Appeni A) Supplemenary Soluions (E) Pages D D4 (Appeni D) Limis an Coninuiy (Aciviies 4-6) Lab Aciviies Pages 7-4 Supplemenary Graps Pages B - B (Appeni B) Limi Laws an Raional Limi Forms Pages C C4 (Appeni C) Supplemenary Eercises (E) Pages A A6 (Appeni A) Supplemenary Soluions (E) Pages D5 D0 (Appeni D) Inroucion o e Firs Derivaive (Aciviies 7-0) Lab Aciviies Pages 5 - Supplemenary Eercises (E) Pages A7 A8 (Appeni A) Supplemenary Soluions (E) Pages D D4 (Appeni D) Funcions, Derivaives, an Anierivaives (Aciviies -6) Lab Aciviies Pages - 46 Supplemenary Grap Page B (Appeni B) Supplemenary Eercises (E4) Pages A9 A4 (Appeni A) Supplemenary Soluions (E4) Pages D5 D8 (Appeni D) Derivaive Formulas (Aciviies 7-7) Lab Aciviies Pages Rules of Differeniaion Pages C5 C6 (Appeni C) Supplemenary Eercises (E5) Pages A5 A8 (Appeni A) Supplemenary Soluions (E5) Pages D9 D8 (Appeni D) Te Cain Rule (Aciviies 8-4) Lab Aciviies Pages 57-6 Rules of Differeniaion Pages C5 C6 (Appeni C) Supplemenary Eercises (E6) Pages A9 A0 (Appeni A) Supplemenary Soluions (E6) Pages D9 D6 (Appeni D) Preface i

3 Porlan Communiy College MTH 5 Lab Manual Implici Differeniaion (Aciviies 4-4) Lab Aciviies Pages 6-66 Rules of Differeniaion Pages C5 C6 (Appeni C) Supplemenary Eercises (E7) Pages A A (Appeni A) Supplemenary Soluions (E7) Pages D7 D40 (Appeni D) Relae Raes (Aciviies 44-47) Lab Aciviies Pages 67-7 Supplemenary Eercises (E8) Pages A A4 (Appeni A) Supplemenary Soluions (E8) Pages D4 D44 (Appeni D) Criical Numbers an Graping from Formulas (Aciviies 48-54) Lab Aciviies Pages 7-84 Supplemenary Eercises (E9) Pages A5 A6 (Appeni A) Supplemenary Soluions (E9) Pages D45 D5 (Appeni D) ii P reface

4 Porlan Communiy College MTH 5 Lab Manual To e Suen MTH 5 is aug a Porlan Communiy College using a lecure/lab forma. Te laboraory ime is se asie for suens o invesigae e opics an pracice e skills a are covere uring eir lecure perios. Te lab aciviies ave been wrien uner e presumpion a suens will be working in groups an will be acively iscussing e eamples an problems inclue in eac aciviy. Many of e eercises an problems len emselves quie naurally o iscussion. Some of e more algebraic problems are no so muc iscussion problems as ey are pracice an elp problems. You o no nee o fully unersan an eample before saring on e associae problems. Te inen is a your unersaning of e maerial will grow wile you work on e problems. Wen working roug e lab aciviies e suens in a given group soul be working on e same aciviy a e same ime. Someimes is means an iniviual suen will ave o go a lile more slowly an e or se may like an someimes i means an iniviual suen will nee o move on o e ne aciviy before e or se fully grasps e curren aciviy. Many insrucors will wan you o focus some of your energy on e way you wrie your maemaics. I is imporan a you o no rus roug e aciviies. Wrie your soluions as if ey are going o be grae; a way you will know uring lab ime if you unersan e proper way o wrie an organize your work. If your lab secion mees more an once a week, you soul no work on lab aciviies beween lab secions a occur uring e same week. I is OK o work on lab aciviies ousie of class once e enire classroom ime alloe for a lab as passe. Tere are no wrien soluions for e lab aciviy problems. Beween your group maes, your insrucor, an (if you ave one) your lab assisan, you soul know weer or no you ave e correc answers an proper wriing sraegies for ese problems. Eac lab as a secion of supplemenary eercises; ese eercises are fully keye. Te supplemenary eercises are no simply copies of e problems in e lab aciviies. Wile some quesions will look familiar, many oers will callenge you o apply e maerial covere in e lab o a new ype of problem. Tese quesions are mean o supplemen your ebook omework, no replace your ebook omework. Te MTH 5 Laboraory Manual was wrien by Seve Simons. Te cover ar for e manual was esigne by Pil Turber. Te cover inclues a page from Isaac Newon s Pilosopiæ Nauralis Principia Maemaica. Te cover inclues William Blake s Newon. Preface iii

5 Porlan Communiy College MTH 5 Lab Manual To e Insrucor Tis manual is significanly ifferen from earlier versions of e lab manual. Te opics ave been arrange in a evelopmenal orer. Because of is, suens wo work eac aciviy in e orer ey appear may no ge o all of e opics covere in a paricular week. I is srongly recommene a e insrucor pick an coose wa ey consier o be e mos vial aciviies for a given week an a e insrucor ave e suens work ose aciviies firs; for some aciviies you mig also wan o ave e suens only work selece problems in e aciviy. Suens wo complee e ig prioriy aciviies an problems can en go back an work e aciviies a ey iniially skippe. Tere are also fully keye problems in e supplemenary eercises a e suens coul work on bo uring lab ime an ousie of class. A suggese sceule for e labs is sown in Table P. Again, e insrucor soul coose wa ey feel o be e mos relevan aciviies an problems for a given week an ave e suens work ose aciviies an problems firs. Table P: Possible 0 week sceule for e labs. (Suens soul consul eir syllabus for eir sceule.) Week Labs (Lab Aciviies) Supplemenary Eercises Raes of Cange/Limis an Coninuiy (-4) E (all) Limis an Coninuiy (5-6) E (all) Inroucion o e Firs Derivaive (7-0) E (all) 4 Funcions, Derivaives, an Anierivaives (-4) E4.-E4.5 5 Funcions, Derivaives, an Anierivaives (5-6) E4.6-E4.0 6 Derivaive Formulas (7-7) E5 (all) 7 Te Cain Rule (8-4) E6 (all) 8 Implici Differeniaion/Relae Raes (4-47) E7 (all), E8 (all) 9 an 0 Criical Numbers an Graping from Formulas (48-54) E9 (all) iv P reface

6 Porlan Communiy College MTH 5 Lab Manual Raes of Cange Aciviy Moion is frequenly moele using calculus. A builing block for is applicaion is e concep of average velociy. Average velociy is efine o be ne isplacemen ivie by elapse ime. More precisely, if p is a posiion funcion for someing moving along a numbere line, en we efine e average velociy over e ime inerval, o be: Epression.: 0 p p 0 o Problem. Accoring o simplifie Newonian pysics, if an objec is roppe from a eig of 00 m an allowe o freefall o e groun, en e eig of e objec (measure in m) is given by e posiion funcion p were is e amoun of ime a as passe since e objec was roppe (measure in s)... Wa, incluing uni, are e values of p ree secons an five secons ino e objec s fall? Use ese values wen working problem..... Calculae p 5s ps 5ss resul ell you in e cone of is problem? ; inclue unis wile making e calculaion. Wa oes e.. Use Epression. o fin a formula for e average velociy of is objec over e general ime inerval,. Te firs couple of lines of is process are sown below. Copy ese 0 lines ono your paper an coninue e simplificaion process. p p Ceck e formula you erive in problem.. using 0 an 5 ; a is, compare e value generae by e formula o a you foun in problem Using e formula foun in problem.., replace 0 wi bu leave as a variable; simplify e resul. Ten copy Table. ono your paper an fill in e missing enries. Table.: Hin In e ne sep you soul facor 4.9 from e numeraor; e remaining facor will facor furer. y (s) p p y (m/s) Lab Aciviies

7 Porlan Communiy College MTH 5 Lab Manual..6 As e value of ges closer o, e values in e y column of Table. appear o be converging on a single number; wa is is number an wa o you ink i ells you in e cone of is problem? Aciviy One of e builing blocks in ifferenial calculus is e secan line o a curve. I is very easy for a line o be consiere a secan line o a curve; e only requiremen a mus be fulfille is a e line inersecs e curve in a leas wo poins. y f as been In Figure., a secan line o e curve rawn roug e poins 0, an 4, 5. You soul verify a e slope of is line is. Te formula for f is f. We can use is formula o come up wi a generalize formula for e slope of secan lines o is curve.,, f is Specifically, e slope of e line connecing e poin f o e poin erive in Eample Figure.: f Eample. m sec f f for 0 0 Tis facoring ecnique is calle facoring by grouping. We can ceck our formula using e line in Figure.. If we le 0 0 an 4 en our simplifie slope formula gives us: 0 40 L ab Aciviies

8 Porlan Communiy College MTH 5 Lab Manual Problem. g Le 5... Following Eample., fin a formula for e slope of e, g an secan line connecing e poins 0 0, g. Please noe a facoring by grouping will no be necessary wen simplifying e epression,.. Ceck your slope formula using e wo poins inicae in Figure.. Ta is, use e grap o fin e slope beween e wo poins an en use your formula o fin e slope; make sure a e wo values agree! Aciviy Wile i s easy o see a e formula f f 0 0 Figure.: g gives e slope of e line connecing wo poins on e funcion f, e resulan epression can a imes be awkwar o work wi. We acually alreay saw a wen we a o use slig-of-an facoring in Eample.. Te algebra associae wi secan lines (an average velociies) can someimes be simplifie if we esignae e variable o be e run beween e wo poins (or e leng of e ime inerval). Wi is esignaion we ave 0 wic gives us 0. Making ese subsiuions we ge Equaion.. Te epression on e rig sie of Equaion. is calle e ifference quoien for f. Equaion. f f f f Le s revisi e funcion funcion is erive in Eample.. f from Eample.. Te ifference quoien for is Eample. f 0 f for Lab Aciviies

9 Porlan Communiy College MTH 5 Lab Manual Please noice a all of e erms wiou a facor of subrace o zero. Please noice, oo, a we avoie all of e ricky facoring a appeare in Eample.! For simpliciy s sake, we generally rop e variable subscrip wen applying e ifference quoien. So for fuure reference we will efine e ifference quoien as follows: Definiion. Te ifference quoien for e funcion y f is e epression f f. Problem. Compleely simplify e ifference quoien for eac of e following funcions. Please noe a e emplae for e ifference quoien nees o be aape o e funcion name an inepenen variable in eac given equaion. For eample, e ifference quoien for e funcion v v in problem.. is. Please make sure a you lay ou your work in a manner consisen wi e way e work is sown in eample. (ecluing e subscrips, of course). v g 7.. w Problem. Suppose a an objec is osse ino e air in suc a way a e elevaion of e objec (measure in f) is given by e funcion s were is e amoun of ime a as passe since e objec was osse (measure in s)... Simplify e ifference quoien for s... Ignoring e uni, use e ifference quoien o eermine e average velociy over e inerval.6,.8. (Hin: Use.6 an.. Make sure a you unersan wy!).. Wa, incluing uni, are e values of s.6 an s.8 problem Use e epression.8 s.6 s.8.6 e uni wile making is calculaion.? Use ese values wen working o verify e value you foun in problem... Inclue..5 Ignoring e uni, use e ifference quoien o eermine e average velociy over e inerval 0.4,.4. 4 L ab Aciviies

10 Porlan Communiy College MTH 5 Lab Manual Problem. Moose an squirrel were aving casual conversaion wen suenly, wiou any apparen provocaion, Boris Baenov launce ani-moose missile in eir irecion. Forunaely, squirrel a abiliy o fly as well as grea knowlege of missile ecnology, an e was able o isarm missile well before i i groun. Te elevaion (f) of e ip of e missile secons afer i was launce is given by e funcion Wa, incluing uni, is e value of an wa oes e value ell you abou e flig of e missile? 0 s 0 s.. Wa, incluing uni, is e value of abou e flig of e missile? 0 s.. Te velociy (f/s) funcion for e missile is v 94.4 v 0 v 0 e value of missile? 0 s an wa oes is value ell you. Wa, incluing uni, is an wa oes is value ell you abou e flig of e Problem.4 Timmy live a long life in e 9 cenury. Wen Timmy was seven e foun a rock a weige eacly alf a sone. (Timmy live in jolly ol Englan, on you know.) Ta rock sa on Timmy s winow sill for e ne 80 years an wouln you know e weig of a rock i no cange even one smige e enire ime. In fac, e weig funcion for is rock was w 0.5 were w was e weig of e rock (sones) an was e number of years a a passe since a ay Timmy broug e rock ome..4. Wa was e average rae of cange in e weig of e rock over e 80 years i sa on Timmy s winow sill?.4. Ignoring e uni, simplify e epression e cone of is problem? w w 0 0. Does e resul make sense in.4. Sowing eac sep in e process an ignoring e uni, simplify e ifference quoien for w. Does e resul make sense in e cone of is problem? Lab Aciviies 5

11 Porlan Communiy College MTH 5 Lab Manual Problem.5 Tru be ol, ere was one ay in 84 wen Timmy s miscievous son Nigel ook a rock ousie an cucke i ino e air. Te velociy of e rock (f/s) was given by v 60 were was e number of secons a a passe since Nigel cucke e rock..5. Wa, incluing uni, are e values of v 0, v, an v an wa o ese values ell you in e cone of is problem? Don jus wrie a e values ell you e velociy a cerain imes; eplain wa e velociy values ell you abou e moion of e rock..5. Ignoring e uni, simplify e ifference quoien for v..5. Wa is e uni for e ifference quoien for v? Wa oes e value of e ifference quoien (incluing uni) ell you in e cone of is problem? Problem.6 Suppose a a va was unergoing a conrolle rain an a e amoun of flui lef in e va / (gal) was given by e formula V 00 were is e number of minues a a passe since e raining process began..6. Wa, incluing uni, is e value of V 4 an wa oes a value ell you in e cone of is problem?.6. Ignoring e uni, wrie own e formula you ge for e ifference quoien of V wen 4. Copy Table. ono your paper an fill in e missing values. Look for a paern in e oupu an wrie own enoug igis for eac enry so a e paern is clearly illusrae ; e firs wo enries in e oupu column ave been given o elp you unersan wa is mean by is irecion..6. Wa is e uni for e y values in Table.? Wa o ese values (wi eir uni) ell you in e cone of is problem? Table.: 4 4 V V y y As e value of ges closer o 0, e values in e y column of Table. appear o be converging o a single number; wa is is number an wa o you ink a value (wi is uni) ells you in e cone of is problem? 6 L ab Aciviies

12 Porlan Communiy College MTH 5 Lab Manual Limis an Coninuiy Aciviy 4 Wile working problem.6 you complee Table 4. (formerly Table.). In e cone of a problem e ifference quoien being evaluae reurne e average rae of cange in e volume of flui remaining in a va beween imes 4 an 4. As e elapse ime closes in on 0 is average rae of cange converges o 6. From a we euce a e rae of cange in e volume 4 minues ino e raining process mus ave been 6 gal/min. Te cone for Problem.6 Suppose a a va was unergoing a conrolle rain an a e amoun of flui lef in e va / (gal) was given by e formula V 00 were is e number of minues a a passe since e raining process began. Table 4.: 4 4 V V y y Please noe a we coul no euce e rae of cange 4 minues ino e process by replacing wi 0 ; in fac, ere are a leas wo ings prevening us from oing so. From a sricly maemaical perspecive, we canno replace wi 0 because a woul lea o ivision by zero in e ifference quoien. From a more pysical perspecive, replacing wi 0 woul in essence sop e clock. If ime is frozen, so is e amoun of flui in e va an e enire concep of rae of cange becomes moo. I urns ou a i is frequenly more useful (no o menion ineresing) o eplore e ren in a funcion as e inpu variable approaces a number raer an e acual value of e funcion a a number. Maemaically we escribe ese rens using limis. If we call e ifference quoien in e eaing for Table 4. ren evience in e able by saying e limi of a as canges value, e value of fie number o wic e value of f, en we coul escribe e f as approaces zero is 6. Please noe f canges, no e value of e limi. Te limi value is a f converges. Symbolically we wrie lim f 6 Mos of e ime e value of a funcion a e number a an e limi of e funcion as approaces a are in fac e same number. Wen is occurs we say a e funcion is coninuous a a. However, o elp you beer unersan e concep of limi we nee o ave you confron siuaions were e funcion value an limi value are no equal o one anoer. Graps can be useful for elping isinguis funcion values from limi values, so a is e perspecive you are going o use in e firs couple of problems in is lab. 0 Lab Aciviies 7

13 Porlan Communiy College MTH 5 Lab Manual Problem 4. Several funcion values an limi values for e funcion in Figure 4. are given below. You an your group maes soul ake urns reaing e equaions alou. Make sure a you rea e symbols correcly, a s par of wa you are learning! Also, iscuss wy e values are wa ey are an make sure a you ge elp from your insrucor o clear up any confusion. f 6 bu f lim f 4 is unefine bu f lim 4 f bu lim f oes no eis Figure 4.: f lim f bu lim f Te limi of f as approaces from e lef. Te limi of f as approaces from e rig. Problem 4. Copy eac of e following epressions ono your paper an eier sae e value or sae a e value is unefine or oesn eis. Make sure a wen iscussing e values you use proper erminology. All epressions are in reference o e funcion g sown in Figure g lim g lim g 4..5 lim g 4..7 g 4..8 lim g lim g 4.. lim g g 4..6 lim g 4..9 g 4.. lim g Figure 4.: g 8 L ab Aciviies

14 Porlan Communiy College MTH 5 Lab Manual Problem 4. Values of e funcion f 65 are sown 5 in Table 4.. Bo of e quesions below are in reference o is funcion. 4.. Wa is e value of f 5? 4.. Wa is e value of lim? 5 Table 4.: f f () Problem 4.4 Values of e funcion p are sown in Table 4.. Bo of e quesions below are in reference o is funcion Wa is e value of p? 4.4. Wa is e value of Problem 4.5 lim? Table 4.: p p() Creae ables similar o ables 4. an 4. from wic you can euce eac of e following limi values. Make sure a you inclue able numbers, able capions, an meaningful column eaings. Make sure a your inpu values follow paerns similar o ose use in ables 4. an 4.. Make sure a you roun your oupu values in suc a way a a clear an compelling paern in e oupu is clearly emonsrae by your sae values. Make sure a you sae e limi value! 0 4 sin 4.5. lim lim lim Aciviy 5 Wen proving e value of a limi we frequenly rely upon laws a are easy o prove using e ecnical efiniions of limi. Tese laws can be foun in Appeni C (pages C an C). Te firs of ese ype laws are calle replacemen laws. Replacemen laws allow us o replace limi epressions wi e acual values of e limis. Problem 5. Te value of eac of e following limis can be esablise using one of e replacemen laws. Copy eac limi epression ono your own paper, sae e value of e limi (e.g. lim 5 5 ), an sae e replacemen law (by number) a esablises e value of e limi lim 5.. lim lim 4 Lab Aciviies 9

15 Porlan Communiy College MTH 5 Lab Manual Problem 5. Te algebraic limi laws allow us o replace limi epressions wi equivalen limi epressions. Wen applying limi laws our firs goal is o come up wi an epression in wic every limi in e epression can be replace wi is value base upon one of e replacemen laws. Tis process is sown in eample 5.. Please noe a all replacemen laws are save for e secon o las sep an a eac replacemen is eplicily sown. Please noe also a eac limi law use is reference by number. Eample 5. lim 4 lim 4 lim Limi Law A lim lim Limi Law A 4 lim lim Limi Law A Limi Laws R an R 99 Use e limi laws o esablis e value of eac of e following limis. Make sure a you use e sep-by-sep, verical forma sown in eample 5.. Make sure a you cie e limi laws use in eac sep. To elp you ge sare, e seps necessary in problem 5.. are ouline below. To elp you ge sare, e seps necessary in problem 5.. are ouline below. Sep : Apply Law A6 Sep : Apply Law A Sep : Apply Law A Sep 4: Apply Laws R an R 5.. lim lim y 7 y y y lim cos Aciviy 6 Many limis ave e form 0 0 wic means e epressions in bo e numeraor an enominaor 6 limi o zero (e.g. lim ). Te form 0 is calle ineerminae because we o no know e 0 value of e limi (or even if i eiss) so long as e limi as a form. Wen confrone wi limis of form 0 0 we mus firs manipulae e epression so a common facors causing e zeros in e numeraor an enominaor are isolae. Limi law A7 can en be use o jusify eliminaing e common facors an once ey are gone we may procee wi e applicaion of e remaining limi laws. Eamples 6. an 6. illusrae is siuaion. 0 L ab Aciviies

16 Porlan Communiy College MTH 5 Lab Manual Eample 6. lim Tis limi oes no ave ineerminae form, so we may apply e remaining limi laws lim lim 5 Limi Law A7 lim lim 5 Limi Law A 5 Limi Laws R an R Tis limi oes no ave ineerminae form, so we may apply e remaining limi laws. Eample cos sin cos cos cos lim lim sin sin cos lim 0 sin cos lim 0 sin cos lim 0 cos lim 0 lim cos lim 0 lim lim cos 0 0 lim 0 lim cos lim cos 0 Limi Law A7 Limi Law A5 Limi Law A Limi Law A6 Limi Laws R an R As seen in eample 6., rigonomeric ieniies can come ino play wile rying o eliminae e form 0. Elemenary rules of logarims can also play a role in is process. Before you begin 0 evaluaing limis wose iniial form is 0, you nee o make sure a you recall some of ese basic 0 rules. Ta is e purpose of problem 6.. Lab Aciviies

17 Porlan Communiy College MTH 5 Lab Manual Problem 6. Complee eac of e following ieniies (over e real numbers). Make sure a you ceck wi your lecure insrucor so a you know wic of ese ieniies you are epece o memorize. Te following ieniies are vali for all values of an y. cos an sin an sin y cos y sin cos Tere are ree versions of e following ieniy; wrie em all. cos cos cos Te following ieniies are vali for all posiive values of an y an all values of n. ln y ln y n ln ln e n L ab Aciviies

18 Porlan Communiy College MTH 5 Lab Manual Problem 6. Use e limi laws o esablis e value of eac of e following limis afer firs manipulaing e epression so a i no longer as form 0. Make sure a you use e sep-by-sep, verical 0 forma sown in eamples 6. an 6.. Make sure a you cie eac limi law use lim 4 5 cos lim 0 cos sin lim 0 sin 4ln lim ln ln ln sin lim sin lim w 9 9 w w Aciviy 7 We are frequenly inerese in a funcion s en beavior; a is, wa is e beavior of e funcion as e inpu variable increases wiou boun or ecreases wiou boun. Many imes a funcion will approac a orizonal asympoe as is en beavior. Assuming a e orizonal asympoe y L represens e en beavior of e funcion f bo as increases wiou boun an as ecreases wiou boun, we wrie lim f L lim f L. Te formalisic way o rea lim is e limi of f f L an as approaces infiniy equals L. Wen rea a way, owever, e wors nee o be aken anying bu lierally. In e firs place, isn approacing anying! Te enire poin is a is increasing wiou any boun on ow large is value becomes. Seconly, ere is no place on e real number line calle infiniy; infiniy is no a number. Hence cerainly can be approacing someing a isn even ere! Problem 7. For e funcion in Figure 7. (Appeni B, page B) we coul (correcly) wrie lim f figures (pages B an B). lim f. Go aea an wrie (an say alou) e analogous limis for e funcions in an Problem 7. Values of e funcion f 6 5 are sown 5 in Table 7.. Bo of e quesions below are in reference o is funcion. 7.. Wa is e value of lim f 7.. Wa is e orizonal asympoe for e grap of y f?? Table 7.: f f (), , , ,000, Lab Aciviies

19 Porlan Communiy College MTH 5 Lab Manual Problem 7. Jorge an Vanessa were in a eae iscussion abou orizonal asympoes. Jorge sai a funcions never cross orizonal asympoes. Vanessa sai Jorge was nus. Vanessa wippe ou er rusy calculaor an generae e values in Table 7. o prove er poin. 7.. Wa is e value of lim g? 7.. Wa is e orizonal asympoe for e grap of y g? 7.. Jus ow many imes oes e curve y g is orizonal asympoe? cross Table 7.: g sin g () Aciviy 8 Wen using limi laws o esablis limi values as or, limi laws A-A6 an R are sill in play (wen applie in a vali manner), bu limi law R canno be applie. (Te reason limi law R canno be applie is iscusse in eail in problem.4) Tere is a new replacemen law a can only be applie wen or ; is is replacemen law R. Replacemen law R essenially says a if e value of a funcion is increasing wiou any boun on large i becomes or if e funcion is ecreasing wiou any boun on ow large is absolue value becomes, en e value of a consan ivie by a funcion mus be approacing zero. An analogy can be foun in eremely poor pary planning. Le s say a you plan o ave a pizza pary an you buy five pizzas. Suppose a as e our of e pary approaces more an more guess come in e oor in fac e guess never sop coming! Clearly as e number of guess coninues o rise e amoun of pizza eac gues will receive quickly approaces zero (assuming e pizzas are equally ivie among e guess). Problem 8. Consier e funcion f. Complee Table 8. wiou e use of your calculaor. Wa limi value an limi law are being illusrae in e able? Table 8.: f,000 0,000 00,000,000,000 f () 4 L ab Aciviies

20 Porlan Communiy College MTH 5 Lab Manual Aciviy 9 Many limis ave e form wic we ake o mean a e epressions in bo e numeraor an enominaor are increasing or ecreasing wiou boun. Wen confrone wi a limi of ype lim f g or lim f g a as e form, we can frequenly resolve e limi if we firs ivie e ominan facor of e ominan erm of e enominaor from bo e numeraor an e enominaor. Wen we o is, we nee o compleely simplify eac of e resulan fracions an make sure a e resulan limi eiss before we sar o apply limi laws. We en apply e algebraic limi laws unil all of e resulan limis can be replace using limi laws R an R. Tis process is illusrae in eample 9.. Eample 9. lim lim lim 5 5 lim lim 5 5 lim lim lim lim Te form of e limi is now 0, so we can 0 5 begin o apply e limi laws because e limis will all eis. Limi Law A5 Limi Laws A an A Limi Laws R an R Problem 9. Use e limi laws o esablis e value of eac limi afer iviing e ominan erm-facor in e enominaor from bo e numeraor an enominaor. Remember o simplify eac resulan epression before you begin o apply e limi laws lim lim 6e 0e e 9.. 4y 5 lim y 5 9 y Lab Aciviies 5

21 Porlan Communiy College MTH 5 Lab Manual Aciviy 0 Many limi values o no eis. Someimes e non-eisence is cause by e funcion value eier increasing wiou boun or ecreasing wiou boun. In ese special cases we use e symbols an o communicae e non-eisence of e limis. Figures can be use o illusrae some ways in wic we communicae e non-eisence of ese ype of limis. In Figure 0. we coul (correcly) wrie lim k, lim k, an lim k. In Figure 0. we coul (correcly) wrie 4 lim w, 4 lim w, an 4 lim w. In Figure 0. we coul (correcly) wrie lim T an lim T soran way of communicaing e non-eisence of e wo sie limi. Tere is no lim T. Figure 0.: k Figure 0.: w 4 Figure 0.: T Problem 0. Draw ono Figure 0.4 a single funcion, f, a saisfies eac of e following limi saemens. Make sure a you raw e necessary asympoes an a you label eac asympoe wi is equaion. lim lim f f lim f 0 lim f Figure 0.4: f 6 L ab Aciviies

22 Porlan Communiy College MTH 5 Lab Manual Aciviy Wenever bu lim f 0 a sie of a e value of f g ese siuaions e line lim g f 0, en lim a a g oes no eis because from eier eier increases wiou boun or ecreasing wiou boun. In a is a verical asympoe for e grap of f y. g 5 For eample, e line is a verical asympoe for e funcion. We say a 5 5 lim as e form no zero over zero. (Specifically, e form of lim is 7.) Every 0 limi wi form no zero over zero oes no eis. However, we frequenly can communicae e 5 non-eisence of e limi using an infiniy symbol. In e case of i s prey easy o see a.99 infer a is a posiive number wereas.0 lim an lim is a negaive number. Consequenly, we can. Remember, ese equaions are communicaing a e limis o no eis as well as e reason for eir non-eisence. Tere is no sor-an way o communicae e non-eisence of e wo-sie limi lim. Problem. Suppose a g 4... Wa is e verical asympoe on e grap of y g.. Wrie an equaliy abou lim g.. Wrie an equaliy abou lim g..4 Is i possible o wrie an equaliy abou lim g...? If so, wrie i...5 Wic of e following limis eis? lim g, lim g Problem. Suppose a z 7... Wa is e verical asympoe on e grap of y z.. Is i possible o wrie an equaliy abou lim z.? If so, wrie i., an lim g.. Wa is e orizonal asympoe on e grap of y z...4 Wic of e following limis eis? lim z, lim z, an lim z Lab Aciviies 7

23 Porlan Communiy College MTH 5 Lab Manual Problem. Consier e funcion f 7. Complee Table. wiou e use of your calculaor. 8 Use is as an opporuniy o iscuss wy limis of form no zero over zero are infinie limis. Wa limi equaion is being illusrae in e able? Table.: f f () Problem.4 Hear me, an ear me lou oes no eis. Tis, in par, is wy we canno apply Limi Law R o an epression like lim. Wen we wrie, say, lim 7, we are replacing e limi epression wi is value a s wa e replacemen laws are all abou! Wen we wrie lim, we are no replacing e limi epression wi a value! We are eplicily saying a e limi as no value (i.e. oes no eis) as well as saying e reason e limi oes no eis. Te limi laws (R-R an A-A6) can only be applie wen all of e limis in e equaion eis. Wi is in min, iscuss an ecie weer eac of e following equaions are rue or false. 7 e lim e 0 lim e 0 lim e 0 T or F lim e e lim lim e e T or F e lim ln lim e T or F lim ln lim ln lim ln 0 0 T or F ln lim e sin lim sin lim ln T or F lim e lim sin lim sin T or F e e lim ln lim lim ln T or F lim e / lim e T or F 8 L ab Aciviies

24 Porlan Communiy College MTH 5 Lab Manual Problem.5 Miny rie o evaluae lim using e limi laws. Tings wen orribly wrong for Miny (er work is sown below). Ienify wa is wrong in Miny s work an iscuss wa a more reasonable approac mig ave been lim lim 6 6 Tis soluion is no correc! Do no emulae Miny s work!! 4 lim 6 6 lim 4 lim 6 lim 4 lim lim Limi Law A5 Limi Law A Limi Laws R an R Aciviy Many saemens we make abou funcions are only rue over inervals were e funcion is coninuous. Wen we say a funcion is coninuous over an inerval, we basically mean a ere are no breaks in e funcion over a inerval; a is, ere are no verical asympoes, oles, jumps, or gaps along a inerval. To make is efiniion more precise, we begin by efining wa we mean we say a a funcion is Definiion. coninuous a e number a. Te funcion f is coninuous a e number a if an only if lim f f a. a Tere are ree ways a e efining propery can fail o be saisfie a a given value of a. To faciliae eploraion of ese ree manner of failure, we can break e efining propery ino a specrum of ree properies. i. f a mus be efine ii. lim f a mus eis iii. lim f a mus equal f a Please noe a if eier propery i or propery ii fails o be saisfie a a given value of a, en propery iii also fails o be saisfie a a. Lab Aciviies 9

25 Porlan Communiy College MTH 5 Lab Manual Problem. Sae e values of a wic e funcion sown in Figure. is isconinuous. For eac insance of isconinuiy, sae (by number) all of e sub-properies in Definiion. a fail o be saisfie. Figure.: 5 Aciviy Wen a funcion as a isconinuiy a a, e funcion is someimes coninuous from only e rig or only e lef a a. (Please noe a wen we say e funcion is coninuous a a we mean a e funcion is coninuous from bo e rig an lef a a.) Definiion. Te funcion f is coninuous from e lef a a if an only if lim f f a coninuous from e rig a a if an only if lim f f a a. a an f is Some isconinuiies are classifie as removable isconinuiies. Specifically, isconinuiies a are oles or skips (oles wi a seconary poin) are calle removable. Definiion. We say a f as a removable isconinuiy a a if f is isconinuous a a bu lim f a eiss. 0 L ab Aciviies

26 Porlan Communiy College MTH 5 Lab Manual Problem. Referring o e funcion sown in Figure., sae e values of were e funcion is coninuous from e rig bu no e lef. Ten sae e values of were e funcion is coninuous from e lef bu no e rig. Problem. Referring again o e funcion sown in Figure., sae e values of were e funcion as removable isconinuiies. Aciviy 4 Figure.: Now a we ave a efiniion for coninuiy a a number, we can go aea an efine wa we mean wen we say a funcion is coninuous over an inerval. 5 Definiion 4. Te funcion f is coninuous over an open inerval if an only if i is coninuous a eac an every number on a inerval. Te funcion is coninuous over e close inerval ab, if an only if i is coninuous on ab,, coninuous from e rig a a, an coninuous from e lef a b. Similar efiniions apply o alf-open inervals. Problem 4. Wrie a efiniion for coninuiy over e alf-open inerval ab,. Problem 4. Referring o e funcion in Figure 4., ecie weer eac of e following saemens are rue or false. 4.. is coninuous on 4, 4.. is coninuous on 4, 4.. is coninuous on 4, 4..4 is coninuous on, 4..5 is coninuous on, 4..6 is coninuous on, is coninuous on, 4 Figure 4.: 5 Lab Aciviies

27 Porlan Communiy College MTH 5 Lab Manual Problem 4. Several funcions are escribe below. Your ask is o raw eac funcion on is provie ais sysem. Do no inrouce any unnecessary isconinuiies or inerceps a are no irecly implie by e sae properies. Make sure a you raw all implie asympoes an label em wi eir equaions. 4.. Draw ono Figure 4. a funcion a saisfies all of e following properies. 4 lim f 4 lim f 5 f 0 4 an f f f 4 5 lim lim 4 Figure 4.: f 4.. Draw ono Figure 4. a funcion a saisfies all of e following properies. lim lim g g g 0 4, g, an g 6 0 g is coninuous an as consan slope on 0,. 4.. Draw ono Figure 4.4 a funcion a saisfies all of e following properies. Figure 4.: g Te only isconinuiies on m occur a 4 an m as no -inerceps m lim m lim m lim m m as a consan slope of over, 4 m is coninuous over 4, Figure 4.4: m L ab Aciviies

28 Porlan Communiy College MTH 5 Lab Manual Aciviy 5 Disconinuiies are a lile more callenging o ienify wen working wi formulas an wen working wi graps. One reason for e ae ifficuly is a wen working wi a funcion formula you ave o ig ino your memory bank an rerieve funamenal properies abou cerain ypes of funcions. Problem Wa woul cause a isconinuiy on a raional funcion (a polynomial ivie by anoer polynomial)? 5.. Wa is always rue abou e argumen of e funcion, u, over inervals were e funcion y ln u is coninuous? 5.. Name ree values of were e funcion y an 5..4 Wa is e omain of e funcion k 4? 5..5 Wa is e omain of e funcion g 4? is isconinuous. Aciviy 6 Piece-wise efine funcions are funcions were e formula use epens upon e value of e inpu. Wen looking for isconinuiies on piece-wise efine funcions, you nee o invesigae e beavior a values were e formula canges as well as values were e issues iscusse in Aciviy 5 mig pop up. Problem 6. Tis quesion is all abou e funcion f sown o e rig. Answer quesion 6.. a eac of e values,, 4, 5, 7, an 8. A e values were e answer o quesion 6.. is yes, go aea an answer quesions ; skip quesions a e values were e answer o quesion 6.. is no. 6.. Is f isconinuous a e given value? 6.. Is f coninuous only from e lef a e given value? 6.. Is f coninuous only from e rig a e given value? 6..4 Is e isconinuiy removable? f 4 if 5 if 4 if if 7 8 Lab Aciviies

29 Porlan Communiy College MTH 5 Lab Manual Problem 6. Consier e funcion g sown o e rig. Te leer C represens e same real number in all ree of e piece-wise formulas. 6.. Fin e value for C a makes e funcion coninuous on,0. Make sure a your reasoning is clear. 6.. Is i possible o fin a value for C a makes e funcion coninuous over,? Eplain. C if 0 7 g C if 0 C 4 if 0 Problem 6. Consier e funcion f sown o e rig. Sae e values of were eac of e following occur. If a sae propery oesn occur, make sure a you sae a (as oppose o simply no responing o e quesion). No eplanaion necessary. 6.. A wa values of is f isconinuous? 6.. A wa values of is f coninuous from e lef bu no e rig? 5 if f if if A wa values of is f coninuous from e rig bu no e lef? 6..4 A wa values of oes f ave removable isconinuiies? 4 L ab Aciviies

30 Porlan Communiy College MTH 5 Lab Manual Inroucion o e Firs Derivaive Aciviy 7 Mos of e focus in e Raes of Cange lab was on average raes of cange. Wile e iea of raes of cange a one specific insan was allue o, we couln eplore a iea formally because we an ye alke abou limis. Now a we ave covere average raes of cange an limis we can pu ose wo ieas ogeer o iscuss raes of cange a specific insances in ime. Suppose a an objec is osse ino e air in suc a way a e elevaion of e objec (measure in f) is given by e funcion s were is e amoun of ime a as passe since e objec was osse (measure in s). Le s eermine e velociy of e objec secons ino is flig. s s Recall a e ifference quoien gives us e average velociy for e objec beween e imes an. So long as is posiive, we can ink of as e leng of e ime inerval. To infer e velociy eacly secons ino e flig we nee e ime inerval as close o 0 as possible; is is one using e appropriae limi in eample 7.. Eample 7. v s s lim lim lim lim lim 0 lim From is we can infer a e velociy of e objec secons ino is flig is 4 f/s. From a we know a e objec is falling a a spee of 4 f/s. Lab Aciviies 5

31 Porlan Communiy College MTH 5 Lab Manual Problem 7. Suppose a an objec is osse ino e air so a is elevaion (measure in m) is given by e funcion p were is e amoun of ime a as passe since e objec was osse (measure in s). 4 4 p p 7.. Evaluae lim 0 illusrae in eample 7.). sowing eac sep in e simplificaion process (as 7.. Wa is e uni for e value calculae in problem 7.. an wa oes e value (incluing uni) ell you abou e moion of e objec? 7.. Copy Table 7. ono your paper an compue an recor e missing values. Do ese values suppor your answer o problem 7..? Table 7.: Average Velociies p p 4 4 Aciviy 8 In previous aciviies we saw a if p is a posiion funcion, en e ifference quoien for p can p0 p0 be use o calculae average velociies an e epression lim calculaes e 0 insananeous velociy a ime 0. Grapically, e ifference quoien of a funcion f can be use o calculae e slope of secan lines o f. Wa appens wen we ake e run of e secan line o zero? Basically, we are connecing wo poins on e line a are really, really, (really), close o one anoer. As menione above, sening o zero urns an average velociy ino an insananeous velociy. Grapically, sening o zero urns a secan line ino a angen line. Te angen line o e funcion of is line is sown in Eample 8.. f a 4 is sown in Figure 8.. A calculaion of e slope 6 L ab Aciviies

32 Porlan Communiy College MTH 5 Lab Manual Eample 8. m an 4 4 f f lim lim lim lim 0 4 lim 0 4 lim 0 4 Figure 8.: f 40 4 You soul verify a e slope of e angen line sown in Figure 8. is inee. You soul 4 also verify a e equaion of e angen line is y. 4 Problem 8. Consier e funcion g Fin e slope of e angen line sown in Figure 8. g g using man lim. Sow work 0 consisen wi a illusrae in eample Use e line in Figure 8. o verify your answer o problem Sae e equaion of e angen line o g a. Figure 8.: g Lab Aciviies 7

33 Porlan Communiy College MTH 5 Lab Manual Aciviy 9 f a f a So far we ve seen wo applicaions of epressions of form lim. I urns ou a 0 is epression is so imporan in maemaics, e sciences, economics, an many oer fiels a i eserves a name in an of is own rig. We call e epression e firs erivaive of f a a. So far we ve always fie e value of a before making e calculaion. Tere s no reason wy we couln use a variable for a, make e calculaion, an en replace e variable wi specific values; in fac, i seems like is mig be a beer plan all aroun. Tis leas us o a efiniion of e firs erivaive funcion. Definiion 9. Te Firs Derivaive Funcion If f is a funcion of, en we efine e firs erivaive funcion, f, as: f f f lim. 0 Te symbols f are rea alou as f prime of or f prime a. As we ve alreay seen, a f gives us e slope of e angen line o f a a. We ve also seen a if s is a posiion funcion, en s a gives us e insananeous velociy a a. I s no oo muc of a srec o infer a e velociy funcion for s woul be v s Problem 9. A grap of e funcion in Eample 9.. f is sown in Figure 9. an e formula for f is erive 9.. Use e formula f o calculae 9.. Draw ono Figure 9. a line roug e poin oug e poin 5, wi a slope of f 5 jus rew? Wa are eir equaions? f an 5 f., wi a slope of f. Also raw a line. Wa are e names for e wo lines you 9.. Sowing work consisen wi a sown in eample 9., fin e formula for g g 5. were 8 L ab Aciviies

34 Porlan Communiy College MTH 5 Lab Manual Eample 9. f f f lim 0 lim 0 lim lim 0 lim 0 lim 0 0 Figure 9.: f y 0 Problem 9. Suppose a e elevaion of an objec (measure in f) is given by s 6 5 were is e amoun of ime a as passe since e objec was launce ino e air (measure in s). 9.. Use Equaion 9. o fin e formula for e velociy funcion associae wi is moion. Te firs wo lines of your presenaion soul be an eac copy of Equaion 9.. Equaion 9. v s lim s s Fin e values of v an v 5. Wa is e uni on eac of ese values? Wa o e values ell you abou e moion of e objec; on jus say e velociy escribe wa is acually appening o e objec secons an 5 secons ino is ravel. Lab Aciviies 9

35 Porlan Communiy College MTH 5 Lab Manual 9.. Use e velociy funcion o eermine wen e objec reaces is maimum elevaion. (Tink abou wa mus be rue abou e velociy a a insan.) Also, wa is e common maemaical erm for e poin on e parabola y s a occurs a a value of? 9..4 Use Equaion 9. o fin e formula for v. Te firs line of your presenaion soul be an eac copy of Equaion 9.. Equaion 9. v v v lim Wa is e common name for e funcion v? Is is formula consisen wi wa you know abou objecs in freefall on Ear? Problem 9. Wa is e consan slope of e funcion w? Verify is by using Definiion 9. o fin e formula for e funcion w. Aciviy 0 We can ink abou e insananeous velociy as being e insananeous rae of cange in posiion. In general, wenever you see e prase rae of cange you can assume a e rae of cange a one insan is being iscusse. Wen we wan o iscuss average raes of cange over a ime inerval we always say average rae of cange. In general, if f is any funcion, en f (a) ells us e rae of cange in f a a. Aiionally, if f is an applie funcion wi an inpu uni of i uni an an oupu uni of uni f a is funi. Please noe a is uni loses all meaning if i is simplifie in any way. Consequenly, we o iuni no simplify erivaive unis in any way, sape, or form. f, en e uni on For eample, if v is e velociy of your car (measure in mi/r) were is e amoun of ime v is mi/r min. a as passe since you i e roa (measure in minues), en e uni on 0 L ab Aciviies

36 Porlan Communiy College MTH 5 Lab Manual Problem 0. Deermine e uni for e firs erivaive funcion for eac of e following funcions. Remember, we o no simplify erivaive unis in any way, sape, or form. 0.. Vr is e volume of a spere (measure in ml) wi raius r (measure in mm). 0.. A is e area of a square (measure in f ) wi sies of leng (measure in f). 0.. V is e volume of waer in a baub (measure in gal) were is e amoun of ime a as elapse since e ub began o rain (measure in minues) R is e rae a wic a baub is raining (measure in gal/min) were is e amoun of ime a as elapse since e ub began o rain (measure in minues). Problem 0. Akbar was given a formula for e funcion escribe in problem 0... Akbar i some calculaions an ecie a e value of V 0 was (wiou uni).5. Nguyen ook one look an Akbar s value an sai a s wrong. Wa is i abou Akbar s value a cause Nguyen o ismiss i as wrong? Problem 0. Afer a wile Nguyen convince Akbar a e was wrong, so Akbar se abou oing e calculaion over again. Tis ime Akbar came up wi a value of,58. Nguyen ook one look a Akbar s value an eclare sill wrong. Wa s e problem now? Problem 0.4 Consier e funcion escribe in problem Wa woul i mean if e value of R was zero for all Wa woul i mean if e value of R was zero for all 0.5. Lab Aciviies

37 Porlan Communiy College MTH 5 Lab Manual L ab Aciviies

38 Porlan Communiy College MTH 5 Lab Manual Funcions, Derivaives, an Anierivaives Aciviy Funcions, erivaives, an anierivaives ave many enangle properies. For eample, over inervals were e firs erivaive of a funcion is always posiive, we know a e funcion iself is always increasing. (Do you unersan wy?) Many of ese relaionsips can be epresse grapically. Consequenly, i is imperaive a you fully unersan e meaning of some commonly use grapical epressions. Tese epressions are loosely efine in Table.. Table.: Some Common Grapical Prases Te funcion is posiive Tis means a e verical-coorinae of e poin on e funcion is posiive. As suc, a funcion is posiive wenever i lies above e orizonal ais. Te funcion is increasing Tis means a e verical-coorinae of e funcion consisenly increases as you move along e curve from lef o rig. Linear funcions wi posiive slope are always increasing. Te funcion is concave up A funcion is concave up a a if e angen line o e funcion a a lies below e curve. An upwar opening parabola is everywere concave up. Te funcion is negaive Tis means a e verical-coorinae of e poin on e funcion is negaive. As suc, a funcion is negaive wenever i lies below e orizonal ais. Te funcion is ecreasing Tis means a e verical-coorinae of e funcion consisenly ecreases as you move along e curve from lef o rig. Linear funcions wi negaive slope are always ecreasing. Te funcion is concave own A funcion is concave own a a if e angen line o e funcion a a lies above e curve. A ownwar opening parabola is everywere concave own. Problem. Answer eac of e following quesions in reference o e funcion sown in Figure.. Eac answer is an inerval (or inervals) along e -ais. Use inerval noaion wen epressing your answers. Make eac inerval as wie as possible; a is, o no break an inerval ino pieces if e inerval oes no nee o be broken up. Assume a e slope of e funcion is consan on, 5 4,.,, 4, an.. Over wa inervals is e funcion posiive?.. Over wa inervals is e funcion negaive?.. Over wa inervals is e funcion increasing?..4 Over wa inervals is e funcion ecreasing?..5 Over wa inervals is e funcion concave up?..6 Over wa inervals is e funcion concave own?..7 Over wa inervals is e funcion linear?..8 Over wa inervals is e funcion consan? Figure.: f Lab Aciviies

39 Porlan Communiy College MTH 5 Lab Manual Aciviy Muc informaion abou a funcion s firs erivaive can be gleane simply by looking a a grap of e funcion. In fac, a person wi goo visual skills can see e grap of e erivaive wile looking a e grap of e funcion. Tis aciviy focuses on elping you evelop a skill. Problem. A parabolic funcion is sown in Figure.. Eac quesion in is problem is in reference o a funcion... Several values of e funcion g are given in Table.. For eac given value raw a nice long line segmen a e corresponing poin on g wose slope is equal o e value of g. If we ink of ese line segmens as acual lines, wa o we call e lines?.. Wa is e value of g a? How o you know? Go aea an ener a value ino Table.... Te funcion g is symmeric across e line ; a is, if we move equal isance o e lef an rig from is line e corresponing y-coorinaes on g are always equal. Noice a e slopes of e angen lines are equal bu opposie a poins a are equally remove from e ais of symmery; is is reflece in e values of g an g. Use e iea of equal bu opposie slope equiisance from e ais of symmery o complee Table....4 Plo e poins from Table. ono Figure. an connec e os. Deermine e formula for e resulan linear funcion...5 Te formula for g is g Use Definiion 9. o eermine e formula for g...6 Te line you rew ono Figure. is no a angen line o g. Jus wa eacly is is line? Table.: y g y Figure.: g Figure.: g 4 L ab Aciviies

40 Porlan Communiy College MTH 5 Lab Manual Problem. A funcion f is sown in Figure. an e corresponing firs erivaive funcion f is sown in Figure.4. Answer eac of e following quesions referencing ese wo funcions... Draw e angen line o f a e ree poins inicae in Figure. afer firs using e grap of f o eermine e eac slope of e respecive angen lines... Wrie a senence relaing e slope of e angen line o f wi e corresponing y- coorinae on f... Copy eac of e following prases ono your paper an supply e wors or prases a correcly complee eac senence. Over e inerval were f is always negaive f is always. Over e inervals were f is always posiive f is always. Over e inerval were f is always increasing f is always. Over e inerval were f is always ecreasing f is always. Figure.: f Figure.4: f Problem. In eac of figures.5 an.6 a funcion (e in curve) is given; bo of ese funcions are symmeric abou e y-ais. Te firs erivaive of eac funcion (e ick curves) ave been rawn over e inerval 0,7. Use e given porion of e firs erivaive ogeer wi e symmery of e funcion o elp you raw eac firs erivaive over e inerval 7,0. y 0.5 Figure.5: g an g 0 Figure.6: k an k Lab Aciviies 5

41 Porlan Communiy College MTH 5 Lab Manual Problem.4 A grap of e funcion y is sown in Figure.7; call is funcion f..4. Ecep a 0, ere is someing a is always rue abou e value of f ; wa is e common rai?..4. Use Definiion 9. o fin e formula for f.4. Does e formula for f.4.4 Use e formula for f grap of y f. suppor your answer o problem.4.? o eermine e orizonal an verical asympoes for e.4.5 Keeping i simple, raw ono Figure.8 a curve wi e asympoes foun in problem.4.4 an e propery eermine in problem.4.. Does e curve you rew ave e properies you woul epec o see in e firs erivaive of f? For eample, f is concave own over,0 an concave up over 0, ; wa are e corresponing ifferences in e beavior of f over ose wo inervals? y 0 Figure.7: f 0 Figure.8: f Problem.5 A grap of e funcion g is sown in Figure.9. Te absolue minimum value ever obaine by g is. Wi a in min, raw g ono Figure.0. Make sure a you raw an label any an all necessary asympoes. Make sure a your grap of g aequaely reflecs e symmery in e grap of g. y 5 y Figure.9: g Figure.0: g 6 L ab Aciviies

42 Porlan Communiy College MTH 5 Lab Manual Problem.6 A funcion, w, is sown in Figure.. A larger version of Figure. is available in Appeni B (page B). Answer eac of e following quesions in reference o is funcion..6. An inflecion poin is a poin were e funcion is coninuous an e concaviy of e funcion canges. Te inflecion poins on w occur a,.5, an 6. Wi a in min, over eac inerval sae in Table. eacly wo of e wors in Table. apply o w. Complee Table. wi e appropriae pairs of wors. Table.: Properies posiive negaive increasing ecreasing Figure.: w Table.: Properies of w Inerval,,.5.5,4 4,6 6,7 7, Properies.6. In Table.4, ree possible values are given for w a several values of. In eac case, one of e values is correc. Use angen lines o w o eermine eac of e correc values. (Tis is were you probably wan o use e grap on page B.).6. Te value of w is e same a, 4, an 7. Wa is is common value?.6.4 Pu i all ogeer an raw w ono Figure.. Table.4: Coose e correc values for w Propose values 0 or 8 or 8 or or 5 or or 5 or or 6 4 or 8 or 4 8 or 6 or Figure.: w Lab Aciviies 7

43 Porlan Communiy College MTH 5 Lab Manual Aciviy A funcion is sai o be noniffereniable a any value is firs erivaive is unefine. Tere are ree grapical beaviors a lea o non-iffereniabiliy. f is noniffereniable a a if f is isconinuous a a. f is noniffereniable a a if e slope of f is ifferen from e lef an rig a a. f is noniffereniable a a if f as a verical angen line a a. Problem. Consier e funcion k sown in Figure.... Tere are four values were k is noniffereniable; wa are ese values?.. Draw k ono Figure.. Figure.: k Figure.: k Problem. Consier e funcion g sown in Figure.... g as been rawn ono Figure.4 over e inerval 5,.5. Use e piece-wise symmery an perioic beavior of g o elp you raw e remainer of g over.. Wa si syllable wor applies o g a 5, 0, an 5?.. Wa five syllable an si syllable wors apply o g a 5, 0, an 5? 7,7 Figure.: g Figure.4: g 8 L ab Aciviies

44 Porlan Communiy College MTH 5 Lab Manual Aciviy 4 Seeing as e firs erivaive of f is a funcion in is own rig, f mus ave is own firs erivaive. Te firs erivaive of f is e secon erivaive of f an is symbolize as f (f ouble-prime). Likewise, f (f riple-prime) is e firs erivaive of f, e secon erivaive of f, an e ir erivaive of f. Table 4.: f an f Wen f is f is Posiive Increasing Negaive Decreasing Consanly Zero Consan Increasing Concave Up Decreasing Concave Down Consan Linear Table 4.: f an f Wen f is f is Posiive Increasing Negaive Decreasing Consanly Zero Consan Increasing Concave Up Decreasing Concave Down Consan Linear All of e grapical relaionsips you ve esablise beween f an f work eir way own e erivaive cain; is is illusrae in ables 4., 4., an 4.. Problem 4. Erapolaing from ables 4. an 4., wa mus be rue abou f over inervals were f is, respecively, posiive, negaive, an consanly zero? Problem 4. A funcion, g, an is firs ree erivaives are sown in figures , aloug no in a orer. Deermine wic curve is wic funcion ( g, g, g, an g ). Table 4.: f an f Wen f is f is Posiive Increasing Negaive Decreasing Consanly Zero Consan Increasing Concave Up Decreasing Concave Down Consan Linear Figure 4. Figure 4. Figure 4. Figure 4.4 Lab Aciviies 9

45 Porlan Communiy College MTH 5 Lab Manual Problem 4. Tree conainers are sown in figures Eac of e following quesions are in reference o ese conainers. 4.. Suppose a waer is being poure ino eac of e conainers a a consan rae. Le 5, 6, an 7 be e eigs (measure in cm) of e liqui in conainers , respecively, secons afer e waer began o fill e conainers. Wa woul you epec e sign o be on e secon erivaive funcions 5, 6, 7 wile e conainers are being fille? (Hin: Tink abou e sape of e curves y y, an 7 y.), Suppose a waer is being raine from eac of e conainers a a consan rae. Le 5, 6, an 7 be e eigs (measure in cm) of e liqui remaining in e conainers secons afer e waer began o rain. Wa woul you epec e sign o be on e secon erivaive funcions 5, 6, 7 wile e conainers are being raine? Figure 4.5 Figure 4.6 Figure 4.7 Problem 4.4 During e recession of , e oal number of employe Americans ecrease every mon. One mon a alking ea on e elevision mae e observaion a a leas e secon erivaive was posiive is mon. Wy was i a goo ing a e secon erivaive was posiive? Problem 4.5 During e early 980s e problem was inflaion. Every mon e average price for a gallon of milk was iger an e mon before. Was i a goo ing wen e secon erivaive of is funcion was posiive? Eplain. 40 L ab Aciviies

46 Porlan Communiy College MTH 5 Lab Manual Aciviy 5 Te erivaive coninuum can be epresse backwars as well as forwars. Wen you move from funcion o funcion in e reverse irecion e resulan funcions are calle anierivaives an e process is calle aniiffereniaion. Tese relaionsips are sown in figures 5. an 5.. f f f f iffereniae iffereniae iffereniae iffereniae Figure 5.: Differeniaing f f f f F aniiffereniae aniiffereniae aniiffereniae aniiffereniae aniiffereniae Figure 5.: Aniiffereniaing Tere are (a leas) wo imporan ifferences beween e iffereniaion cain an e aniiffereniaion cain (besies eir reverse orer). Wen you iffereniae, e resulan funcion is unique. Wen you aniiffereniae, you o no ge a unique funcion - you ge a family of funcions; specifically, you ge a se of parallel curves. We inrouce a new funcion in e aniiffereniaion cain. We say a F is an anierivaive of f. Tis is were we sop in a irecion; we o no ave a variable name for an anierivaive of F. Since F is consiere an anierivaive of f, i mus be e case a f is e firs erivaive of F. Hence we can a F o our erivaive cain resuling in Figure 5. F f f f f iffereniae iffereniae iffereniae iffereniae iffereniae Figure 5.: Differeniaing Problem 5. Eac of e linear funcions in Figure 5.4 ave e same firs erivaive funcion. 5.. Draw is common firs erivaive funcion ono Figure 5.4 an label i g. 5.. Eac of e given lines Figure 5.4 is calle wa in relaion o g? Figure 5.4 Lab Aciviies 4

47 Porlan Communiy College MTH 5 Lab Manual Problem 5. Te funcion f is sown in Figure 5.5. Reference is funcion in e following quesions. 5.. A wa values of is f noniffereniable? 5.. A wa values of are anierivaives of f noniffereniable? 5.. Draw ono Figure 5.6 e coninuous anierivaive of f a passes roug e poin,. Please noe a every anierivaive of f increases eacly one uni over e inerval, Because f is no coninuous, ere are oer anierivaives of f a pass roug e poin,. Specifically, anierivaives of f may or may no be coninuous a. Draw ono figures 5.7 an 5.8 ifferen anierivaives of f a pass roug e poin,. Figure 5.5: f Figure 5.6 Figure 5.7 Figure 5.8 Problem 5. Te funcion y sin is an eample of a perioic funcion. Specifically, e funcion as a perio of because over any inerval of leng e beavior of e funcion is eacly e same as i was e previous inerval of leng. A lile more precisely, sin sin regarless of e value of. Jasmine was inking an ol er lab assisan a erivaives an anierivaives of perioic funcions mus also be perioic. Jasmine s lab assisan ol er a se was alf rig. Wic alf i Jasmine ave correc? Also, raw a funcion a illusraes a e oer alf of Jasmine s saemen is no correc. 4 L ab Aciviies

48 Porlan Communiy College MTH 5 Lab Manual Problem 5.4 Consier e funcion g sown in Figure Le G be an anierivaive of g. Suppose a G is coninuous on 6,6, a e greaes value G ever acieves is 6. Draw G ono Figure A wa values of is G noniffereniable? 5.4. A wa values of is g noniffereniable? G 6, an Figure 5.9 g Figure 5.0 G Problem 5.5 Answer e following quesion in reference o a coninuous funcion g wose firs erivaive is sown in Figure 5.. You o no nee o sae ow you mae your eerminaion; jus sae e inerval(s) or values of a saisfy e sae propery. Noe: Te correc answer o one or more of ese quesions may be Tere is no way of knowing Over ese inervals, g is posiive an increasing A ese values of, g is noniffereniable Over ese inervals, g is never negaive A ese values of, every anierivaive of g is noniffereniable. Figure 5. g Over ese inervals, e value of g is consan Over ese inervals, g is linear Over ese inervals, anierivaives of g are linear Over ese inervals, g is never negaive. Lab Aciviies 4

49 Porlan Communiy College MTH 5 Lab Manual Aciviy 6 Wen given a funcion formula, we ofen fin e firs an secon erivaive formulas o eermine beaviors of e given funcion. In a laer lab we will use e firs an secon erivaive formulas o elp us grap a funcion given e formula for e funcion. One ing we o wi e erivaive formulas is eermine were ey are posiive, negaive, zero, an unefine. Tis elps us eermine were e given funcion is increasing, ecreasing, concave up, concave own, an linear. Problem Draw ono Figure 6. a coninuous funcion f a as a orizonal angen line a e poin, along wi e properies sae in Table Given e coniions sae in problem 6.., oes i ave o be e case a f 0? 6.. Is f increasing a? How o you know? 6..4 Can a coninuous, everywere iffereniable funcion saisfy e properies sae in Table 6. an no ave a slope of zero a? Draw a picure a suppors your answer. Table 6.: Signs on f an f Inerval f f, Posiive Negaive, Posiive Posiive Figure 6. f Problem Draw ono Figure 6. a coninuous funcion g a as a verical angen line a e poin, along wi e properies sae in Table Given e coniions sae in problem 6.., oes i ave o be e case a g is unefine? 6.. Is g increasing a? How o you know? 6..4 Can a coninuous, everywere iffereniable funcion saisfy e properies sae in Table 6. an no ave a verical angen line a? Draw a picure a suppors your answer. Table 6.: Signs on g an g Inerval g g, Posiive Posiive, Posiive Negaive 44 L ab Aciviies Figure 6. g

50 Porlan Communiy College MTH 5 Lab Manual Problem Draw ono Figure 6. a coninuous funcion k a passes roug e poin, also saisfies e properies sae in Table A wa values of is k noniffereniable? an Table 6.: Signs on k an k Inerval k k, Negaive Posiive, Posiive Negaive Problem 6.4 Figure 6. k You soul know by now a e firs erivaive coninually increases over inervals were e secon erivaive is consanly posiive an a e firs erivaive coninually ecreases over inervals were e secon erivaive is consanly negaive. A person mig infer from is a a funcion canges more an more quickly over inervals were e secon erivaive is consanly posiive an a a funcion canges more an more slowly over inervals were e secon erivaive is consanly negaive. We are going o eplore a iea in is problem Suppose a V is e volume of waer in an ice cube (ml) were is e amoun of ime a as passe since noon (measure in minues). Suppose a V 6 0 ml an a min ml V as a consan value of. min over e inerval 6,. Wa is e value of min V? Wen is V canging more quickly, a :06 pm or a : pm? 6.4. Referring o e funcion V in problem 6.4., wa woul e sape of V be over e inerval 6,? (Coose from opions a.) a b c 6.4. Again referring o opions a-, wic funcions are canging more an more rapily from lef o rig an wic funcions are canging more an more slowly from lef o rig? Wic funcions ave posiive secon erivaives an wic funcions ave negaive secon erivaives? Do e funcions wi posiive secon erivaive values bo cange more an more quickly from lef o rig? Lab Aciviies 45

51 Porlan Communiy College MTH 5 Lab Manual Consier e signs on bo e firs an secon erivaives in opions a-. Is ere someing a e wo funcions a cange more an more quickly ave in common a is ifferen in e funcions a cange more an more slowly? Problem 6.5 Resolve eac of e following ispues One ay Sara an Jermaine were working on an assignmen. One quesion aske em o raw a funcion over e omain, wi e properies a e funcion is always increasing an always concave own. Sara insise a e curve mus ave a verical asympoe a an Jermaine insise a e funcion mus ave a orizonal asympoe somewere. Were eier of ese suens correc? Te ne quesion Sara an Jermaine encounere escribe e same funcion wi e ae coniion a e funcion is never posiive. Sara an Jermaine mae e same conenions abou asympoes. Is one of em now correc? 6.5. A anoer able Pero an Yosi were aske o raw a coninuous curve a, among oer properies, was never concave up. Pero sai OK, so e curve is always concave own o wic Yosi replie Pero, you nee o open your min o oer possibiliies. Wo s rig? In e ne problem Pero an Yosi were aske o raw a funcion a is everywere coninuous an a is concave own a every value of ecep. Yosi eclare impossible an Pero respone ave some fai, Yos-man. Pero en began o raw. Is i possible a Pero came up wi suc a funcion? Problem 6.6 Deermine e correc answer o eac of e following quesions. Picures of e siuaion may elp you eermine e correc answers Wic of e following proposiions is rue? If a given proposiion is no rue, raw a grap a illusraes is unru. 6.6.a If e grap of f as a verical asympoe, en e grap of f mus also ave a verical asympoe. 6.6.b If e grap of f as a verical asympoe, en e grap of f mus also ave a verical asympoe Suppose a e funcion f is everywere coninuous an concave own. Suppose furer a f 7 5 an f 7. Wic of e following is rue? 9 f 9 a. f 9 b. f c.. Tere is no enoug informaion o eermine e relaionsip beween f 9 an. 46 L ab Aciviies

52 Porlan Communiy College MTH 5 Lab Manual Derivaive Formulas Aciviy 7 Wile e primary focus of is lab is o elp you evelop sorcu skills for fining erivaive formulas, ere are ineviable noaional issues a mus be aresse. I urns ou a e laer issue is e one we are going o aress firs. If y f, we say a e erivaive of y wi respec o is equal o f we wrie: y f. Symbolically Wile e symbol y cerainly looks like a fracion, i is no a fracion. Te symbol is Leibniz noaion for e firs erivaive of y wi respec o. Te sor way of reaing e symbol alou is -y-- (on enunciae e ases). If z we wrie: g, we say a e e erivaive of z wi respec o is equal o g z g. (Rea alou as -z-- equals g-prime of. ). Symbolically Problem 7. Take e erivaive of bo sies of eac equaion wi respec o e inepenen variable as inicae in e funcion noaion. Wrie an say e erivaive using Leibniz noaion on e lef sie of e equal sign an funcion noaion on e rig sie of e equal sign. Make sure a every one in your group says a leas one of e erivaive equaions alou using bo e formal reaing an informal reaing of e Leibniz noaion. 7.. y k 7.. V f r 7.. T gp Aciviy 8 y is e name of a erivaive in e same way a f is e name of a erivaive. We nee a ifferen symbol a ells us o ake e erivaive of a given epression (in e same way a we ave symbols a ell us o ake a square roo, sine, or logarim of an epression). Te symbol is use o ell us o ake e erivaive wi respec o of someing. Te symbol iself is an incomplee prase in e same way a e symbol is an incomplee prase; in bo cases we nee o inicae e objec o be manipulae wa number or formula are we aking e square roo of? wa number or formula are we iffereniaing? Lab Aciviies 47

53 Porlan Communiy College MTH 5 Lab Manual One ing you can o o elp you remember e ifference beween e symbols y an is o ge in e abi of always wriing grouping symbols afer. In is way e symbols sin mean e erivaive wi respec o of e sine of. Similarly, e symbols erivaive wi respec o of -square. Problem 8. Wrie e Leibniz noaion for eac of e following epressions. 8.. Te erivaive wi respec o of cos. 8.. Te erivaive wi respec o of y. 8.. Te erivaive wi respec o of ln. (Yes, we will o suc ings.) 8..4 Te erivaive of z wi respec o. g 8. (Yes, we will also o suc ings.) 8..5 Te erivaive wi respec o of mean e Aciviy 9 Te firs iffereniaion rule we are going o eplore is calle e power rule of iffereniaion. n n n Equaion 9. for n 0 Wen n is a posiive ineger, i is fairly easy o esablis is rule using Definiion 9.. Te proof of e rule ges a lile more complicae wen n is negaive, fracional, or irraional. For purposes of is lab, we are going o jus accep e rule as vali. Tis rule is one you jus o in your ea an en wrie own e resul. Tree eamples of wa you woul be epece o wrie wen iffereniaing power funcions are sown below. Given Funcion y 7 7 f z 5 y Wa you soul ink or wrie (as necessary) Wa you soul wrie y y f 7 7/ f 7 4/4 z y 5 z 5y y 5 6 y 6 48 L ab Aciviies

54 Porlan Communiy College MTH 5 Lab Manual Noice a e ype of noaion use wen naming e erivaive is icae by e manner in 7 wic e original funcion is epresse. For eample, y is elling us e relaionsip beween wo variables; in is siuaion we name e erivaive using e noaion y. On e oer an, 7 funcion noaion is being use o name e rule in f ; in is siuaion we name e erivaive using e funcion noaion f. Problem 9. Fin e firs erivaive formula for eac of e following funcions. In eac case ake e erivaive wi respec o e inepenen variable as implie by e epression on e rig sie of e equal sign. Make sure a you use e appropriae name for eac erivaive f 9.. Aciviy 0 z P 9..4 Te ne rule you are going o pracice is e consan facor rule of iffereniaion. Tis is anoer rule you o in your ea. Equaion 0. kf k f for k In e following eamples an problems several erivaive rules are use a are sown in Appeni C (pages C5 an C6). Wile working is lab you soul refer o ose rules; you may wan o cover up e cain rule an implici erivaive columns is week! You soul make a goal of aving all of e basic formulas memorize wiin a week. Given Funcion Wa you soul wrie 9 6 f cos z ln f 8 54 f sin f z Problem 0. Fin e firs erivaive formula for eac of e following funcions. In eac case ake e erivaive wi respec o e inepenen variable as implie by e epression on e rig sie of e equal sign. Make sure a you use e appropriae name for eac erivaive. 0.. z P 7sin 0..4 z an 0..5 ln P T 4 Lab Aciviies 49

55 Porlan Communiy College MTH 5 Lab Manual Aciviy Wen an epression is ivie by e consan k, we can ink of e epression as being muliplie by e fracion k formula is muliplie or ivie by a consan.. In is way, e consan facor rule of iffereniaion can be applie wen a Given Funcion Wa you soul ink Wa you soul wrie 4 an f y z 8 f 8 an ln y 4 y z y f ln y sec z y y Problem. Fin e firs erivaive formula for eac of e following funcions. In eac case ake e erivaive wi respec o e inepenen variable as implie by e epression on e rig sie of e equal sign. Make sure a you use e appropriae name for eac erivaive... z Aciviy sin r Gmm.. V r.. f r 6 r G, m, an m are consans. Wen aking e erivaive of wo or more erms, you can ake e erivaives erm by erm an inser plus or minus signs as appropriae. Collecively we call is e sum an ifference rules of iffereniaion. Equaion. f g f g In e following eamples an problems we inrouce linear an consan erms ino e funcions being iffereniae. k k k k k Equaion. 0 for Equaion. for Problem. Eplain wy bo e consan rule (Equaion.) an linear rule (Equaion.) are obvious. Hin: Tink abou e graps y k an y k. Wa oes a firs erivaive ell you abou a grap? 50 L ab Aciviies

56 Porlan Communiy College MTH 5 Lab Manual A couple of eamples using e sum an ifference rule are sown below. Given Funcion Wa you soul ink or wrie (as necessary) Wa you soul efiniely wrie y P 5 6 y 6 y sin cos 4 sin cos 4 6/5 / 4 8 P 4 5 /5 / P cos sin Problem. Fin e firs erivaive formula for eac of e following funcions. In eac case ake e erivaive wi respec o e inepenen variable as implie by e epression on e rig sie of e equal sign. Make sure a you use e appropriae name for eac erivaive... T sin cos.. k r Aciviy 4sec csc 4 ln r ln 9 Te ne rule we are going o eplore is calle e prouc rule of iffereniaion. We use is rule wen ere are wo or more variable facors in e epression we are iffereniaing. (Remember, we alreay ave e consan facor rule o eal wi wo facors wen one of e wo facors is a consan.) Equaion. f g f g f g Inuiively, wa is appening in is rule is a we are alernaely reaing one facor as a consan (e one no being iffereniae) an e oer facor as a variable funcion (e one a is being iffereniae). We en a ese wo raes of cange ogeer. Ulimaely, you wan o perform is rule in your ea jus like all of e oer rules. Your insrucor, owever, may iniially wan you o sow seps; uner a presumpion, seps are going o be sown in eac an every eample of is lab wen e prouc rule is applie. Two simple eamples of e prouc rule are sown a e op of e ne page. Lab Aciviies 5

57 Porlan Communiy College MTH 5 Lab Manual Given Funcion Derivaive sin P e cos y y sin cos sin sin P e cos e sin e cos e cos A ecision you ll nee o make is ow o anle a consan facor in a erm a requires e f 5 ln are sown below. prouc rule. Two opions for aking e erivaive of Opion A f 5ln 0 ln 5 5 ln ln Opion B f 0 ln 5 0 ln 5 5 ln 5 ln In Opion B we are reaing e facor of 5 as a par of e firs variable facor. In oing so, e facor of 5 isribues iself. Tis is e preferre reamen of e auor, so is is wa you will see illusrae in is lab. Problem. Fin e firs erivaive formula for eac of e following funcions. In eac case ake e erivaive wi respec o e inepenen variable as implie by e epression on e rig sie of e equal sign. Make sure a you use e appropriae name for eac erivaive... T sec an.. e k f co co.. y 4ln..4 Problem. Fin eac of e following erivaives wiou firs simplifying e formula; a is, go aea an use e prouc rule on e epression as wrien. Simplify eac resulan erivaive formula. For eac erivaive, ceck your answer by simplifying e original epression an en aking e erivaive of a simplifie epression L ab Aciviies

58 Porlan Communiy College MTH 5 Lab Manual Aciviy 4 Te ne rule we are going o eplore is calle e quoien rule of iffereniaion. We never use is rule unless ere is a variable facor in e enominaor of e epression we are iffereniaing. (Remember, we alreay ave e consan facor rule o eal wi consan facors in e enominaor.) Equaion 4. f g f g f ; g 0 g g Like all of e oer rules, you ulimaely wan o perform e quoien rule in your ea. Your insrucor, owever, may iniially wan you o sow seps wen applying e quoien rule; uner a presumpion, seps are going o be sown in eac an every eample of is lab wen e quoien rule is applie. Two simple eamples of e quoien rule are sown below. Given Funcion Derivaive y y ln ln 4 ln 4 csc V ln an 4 ln4 ln ln 4 ln V csc an csc an an an an csc co an csc sec csc sec Problem 4. Fin e firs erivaive formula for eac of e following funcions. In eac case ake e erivaive wi respec o e inepenen variable as implie by e epression on e rig sie of e equal sign. Make sure a you use e appropriae name for eac erivaive. 4ln 4.. g 4.. jy 4.. y sin 4sec 4..4 f y cos Problem 4. Fin eac of e following erivaives wiou firs simplifying e formula; a is, go aea an use e quoien rule on e epression as wrien. For eac erivaive, ceck your answer by simplifying e original epression an en aking e erivaive of a simplifie epression. 4.. sin sin e y 0 Lab Aciviies 5

59 Porlan Communiy College MTH 5 Lab Manual Aciviy 5 In problems. an 4. you applie e prouc an quoien rules o epressions were e erivaive coul ave been foun muc more quickly a you simplifie e epression before aking e erivaive. For eample, wile you can fin e correc erivaive formula using e quoien rule wen working problem 4.., e erivaive can be foun muc more quickly if you simplify e epression before applying e rules of iffereniaion. Tis is illusrae in Eample 5.. Eample Par of learning o ake erivaives is learning o make goo coices abou e meoology o employ wen aking erivaives. In eamples 5. an 5., e nee o use e prouc rule or quoien rule is obviae by firs simplifying e epression being iffereniae. Fin y Eample 5. Problem if y seccos. Soluion y sec cos cos cos y 0 Problem Eample 5. Soluion Fin f if f 5 4. f f 54 L ab Aciviies

60 Porlan Communiy College MTH 5 Lab Manual Problem 5. Fin e erivaive wi respec o for eac of e following funcions afer firs compleely simplifying e formula being iffereniae. In eac case you soul no use eier e prouc rule or e quoien rule wile fining e erivaive formula. 5.. y 4 4sin 5.. g 5.. cos zsin cos 5..5 z T 4 6 ln ln Aciviy 6 Someimes bo e prouc rule an quoien rule nee o be applie wen fining a erivaive formula. Problem 6. Consier e funcions sin an g f e sin e. 6.. Discuss wy f an g are in fac wo represenaions of e same funcion. 6.. Fin f necessary). 6.. Fin g necessary). by firs applying e prouc rule an en applying e quoien rule (were by firs applying e quoien rule an en applying e prouc rule (were 6..4 Rigorously esablis a e formulas for f an g are inee e same. Aciviy 7 Derivaive formulas can give us muc informaion abou e beavior of a funcion. For eample, e erivaive formula for f is f. Clearly f is negaive wen is negaive an f is posiive wen is posiive. Tis ells us a f is ecreasing wen is negaive an a f is increasing wen is posiive. Tis maces e beavior of e parabola y. Lab Aciviies 55

61 Porlan Communiy College MTH 5 Lab Manual Problem 7. Answer eac of e following quesions abou applie funcions. 7.. Te amoun of ime (secons), T, require for a penulum o complee one perio is a L funcion of e penulum s leng (meers), L. Specifically, T were g is e g acceleraion consan for Ear (rougly 9.8 m/s ). 7...a 7...b Fin T L Te sign on T afer firs rewriing e formula for T as a consan imes L. L is e same regarless of e value of L. Wa is is sign an wa oes i ell you abou e relaive perios of wo penulums wi ifferen lengs? 7.. Te graviaional force (Newons) beween wo objecs of masses m an m (kg) is a funcion of e isance (meers) beween e objecs ceners of mass, r. Specifically, Gmm Fr were G is e universal graviaional consan (wic is approimaely r 6.70 n m/kg ). 7...a Leaving G, m, an m as consans, fin for F as a consan imes a power of r. 7...b Te sign on F r r F afer firs rewriing e formula is e same regarless of e value of F. Wa is is sign an wa oes i ell you abou e effec on e graviaional force beween wo objecs wen e isance beween e objecs is cange? 7...c Leaving G, m, an 7... F.0 0. Calculae F m as consans, evaluae F.00 0, F F,. Wic of e quaniies foun in par c comes closes o is value? Draw a skec of F an iscuss wy is resul makes sense. 56 L ab Aciviies

62 Porlan Communiy College MTH 5 Lab Manual Te Cain Rule Aciviy 8 Te funcions f sin an k sin 8.. Since f cos k cos. Bu is woul imply a k f are sown in Figure, i is reasonable o speculae a 0 0, an a quick glance of e wo funcions a 0 soul convince you a is k 0 f 0. is no rue; clearly Te funcion k moves roug ree perios for every one perio Figure 8.: f an k generae by e funcion f. Since e ampliues of e wo funcions are e same, e only way k can generae perios a a rae of : (compare o f ) is if is k cos ; please noe a is e firs rae of cange is ree imes a of f. In fac, erivaive of. Tis means a e formula for k e ousie funcion ( sin u cos u ) an e insie funcion ( u is e prouc of e raes of cange of ). k is an eample of a composie funcion (as illusrae in Figure 8.). If we efine g by e rule g k f g., en sin Noe a k0 g0 f g0 g Figure 8.: g, f u sinu, an k f g Taking e oupu from g an processing i roug a secon funcion, f, is e acion a caracerizes k as a composie funcion.. Tis las equaion is an eample of wa we call e cain rule for iffereniaion. Loosely, e cain rule ells us a wen fining e rae of cange for a composie funcion (a 0), we nee o muliply e rae of cange of e ousie funcion, f g0, wi e rae of cange of e insie funcion, g 0. Tis is symbolize for general values of in Equaion 8. were u represens a funcion of (e.g. u g ). f Equaion 8. f u fu u Tis rule is use o fin erivaive formulas in eamples 8. an 8.. Lab Aciviies 57

63 Porlan Communiy College MTH 5 Lab Manual Problem Eample 8. Soluion Te facor of a cain rule facor. is calle Fin y if y sin. y cos cos cos Fin f Eample 8. Te facor of Problem Soluion 9 if f sec. sec is calle a cain rule facor. 9 f sec 8 f 9sec sec 8 9sec sec an 9sec 9 an Problem Eample 8. Soluion Fin y if y 4. y ln 4 4 Please noe a e cain rule was no applie ere because e funcion being iffereniae was no a composie funcion. Wile you ulimaely wan o perform e cain rule sep in your ea, your insrucor may wan you o illusrae e sep wile you are firs pracicing e rule. For is reason, e sep will be eplicily sown in every eample given in is lab. Problem 8. Fin e firs erivaive formula for eac funcion. In eac case ake e erivaive wi respec o e inepenen variable as implie by e epression on e rig sie of e equal sign. Make sure a you use e appropriae name for eac erivaive (e.g. ). 8.. cos P sin 4 z 7 ln 8..5 z sin 8.. w co 8..6 P an y sin ln T 8..9 y sec e 58 L ab Aciviies

64 Porlan Communiy College MTH 5 Lab Manual Problem 8. A funcion, f, is sown in Figure 8.. Answer eac of e following quesions in reference o is funcion. 8.. Use e grap o rank e following in ecreasing orer: 0 f. f, f, f, f, an 8.. Te formula for f is f e e. Fin e formulas for f an f an use e formulas o verify your answer o problem Fin e equaion of e angen line o f a Fin e equaion of e angen line o f a 0. Figure 8.: f 8..5 Tere is an anierivaive of f a passes roug e poin 0,7. Fin e equaion of e angen line o is anierivaive a 0. Aciviy 9 Wen fining erivaives of comple formulas you nee o apply e rules for iffereniaion in e reverse of orer of operaions. For eample, wen fining sine e firs rule you nee o apply is e erivaive formula for nees o be applie is e prouc rule. sinu bu wen fining sin e e firs rule a Problem 9. Fin e firs erivaive formula for eac funcion. In eac case ake e erivaive wi respec o e inepenen variable as implie by e epression on e rig sie of e equal sign. Make sure a you use e appropriae name for eac erivaive (e.g. f ). 9.. f sine 9.. g sin e an ln cos 9.. y f y ln y sin y z G sin ln Aciviy 40 As always, you wan o simplify an epression before jumping in o ake is erivaive. Never-eless, i can buil confience o see a e rules work even if you on simplify firs. Lab Aciviies 59

65 Porlan Communiy College MTH 5 Lab Manual Problem 40. Consier e funcions f g an Assuming a is no negaive, ow oes eac of ese formulas simplify? Use e simplifie formula o fin e formulas for f g. an 40.. Use e cain rule (wiou firs simplifying) o fin e formulas for f an g simplify eac resul (assuming a is posiive) Are f an g e same funcion? Eplain wy or wy no. Problem 40. So long as falls on e inerval, an an Problem How oes e formula g ln e g an an. Use e cain rule o fin, an sow a i simplifies as i soul. ; simplify an wa oes is ell you abou e formula for? Afer answering ose quesions use e cain rule o fin e formula for g a i simplifies as i soul. Problem 40.4 Consier e funcion g Fin e formula for g 5 ln. sec wiou firs simplifying e formula for g. an sow Use e quoien, prouc, an power rules of logarims o epan e formula for g ino ree logarimic erms. Ten fin g by aking e erivaive of e epane version of g Sow a e wo resulan formulas are in fac e same. Also, reflec upon wic process of iffereniaion was less work an easier o clean up. Aciviy 4 So far we ave worke wi e cain rule as epresse using funcion noaion. In some applicaions i is easier o ink of e cain rule using Leibniz noaion. Consier e following eample During e 990s, e amoun of elecriciy use per ay in Eown increase as a funcion of populaion a e rae of 8 kw/person. On July, 997, e populaion of Eown was 00,000 an e populaion was ecreasing a a rae of 6 people/ay. In Equaion 4. (page 6) we use ese values o eermine e rae a wic elecrical usage was canging (wi respec o ime) in Eown on 7//997. Please noe a in is eremely simplifie eample we are ignoring all facors a conribue o ciywie elecrical usage oer an populaion (suc as emperaure). 60 L ab Aciviies

66 Porlan Communiy College MTH 5 Lab Manual Equaion 4. kw people kw person ay ay Le s efine g() as e populaion of Eown years afer January, 990 an f ( u ) as e aily amoun of elecriciy use in Eown wen e populaion was u. From e given informaion, ,000 g 7.5 6, an f ( u) 8 y f u were g ( ), ( ) u g(), en we ave (from Equaion 4.): for all values of u. If we le ( ) Equaion 4. y u y u u 00, You soul noe a Equaion 4. is an applicaion of e cain rule epresse in Leibniz noaion; f g 7.5 g 7.5. specifically, e epression on e lef sie of e equal sign represens ( ( )) ( ) In general, we can epress e cain rule as sown in Equaion 4.. Equaion 4. y y u u Problem 4. is Suppose a Carla is jogging in er swee new running soes. Suppose furer a r f ( ) Carla s pace (mi/r) ours afer pm an y ( r) is Carla s ear rae (bpm) wen se jogs a a rae of r mi/r. In is cone we can assume a all of Carla s moion was in one irecion, so e wors spee an velociy are compleely inercangeable. 4.. Wa is e meaning of f (.75) 7? 4.. Wa is e meaning of ( 7) 5? 4.. Wa is e meaning of ( )( ) 4..4 Wa is e meaning of 4..5 Wa is e meaning of f.75 5? r.75 y r r ? 8? 4..6 Assuming a all of e previous values are for real, wa is e value of wa oes is value ell you abou Carla? y.75 an Lab Aciviies 6

67 Porlan Communiy College MTH 5 Lab Manual Problem 4. Porions of SW 5 Avenue are eremely illy. Suppose a you are riing your bike along SW 5 Ave from Vermon Sree o Capiol Higway. Le u be e isance you ave ravelle (f) were is e number of secons a ave passe since you began your journey. Suppose a y eu is e elevaion (m) of SW 5 Ave were u is e isance (f) from Vermon S eae owars Capial Higway. 4.. Wa, incluing unis, woul be e meanings of 5 00, e 540? e 00 40, an 4.. Wa, incluing unis, woul be e meanings of u 5 4 an y 0. u u Suppose a e values sae in problem 4.. are accurae. Wa, incluing uni, is e y value of? Wa oes is value ell you in e cone of is problem? 5 Problem 4. Accoring o Hooke s Law, e force (lb), F, require o ol a spring in place wen is isplacemen from e naural leng of e spring is (f), is given by e formula F k were k is calle e spring consan. Te value of k varies from spring o spring. Suppose a i requires 0 lb of force o ol a given spring.5 f beyon is naural leng. 4.. Fin e spring consan for is spring. Inclue unis wen subsiuing e values for F an ino Hooke s Law so a you know e uni on k. 4.. Wa, incluing uni, is e consan value of F? 4.. Suppose a e spring is srece a a consan rae of.0 f/s. If we efine o be e amoun of ime (s) a passes since e srecing begins, wa, incluing uni, is e consan value of? 4..4 Use e cain rule o fin e consan value (incluing uni) of F. Wa is e coneual significance of is value? 6 L ab Aciviies

68 Porlan Communiy College MTH 5 Lab Manual Implici Differeniaion Aciviy 4 Some poins a saisfy e equaion y 4y are grape in Figure 4.. Clearly is se of poins oes no consiue a funcion were y is a funcion of ; for eample, ere are ree poins a ave an -coorinae of. Never-e-less, ere is a unique angen line o e curve a eac poin on e curve an so long as e angen line is no verical i as a unique slope. We sill ienify e value of is slope using e symbol y, so i woul be elpful if we a a Figure 4.: formula for y. If we coul solve y 4y snap. Hopefully you quickly see a suc an approac is jus no possible. for y, fining e formula for y woul be a To ge aroun is problem we are going o employ a ecnique calle implici iffereniaion. We use is ecnique o fin e formula for y wenever e equaion relaing an y is no eplicily solve for y. Wa we are going o o is rea y as if i were a funcion of an se e erivaives of e wo sies of e equaion equal o one anoer. Tis is acually a reasonable ing o o because so long as we are a a poin on e curve were e angen line is no verical, we coul make y a funcion of using appropriae resricions on e omain an range. Since we are reaing y as a funcion of, we nee o make sure a we use e cain rule wen iffereniaing erms like Since e name we give y. Wen u is a funcion of, we know a u u u y is y Te erivaion of y for e equaion y y., i follows a wen y is a funcion of, y 4 is sown in eample 4.. y 4y y y. Eample 4. y 4y y y y y y 4 y y 4y 4 y 4 Begin by iffereniaing bo sies of e equaion wi respec o. Te cain rule only comes ino play on e erms involving y. We now solve e equaion for y. Lab Aciviies 6

69 Porlan Communiy College MTH 5 Lab Manual A firs i mig be unseling a e formula for y conains bo e variables an y. However, if you ink i roug you soul conclue a e formula mus inclue e variable y; oerwise, ow coul e formula generae ree ifferen slopes a e poins,,, 0, an,? Tese slopes are given below. Te reaer soul verify eir values by rawing lines ono Figure 4. wi e inicae slopes a e inicae poins. y, 4 4 y, y, 4 4 Problem 4. Use e process of implici iffereniaion o fin a formula for y for e curves generae by eac of e following equaions. Do no simplify e equaions before aking e erivaives. You will nee o use e prouc rule for iffereniaion in problems sin sin y 6y y y y 4..4 y e 4..6 ye y e y y Problem 4. Several poins a saisfy e equaion y y e are grape in Figure 4.. Fin e slope an equaion of e angen line o is curve a e origin. (Please noe a you alreay foun e formula for y in problem 4..5.) Problem 4. Consier e se of poins a saisfy e equaion y 4. Figure 4.: y e 4.. Use implici iffereniaion o fin a formula for y. 4.. Fin a formula for y afer firs solving e equaion y 4 for y. 4.. Sow a e wo formulas are in fac equivalen so long as y 4. y 64 L ab Aciviies

70 Porlan Communiy College MTH 5 Lab Manual Problem 4.4 cos y 4 y A se of poins a saisfy e equaion is grape in Figure 4.. Fin e slope an equaion of e angen line o is curve a e poin 0, 4. Aciviy 4 Using e efiniion f f f lim, i is easy o esablis a 0 can use is formula an implici iffereniaion o fin e formula for cos y 4 y Figure 4.:. We. If y, en y (an y 0 ). Using implici iffereniaion we ave: y y y y y y Bu y, so we ave: y y In a similar manner, we can use e fac a sin. sin cos o come up wi a formula for Lab Aciviies 65

71 Porlan Communiy College MTH 5 Lab Manual If y sin, en sin y an y. Tis gives us: y y sin y cosy y sin y y cosy sin y Tis epression comes from e rigonomeric ieniy sin cos. If you solve a equaion for cos you ge cos sin funcion y sin. Because e never as negaive slope we can iscar e negaive soluion o e equaion. Here we use a fac a sin y. Problem 4. e e Use e fac a Begin by using e fac a ogeer wi implici iffereniaion o sow a ln iffereniae bo sies of e equaion y ln implies a y e y e (an y 0 ). Your firs sep is o wi respec o.. Problem 4. Use e fac a ln e e ogeer wi implici iffereniaion o sow a. Begin by using e fac a bo sies of e equaion y ln y e implies a ln y wi respec o.. Your firs sep is o iffereniae Problem 4. Use e fac a an sec ogeer wi implici iffereniaion o sow a an. Begin by using e fac a y an implies a an y (an y ). Your firs sep is o iffereniae bo sies of e equaion an y wi respec o. Please noe a you will nee o use e Pyagorean ieniy a relaes e angen an secan funcions wile working is problem. 66 L ab Aciviies

72 Porlan Communiy College MTH 5 Lab Manual Relae Raes Aciviy 44 Consier e oy rocke sown in Figure 44.. As e rocke s elevaion (, measure in fee) canges e angle of elevaion from e groun o e base of e rocke (, measure in raians) also canges. Consequenly, ere is a relaionsip beween e raes a wic e elevaion an angle of elevaion cange. Tis is an eample of e opic we call relae raes. If we assume a e rocke flies sraig upwar (an subsequenly falls sraig ownwar) an we efine an as funcions of ime,, were is e amoun of ime a as passe (s) since e rocke was launce, en e raes of cange in e elevaion an angle of elevaion are, respecively, (measure in f/s) an (measure in ra/s). Using simple rig riangle rigonomery we eermine e relaion equaion beween an (Equaion 44.). Since an bo vary as funcions of, we can iffereniae bo sies of Equaion 44. wi respec o resuling in e rae equaion (Equaion 44.) 60 f Figure 44.: Jonny s Rocke Equaion 44.: an 60 Equaion 44.: sec 60 Problem 44. Raes of cange in e rocke s elevaion are given for wo elevaions in Table Use Equaion 44. o eermine e values of a e given elevaions. Do no approimae ese value sae eir eac values Use Equaion 44. o eermine e corresponing values of. Roun eac of ese values o e neares unre Wrie wo complee senences a fully communicae wa is appening o e rocke s elevaion an angle of elevaion as implie by e values in Table 44.. Eac senence soul inclue e elevaion of e rocke, a clear inicaion of weer e rocke is rising or falling, e spee a wic e rocke is moving, a clear inicaion of weer e angle of elevaion is increasing or ecreasing, an e rae a wic e angle is increasing or ecreasing. All values soul be sae using appropriae unis. Table 44.: Tracking Jonny s Rocke (f) 60 6 (f/s) (ra) (ra/s) 60 8 Lab Aciviies 67

73 Porlan Communiy College MTH 5 Lab Manual Aciviy 45 Tere is someing quie ifferen abou e manner in wic variables are efine wile working relae raes problems from e way in wic you efine variables wen working applie problems in earlier classes. In mos applicaions, you efine your variables o eplicily represen e quaniy (or quaniies) you are aske o fin. Wen efining variables for a relae raes problem, owever, you efine a variable from wic you can infer e value you are rying o fin an, more uniquely, you efine variables for quaniies wose values you are acually given! Problem 45. A PM one ay, Muffin an Re were sleeping aop one anoer. A lou noise sarle e animals an Muffin began o run ue nor a a consan rae of.7 f/s wile Re ran ue eas a a consan rae of. f/s. Te criers mainaine ese pas an raes for several secons. Suppose a you wane o calculae e rae a wic e isance beween e ca an og was canging ree secons ino eir run A rig riangle represening is problem as been rawn in Figure 45.. Te leng of eac sie an e measure of eac angle ave (emporarily) been assigne variable names; ese measuremens are e poenial variables for e problem. Circle e variable wose rae you are rying o eermine an e wo variables wose rae you are given. Tese ree measuremens are e acual variables for e problem One of e remaining pieces in Figure 45. oes no cange value as e animals run. Cross ou e corresponing variable an replace i wi is fie value Cross ou any poenial variable no aresse in problems 45.. an 45..; aloug canging value, ese pieces of e picure are irrelevan o e quesion a an Copy e new an improve iagram (wi e relevan variables an fie value) ono your own paper; is is your working iagram for e problem. Eplicily efine a ime variable an en efine your ree leng variables making sure a you esablis eir epenence upon ime. Te rae variables will emerge wen you iffereniae e leng variables wi respec o ime Use your iagram o eermine e relaion equaion an en iffereniae bo sies of a equaion wi respec o ime. Te resulan equaion is your rae equaion Wa are e values of e leng variables ree secons ino e animals panicke runs? Wic rae values in e rae equaion o you know an wa are eir values? Subsiue e known values ino e rae equaion an solve for e unknown rae Wrie a coneual conclusion senence a clearly inicaes weer e isance beween e ca an og is increasing or ecreasing an clearly communicaes e rae a wic is cange is appening. a c Figure 45.: Re an Muffin an e Poenial Variables b 68 L ab Aciviies

74 Porlan Communiy College MTH 5 Lab Manual Problem 45. Scuyler s clock is kapu; e minue an funcions as i soul bu e our an is suck a 4. Te minue an on e clock is 0 cm long an e our an is 0 cm long. In is problem you are going o eermine e rae of cange beween e ips of e ans every ime e minue an poins irecly a A riangle represening is problem as been rawn in Figure 45.; e minue an as eliberaely been rawn so a i poins a a number oer an. Tis iagram represens e moion of e minue an, no is posiion a one specific ime. Te our an, owever, as been rawn is in is fie posiion. Te leng of eac sie of e riangle an e measure of eac angle of e riangle ave (emporarily) been assigne variable names; ese are e poenial variables for e problem. Circle e variable wose rae you are rying o eermine. One of e variables represens a piece wose rae of cange is consan wen e minue an falls beween 0 an 4 (in e clock-wise irecion). Ienify is piece an circle e corresponing variable. Te wo circle variables are e acual variables for e problem Two of e remaining pieces in Figure 45. o no cange value as e ime passes. Cross ou e corresponing variables an replace em wi eir fie values Cross ou any poenial variable no aresse in problems 45.. an 45..; aloug canging value, ese pieces of e picure are irrelevan o e quesion a an Copy e new an improve iagram (wi e relevan variables an fie values) ono your own paper; is is your working iagram for e problem. Eplicily efine a ime variable an en efine your angular an leng variables making sure a you esablis eir epenence upon ime. Make sure a you use raians as e measure of e angular variable. Te rae variables will emerge wen you iffereniae e angular an leng variables wi respec o ime Use your iagram o eermine e relaion equaion an en iffereniae bo sies of a equaion wi respec o ime. Te resulan equaion is your rae equaion. You mig wan o look up e law of cosines before wriing own your relaion equaion Wa are e values of e variables wen e minue an poins o? Wa is e known rae value in your rae equaion? (Careful is is rae posiive or negaive?) Subsiue e known values ino e rae equaion an solve for e unknown rae. Roun your soluion o e neares Wrie a coneual conclusion senence a clearly inicaes weer e isance beween e ip of e clock ans is increasing or ecreasing an clearly communicaes e rae a wic is cange is appening. a b c Figure 45.: Scuyler s clock Figure 45.: Poenial Variables Lab Aciviies 69

75 Porlan Communiy College MTH 5 Lab Manual Aciviy 46 Someimes e relaion equaion is base upon a given or known formula. Problem 46. Slusy is flowing ou of e boom of a cup a e consan rae of 0. cm /s. Te cup is e sape of a rig circular cone. Te eig of e cup is cm an e cup (wen full) ols a oal of 6 cm of slusy. Deermine e rae a wic e eig of e remaining slusy canges a e insan ere are eacly 8 cm of slusy remaining in e cup Te volume formula for a rig circular cyliner is V r were V is e volume of e cone, is e eig of e cone, an r is e raius a e op of e cone. We ulimaely are going o efine wo of ese variables in erms of e amoun of slusy remaining in e cup. Wic are e wo variables relevan o e quesion a an? Ta is, wic quaniy s rae of cange are you given an wic quaniy s rae of cange are you rying o eermine? 46.. Hopefully you eermine a V an are e relevan variables. Tis means a r nees o be eliminae from e volume formula. A cross secion of e cup an slusy is sown in Figure 46.. Wa is e raius a e op of e cup? Use e concep of similar riangle o epress r in erms of. Subsiue is epression ino e volume formula an simplify. Te resulan equaion is e relaion equaion for e problem Eplicily efine an V (incluing unis) in erms of e amoun of slusy remaining in e cup. Make sure a you communicae a eac variable is epenen upon ime an a you eplicily efine your ime variable. Te rae variables will emerge wen you iffereniae e eig an volume variables wi respec o ime Go aea an complee e problem in a manner consisen wi a implie in problems 45. an 45..? r Figure 46.: Slusy cups Figure 46.: Slusy cross-secion 70 L ab Aciviies

76 Porlan Communiy College MTH 5 Lab Manual Problem 46. In e classic TV series Baman, sory lines almos always lase for wo episoes. A e en of e firs episoe, Baman an Robin were invariably caug in some iabolical rap inene o en eir lives. In one episoe, Eggea ecie o bo roas an crus e ynamic uo. He place e cape crusaers in a room wose emperaure seaily rose as wo of e walls close in on e men in igs. Accoring o e ieal gas law, e pressure (P), emperaure (T), an volume (V) insie e room were relae by e equaion PV gas a were in e room. T k were k is a consan specific o e ypes an amouns of Wen e eroes were place in e room, e room was m m m, e emperaure in e room was 0 C, an e air pressure in e room was 00 kpa. Once Eggea acivae e rap, wo of e walls move owar eac oer a e consan rae of. m/min (in oal) an e emperaure in e room rose a a consan rae of C/min. Since e volume an emperaure bo cange a consan raes, i mig seem inuiive o you a e pressure also cange a a consan rae. Le s see if a is in fac e case Use e iniial coniions in e room o eermine e value of k (wiou uni) an rewrie e ieal gas law using is value for k Eplicily efine P, T, an V (incluing unis) in erms of e amoun of ime a a passe since e rap was se in moion Your rae equaion comes from iffereniaing e ieal gas law equaion wi respec o ime. Tere is a simple algebraic ajusmen you can make o e equaion a will grealy simplify is ask. Go aea an ajus e equaion an en fin your rae equaion Deermine e values of P, T, an V a bo five minues an eig minues ino e moion of e rap. Sae e known rae vales (remembering o ink abou e sign on eac rae). You mig wan o ink wice abou e value of V. Use all of ese values along wi e rae equaion o eermine e value of P e moion of e rap. a bo five minues an eig minues ino Figure 46.: Eggea Figure 46.4: Te Dynamic Duo Lab Aciviies 7

77 Porlan Communiy College MTH 5 Lab Manual Aciviy 47 Solve eac of e following problems using proceures similar o ose suggese in e previous problems. Problem 47. I was a ark nig an 5.5 f all Baram was walking owars a sree lamp wose lig was perce 40 f ino e air. As e walke, e lig cause a saow o fall bein Baram. Wen Baram was 80 fee from e base of e lamp e was walking a a pace of f/s. A wa rae was e leng of Baram s saow canging a a insan? Make sure a eac sae an calculae rae value as e correc sign. Make sure a your conclusion senence clearly communicaes weer e leng of e saow was increasing or ecreasing a e inicae ime. Problem 47. Te graviaional force, F, beween wo objecs in space wi masses m an m is given by e Gmm formula F were r is e isance beween e objecs ceners of mass an G is e r universal graviaional consan. Two pieces of space junk, one wi mass 500 kg an e oer wi mass 000 kg, were rifing irecly owar one anoer. Wen e objecs ceners of mass were 50 km apar e liger piece was moving a a rae of 0.5 km/our an e eavier piece was moving a a rae of 0.9 km/our. Leaving G as a consan, eermine e rae a wic e graviaional force beween e wo objecs was canging a a insan. Make sure a eac sae an calculae rae value as e correc sign. Make sure a your conclusion senence clearly communicaes weer e force was increasing or ecreasing a a given ime. Te uni for F is Newons (N). Problem 47. An eigeen inc penulum siing aop a able is in e ownwar par of is moion. Te pivo poin for e penulum is 0 inces above e able op. Wen e penulum is 45 away from verical, e angle forme a e pivo is ecreasing a e rae of 5 /s. A wa rae is e en of e penulum approacing e able op a is insan? Hin: Draw a rig riangle wi one acue angle vere a e pivo poin an e oer acue angle vere a e en of e penulum. Te rae of cange of one piece of is riangle is e same as e rae you are rying o fin. Make sure a eac sae an calculae rae value as e correc sign. Make sure a you use appropriae unis wen efining your variables. 7 L ab Aciviies

78 Porlan Communiy College MTH 5 Lab Manual Criical Numbers an Graping from Formulas Aciviy 48 In e firs aciviy of is lab you are going o iscuss a few quesions wi your group maes a will opefully moivae you for one of e opics covere in e lab. Problem 48. Discuss ow you coul use e firs erivaive formula o elp you eermine e vere of e parabola y 8 7 an en eermine e vere. Remember a e vere is a poin in e y-plane an as suc is ienifie using an orere pair. Problem 48. Te curves in figures 48. an 48. were generae by wo of e four funcions given below. Use e given funcions along wi eir firs erivaives o eermine wic funcions generae e curves. Please noe a e y-scales ave eliberaely been omie from e graps an a ifferen scales were use o generae e wo graps. Resis any empaion o use your calculaor; use of your calculaor oally obviaes e poin of e eercise. Poenial Funcions: f C 0/7 f C /7 C C 4 are unknown consans /7 0/7 f C f C 4 4 Figure 48.: mysery curve Figure 48.: mysery curve Aciviy 49 Te vere of e parabola in problem 48. is calle a local maimum poin an e poins,c an,c in problem 48. are calle local minimum poins. Collecively, ese poins are calle 4 local ereme poins. Lab Aciviies 7

79 Porlan Communiy College MTH 5 Lab Manual Wile working Aciviy 48 you opefully came o e conclusion a e local ereme poins a cerain caracerisics in common. In e firs place, ey mus occur a a number in e omain of e funcion (wic eliminae f an f from conenion in problem 48.). Seconly, one of wo ings mus be rue abou e firs erivaive wen a funcion as a local ereme poin; i eier as a value of zero or i oes no eis. Tis leas us o e efiniion of a criical number of a funcion. Definiion 49. Criical Numbers If f is a funcion, en we efine e criical numbers of f as e numbers in e omain of f were e value of f is eier zero or oes no eis. Problem 49. Te funcion g sown in Figure 49. as a verical angen line a. Veronica says a is a criical number of g bu Tio isagrees. Tio conens a is no a criical number because g oes no ave a local ereme poin a. Wo s rig? Figure 49.: g Problem 49. Answer eac of e following quesions (using complee senences) in reference o e funcion f sown in Figure Wa are e criical numbers of f? 49.. Wa are e local ereme poins on f? Classify e poins as local minimums or local maimums an remember a poins on e plane are represene by orere pairs Wa is e absolue maimum value of f over e inerval 7,7. Please noe a e funcion value is e value of e y-coorinae a e poin on e curve an as suc is a number. 5 Figure 49.: f 74 L ab Aciviies

80 Porlan Communiy College MTH 5 Lab Manual Problem 49. Decie weer eac of e following saemens is rue or false A funcion always as a local ereme poin a eac of is criical numbers If e poin, 49.. If g.7 0, en g mus ave a local ereme poin a.7. is a local minimum poin on, en mus be a criical number of If g.7 0, en.7 mus be a criical number of g If g 9 oes no eis, en 9 mus be a criical number of g. Aciviy 50 Wen fining criical numbers base upon a funcion formula, ere are ree issues a nee o be consiere; e omain of e funcion, e zeros of e firs erivaive, an e numbers in e omain of e funcion were e firs erivaive is unefine. Wen wriing a formal analysis of is process eac of ese quesions mus be eplicily aresse. Te following ouline sows e work you nee o sow wen you are aske o wrie a formal eerminaion of criical numbers base upon a funcion formula. A process for formal eerminaion of criical numbers Please noe a you soul presen your work in narraive form using complee senences a esablis e significance of all sae inervals an values. For eample, for e funcion 7 e firs senence you soul wrie is Te omain of g is g 7,.. Using inerval noaion, sae e omain of e funcion.. Differeniae e funcion an compleely simplify e resulan formula. Simplificaion inclues: a. Manipulaing all negaive eponens ino posiive eponens. b. If any raional epression occurs in e formula, all erms mus be wrien in raional form an a common enominaor mus be esablise for e erms. Te final epression soul be a single raional epression. c. Te final epression mus be compleely facore.. Sae e values were e firs erivaive is equal o zero; sow any non-rivial work necessary in making is eerminaion. Make sure a you wrie a senence aressing is issue even if e firs erivaive as no zeros. 4. Sae e values in e omain of e funcion were e firs erivaive oes no eis; sow any non-rivial work necessary in making is eerminaion. Make sure a you wrie a senence aressing is issue even if no suc numbers eis. 5. Sae e criical numbers of e funcion. Make sure a you wrie is conclusion even if e funcion as no criical numbers. Lab Aciviies 75

81 Porlan Communiy College MTH 5 Lab Manual Because e omain of a funcion is suc an imporan issue wen eermining criical numbers, ere are some ings you soul keep in min wen eermining omains. Division by zero is never a goo ing. Over e real numbers, you canno ake even roos of negaive numbers. Over e real numbers, you can ake o roos of negaive numbers. n 0 0 for any posiive ineger n. Over e real numbers, you can only ake logarims of posiive numbers. Te sine an cosine funcion are efine a all real numbers. You soul ceck wi your lecure insrucor o see if you are epece o know e omains of e oer four rigonomeric funcions an/or e inverse rigonomeric funcions. Problem 50. Formally esablis e criical numbers for eac of e following funcions following e proceure ouline on page 75. f 9 4 g p 8 / z ln cos y e T Problem 50. Te firs erivaive of e funcion m 5 7 is m Rolan says a 5 is a criical number of m bu Yuna isagrees. Wo is correc an wy? 50.. Rolan says a 7 is a criical number of m bu Yuna isagrees. Wo is correc an wy? 50.. Rolan says a is a criical number of m bu Yuna isagrees. Wo is correc an wy? Aciviy 5 Once you ave eermine e criical numbers of a funcion, e ne ing you mig wan o eermine is e beavior of e funcion a eac of is criical numbers. One way you coul o a involves a sign able for e firs erivaive of e funcion Problem 5. Te firs erivaive of e funcion f 5 4 is f 5 9 Te criical numbers of f are rivially sown o be 5 an 9.. Copy Table 5. ono your paper an fill in e missing informaion. Ten sae e local minimum an maimum poins on f. Specifically aress bo minimum an maimum poins even if one an/or e oer oes no eis. Remember a poins on e plane are represene by orere pairs. Make sure a you sae poins on f an no f! 76 L ab Aciviies

82 Porlan Communiy College MTH 5 Lab Manual Table 5.: f5 9 Inerval Sign of f Beavior of f,5 5,9 9, Problem 5. Te firs erivaive of e funcion g 4 is g Sae e criical numbers of g; you o no nee o sow a formal eerminaion of e criical numbers. You o nee o wrie a complee senence. 5.. Copy Table 5. ono your paper an fill in e missing informaion. Table 5.: g 5 4 Inerval Sign of g Beavior of g 4,5 5, 5.. Wy i we no inclue any par of e inerval,4 in Table 5.? 5..4 Formally we say a g 0 is a local minimum value of g if ere eiss an open inerval over wic g g cenere a 0 0 for every value of on a inerval (oer an 0, of course). Since g is no efine o e lef of 4, i is impossible for is efiniion o be saisfie a 4; ence g oes no ave a local minimum value a 4. Sae e local minimum an maimum poins on g. Specifically aress bo minimum an maimum poins even if one an/or e oer oes no eis Wrie a formal efiniion for a local maimum poin on g. Lab Aciviies 77

83 Porlan Communiy College MTH 5 Lab Manual Problem 5. Te firs erivaive of e funcion k is k Sae e criical numbers of k; you o no nee o sow a formal eerminaion of e criical numbers. You o nee o wrie a complee senence. 5.. Copy Table 5. ono your paper an fill in e missing informaion. Ten sae e local minimum an maimum poins on k. Specifically aress bo minimum an maimum poins even if one an/or e oer oes no eis. Table 5.: 4 k Inerval Sign of k Beavior of k,,, 4 4, 5.. Te number was inclue as an enpoin in Table 5. even oug is no a criical number of k. Wy i we ave o inclue e inervals, an, in e able as oppose o jus using e single inerval,? Problem 5.4 Perform eac of e following for e funcions in problems Formally esablis e criical numbers of e funcion. Creae a able similar o ables Number e ables, in orer, Don forge o inclue able eaings an column eaings. Sae e local minimum poins an local maimum poins on e funcion. Make sure a you eplicily aress bo ypes of poins even if ere are none of one ype an/or e oer k g F ln 78 L ab Aciviies

84 Porlan Communiy College MTH 5 Lab Manual Problem 5.5 f 9. Consier a funcion f wose firs erivaive is Is 9 efiniely a criical number of f? Eplain wy or wy no. f 5.5. Oer an a 9, wa is always e sign of? Wa oes is sign ell you abou e funcion f? 5.5. Wa ype of poin oes f ave a 9? (Hin, raw a freean skec of e curve.) How coul e secon erivaive of f be use o confirm your conclusion in problem 5.5.? Go aea an o i. Aciviy 5 Wen searcing for inflecion poins on a funcion, you can narrow your searc by ienifying numbers were e funcion is coninuous (from bo irecions) an e secon erivaive is eier zero or unefine. (By efiniion an inflecion poin canno occur a a number were e funcion is no coninuous from bo irecions.) You can en buil a sign able for e secon erivaive a implies e concaviy of e given funcion. Wen performing is analysis, you nee o simplify e secon erivaive formula in e same way you simplify e firs erivaive formula wen looking for criical numbers an local ereme poins. Problem 5. Ienify e inflecion poins for e funcion sown in Figure 49. (page 74). Problem 5. Te firs wo erivaives of e funcion y 5. y are y 4 an 5.. Yolana was given is informaion an aske o fin e inflecion poins on y. Te firs ing Yolana wroe was Te criical numbers of y are an. Eplain o Yolana wy is is no rue 5.. Wa are e criical numbers of y an in wa way are ey imporan wen aske o ienify e inflecion poins on y? 5.. Copy Table 5. (page 80) ono your paper an fill in e missing informaion Sae e inflecion poins on y; you may roun e epenen coorinae of eac poin o e neares unre Te funcion y as a verical asympoe a. Given a fac, i was impossible a y woul ave an inflecion poin a. Wy, en, i we never-e-less break e inerval, a wen creaing our concaviy able? Lab Aciviies 79

85 Porlan Communiy College MTH 5 Lab Manual Table 5.: y ( ) ( + )( ) ( + ) 5 Inerval Sign of y Beavior of y (, ) (, ) (, ) (, ) Problem 5. Perform eac of e following for e funcions in problems Sae e omain of e funcion. Fin, an compleely simplify, e formula for e secon erivaive of e funcion. I is no necessary o simplify e formula for e firs erivaive of e funcion. Sae e values in e omain of e funcion were e secon erivaive is eier zero or oes no eis. Creae a able similar o Table 5.. Number e ables, in orer, Don forge o inclue able eaings an column eaings. Sae e inflecion poins on e funcion. Make sure a you eplicily aress is quesion even if ere are no inflecion poins. f ( ) ( ) ( ) G ( ) g e Problem Te secon erivaive of e funcion w ( ) 9 is w () inflecion poins. Wy is a? ( + ) ye w as no.5 4 Aciviy 5 We are frequenly inerese in a funcion s en beavior; a is, wa is e beavior of e funcion as e inpu variable increases wiou boun or ecreases wiou boun. 80 L ab Aciviies

86 Porlan Communiy College MTH 5 Lab Manual Many imes a funcion will approac a orizonal asympoe as is en beavior. Assuming a e orizonal asympoe y L represens e en beavior of e funcion f bo as increases wiou boun an as ecreases wiou boun, we wrie lim an lim f L f L. Wile working e Limis an Coninuiy lab you invesigae sraegies for formally esablising limi values as or. In is aciviy you are going o invesigae a more informal sraegy for eermining ese ype limis. 4 Consier lim. Wen e value of is really large, we say a e erm 4 ominaes 0 e numeraor of e epression 4 an e erm 0 ominaes e enominaor. We 0 acually call ose erms e ominan erms of e numeraor an enominaor. Te ominan erms are significan because wen e value of is really large, e oer erms in e epression conribue almos noing o e value of e epression. Ta is, for really large values of : For eample, even if as e palry value of,000, Tis ells us a 4 y. 0 4 lim 0 5 an a y is a orizonal asympoe for e grap of 5 Problem 5. Te formulas use o grap figures (pages B an B) are given below. Focusing firs on e ominan erms of e epressions, mac e formulas wi e funcions (f roug f 5) y y y 6 5 y y Lab Aciviies 8

87 Porlan Communiy College MTH 5 Lab Manual Problem 5. Use e concep of ominan erms o informally eermine e value of eac of e following limis lim 4 8 lim lim lim e 8e lim e e 5..6 lim 4e 8e e e Aciviy 54 Le s pu i all ogeer an prouce some graps. Problem 54. Consier e funcion f Evaluae eac of e following limis: lim f, lim f, lim f, an lim f 54.. Wa are e orizonal an verical asympoes for a grap of f? 54.. Wa are e orizonal an verical inerceps for a grap of f? Use e formulas f of e following. 4 an f 4 44 o elp you accomplis eac Sae e criical numbers of f. Creae well-ocumene increasing/ecreasing an concaviy ables for f. Sae e local minimum, local maimum, an inflecion poins on f. Make sure a you eplicily aress all ree ypes of poins weer ey eis or no Grap y f ono Figure 54.. Make sure a you coose a scale a allows you o clearly illusrae eac of e feaures foun in problems Make sure a all aes an asympoes are well labele an also wrie e coorinaes of eac local ereme poin an inflecion poin ne o e poin on e grap.. 8 L ab Aciviies

88 Porlan Communiy College MTH 5 Lab Manual Ceck your grap using a graping calculaor. Figure 54.: f Problem 54. Consier e funcion g e Evaluae eac of e following limis: lim g. an lim g Wa are e orizonal an verical asympoes for a grap of g? 54.. Wa are e orizonal an verical inerceps for a grap of g? Use e formulas g of e following. e e 4 an g e e 4 4 e o elp you accomplis eac Sae e criical numbers of g. Creae well-ocumene increasing/ecreasing an concaviy ables for g. Sae e local minimum, local maimum, an inflecion poins on g. Make sure a you eplicily aress all ree ypes of poins weer ey eis or no. Lab Aciviies 8

89 Porlan Communiy College MTH 5 Lab Manual Grap y g ono Figure 54.. Make sure a you coose a scale a allows you o clearly illusrae eac of e feaures foun in problems Make sure a all aes an asympoes are well labele an also wrie e coorinaes of eac local ereme poin an inflecion poin ne o e poin on e grap Ceck your grap using a graping calculaor. Figure 54.: g 84 L ab Aciviies

90 Porlan Communiy College MTH 5 Lab Manual Supplemenal Eercises for e Raes of Cange Lab Eercise. Te funcion z sown in Figure E. was generae by e formula y 4. E.. Simplify e ifference quoien for z. E.. Use e grap o fin e slope of e secan line o z beween e poins were an. Ceck your simplifie ifference quoien for z by using i o fin e slope of e same secan line. E.. Replace wi 4 in your ifference quoien formula an simplify e resul. Ten copy Table E. ono your paper an fill in e missing values. E..4 As e value of ges closer o 0, e values in e y column of Table. appear o be converging on a single number; wa is is number? E..5 Te value foun in problem..4 is calle e slope of e angen line o z a 4. Draw ono Figure e line a passes roug e poin 4, wi is slope. Te line you jus rew is calle e angen line o z a 4. Table E.: y 4 4 z z y Figure E.: y 4 Eercise. Fin e ifference quoien for eac funcion sowing all relevan seps in an organize manner. E.. f 7 E.. g 7 4 E.. E..5 z k E..4 8 s 9 Appeni A A

91 Porlan Communiy College MTH 5 Lab Manual Eercise. Suppose a an objec is osse ino e air in suc a way a e elevaion of e objec (measure in f) is given by e funcion s were is e amoun of ime a as passe since e objec was osse (measure in secons). E.. Fin e ifference quoien for s. E.. Use e ifference quoien o eermine e average velociy of e objec over e s4. s4 inerval 4, 4. an en verify e value by calculaing. Eercise Several applie funcions are given below. In eac case, fin e inicae quaniy (incluing uni) an inerpre e value in e cone of e applicaion. E.4. Te velociy of a roller coaser (in f/s) is given by 00sin 5 v were is e amoun of ime (s) a as passe since e coaser oppe e firs ill. Fin an inerpre 7.5 v0 v 7.5s 0s. E.4. Te elevaion of a ping pong ball relaive o e able op (in m) is given by e funcion. cos wen ino play. Fin an inerpre were is e amoun of ime (s) a as passe since e ball.5 s.5s 6 E.4. Te perio of a penulum (s) is given by P were is e number of swings e P9 P penulum as mae. Fin an inerpre. 9swing swing E.4.4 Te acceleraion of a rocke (mp/s) is given by ime (s) a as passe since lif-off. Fin an inerpre. were is e amoun of 0s 60s. A A ppeni A

92 Porlan Communiy College MTH 5 Lab Manual Supplemenal Eercises for e Limis an Coninuiy Lab Eercise. For eac of ables E.-E.5, sae e limi suggese by e values in e able an sae weer or no e limi eiss. Te correc answer for Table E. as been given o elp you unersan e irecions o e quesion. Table E.: y g Limi Limi eis? y 0, , ,000, Table E.: lim g No y f Limi Limi eis? y 5, , ,00, Table E.: y z Limi Limi eis? y Table E.4: y g Limi Limi eis? y.9,999,990.99,999, ,999,999.9 Table E.5: y T y.778 9, , ,999,999.9 Limi Limi eis? Appeni A A

93 Porlan Communiy College MTH 5 Lab Manual Eercise. Skec ono Figure E. a funcion, f, wi eac of e following properies. Make sure a your grap inclues all of e relevan feaures aresse in lab. Te only isconinuiies on f occur a 4 an f as no -inerceps f is coninuous from e rig a 4 lim f an lim f 4 lim f lim f 4 f as a consan slope of over, 4 Eercise. Figure E.: f Skec ono Figure E. a funcion, f, a saisfies eac of e properies sae below. Assume a ere are no inerceps or isconinuiies oer an ose irecly implie by e given properies. Make sure a your grap inclues all of e relevan feaures aresse in lab. f f lim lim 4 an f lim f lim f an lim 5 4 lim f f 0, f 0, an f 4 5 Eercise.4 Figure E. f Deermine all of e values of were e funcion f (given below) as isconinuiies. A eac value were f as a isconinuiy, eermine if f is coninuous from eier e rig or lef a an also sae weer or no e isconinuiy is removable. f if sin if 4 sin if 4 7 sin 8 if 7 4 A4 A ppeni A

94 Porlan Communiy College MTH 5 Lab Manual Eercise.5 Deermine e value of k a makes e funcion,. g k k if 4 if k coninuous over Eercise.6 Deermine e appropriae symbol o wrie afer an equal sign following eac of e given limis. In eac case, e appropriae symbol is eier a real number,, or. Also, sae weer or no eac limi eiss an if e limi eiss prove is eisence (an value) by applying e appropriae limi laws. Te Raional Limi Form able in Appeni C (Pages C an C4) summarizes sraegies o be employe base upon e iniial form of e limi. E.6. lim E.6. e lim e / / E.6. lim 4 4 E.6.4 lim E.6.5 lim ln 6 ln 7ln E.6.6 lim E.6.7 e e lim sin 4 e E.6.8 ln lim ln E.6.9 lim E lim 0 E.6. 5 lim 0 E.6. lim 0 9 E.6. sin lim sin E.6.4 ln lim 0 ln e e Appeni A A5

95 Porlan Communiy College MTH 5 Lab Manual Eercise.7 Draw skeces of e funcions y e, y e, an y. Noe e coorinaes of any an all inerceps. Tis is someing you soul be able o o from memory/inuiion. If you canno alreay o so, spen some ime reviewing waever you nee o review so a you can o so., y ln Once you ave e graps rawn, fill in eac of e blanks below an also circle weer or no eac limi eiss. Te appropriae symbol for some of e blanks is eier or Does limi eis? Does limi eis? lim e yes or no lim e yes or no lim e 0 yes or no lim e yes or no lim e lim yes or no 0 e yes or no lim ln yes or no lim ln yes or no lim ln 0 yes or no lim yes or no lim yes or no 0 lim yes or no lim 0 yes or no lim e / yes or no lim e yes or no lim e yes or no lim e yes or no lim e yes or no A6 A ppeni A

96 Porlan Communiy College MTH 5 Lab Manual Supplemenal Eercises for e Inroucion o e Firs Derivaive Lab Eercise. Fin e firs erivaive formula for eac of e following funcions wice: firs by evaluaing f ( + ) f ( ) f ( ) f ( ) lim an en by evaluaing lim. 0 E.. f ( ) E.. f ( ) E.. f ( ) 7 Eercise. I can be sown a 0 ( ) sin lim an 0 esablis e firs erivaive formula for ( ) ( ) cos lim 0. Use ese limis o elp you o sin. Hin: Begin wi Eercise. sin lim 0 ( + ) sin ( ) +. an use e sum formula for sin ( ) Suppose a an objec is osse ino e air in suc a way a e elevaion of e objec (measure in f) is given by e funcion s( ) were is e amoun of ime a as passe since e objec was osse (measure in secons). E.. Fin e velociy funcion for is moion an use a funcion o eermine e velociy of e objec 4. s ino is moion. E.. Fin e acceleraion funcion for is moion an use a funcion o eermine e acceleraion of e objec 4. s ino is moion. Eercise.4 Deermine e uni on e firs erivaive funcion for eac of e following funcions. Remember, we o no simplify erivaive unis in any way, sape, or form. E.4. R ( p ) is Carl s ear rae (beas/min) wen e jogs a a rae of p (measure in f/min). E.4. F( v ) is e fuel consumpion rae (gal/mi) of Han s pick-up wen se rives i on level groun a a consan spee of v (measure in mi/r). E.4. v() is e velociy of e space sule (mi/r) were is e amoun of ime a as passe since lif-off (measure in secons). E.4.4 () is e elevaion of e space sule (mi) were is e amoun of ime a as passe since lif-off (measure in secons). Appeni A A7

97 Porlan Communiy College MTH 5 Lab Manual Eercise.5 Referring o e funcions in Eercise.4, wrie senences a eplain e meaning of eac of e following funcion values. E.5. R ( 00) 84 E.5. R ( 00 ).0 E.5. ( ) F 50.0 E.5.4 F ( 50 ).0006 E.5.5 v ( 0) 66 E.5.6 ( ) E.5.7 ( 0 ).7 E.5.8 ( ) v Eercise.6 f ln + is I can be sown a e erivaive formula for e funcion ( ) ( ) + f ( ). E.6. Deermine e equaion of e angen line o f a. E.6. A grap of f is sown in Figure E.; ais scales ave eliberaely been omie from e grap. Te grap sows a f quickly resembles a line. In a eaile skec of f we woul reflec is apparen linear beavior by aing a skew asympoe. Wa is e slope of is skew asympoe? f ln + Figure E.: ( ) ( ) A8 A ppeni A

98 Porlan Communiy College MTH 5 Lab Manual Supplemenal Eercises for e Funcions, Derivaives, an Anierivaives Lab Eercise 4. Skec e firs erivaives of e funcions sown in figures E4.a, E4.a, an E4.a ono, respecively, figures E4.b, E4.b, an E4.b. y y 5 Figure E4.a Figure E4.b Figure E4.a Figure E4.b Figure E4.c Figure E4.b Appeni A A9

99 Porlan Communiy College MTH 5 Lab Manual Eercise 4. A e boom of e page seven curves are ienifie as Figures A-G. For eac saemen below, ienify e figure leers a go along wi e given saemen. Assume eac saemen is being mae only in reference o e porion of e curves sown in eac of e figures. Assume in all cases a e aes ave e raiional posiive/negaive orienaion. Te firs quesion as been answere for you o elp you ge sare. C, D, an E E4.. Eac of ese funcions is always increasing. E4.. Te firs erivaive of eac of ese funcions is always increasing. E4.. Te secon erivaive of eac of ese funcions is always negaive. E4..4 Te firs erivaive of eac of ese funcions is always posiive. E4..5 Any anierivaive of eac of ese funcions is always concave up. E4..6 Te secon erivaive of eac of ese funcions is always equal o zero. E4..7 Any anierivaive of eac of ese funcions graps o a line. E4..8 Te firs erivaive of eac of ese funcions is a consan funcion. E4..9 Te secon erivaive of eac of ese funcions is never posiive. E4..0 Te firs erivaive of eac of ese funcions mus be linear. Figure A Figure B Figure C Figure D Figure E Figure F Figure G A0 A ppeni A

100 Porlan Communiy College MTH 5 Lab Manual Eercise 4. As waer rains from a ank e rainage rae coninually ecreases over ime. Suppose a a ank a iniially ols 60 gal of waer rains in eacly 6 minues. Answer eac of e following quesions abou is problem siuaion. E4.. Suppose a V() is e amoun of waer (gal) lef in e ank were is e amoun of ime a as elapse (s) since e rainage began. Wa are e unis on V ( 45) an V ( 45)? Wic of e following is e mos realisic value for V ( 45)? a 0.8 b. c 0.8. E4.. Suppose a R() is e waer s flow rae (gal/s) were is e amoun of ime a as elapse (s) since e rainage began. Wa are e unis on R ( 45) an R ( 45)? Wic of e following is e mos realisic value for R ( 45)? a.0 b 0.00 c E4.. Suppose a V() is e amoun of waer (gal) lef in e ank were is e amoun of ime a as elapse (s) since e rainage began. Wic of e following is e mos V 45? realisic value for ( ) a.0 b 0.00 c Eercise 4.4 For eac saemen (E4.4.-E4.4.5), ecie wic of e following is rue. i. Te saemen is rue regarless of e specific funcion f. ii. Te saemen is rue for some funcions an false for oer funcions. iii. Te saemen is false regarless of e specific funcion f. f 0 > 0. E4.4. If a funcion f is increasing over e enire inerval (, 7), en ( ) f 4 0. E4.4. If a funcion f is increasing over e enire inerval (, 7), en ( ) f < 0. E4.4. If a funcion f is concave own over e enire inerval (, 7), en ( ) f 0 > 0. E4.4.4 If e slope of f is increasing over e enire inerval (,7), en ( ) E4.4.5 If a funcion f as a local maimum a, en f ( ) 0. Appeni A A

101 Porlan Communiy College MTH 5 Lab Manual Eercise 4.5 For a cerain funcion g, g () 8 a all values of on (, ) an g () on (, ). a all values of E4.5. Janice inks a g mus be isconinuous a bu Lisa isagrees. Wo s rig? E4.5. Lisa inks a e grap of g is e line y 0 bu Janice isagrees. Wo s rig? Eercise 4.6 Eac saemen below is in reference o e funcion sown in Figure E4.4. Decie weer eac saemen is rue or false. A capion as been omie from Figure E4.4 because e ieniy of e curve canges from quesion o quesion. E4.6. If e given funcion is f, en () f () f >.. E4.6. If e given funcion is f, en f () > f () E4.6. If e given funcion is f, en f () > f () E4.6.4 E If e given funcion is f, en e angen line o f a 6 is orizonal. If e given funcion is f, en f is always increasing. Figure E4.4 E4.6.6 If e given funcion is f, en f is noniffereniable a 4. E4.6.7 If e given funcion is f, en f is noniffereniable a 4. E4.6.8 Te firs erivaive of e given funcion is perioic (saring a 0). E4.6.9 Anierivaives of e given funcion are perioic (saring a 0). E4.6.0 E4.6. E4.6. If e given funcion was measuring e rae a wic e volume of air in your lungs was canging secons afer you were frigene, en ere was e same amoun of air in your lungs 9 secons afer e frig as was ere 7 secons afer e frig. If e given funcion is measuring e posiion of a weig aace o a spring relaive o a able ege (wi posiive an negaive posiions corresponing, respecively, o e weig being above an below e ege of e able), en e weig is in e same posiion 9 secons afer i begins o bob as i is 7 secons afer e bobbing commence (assuming a represens e number of secons a pass afer e weig begins o bob). If e given funcion is measuring e velociy of a weig aace o a spring relaive o a able ege (wi posiive an negaive posiions corresponing, respecively, o e weig being above an below e ege of e able), en e spring was moving ownwar over e inerval (,) (assuming a represens e number of secons a pass afer e weig begins o bob). A A ppeni A

102 Porlan Communiy College MTH 5 Lab Manual Eercise 4.7 Complee as many cells in Table E4. as possible so a wen rea lef o rig e relaionsips beween e erivaives, funcion, an anierivaive are always rue. Please noe a several cells will remain blank. Table E4. f f f F Posiive Negaive Consanly Zero Increasing Decreasing Consan Concave Up Concave Down Linear Eercise 4.8 Answer eac quesion abou e funcion f wose firs erivaive is sown in Figure E4.5. Te correc answer o one or more of e quesions may be ere s no way of knowing. E4.8. Were, over ( 6,6), is f noniffereniable? E4.8. Were, over ( 6,6), are anierivaives of f noniffereniable? E4.8. Were, over ( 6,6), is f ecreasing? E4.8.4 Were, over ( 6,6), is f concave own? E4.8.5 Were, over ( 6,6), is f ecreasing? E4.8.6 Were, over ( 6,6), is f posiive? E4.8.7 Were, over ( 6,6), are anierivaives of f concave up? E4.8.8 Were, over ( 6,6), are anierivaives of f increasing? E4.8.9 Were, over ( 6,6), oes f ave is maimum value? Figure E4.5: f E4.8.0 Suppose a f ( ) 4. Wa, en, is e equaion of e angen line o f a? Appeni A A

103 Porlan Communiy College MTH 5 Lab Manual Eercise 4.9 A cerain anierivaive, F, of e funcion f sown in Figure E4.6a passes roug e poins,6, 6,6 ) ono Figure E4.6b. ( ) an ( ). Draw is anierivaive (over e inerval ( ) Figure E4.6a: f Figure E4.6b: F Eercise 4.0 Draw ono Figure E4.7 e funcion f given e following properies of f. f ( ) f ( ) 0 4 Te only isconinuiy on f occurs a e verical asympoe. f ( ) 0 f ( ) > 0 on (, ), (, ), an ( 4, ) f ( ) < 0 on (, 4 ) f ( ) > 0 on (, ) an (, 4 ) f ( ) < 0 on (, ) f ( ) 0 on ( 4, ) lim f ( ) an lim f ( ) Figure E4.7 f A4 A ppeni A

104 Porlan Communiy College MTH 5 Lab Manual Supplemenal Eercises for e Derivaive Formulas Lab Eercise 5. Apply e consan facor an power rules of iffereniaion o fin e firs erivaive of eac of e following funcions. In eac case ake e erivaive wi respec o e inepenen variable suggese by e epression on e rig sie of e equal sign. Make sure a you assign e proper name o e erivaive funcion. Please noe a ere is no nee o employ eier e quoien rule or e prouc rule o fin any of ese erivaive formulas. E5.. ( ) E5..4 ( α) 6 f E5.. 6 z eα π E5..5 y z 5 7 y u E5.. ( ) 8 E T 7 4 u Eercise 5. Apply e consan facor an prouc rules of iffereniaion o fin e firs erivaive of eac of e following funcions. Wrie ou e Leibniz noaion for e prouc rule as applie o e given funcion. In eac case ake e erivaive wi respec o e inepenen variable suggese by e epression on e rig sie of e equal sign. Make sure a you assign e proper name o e erivaive funcion. E5.. y 5sin( ) cos( ) E5.. 5 e y E5.. F( ) 4ln( ) 7 E5..4 z sin ( ) E5..5 () ( T ) an () + E5..6 T 7 7 Eercise 5. Apply e consan facor an quoien rules of iffereniaion o fin e firs erivaive of eac of e following funcions. Wrie ou e Leibniz noaion for e quoien rule as applie o e given funcion. In eac case ake e erivaive wi respec o e inepenen variable suggese by e epression on e rig sie of e equal sign. Make sure a you assign e proper name o e erivaive funcion. E5.. q ( ) E5..4 F( ) θ 4e θ θ e + an an ( ) ( ) E5..5 E5.. u( ) ln an p + ( ) E5.. 4 () E5..6 F 5 y sin 4 ( β) cos ( β) Appeni A A5

105 Porlan Communiy College MTH 5 Lab Manual Eercise 5.4 Muliple applicaions of e prouc rule are require wen fining e erivaive formula of e prouc of ree of more funcions. Afer simplificaion, owever, e preicable paern sown below emerges. ( f g) ( f ) g + f ( g) ( f g) ( f) g+ f ( g) + f g ( ) ( f gk) ( f ) gk + f ( g) k + f g ( ) k + f g ( k) ( ) o verify e formula sae above for e prouc of ree funcions. Te firs applicaion of e prouc rule is sown below o elp you ge you sare. E5.4. Apply e prouc rule wice o e epression f ( ) g( ) ( ) ( f ( ) g( ) ( ) ) ( f ( ) ) g( ) ( ) + f ( ) g( ) ( ) E5.4. Apply e formula sown for e prouc of four funcions o eermine e erivaive f e sin cos wi respec o of e funcion ( ) ( ) ( ) Eercise 5.5 Fin e firs erivaive wi respec o for eac of e following funcions an for eac funcion fin e equaion of e angen line a e sae value of. Make sure a you use appropriae ecniques of iffereniaion. + + E5.5. f ( ) ; angen line a 5 E5.5. ( ) E5.5. K( ) r + ; angen line a 8 E5.5.4 ( ) ; angen line a + e ; angen line a 0 Eercise 5.6 Fin e secon erivaive of y wi respec o if y 5 4. Make sure a you use appropriae ecniques of iffereniaion an a you use proper noaion on bo sies of e equal sign. A6 A ppeni A

106 Porlan Communiy College MTH 5 Lab Manual Eercise 5.7 A funcion f is sown in Figure E5.. Eac quesion below is in reference o is funcion. Please noe a e numeric enpoins of eac inerval answer are all inegers. E5.7. E5.7. E5.7. Over wa inervals is f posiive an over wa inervals is f negaive? Over wa inervals is f increasing an over wa inervals is f ecreasing? Over wa inervals is f posiive an over wa inervals is f negaive? E5.7.4 Te formula for f is ( ) f + 6. Fin e formulas for f an f, an en grap e erivaives an verify your answers o pars a-c. Figure E5. Eercise 5.8 A funcion g is sown in Figure E5.. Eac quesion below is in reference o is funcion. E5.8. Were over e inerval [ 0, π ] oes g () 0 π noe a e -scale in Figure E5. is.? Please E5.8. Te formula for g is g() () () sin + sin 5. Fin e formula for g an use a formula o solve e equaion g () 0 over e inerval [ ] your answers o ose foun in par a. 0, π. Compare Figure E5. Hin: sin () sin () sin () Eercise 5.9 Fin e equaion of e angen line o f a π if f ( ) sin ( ). Hin: Begin by applying a ouble angle ieniy. Also, make sure a you rea e quesion carefully. Appeni A A7

107 Porlan Communiy College MTH 5 Lab Manual Eercise 5.0 In Table E5. several funcion an erivaive values are given for e funcions f an g. Tis enire problem is base upon ese wo funcions. Suppose a you were aske o fin e value of ( ) were ( ) f ( ) g( ) you woul nee o o is fin a formula for ( ). Te firs ing. You woul en replace wi, subsiue e appropriae funcion an erivaive values, an simplify. Tis is illusrae in eample E5.. Following e process ouline in eample E5., fin eac of e following. E5.0. Fin ( 4) were ( ) f ( ) g( ). E5.0. Fin ( ) were ( ) f ( ) g( ) E5.0. Fin k ( ) were k( ). ( ) ( ) g. E5.0.4 p ( 4) were p ( ) 6 f ( ) f. E5.0.5 r () were r( ) g( ). E5.0.6 s ( ) were s( ) f ( ) g( ). E5.0.7 F ( 4) were F( ) g( 4) E5.0.8 T ( 0) were T( ). f ( ). e Eample E5. ( ) f ( ) g + f g f g + f g ( ) ( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) ( 7)( ) + ( 4)( 7) f g f g 5 Table E5. f ( ) f ( ) f ( ) g( ) g ( ) g ( ) A8 A ppeni A

108 Porlan Communiy College MTH 5 Lab Manual Supplemenal Eercises for e Cain Rule Lab Eercise 6. Te funcions f ( ) e, g( ) ( e ), an ( ) ( e ) are equivalen. Fin e firs erivaive formulas for eac of e funcions (wiou alering eir given forms) an en eplicily esablis a e erivaive formulas are e same. Eercise 6. Consier e funcion k ( θ) sin ( sin( θ) ). π π E6.. Over e inerval,, k ( θ) θ. Wa oes is ell you abou e formula for π π k ( θ ) over,? E6.. Use e cain rule an an appropriae rigonomeric ieniy o verify your answer o π π quesion E6... Please noe a cos( θ ) > 0 over,. E6.. Wa is e consan value of k ( θ ) over E6..4 Eercise 6. over a inerval? Wa is e value of π k? Consier e funcion f sown in Figure E6.. π π,. Hin: Wa is e sign on ( ) cos θ g f E6.. Suppose a ( ) ( ) 4 posiive? r f ( ) E6.. Suppose a ( ) posiive? w f ( ) E6.. Suppose a ( ) posiive? E6..4 Suppose a ( ). Over wa inervals is g e. Over wa inervals is r e. Over wa inervals is w f. Is noniffereniable a? ( ) (Please noe a f as a orizonal angen line a.) Figure E6.: f Appeni A A9

109 Porlan Communiy College MTH 5 Lab Manual Eercise 6.4 Decie weer or no i is necessary o use e cain rule wen fining e erivaive wi respec o of eac of e following funcions. E6.4. f ( ) ln ( + ) E6.4. f ( ) E6.4. f ( ) cos( π ) 5 E6.4.4 f ( ) cos( ) E6.4.5 f ( ) cos( π ) E6.4.6 f ( ) Eercise 6.5 cos π ( π ) Fin e firs erivaive wi respec o of eac of e following funcions. In all cases, look for appropriae simplificaions before aking e erivaive. Please noe a some of e funcions will be simpler o iffereniae if you firs use e rules of logarims o epan an simplify e logarimic epression. E6.5. f ( ) an ( ) E6.5. ( ) ( ) f e E6.5. f ( ) sin cos( ) E6.5.4 f ( ) an sec( ) E6.5.5 f ( ) an ( ) sec sec( ) sin( e ) ( ) ( ) ( ) E6.5.6 f ( ) sin ( ) ( ) E6.5.7 f ( ) 4sin ( ) E6.5.8 f ( ) ln ln ( ) E6.5.9 f ( ) E6.5. f ( ) 5 ln e + e ln + ( ) E6.5.0 f ( ) ln an ( ) e E6.5. f ( ) ln ( + e) 4 E6.5. f ( ) sec ( e ) E6.5.4 f ( ) sec ( e ) E6.5.5 f ( ) csc E6.5.6 f ( ) csc ( ) an ( ) E6.5.7 f ( ) E6.5.8 ( ) f sin E6.5.9 f ( ) 4 7 E6.5.0 f ( ) 5 sin ( ) sin( ) E6.5. f ( ) e e E6.5. f ( ) sin( ) f 4sin cos E6.5.4 f ( ) sin cos ( ) E6.5. ( ) ( ) ( ) e e ( ) A0 A ppeni A

110 Porlan Communiy College MTH 5 Lab Manual Supplemenal Eercises for e Implici Differeniaion Lab Eercise 7. Te curve ( y) y sin is sown in Figure E7.. Fin a formula for y an use a formula o eermine e -coorinae a eac of e wo poins e curve crosses e -ais. (Noe: Te angen line o e curve is verical a eac of ese poins.) Scales ave eliberaely been omie in Figure E7.. Eercise 7. Soluions o e equaion ln ( y ) + y are grape in Figure E7.. Deermine e equaion of e angen line o is,. curve a e poin ( ) Figure E7.: sin ( y) y Hin: I is easier o iffereniae if you firs use rules of logarims o compleely epan e logarimic epression. Figure E7.: ln ( y ) + y Eercise 7. You ave formulas a allow you o iffereniae,, an. You on, owever, ave a formula o iffereniae. In is eercise you are going o use a process calle logarimic iffereniaion o eermine e erivaive formula for e funcion y. Eample E7. (page A) sows is process for a ifferen funcion. Te funcion y ln is only efine for posiive values of (wic in urn means y is also posiive), so we can say a ( y) ln( ) formula for y. Wa you nee o o is use implici iffereniaion o fin a afer firs applying e power rule of logarims o e logarimic epression on e rig sie of e equal sign. Once you ave your formula for y, subsiue Voila! You will ave e erivaive formula for. So go aea an o i. for y. Appeni A A

111 Porlan Communiy College MTH 5 Lab Manual Eercise 7.4 In e olen ays (pre-symbolic calculaors) we woul use e process of logarimic iffereniaion o fin erivaive formulas for complicae funcions. Te reason is process is simpler an sraig forwar iffereniaion is a we can obviae e nee for e prouc an quoien rules if we compleely epan e logarimic epression before aking e erivaive. Use e process of logarimic iffereniaion o fin a firs erivaive formula for eac of e following funcions. Te process of logarimic iffereniaion is illusrae in y sin( ) 7.4. y sin e 7.4. ( ) ln 4 y 5 ln ( ) ( ) Eample E7. e y 4 + e ln ( y) ln 4 + ln ln ln 4 e ( y) ( ) ( + ) ln ( y) e ln ( ) ln ( 4 + ) ( ln ( y) ) e ln ( ) ln ( 4 + ) ( ) Se e naural logarims of e wo epressions equal o one anoer. Compleely epan e logarimic epression on e rig sie of e equal sign. y + + y 4+ y e 4 e ln ( ) + y 4+ Solve for y y e 4 y e ln ( ) e y e 4 e ln ( ) ( e ) ln ( ) e ( ln ( ) ) ( 4 ) afer going roug e process of implici iffereniaion. Replace y wi is original formula in e formula for y. A A ppeni A

112 Porlan Communiy College MTH 5 Lab Manual Supplemenal Eercises for e Relae Raes Lab Eercise 8. A weaer balloon is rising verically a e rae of 0 f/s. An observer is saning on e groun 00 f orizonally from e poin were e balloon was release. A wa rae is e isance beween e observer an e balloon canging wen e balloon is 400 f eig? Eercise 8. Imagine a railroa-crossing gae. For purposes of is problem we are going o rea e arm of e gae as a line a pivos on e verical pole via a roun gear wose cener is eacly 4 fee off e groun. Te isance beween e ip of e arm an e cener of e gear is eacly 8 f. Wen e arm is being lowere e angle of elevaion of e arm ecreases a a consan rae of 6 O /sec. Fin e rae a wic e ip of e arm approaces e groun (verically) a e insan e angle of elevaion of e arm is 0 O. Eercise 8. Jimbo is rinking soa from a conical cup; e raius of e cup a is op is 5 cm an e eig of e cup is 0 cm. Jimbo is rinking roug a sraw a a consan rae of 0.5 cm /s. Assuming a e cup remains verical wils Jimbo rinks, fin e rae of cange in e eig of e liqui wen ere are eacly 00 cm of soa lef in e cup Eercise 8.4 A cerain snowball mainains a perfecly sperical sape as i mels; e snowball mels a a consan rae of 5 cm /min. Deermine e rae a wic e surface area of e snowball canges a e insan e raius of e snowball as a raius of 6 cm. Hin: Wa quaniy is measure using cubic cenimeers? Ta quaniy an e surface area are e variables for e problem. Appeni A A

113 Porlan Communiy College MTH 5 Lab Manual A4 A ppeni A

114 Porlan Communiy College MTH 5 Lab Manual Supplemenal Eercises for e Criical Numbers an Graping from Formulas Lab Eercise 9. Te sine an cosine funcion are calle circular funcions because for any given value of e poin cos,sin lies on e circle wi equaion + y. ( () ()) Tere are analogous funcions calle yperbolic sine an yperbolic cosine. As you mig suspec, ese funcions generae poins a lie on a yperbola; specifically, for all values of e poin cos,sin lies on e yperbola y. I urns ou a ere are alernae ( () ()) formulas for e yperbolic funcions. Specifically: cos () e + e an sin () e e E9.. Use e eponenial formulas o verify e ieniy () () E9.. cos sin. Use e eponenial formulas o eermine e firs erivaives (wi respec o ) of cos () an sin (). Deermine, by inference, e secon erivaive formulas for eac of ese funcions. E9.. Deermine e criical numbers of cos () an sin (). E9..4 Deermine e inervals over wic cos () an sin () are increasing/ecreasing an over wic ey are concave up/concave own. E9..5 Use e eponenial formulas o eermine eac of e following limis: lim cos (), E9..6 lim cos (), lim sin (), an lim sin (). Use e informaion you eermine in problems E9..-E9..5 o elp you raw freean skeces of y cos () an y sin (). E9..7 Tere are four more yperbolic funcions a correspon o e four aiional circular sin () funcions; e.g. an (). Fin e eponenial formulas for ese four cos () funcions. E9..8 Wa woul you guess o be e firs erivaive of an ()? Take e erivaive of e eponenial formula for an () o verify your suspicion. E9..9 Le f () an (). Wa is e sign on f () abou e funcion f? Wa are e values of ( 0) a all values of? Wa oes is ell you f? E9..0 Use e eponenial formula o eermine lim an () an lim an (). E9.. f an ( 0) Use e informaion you eermine in problems E9..9-E9..0 o elp you raw a freean skec of y an (). Appeni A A5

115 Porlan Communiy College MTH 5 Lab Manual Eercise 9. Consier e funcion k ( ) 56 8/ /. E9.. Wa are e criical numbers of k? Remember o sow all relevan work! Remember a your formula for k () nees o be a single, compleely facore, fracion! E9.. Creae an increasing/ecreasing able for k. E9.. Sae eac local minimum poin an local maimum poin on k. Eercise 9. Consier e funcion f ( ) cos ( ) sin( ) + over e resrice omain [ ] 0, π. E9.. Wa are e criical numbers of f? Remember o sow all relevan work! Remember a your formula for f ( ) nees o be a single, compleely facore, fracion! E9.. Creae an increasing/ecreasing able for f. E9.. Sae eac local minimum poin an local maimum poin on f. Eercise Consier g(). Fin (an compleely simplify) g () an sae all numbers were g () is zero or unefine. Ten consruc a concaviy able for g an sae all of e inflecion poins on g. Eercise 9.5 For eac of e following funcions buil increasing/ecreasing ables an concaviy ables an en sae all local minimum poins, local maimum poins, an inflecion poins on e funcion. Also eermine an sae all orizonal asympoes an verical asympoes for e funcion. Finally, raw a eaile skec of e funcion. E9.5. f ( ) ( + ) E9.5. ( ) ( 4) g ( + 5 ) E9.5. k ( ) + A6 A ppeni A

116 Porlan Communiy College MTH 5 Lab Manual y 4 Figure 7.: f y 4 6 Figure 7.: f y Figure 7.: f Appeni B B

117 Porlan Communiy College MTH 5 Lab Manual y Figure 7.4: f 4 y 0 Figure 7.5: f 5 5 B A ppeni B

118 Porlan Communiy College MTH 5 Lab Manual Figure.: w Appeni B B

119 Porlan Communiy College MTH 5 Lab Manual B4 A ppeni B

120 Porlan Communiy College MTH 5 Lab Manual Replacemen Limi Laws Te following laws allow you o replace a limi epression wi an acual value. For eample, e epression lim can be replace wi e number 5, e epression lim 6 5 can be replace wi e number 6, an e epression number 0. In all cases bo a an C represen real numbers. 7 lim z e z can be replace wi e Limi Law R lim lim lim a a a a Limi Law R lim C lim C lim C C a a a C lim f lim 0 Limi Law R lim C C an lim C C f f C lim f lim 0 f f C lim lim 0 f C lim lim 0 f Infinie Limi Laws: Limis as e oupu increases or ecreases wiou boun Many limi values o no eis. Someimes e non-eisence is cause by e funcion value eier increasing wiou boun or ecreasing wiou boun. In ese special cases we use e symbols an o communicae e non-eisence of e limis. Some eamples of ese ype of non-eisen limis follow. lim lim n n (if n is a posiive real number) (if n is an even posiive ineger) lim e lim n, lim e (if n is an o posiive ineger), lim ln, an lim ln( ) 0 Appeni C C

121 Porlan Communiy College MTH 5 Lab Manual Algebraic Limi Laws Te following laws allow you o replace a limi epression wi equivalen limi epressions. lim can be replace wi e epression lim lim. For eample, e epression 5 Limi laws A-A6 are vali if an only if every limi in e equaion eiss. 5 5 In all cases a can represen a real number, a real number from e lef, a real number from e rig, e symbol, or e symbol. Limi Law A: f g f g lim lim lim a a a Limi Law A: f g f g lim lim lim a a a Limi Law A: lim Cf Clim f a a were C is any real number. Limi Law A4: f g f g lim lim lim a a a Limi Law A5: Limi Law A6: f lim a g lim f a if an only if lim g g a a lim 0 lim (wen f is coninuous a lim lim f g f g a a a g ) Limi Law A7: If ere eiss an open inerval cenere a a over wic f g for a, en lim f lim g (provie a bo limis eis) a a Aiionally, if ere eiss an open inerval cenere a a over wic f g for a, en lim f if an only if a an, similarly, lim f a if an only if lim g. a lim g a C A ppeni C

122 Porlan Communiy College MTH 5 Lab Manual Raional Limi Forms Form Eample Course of Acion real number non-zero real number non-zero real number zero + 6 lim 5 4 (e form is ) lim (e form is 4 0 ) Te value of e limi is e number o wic e limi form simplifies; for eample, + 6 lim 5 4 You can immeiaely begin applying limi laws if you are proving e limi value. Te limi value oesn eis. Te epression wose limi is being foun is eier increasing wiou boun or ecreasing wiou boun (or possibly bo if you ave a wo-sie limi). You may be able o communicae e non-eisence of e limi using or. For eample, if e value of is a lile less + 7 an 7, e value of is posiive. Hence you coul wrie. lim 7 7. zero zero real number infiniy lim (e form is 0 0 ) 4e lim ln + 0 (e form is ( ) ) Tis is an ineerminae form limi. You o no know e value of e limi (or even if i eiss) nor can you begin o apply limi laws. You nee o manipulae e epression wose limi is being foun unil e resulan limi no longer as ineerminae form. For eample: lim (Te limi form is now limi laws.) ( )( + ) ( )( ) ( ) ( + ) 4 lim lim 4 ; you may begin applying e Te limi value is zero an is is jusifie by logic similar o a use o jusify Limi Law R. Appeni C C

123 Porlan Communiy College MTH 5 Lab Manual Raional Limi Forms Form Eample Course of Acion infiniy real number lim+ 0 ln ( ) 4 e (e form is ) Te limi value oesn eis. Te epression wose limi is being foun is eier increasing wiou boun or ecreasing wiou boun (or possibly bo if you ave a wo-sie limi). You may be able o communicae e non-eisence of e limi using ln ( ) or. For eample, you can wrie lim e Tis is an ineerminae form limi. You o no know e value of e limi (or even if i eiss) nor can you begin o apply limi laws. You nee o manipulae e epression wose limi is being foun unil e resulan limi no longer as ineerminae form. For eample: infiniy infiniy + e lim + e + e + e lim lim e + e + e e + lim e + e (Te limi form is now ; you may begin applying e limi laws.) Tere are oer ineerminae limi forms, aloug e wo menione in Table are e only wo raional ineerminae forms. Te oer ineerminae forms are iscusse in MTH 5. C4 A ppeni C

124 Porlan Communiy College MTH 5 Lab Manual Derivaive Formulas: k, a, an n represen consans; u an y represen funcions of Basic Formulas Cain Rule Forma Implici Derivaive Forma ( k ) 0 n n ( ) n n n n n y ( u ) nu ( u) ( y ) ny ( ) ( u) ( u) u ( ) y y y y ( sin ( ) ) cos( ) ( sin ( u) ) cos( u) ( u) ( sin ( y) ) cos( y) y ( cos ( ) ) sin ( ) ( cos ( u) ) sin( u) ( u) ( cos ( y) ) sin ( y) y ( an ( ) ) sec ( ) ( an ( u) ) sec ( u) ( u) ( an ( y) ) sec ( y) y ( sec( ) ) sec( ) an ( ) ( sec( u) ) sec( u) an ( u) ( u) ( sec ( y) ) sec ( y) an ( y) y ( co ( ) ) csc ( ) ( co ( u) ) csc ( u) ( u) ( co ( y) ) csc ( y) y ( csc( ) ) csc ( ) co ( ) ( csc( u) ) csc( u) co ( u) ( u) ( csc( y) ) csc( y) co ( y) y ( an ( ) ) ( ) + ( an u ) ( u) ( an ( y) ) + u + y ( sin ( ) ) ( sec ( ) ) e e ( sin ( u) ) ( u) u ( sec ( u) ) ( u) u u ( sin ( y) ) ( sec ( y) ) u u y ( ) ( e ) e ( u) ( e ) a a a u u y ( ) ln ( ) ( a ) ln ( a) a ( u) ( ) ln ( ) ( ln ( ) ) ln ( u) u ( ) ( u) y y y y y y y e y y a a a y ( ln y ) y ( ) Consan Facor Rule of Differeniaion: k represens a consan ( kf( ) ) k ( f( ) ) Alernae Form: If y f ( ), en ( ky) y k Appeni C C5

125 Porlan Communiy College MTH 5 Lab Manual Sum/Difference Rule of Differeniaion ( f ( ) ± g( ) ) f ( ) ± g ( ) ( ( )) Prouc Rule of Differeniaion ( f ( ) g( ) ) ( f ( ) ) g( ) + f ( ) ( g( ) ) Quoien Rule of Differeniaion ( ) ( f ( ) ) g( ) f ( ) ( g( f )) ; g ( ) 0 g ( ) g( ) Cain Rule of Differeniaion ( ( ( ))) ( ) ( ) ( ) f g f g g Alernae Form: If u g( ), en ( ) Alernae Form: If y f ( u) were u g( ) ( f u ) f ( u) ( u), en y y u u Some Useful Rules of Algebra For posiive inegers m an n: k For real numbers k an n: n k ; 0 n n m an f ( ) k m/ n f ( ) ; k 0 k For posiive real numbers A an B an all real numbers n: ( AB) ( A) + ( B), ln ln ( A) ln ( B) ln ln ln A B n, an ln ( A ) n ln ( A) C6 A ppeni C

126 Porlan Communiy College MTH 5 Lab Manual Supplemenal Soluions for e Raes of Cange Lab Eercise. E.. Te ifference quoien for z is: z z ; for 0 E.. Te rise beween e wo poins is 9 an e run is, so e slope beween e wo poins is given by 9. Using e ifference quoien, if we le an we ge: 4 4 E.. E..4 Te values are converging on 4. E z z Table E.K: y 4 4 z z y Figure E.K: y 4 Appeni D D

127 Porlan Communiy College MTH 5 Lab Manual Eercise. f f E ; for g g ;for0 E.. E.. 44 z z 0 0; for s s E..4 ; for 0 D A ppeni D

128 Porlan Communiy College MTH 5 Lab Manual E..5 Eercise. E k k ;for 0 s s for 0 E.. Leing 4 an 0. we ave (from e ifference quoien): Cecking s s (Pew!) Appeni D D

129 Porlan Communiy College MTH 5 Lab Manual E.4. v v f/s 7.5s 0s s Tis value ells us a uring e firs 7.5 secons of escen, e average rae of cange f/s in e coaser s velociy is. In oer wors, on average, wi eac passing s f/s less an i was e preceing secon. We coul also say secon e velociy is a e average acceleraion eperience by e coaser over e firs 7.5 secons of f/s escen is s. E.4..5 s.5s 0m/s Tis value ells us a e average velociy eperience by e ball beween e.5 secon of play an e ir secon of play is 0m/s. Please noe a is oes no imply a e ball oesn move; i simply means a e ball is a e same elevaion a e wo imes. E.4. 9 P P 0. s 9swing swing swing Tis value ells us a beween e firs swing an e 9 swing e average rae of cange in e penulum s perio is 0. s. In oer wors, on average, wi eac swing passing swing e perio ecreases by a en of a secon. E mp/s. 0s 60s s Tis value ells us a uring e secon minue of flig, e average rae of cange in e mp/s rocke s acceleraion is.. In oer wors, on average, wi eac passing secon s e acceleraion is. mp/s more an i was e preceing secon. You mig ave noice a since e acceleraion funcion is linear, e rae of cange in e acceleraion is consan. Ta is, wi eac passing secon e acceleraion acually is. mp/s more an i was e preceing secon. D4 A ppeni D

130 Porlan Communiy College MTH 5 Lab Manual Soluions o e Supplemenal Eercises for e Limis an Coninuiy Lab Eercise. Table Limi Eisence? Commens Te relevan paern in e y column is e E. lim f 0 powers of 0. Te firs nonzero igi is Yes moving farer an farer o e rig of e ecimal poin. You soul efiniely recognize ecimals E. approacing common fracions. Goo cac if lim z Yes you noe e was approacing only from e lef. E.4 lim g, 000, 000 Yes Te las enry in e oupu column is e same in ese wo ables. Hopefully you recognize a you nee o look a e paern in e oupu, no jus e las enry. lim T E No Recognizing a approaces 7 9 probably requires some guessing an cecking. Say focuse wen eermining from e lef or from e rig; a can ge ricky especially wen e numbers are negaive. Eercise. Eercise. y Figure E.K: f Figure E.K: f Appeni D D5

131 Porlan Communiy College MTH 5 Lab Manual Eercise.4 f is isconinuous from bo irecions a 0 an e isconinuiy is no removable. f is isconinuous a aloug i is coninuous from e lef a. Te isconinuiy is no removable. (Te problem is a e limis are ifferen from e lef an rig of.) f is isconinuous from bo irecions a 4 bu e isconinuiy is removable ( lim f f is isconinuous from bo irecions a bu e isconinuiy is removable ( Eercise.5 4 ). lim f Because e wo formulas are polynomials, e only number were coninuiy is a issue is. Te op formula is use a, so g is efine regarless of e value of k. Basically all we nee o o is ensure a e limis from e lef an e rig of are bo equal o g 9 k, lim g 9 k, an g lim 9 4 k ). g. We ave: So g will be coninuous a (an, consequenly, over, ) if an only if 9 k 9 4k. Tis gives us k 0. Do! Turns ou e g isn piece-wise a all, i s simply e parabolic funcion g. Eercise.6 E.6. lim Tis limi oes no ei. E.6. e lim e / / lim e lim 0 e e / LL A6 LL R Tis limi oes eis. E.6. lim lim lim 4 lim lim 4 lim lim 4 lim lim 4 lim lim LL A5 LL A an A LL A6 LL R an R Tis limi oes eis. D6 A ppeni D

132 Porlan Communiy College MTH 5 Lab Manual E.6.4 lim lim We can apply limi laws A-A6 ye because lim LL A7 e limi as e ineerminae form ln ln ln E ln lim lim 7ln 7ln 7ln lim We can apply limi laws A- 4ln A6 ye because e limi as lim e ineerminae form. E.6.6 lim lim lim We can apply limi laws A- A6 ye because e limi as e ineerminae form. lim lim lim lim lim lim 0 0 LL A5 LL A an A LL R an R E.6.7 e e lim sin lim sin e e 4e e 4e e lim sin We can apply limi laws A- 4 A6 ye because e limi as e e ineerminae form. sinlim 4 e LL A6 lim sin LL A5 lim 4 e lim sin lim lim 4 e LL A sin 0 4 LL R an R LL A7 LL R Tis limi oes eis. Tis limi oes no eis. Tis limi oes eis. Tis limi oes eis. Appeni D D7

133 Porlan Communiy College MTH 5 Lab Manual E.6.8 ln lim ln oes no eis because e omain of e funcion ln y is. ln E.6.9 E.6.0 E.6. lim We can apply limi laws A-A6 ye because e insie limi as e ineerminae form 0. 0 We can apply limi laws A- A6 ye because e limi as e ineerminae form lim 5 5 lim 7 lim 7 lim 7 lim 7 lim LL A7 LL A6 LL A LL R an R lim lim lim lim 5 0 lim lim lim lim 0 0 5lim lim 0 0 lim lim lim lim 0 lim lim 5 lim 5 lim lim 0 0 lim lim 50 0 lim 9 lim lim 9 4 lim LL A5 LL A an A LL A LL A6 LL R an R Tis limi oes eis. LL A7 LL A LL A LL R an R Tis limi oes eis. Tis limi oes eis. D8 A ppeni D

134 Porlan Communiy College MTH 5 Lab Manual E.6. lim lim lim 0 9 We can apply limi laws A- A6 ye because e limi as lim 0 9 e ineerminae form 0. 0 lim LL A7 0 9 lim 0 LL A5 lim 9 E.6. We can apply limi laws A- A6 ye because e limi as e ineerminae form lim 0 lim 9 lim lim lim 9 lim lim 0 lim 9 lim lim sin sin cos cos sin lim lim sin sin cos cos sin lim cos sin cos lim sin cos lim sin lim lim sin lim limsin lim sin lim sin LL A LL A6 LL A LL R an R LL A7 LL A5 LL A LL A6 LL R an R Tis limi oes eis. Tis limi oes eis. Appeni D D9

135 Porlan Communiy College MTH 5 Lab Manual E.6.4 ln lim 0 ln e e Tis limi oes no eis. Eercise.7 Does limi eis? Does limi eis? lim e No lim e 0 Yes lim e Yes lim e 0 Yes 0 lim e No e 0 lim Yes lim ln No lim ln 0 Yes lim ln 0 lim 0 lim 0 lim 0 e lim e No Yes No Yes No lim 0 lim 0 / lim e lim e lim 0 e Yes No Yes No Yes D0 A ppeni D

136 Porlan Communiy College MTH 5 Lab Manual Supplemenal Soluions for e Inroucion o e Firs Derivaive Lab Eercise. E.. f 0 f f lim 0 lim 0 lim 0 lim 0 lim 0 lim 0 f lim lim f f lim lim E.. E.. f f f f lim 0 lim 0 lim 0 lim 0 lim f f lim lim 0 0 lim 0 lim 0 Tis sep is no opional. On e preceing line e limi as e ineerminae form 0. 0 f f f f lim lim lim lim f f lim 7 7 lim 0 lim lim 0 0 Appeni D D

137 Porlan Communiy College MTH 5 Lab Manual Eercise. If f sin Eercise. f, en: E.. Te velociy funcion is: f f lim 0 sin sin lim 0 sin cos cos sin sin lim 0 sin cos cos sin lim 0 cos sin sin lim cos lim 0 0 sin 0 cos cos Here we ve use e wo given limi values. s s v lim lim lim lim lim 0 lim 60 6 Here we ve applie limi laws A an A5. Remember, from is limi symbol s perspecive e variable is ; so e facors of sin an cos are consans from e perspecive of e limi symbol. Te velociy of e objec 4. s ino is moion is (incluing uni): v 7. f s f s D A ppeni D

138 Porlan Communiy College MTH 5 Lab Manual E.. Te acceleraion funcion is: a lim 0 lim 0 v v lim lim lim 0 Te acceleraion of e objec 4. s ino is moion is (incluing uni): f s a 4. s Eercise.4 E.4. Te uni for R is beas min f min. E.4. Te uni for F is gal mi mi r. E.4. Te uni for v is mi r s. E.4.4 Te uni for is mi. s Eercise.5 E.5. E.5. E.5. Wen Carl jogs a a pace of 00 f/min is ear rae is 84 beas/min. A e pace of 00 f/min, Carl s ear rae canges relaive o is pace a a rae of 0.0 beas min f min. So, if Carl picks up is pace from 00 f/min o 0 f/min we epec is ear rae o increase o abou 84.0 beas/min. Conversely, if Carl ecreases is pace from 00 f/min o 99 f/min we epec is ear rae o ecrease o abou 8.98 beas/min. Wen Han rives er pick-up on level groun a a consan spee of 50 mp, e ruck burns fuel a e rae of 0.0 gal/mi. Appeni D D

139 Porlan Communiy College MTH 5 Lab Manual E.5.4 (On level groun), a e spee of 50 mp, e rae a wic Han s ruck burns fuel gal mi canges relaive o e spee a a rae of So, if Han increase er spee mi r from 50 mp o 5 mp, we epec e fuel consumpion rae for er ruck o ecrease o abou.094 gal/mi. Conversely, if Han ecrease er spee from 50 mp o 49 mp we epec o fuel consumpion rae for er ruck o increase o abou.006 gal/mi. E.5.5 E.5.6 E.5.7 Tweny secons afer lif-off e space sule is cruising a e rae of 66 mi/r. Tweny secons afer lif-off e velociy of e space sule is increasing a e rae mi r 8.9. So, we epec a 9 secons ino lif-off e velociy was abou 47. s mi/r an secons ino lif-off e velociy will be abou 84.9 mi/r. Tweny secons afer lif-off e space sule is a an elevaion of 0.7 miles. E.5.8 Tweny secons afer lif-off e space sule s elevaion is increasing a a rae of mi/s. So, we epec a 9 secons ino lif-off e elevaion was abou 0.66 miles an secons ino lif-off e elevaion will be abou mile. Eercise.6 E.6. Te poin on f wen is,. Te value of f ells us e slope of e angen line o f roug, ; is value is. Consequenly, e equaion of e angen line o f a is y. E.6. Te long erm slope of f is given by: lim f lim So e slope of e skew asympoe is. D4 A ppeni D

140 Porlan Communiy College MTH 5 Lab Manual Supplemenal Soluions for e Funcions, Derivaives, an Anierivaives Lab Eercise 4. y 0 Figure E4.bK Figure E4.bK Figure E4.bK Eercise 4. E4.. C, D, E E4.. B. D E4.. A, C E4..4 C, D, E E4..5 C, D, E E4..6 E, F, G E4..7 G E4..8 E, F, G E4..9 A, C, E, F, G E4..0 E, F, G Eercise 4. Iniial observaions Te average rainage rae is 60 gal/min wic is equivalen o gal/s. Iniially e ank rains faser an is rae an owars e en of e rainage e ank is raining slower an is rae. Appeni D D5

141 Porlan Communiy College MTH 5 Lab Manual E4.. Te uni for V 45 is gal s an e uni for V gal 45 is s Since e amoun of waer lef in e ank is ecreasing, e value of 45 s. V canno be posiive. Since 45 secons is early in e rainage process e rainage rae is almos cerainly greaer an e average rae of gal/s. Hence e mos realisic value for V 45 is.. E4.. Te uni for R 45 is gal s s an e uni for R 45 is Since e flow rae ecreases over ime, e value of 45 gal s s s. R canno be posiive. Since e average flow rae over e 60 secons is gal/s, i is no believable a 45 secons ino e process e flow rae was ecreasing a a rae gal/s/s. Hence e mos realisic R 45 is value for E4.. Tis quesion is similar o quesion E4.5. wi one key ifference; because V is efine as e amoun of waer lef in e ank, V as negaive values. Since e value of V is geing closer o zero, V is increasing an, consequenly, V canno be negaive. So e V 45 is mos realisic value for Eercise 4.4 Te answer o eac quesion is ii. Eercise 4.5 E4.5. Lisa is rig. Maybe g is isconinuous a, bu maybe i simply is poiny a 4. E4.5. Janice is rig aloug Lisa is prey arn close o being rig. Te grap of g is basically e line y 0, bu because g is noniffereniable a, g as a ole a 4. Eercise 4.6 E4.6. False: f 0 on, f ecreasing on a inerval E4.6. True: f concave up on, f increasing on a inerval E4.6. False: f 0 wile f 0 E4.6.4 False: f 6.7, so e angen line o f a 6 is a ecreasing line. E4.6.5 False: f is perioic; i woul be increasing over eac perio, owever (e.g. 0,4, 4,8, D6 A ppeni D

142 Porlan Communiy College MTH 5 Lab Manual E4.6.6 False: f ( 4) 4.; if f were noniffereniable a 4 en f woul ave no value a 4. E4.6.7 E4.6.8 E4.6.9 E4.6.0 E4.6. E4.6. True True False: Aloug e sape if e anierivaives is perioic, e anierivaives clearly en up iger en ey sare afer eac inerval of leng 4. False: Te amoun of air in your lungs was increasing e enire wo secons. True True: Negaive velociy correspons o ownwar moion (because e posiion is ecreasing wen e velociy is negaive). Eercise 4.7 Table E4.K f f f F Posiive Increasing Negaive Decreasing Consanly Zero Consan Posiive Increasing Concave Up Negaive Decreasing Concave Down Consanly Zero Consan Linear Posiive Increasing Concave Up Negaive Decreasing Concave Down Consanly Zero Consan Linear Eercise 4.8 E4.8.,, an 4 E4.8. nowere E4.8. ( 5,6) E4.8.4 ( 6, ) an (,) E4.8.5 ( 4,6 ) E4.8.6 (, ), (, 4 ), an ( 4,6 ) E4.8.7 ( 6, 5) E4.8.8 ere's no way o ell E E4.8.0 y 4 + Appeni D D7

143 Porlan Communiy College MTH 5 Lab Manual Eercise 4.9 Figure E4.6bK: F Eercise 4.0 Figure E4.7K f D8 A ppeni D

144 Porlan Communiy College MTH 5 Lab Manual Supplemenal Soluions for e Derivaive Formulas Lab Eercise ( ) f 6 f ( ) 6/ 5/ y 5 y y ( u) y ( u) u 4u 4 u / / 5..4 z( α) z ( α ) Eercise 5. eα π eπα π 5.. y 5sin( ) cos( ) 5..5 z 8 7/ z / 5..6 T T 4 4 / 4 4 7/ / y + 5cos cos + 5sin sin ( 5sin( ) ) cos( ) 5sin( ) cos( ) ( ) ( ) ( ) ( ) ( ) ( ) 5sin ( ) 5cos ( ) y e 7 y e + e 7 7 5e ( + ) F( ) 4 ln( ) ( ) e e F + 4ln ( ) + 4 ( ) ( ) ( 4 ) ln( ) 4 ln( ) ( ) 4ln + 4 Appeni D D9

145 Porlan Communiy College MTH 5 Lab Manual 5..4 z sin ( ) z + sin ( ) + ( ) sin ( ) ( sin ( ) ) ( ) sin T() ( + ) an () 5..6 Eercise q( θ ) T an () + ( + ) + () ( ) an () ( ) ( an ()) () + an 7 T 7 T ln( 7) ( 7+ ln( 7) ) ( θ ) q θ 4e θ e + θ ( ) θ θ θ θ ( 4e ) ( e + ) 4e ( e + ) θ θ ( e + ) ( + ) θ ( e + ) ( + ) θ ( e + ) 4e e 4e e θ θ θ θ 4e e e θ θ θ 4e θ θ ( e + ) D0 A ppeni D

146 Porlan Communiy College MTH 5 Lab Manual ln u 5.. u( ) 5.. ( ) ( ) ( ln( ) ) ln( ) ( ) 4 ( ) ln( ) ln( ) 8 8ln 5 4 ( ( ) ) 4ln / F 5 / / 8 ( ) ( 5 ) / / / 5 / / ()( 5) ( 5) / ( 5) ( ) / ( 5 ) 9 ( ) / / / ( 5) 9 / F Appeni D D

147 Porlan Communiy College MTH 5 Lab Manual 5..4 F( ) ( ) F an an ( ) ( ) ( ) ( an ( ) ) an ( ) an ( ) an ( ) ( an ( ) ) sec ( ) an ( ) an ( ) an () P ( an ( ) ) ( + ) sec ( ) an ( ) an ( ) ( + )( an ( ) ) + + P ( an () ) ( + ) an () ( + ) an y sin ( + ) ( ) () + an ( + ) ( + ) 4 () ( β) cos( β) y β β β β β ( 4) ( sin( β ) cos( β) ) 4 ( sin( β) cos( β) ) ( sin ( ) cos( )) ( ( β) ( β) ) ( β) ( ( β) ) ( sin ( β) cos( β) ) 4cos ( ( β) sin( β) ) ( sin ( β) cos( β) ) ( ) 0 sin cos 4 cos sin + D A ppeni D

148 Porlan Communiy College MTH 5 Lab Manual Eercise 5.4 E5.4. E5.4. ( f ( ) g( ) ( ) ) ( f ( ) ) g( ) ( ) f ( ) g( ) ( ) + f ( ) g( ) ( ) + f ( ) ( g( ) ) ( ) + g( ) ( ( ) ) f g + f g + f g ( ) sin ( ) cos( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f e f e + e + e + e ( ) ( ) sin ( ) cos( ) ( ) sin ( ) cos( ) ( sin ( )) cos( ) sin ( ) ( cos( )) e sin ( ) cos( ) + e sin ( ) cos( ) + e cos( ) cos( ) + e sin ( ) ( sin ( ) ) e ( sin ( ) cos( ) + sin ( ) cos( ) + cos ( ) sin ( ) ) Eercise 5.5 E5.5. f ( ) + + ; 0 f + ; 0 ( ) f ( 5) 0 an f ( 5) So e angen line o f a 5 passes roug e poin ( ) as a slope of. Te equaion of is line is y 5. E5.5. ( ) + ( ) ( ) ( + ) ( + ) ( + ) ( ) ( + ) + ( + ) ( ) an ( ) So e angen line o a an as a slope of. Te equaion of is line is y ,0 an passes roug e poin (, ) Appeni D D

149 Porlan Communiy College MTH 5 Lab Manual E5.5. K( ) ( + ) ; y K is simply e orizonal line y 0.5 wi a ole a, e Since e grap of ( ) angen line o K a 8 mus be e line wi equaion y 0.5. E5.5.4 ( ) / r e r e + e / / e + e / / e e + / / e ( + ) / / ( ) ( ) ( ) r ( 0) 0 an ( 0). Tis form for ( 0 ) r as e form 0 line a 0. So e angen o r a 0 is e line 0. Eercise 5.6 Eercise 5.7 y 4 / 4 5 y y r implies a r as a verical angen 9/ 7/ E5.7. f is posiive over (, 8) an (, ) ; f is negaive over ( 8, ) E5.7. f is increasing over an (, ); f is ecreasing over (, ). E5.7. f is posiive over (, ); f is negaive over (, ). f E5.7.4 ( ) f ( ) + 6 Figure E5.5.4K: 7 y e. y 0 Figure E5.7.4Ka: y f ( ) Figure E5.7.4Kb: y f ( ) D4 A ppeni D