Rviw of Eponntials and Logarithms - Classwork In our stud of calculus, w hav amind drivativs and intgrals of polnomial prssions, rational prssions, and trignomtric prssions. What w hav not amind ar ponntial prssions, prssions of th form a. Whil ths ar covrd tnsivl in prcalculus, a littl rviw is in ordr and ths tps of prssions ar vr prvalnt in th calculus thatr. An prssions in th form of a will graph an ponntial. An ponntial graph tnds to plod basd on th valu of, sinc th is in th ponnt.. Th graph of various ponntial curvs ar shown abov. W know that th graph of graphs horiz. lin. If a <, th graph of a will not ist at crtain points for instanc, what is ()? DNE So it onl maks sns to amin functions in form of a, if a >, a?. Whn a >, w gt what is calld a growth curv and th largr a is, th stpr th growth curv is. If < a <, th w gt a dca cuvr as shown in th rd graph abov. No mattr what, ponntial curvs in th form of a hav crtain faturs. What point do th hav in common? (, )What is th domain? (:,:) What is th rang?,: Solving basic ponntial quations can b accomplishd b using th fact that if a Eampls) Solv for. a, thn ) + 8 + ) ) ) 7 + ' *, 9+ 7 7 + 6 No Solution ) 8 8 8 www.mastrmathmntor.com AB Solutions - 8 - Stu Schwartz
Solving ponntial quations lik th ons abov ar as whn ach sid of th quation hav common bass. But problms lik caus problms. With that problm cratd, w introducd th concpt of logarithms. A logarithm is simpl an invrs of an ponntial. Studnts tpicall har th word logarithm and go into a cold swat bcaus th do not undrstand thm. So lts gt it straight onc and for all. Th statmnt b can b writtn in an altrnat wa: log. Th man th sam thing. Whnvr ou ar givn a logarithmic statmnt, writ it ponntiall. You will know th answr Eampls: find th valu of th following: ) log 8 ) log ) log 8 ) log 6 ) log 7 DNE 6 b If th bas is not spcifid, it is assumd to b. log and log ar th sam things. Eampls) Find th valu of th following: 6) log 7) log 8) log 9) log ) log ' * On what sms to b a sid not, lt s amin th prssion ) +, for various valus of. What w ar ( + doing is looking at ' * ) +,, ' * ) +,, ' * ' * ) +,, ) +, Logic would tll us that as gts largr, + gts ( + ( + ( + ( + closr to and this th limit as approachs infinit of + ' * is..thus, logic also dictats that lim) +, <: ( + But whn ou pla with infinit, logic dosn t alwas work. You can s that if ou st up our calculator with th ' prssion Y + * ), ( + and look at a tabl of valus. ' + * ' ), ( + + * ' ), ( + + * ' ), ( + + * ), ( +.788.7699.7868..7688.7767.787.777.779.77889.7877.6.7798.779.7878.88.78668.78.7879 6.667.7979 6.78 6.78796 7.6997 6.7988 7.788769 7.787989 8.678 7.797 8.7896 8.788 9.8779 8.7769 9.788 9.788.976 9.7679.789.7887 www.mastrmathmntor.com AB Solutions - 8 - Stu Schwartz
If should b obvious that as gts largr and largr, th prssion + is not approaching on but th numbr.78. This numbr is a vr spcial numbr in mathmatics and is calld Eulr s numbr. Lonhard Eulr (77-78) discovrd this numbr and it is known as. Th valu of is.78888. is a transcndntal numbr which, lik and, continus on forvr without an pattrn. (Not: th 88 in, although apparing twic conscutivl nar th start dos not appar again for a vr long whil. It is compltl coincidntal that it appars twic) Th numbr is such an important numbr (if ou would hav to dcid what th most important numbrs ar, what would th b?,,, i,,), that it forms th basic of what ar calld natural logarithms of Napirian logs (aftr John Napir, -67, who first usd thm). Just as logarithms (log) us bas, natural logs (ln) us bas. Whn ou wish to find th valu of a log, ou writ th prssion ponntiall. You do th sam thing with a natural log cpt that our bas is now. For instanc, to find ln, ou call it, and ar now solving th quation. Sinc is slightl blow, w pct ln to b btwn th valus of and ". So, givn th function, th domain is (:,:) and th rang is (,: ). Eampls) Find th valu of th following: ) ln ) 9ln ) 8ln ) ln ) ln Thr ar thr basic ruls for opration with logarithms that ou must know. Th ar as follows: +. log log a log. log a > b log a logb Ths ruls work with logs to an bas of th ln function. Eampls) Find th valu of th following prssions: ' a ) *, b. log log ( b+ a b b a ) log+ log log ) log log log 8 ) log ) log log ( + ) 6 ( + ) ( ) ( + 6) + log + - solv for www.mastrmathmntor.com AB Solutions - 8 - Stu Schwartz
Rviw of Eponntials and Logarithms - Homwork For ach curv blow, idntif it b th propr quation numbr. a. b. d.. c. f. g. h.. i. j. + k. + ( ).( ) (. ) + l.. m. n.. o. +... d m i.. 6. k a l 7. 8. 9. b o g... h c j... n f www.mastrmathmntor.com AB Solutions - 86 - Stu Schwartz
Solv for 6. 6 7 7. 8 7 8. 9.. + 6 8. + 8. 7 + 9 7. 6 8 9. 6 8. 9 ' *, 7+ 7 8 6. ' * ), ' ( + * 8+, + 6 7. ' ) * ( +, + 6 8. log 6 9. log 8. log 8. log. log 9 7. log 7. log. log 8 6 6. log 7 6 www.mastrmathmntor.com AB Solutions - 87 - Stu Schwartz
7. log log 9 8. log 9. 9. ln ln.. log ln Solv ach quation in trms of.. log 9 ) log 7 7 6 7 6 9 7 ) ln + ln ln9 9 9 ) log + log ( + ) log ( + ) 8 If a log 6and b log, prss th following in trms of a and b. 6) log( > 6) log + a + log 6 7) 6 log( > 6) log + log 6 + a 8) log log( ) log b www.mastrmathmntor.com AB Solutions - 88 - Stu Schwartz
Diffrntiation of th Natural Log Function - Classwork W hav amind drivativs using th powr rul, product rul, quotint rul, and trig. But what about th drivativ of ln? W hav no rul to covr such functions. Lt s s how our calculator dals with it. In our TI-8 calculator, lt Y ndrviv(ln,,) and st up our tabl with starting at and %. 6 7 8 9 " ERROR......67.... As ou look at th rlationship btwn th valu of and th valu of th drivativ of ln, it should b clar. What is it? Th drivativ is th rciprocal of So w hav a nw diffrntiation rul - th ln rul: d d ln, [ ] > d u d ln du u u d u, u [ ] > " > [ ] > "? d u d ln du u u d u, u What this sas is to tak th drivativ of th ln of som prssions, ou simpl us th rciprocal of th prssion multipld b th drivativ of that prssion. Th prssion must b a positiv numbr. Eampls) Find th drivativ of th following prssions: ) ln ) > ln( ) ln + 8 ) 6 + 8 ) ln Not: - Bttr wa: ln. ln. ln. You now hav sts of ruls - powr, product, quotint, trig (6 ruls), and now ln. Just bcaus thr is an ln in th problm dos not man it uss th ln rul abov. Eampls) Find th drivativ of th following prssions: ln ln ' * ) ), + ( ) ln ln ln 6) 7) ( + ln + ln ln ( ln ) 8) ( ln ) ( ln ) " ln ln cos( ln ) ln( cos ) ln( ln ) 9) " ( ln ) ln ' * ), ( + ) sin ln, + " ' * " sin( ln ) ) ' *,( sin ) cos + tan ) ' *, ' + *, ln + ln www.mastrmathmntor.com AB Solutions - 89 - Stu Schwartz
Rmmbr our log ruls. Th can hlp ou to tak drivativ of hardr prssions. +. log log a log. log a > b log a logb ' a ) *, b. log log ( b+ a b b a Eampls) Find th drivativ d d of th following prssions: ) ln + ln( + ) ) ln Hard wa: Eas wa: ( + ) + + + ' ) * ( +, + ln + + + + ( + ) ln( ) ln ln ln ) ln ln + ln + 6) + ln ln( + ) ln / + + [ ] 7) + ln + ln + ln( + ) ln + 6 + + + 8) ln + " + > " " + > " 9) Find rlativ trma of ln + + 6 + + + 6, rl min b st dr. tst -,ln is a rl. min ) Find th quation of th tangnt lin to ln at, 8, " 7 7 7 A tchniqu not includd on th AP am but hlpful to taking hard drivativs is calld logarithmic diffrntiation. It ssntiall sas to tak th natural log of th hard prssion in ordr to tak advantag of th log ruls. Eampl ) ( ). ln ln( ) ln( + + ) + + d ' * ' + * d ' * ' + * ' * ' + * ( ) ),),. ),), ), ), d ( + ( + + + d ( + ( + + + ( + ( + + + + + www.mastrmathmntor.com AB Solutions - 9 - Stu Schwartz
Diffrntiation of th Natural Log Function - Homwork Find th drivativ d d of th following prssions: ln 6 ln. 6.. 7 ( ln ) 7 ( ln ) " 6. ln + ln. ln ln ln 6. ln ln 8 ln 7. ln 8. ln( ln ) ' *, ' + 6 *, ln + ln 9. lnsin ' *, cos sin + cot. sin( ln ) cos( ln ). ln tan sc. + ln tan tan tan ln tan ln sc ln ln sc tan ln. [( )] + ln + ln + ln 6 + +. + ln ln + ln 6 +. ln 7 ln 7 7 6. ln + ln + 9 + + 7. ln + ln( ) ln + / + [ ] 8. / ln > + ln + ln( ) ln + / + + [ ] www.mastrmathmntor.com AB Solutions - 9 - Stu Schwartz
Us implicit diffrntiation to find d d. 9. + ln + d 8 + + d d d d 8 + + d d d d 8 d +. ln ln + ln d + d + d d d d. ln + ln + d + d + d d d d + + d d d + d Find th quation of th lin tangnt to th graph of th following functions at th indicatd point. + ln at (,-) 6 ln( + ) at (,6) ln at (,) +,. "., "( ) +., " + ( ) 6 ( ) + 6 Find an rlativ trma to th following functions. ln ln. 6. Rl min at Rl ma at not dfind at - 7. ln + ln + ln + ln ln, Rl min at 8. ln ln ln Rl ma at Us logarithmic diffrntiation to find d d. 9. ( )( ) [ ] ln ln( ) + ln( ) + ( ) d / d + + d / d + + ( ) ( ) + ln ln( ) ln +. d / d + [ ] d / d + + www.mastrmathmntor.com AB Solutions - 9 - Stu Schwartz
th Natural Log Function and Intgration - Classwork Th drivativ ruls which w just larnd will now produc th following intgration ruls: d ln + C and if u is a diffrntiabl function of, u du ln u + C Eampls) Find th following: d ) d u, du d ) du ln + C d u ln + C ) 6 d u 6, du 6d 6 6 6 d ln 6 + C du u ) 7 d u, du d 7 7 d 7 ln + C du u Whn ou tak intgrals of fractions, ou usuall think u-substitution with th u bing th dnominator gnrating a ln function. But not alwas. d d 6 u 6, du d u, du d ) 6) ( ) ( 6 ) d ( ) d ( 6 ) + C ( ) + C 7) ln d u ln, du d u du ( ln ) + C 8) + d ' * ) +, ( + d + ln + C sin cos tan d cos d cot d sin d cos sin d + cos tan d u cos, du sin d u sin, du cosd u + sin, du cos d u tan, du sc d 9) ) du du u d u d ) ) du u d du u d ln cos + C ln sin + C ln + sin + C ln tan + C ) d + u +, du d, u,, u 9 du ln u u ln9 ln [ ] 9 ) ln d u ln, du d, u,, u du ln u u ln ") & sin + cos d u + cos, du sin d, u, &, u du u ln ln u www.mastrmathmntor.com AB Solutions - 9 - Stu Schwartz
th Natural Log Function and Intgration - Homwork Find th following: d. + ln + + C. d ln + C. d ln + C. d ln + C. d + ( + ) + C 6. d ln + C 7. d ln + C 8. + d + 8 ln + 8 + C 9. ln d 6 ( ln ) + C 6. ln d ln ln ( ) + C. tan- d- ln cos( -) + C. sin- d cos- - ln cos- + C. d + ln. ( + ln ) d. & sin + cos d '&* ln), ( + www.mastrmathmntor.com AB Solutions - 9 - Stu Schwartz
Drivativs and Intgrals of Eprssions with - Classwork Lt us tr to tak th drivativ of. Again, it sms as if that thr is no rul (powr, product, quotint, trig, ln) to tak it. Lt s amin this b us of th calculator. St Y and Y NDriv(,,). Thn st a tabl to look at ths valus. - - - -.8...68..78 7.89.86.98 d.8...68..78 7.89.86.98 d It should b obvious (and surprising) what is happning. Lt s tr and prov it. Lt s tak th drivativ of using logarithmic diffrntiation. ln ln d d. d d So w nd up with th rsult that quations in th form of C ar th onl quations (with th cption of ) whos drivativ is th sam as th prssion itslf. d d [ ] and if u is a diffrntiabl function of thn d d du u u [ ] d Eampls) Find th drivativ d d of th following prssions: ) ) 8 ) ) W now hav 6 basic ruls for drivativs: powr, product, quotint, trig (6), ln, and now. ) + + 6) + 7) sin cos " 8) sin sin cos " 9) ln + + + ) ) 8 ) www.mastrmathmntor.com AB Solutions - 9 - Stu Schwartz
Find d d b implicit diffrntiation: ) + 8 d d + + 8 d d d + 8 d Find th scond drivativ of th function: ) + "". Find rlativ trma and inflction point(s) for th function. + Nvr quals zro, alwas positiv (alwas incrasing) "" Equals zro whn, concav down if <, concav up if > 6. Find th ara of th largst rctangl that can b inscribd undr th curv in th first quadrant. A A". 68 (Graph and find zros). 68 A. 68. 77 Obviousl, sinc th drivativ of is, it follows that th intgral formula should b as simpl. d + C and if u is a diffrntiabl function of thn u u du + C Eampls) Find th following: d d 7) d 8) d + C + C 9) d + C ) d d + C ) cos > sin + C sin d ) d d ln + C ) + d ln + + C ) + + d + + d + + C www.mastrmathmntor.com AB Solutions - 96 - Stu Schwartz
Find th ara boundd b th curvs and lins.vrif b calculator.,,,,, + A d A 9) ) d + A A ln( + ) A ' + * A ln( + ) ln ln ( + ) sin,,. A sin d. A cos. A cos + cos ) Find th volum whn th first quadrant rgion R boundd b and is rotatd about th -ais. V ' * d ( + V V Finall, occasionall, w hav to tak drivativs of ponntial functions with bass othr than. Using th fact that a a ( ln ), w can tak th drivativ b saing that d d a ( ln a) a a > ln > ln a. You nd to know that: d d a a > ln a and d d a a a du u u > ln > d Eampls) Find th drivativs of: ) ln ) ( ) ( ln) ) 6 6 ln6 6 " ( ) + www.mastrmathmntor.com AB Solutions - 97 - Stu Schwartz
Drivativs and Intgrals of Eprssions with - Homwork Find th drivativs of th following functions:.. 6. " +. 6 6 6 ( 6) 7. tan sc + tan 6. cos sin 7. ln ( ln ) ( ) ln ln " ( ln ) 8. ln ln ln ( ln ) ( ) 9... sc sc sc tan ". + +. ' * ln), ( + + + ln ln +. + sin cos cos + sin sin cos sin. & & ln& 6. sin ( sin ) ( ln) cos Us implicit diffrntiation to find d d Find th scond drivativ of th following 7. d d + d d d d 8. ln ln "" ln www.mastrmathmntor.com AB Solutions - 98 - Stu Schwartz
Find th trma and points of inflction for th following functions: 9. ' * ), ( + ' * ), ( + Rl. ma at,. + ' * Rl. ma at ),, ( +. + ' * Rl. ma at,, rl. min at, ( + Find th following intgrals 6 d.. 6 + C 6 d + C. sin > cos + C cos d. tan cos d tan + C 6. + d + + C 7. d + C 8. + d ln + C 9. + + d + C. d + ln + + C. tan d ln cos + C. 9 d. d www.mastrmathmntor.com AB Solutions - 99 - Stu Schwartz
Find th ara of th rgion boundd b th graphs of th functions:,,,,,,. A d. A d 6. Lt R b th rgion boundd b 6. and and th lins and. Find th volum whn R is rotatd about th -ais.. V & 6 d V. 68& [ ] 7. Find th volum whn th ara btwn th graphs of th following functions is rotatd about th -ais: +,,, V & ' ( + * + d V. 7& www.mastrmathmntor.com AB Solutions - - Stu Schwartz
Invrs Trig Functions - Classwork Th lft hand graph blow shows how th population of a crtain cit ma grow as a function of tim. If ou ar intrstd in finding th tim at which th population rachs a crtain valu, it ma b mor convnint to rvrs th variabls and writ tim as a function of population. Th rlation ou gt b intrchanging th two variabls is calld th invrs of th original function. Th graph of th invrs is shown on th right graph blow. Population Tim Invrs rlation Original Function Tim Population For a linar function such as + 6, intrchanging th variabls givs + 6 for th invrs rlation. Solving for in trms of givs. -. Th smbol f, pronouncd f invrs, is usd for th invrs + function. If f 6, thn f.. If f turns out to b a function (passs th vrtical lin tst), th th original function f is said to b invrtibl. Rmmbr that th - ponnt dos not man th rciprocal of f. Th invrs of a function undos what th ( ) function did to. That is f f. If f, thn f ( ) and. Not that if th sam scals ar usd for th two as, thn th graphs of f and f ar mirror imags with rspct to th o lin. Th invrss of th trigonomtric functions follow from th dfinition. For instanc, if th function is sin, th invrs function is givn b sin. Whn w solv for, w gt sin. Th smbol arcsin is somtims usd to hlp ou distinguish sin from. Hr ar th graphs of sin sin and sin. It is obvious that th invrs sin rlation is not a function. Thr ar man valus of for th sam valu of. To crat a function that is th invrs of sin it is customar to rstrict th rang to & < < &. This includs onl th branch of th graph narst th origin. (rd pictur abov). www.mastrmathmntor.com AB Solutions - - Stu Schwartz
Blow ar picturs of th invrs cosin function and invrs tangnt function. Not that in ordr to nsur that ths rlations ar functions, w hav to rstrict th rang. So, w hav ths dfinitions: / @ & & sin if and onl if sin and, [ ] cos if and onl if cos and @, & ' @ & & * tan if and onl if tan and ),, ( + You must know convrsions of dgrs to radians and spcial triangls. radians 6 o or radians 8 o. Som of th rlationships that ou should know ar: Dgrs 6 9 8 Radians 6 6 In a o -6 o -9 o triangl, th sids ar alwas in th proportion - AB -. In a o - o -9 o triangl, th sids ar alwas in th proportion - - AB. 6 Eampl ) "Evaluat ach of th following: a) arcsin ' ) ( & 6 *, + b. cos & c. tan & d. csc & www.mastrmathmntor.com AB Solutions - - Stu Schwartz
Eampl ) Evaluat th following. Mak a pictur to dscrib th situation. ' a. sin arctan * ' * ), b. tan) arccos, ( + ( + ' c. sc) sin ( *, + 8 ' d. cot cos * ), ( + Eampl ) Evaluat th following. Mak a pictur to dscrib th situation. a. cos sin b. tan cos c. sin cos d. sin tan ( ) + 9 So now w can tak drivativs of invrs trig functions. Find d sin d sin C sin. sin D sin Draw a pictur th angl is, opposit, hpotnus Rmaining sid is Sinc sin, tak th drivativ of ach sid cos d d d or or d d cos d Eampl ) Tak th drivativ of a. cos d d b. tan d d + www.mastrmathmntor.com AB Solutions - - Stu Schwartz
Th drivativs of th thr invrs trig functions ar as follows: d ( sin u) d u d ( cos u) d u d tan u d u + du d du d du d Eampl ) Find th drivativs of sin a. d d 6 b. tan d d + 6 ' * cos, + c. d ' d * cos, + ' * 6 cos, + d. sin + d d + sin + sin Eampl 6) An officr in a patrol car sitting ft from th highwa obsrvs a truck approaching. At a particular instant t sconds, th truck is ft down th road. Th lin of sight to th truck maks an angl of - radians to a prpndicular lin to th road. Truck a. Eprss - as an invrs trig function. b. Find d - dt - tan - ft Patrol Car c. Whn th truck is at ft, th angl is obsrvd to b changing at a rat d- - dgrs/sc. How fast is th car going in ft/sc and mph? dt d- d dt + dt & d d. 9. 77 ft/sc 6.88 mph 8 6 dt dt www.mastrmathmntor.com AB Solutions - - Stu Schwartz
Invrs Trig Functions - Homwork ) "Evaluat ach of th following: a. arccos & b. cot & 6 c. sin & d. sc & 6 ) Evaluat th following. Mak a pictur to dscrib th situation. a. cos ' arcsin * ' ), b. sin arctan * ), ( + ( + c. csc( cot ) 7 d. tan csc ) Evaluat th following. Mak a pictur to dscrib th situation. a. cos( tan ) b. sc sin + c. tan sin ( ) 6 ( ) d. cos tan + + 6 + ) Find th drivativs of a. d d cos ( ) 9 b. sin d d www.mastrmathmntor.com AB Solutions - - Stu Schwartz
c.. ( tan ) d d ( tan ) + arctan d d + ( tan ) + ) Find an rlativ trma of arcsin d arcsin. d d, Critical points,. > E,? d d. f. ( cos ) d d cos d d d dt d dt ( cos ) sin( cos t) cos( cos t) t t No rlativ trma t 6) Th bas of a foot tall it sign is ft abov th drivr s lvl. Whn cars ar far awa, th sign is hard to rad bcaus of th distanc. Whn th ar clos, th sign is hard to rad bcaus th drivr has to look up at a stp angl. Th sign is asist to rad whn th distanc is such that th angl - at th drivr s is as larg as possibl. a) Writ - as th diffrnc of invrs tangnts. - tan tan b) Writ an quation for d- d - d- ft + d + + 9 Car c) Th sign is asist to rad at th valu of whr - stops incrasing and starts dcrasing. This happns whn d-. Find and confirm using th calculator. + + 7. 8. 7 ft. d ft Eit Mil ahad www.mastrmathmntor.com AB Solutions - 6 - Stu Schwartz