The tangent line and the velocity problems. The derivative at a point and rates of change. , such as the graph shown below:
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1 Cpter : Derivtives In tis cpter we will cover: Te tnent line n te velocity problems Te erivtive t point n rtes o cne Te erivtive s unction Dierentibility Derivtives o constnt, power, polynomils n eponentil unctions Te Prouct Rule n te Quotient Rule 3 Derivtives o trionometric unctions 4 Cin Rule Te erivtive o enerl eponentil unctions 5 Implicit ierentition 6 Derivtives o loritmic unctions Loritmic ierentition 9 Relte rtes Cpter Review Te tnent line n te velocity problems Te erivtive t point n rtes o cne A Te tnent problem: Consier te rp o unction, suc s te rp sown below: Fiure : Te rp o eneric unction n clcultion o te tnent line
2 How cn we eine n in te tnent line to tis rp t point on te curve P,? Te einition o te tnent line: We remember tt to in tnent line t te point P, we consier eneric point Q, on te rp, n clculte irst te slope o te secnt line PQ: m PQ, s sown in te iure Ten te tnent line t P, is te line wic psses trou P n wic s te slope: m T mpq, provie tt tis it eists QP Emple : Clculte te eqution o te tnent line to te rp o t, P, P, n P,4 Note: We sometimes reer to te slope o te tnent line to te rp o t point P, s te slope o te unction t P, Tis is becuse i we zoom in enou ner te point P, ten te unction i smoot t te point will pper to be line It is useul to see tt ormul cn lso be written s: m T, by mkin in te substitution Emple : Let unction Clculte te eqution o te tnent line t te point P, on te rp o tis B Te velocity problem: We ve seen in tt or n object wic moves lon strit line wit te position iven by s t, we ve tt te vere velocity o te object over time intervl t,t is iven by: 3 4 v t t t t cne in position ivie by te cne in time over tis time intervl n, Av t t eine te instntneous velocity or instntneous spee t v v, Av Emple 3: Do problem 3 rom te eercise set 7 t s were t is n rbitrry time t
3 Equtions, n 4, wic re very similr, ll ive te instntneous rte o cne o te unction t P, Tis quntity is clle te erivtive o te unction t n is enote by ' Deinition : Te erivtive o unction t point in its omin D is iven by: 5 ', i tis it eists Optionl Note: In enerl, or unction y, te ierence quotient: 5 y rom to represents te vere reltive rte o cne o tis unction, wen cnes by Te reltive instntneous rte o cne o te quntity y t is lso te slope o te tnent line to te rp o Emple 4: Do emple 6 on pe 48 En Optionl Emple 5: t Fin te erivtive o te unction 4 9 t point ', wic is b Fin n eqution o te tnent line to te prbol 4 9 t P 3,6 Emple 5: Clculte,, - n or tese points n interpret tese s te instntneous rte o cne o t Note tt wen te erivtive is lre in mnitue t, ten te y vlues o te unction ner cne rpily, n wen te erivtive is close to, ten te curve is reltively lt ner, n te unction cnes very slowly ner Homework: Problems 3,4, 7, 8, 9,,,5,6,7,,4,7,9,3,3,33,36,37,53,54 rom Eercise set 7 3
4 Te erivtive s unction Dierentibility Note tt in we clculte te erivtive o unction t speciic point in its omin It is oten best to clculte i possible ' t ny point in its omin Tereore, eine: ' wen tis it eists Note: t We cn try to use lso ' s enerliztion o 5 rom ltou te t t ormul bove is more nturl, n tereore lmost lwys use Given ny number, te epression sown in eines new unction ', clle te erivtive o Emple : I t Tis cn be interprete s te slope o te tnent line to te rp o t Te omin o smller tn te omin o is te set D ' eists 3, clculte ' Ten rw te rps o n ' on te sme es n conirm tt ' ives te slope o te tnent line to te rp o t ec point Emple : For, clculte ' Fin te omins o ' n o Grp bot unctions on teir respective omins Deinition : A unction is ierentible t i ' eists b A unction Emple 3: corresponin intervl is ierentible on, b or, or, or, i ' eists t ec point o te ' Were is ierentible n were it is not ierentible? Question: In enerl, ow cn unction il to be ierentible? First, let us estblis te ollowin importnt result: 4
5 Teorem : Consier unction : D R n D I is ierentible t, ten is continuous t n tke te it o tis equlity, rememberin tt Proo: Consier En o proo ' eists Tereore, one wy in wic unction is not ierentible t is wen te unction is iscontinuous t To unerstn best ow unction my il to be ierentible t, eine: ' l n 3 ' r Since ierentibility t mens tt ' eists, ten we cn estblis: Teorem : Consier unction : D R n D is ierentible t i n only i: ' l n ' r eists n i ' l ' r Tereore, unction cn il to be ierentible i one o te ollowin tkes plce: is iscontinuous t tink lso o te etreme cse in wic s te sme slope on eiter, or sie o te point but it is iscontinuous t, suc s :, wic is iscontinuous t, or Sow tt tis unction is not ierentible t, by clcultin ' n ' ' l or it oes not eist or ' or it oes not eist Note tt n ininite semi tnent r line mens tt te semi tnent line is verticl ' l n ' r bot eist but ' l ' r tis cse correspons to corner in te rp o See lso Fiure 7 on pe 59 in te tetbook or rpicl illustrtion o tese cses Homework: Problems,3,4,7,8,9,,6,7,9,,,6,7,8,37,38,44,45,56 rom Eercise Set 8 l r 5
6 Derivtives o constnt, power, polynomils n eponentil unctions From now on n or ew sections we inten to estblis ormuls or erivtives o elementry unctions lebric n trionometric, or trnscenentl unctions n or combintions o tese Tis will llow us to quickly clculte te erivtives o tese unctions, n tese clcultions will be use in mny pplictions s it ives te reltive instntneous rte o cne o tese epressions Finlly, we will collect ll tese ormuls in tble o erivtives, to ornize n elp us wit memorizin tese In tis section, we strt wit erivin ormuls or te erivtives o constnt, power, polynomil n eponentil unction U ormul rom, it is esy to see tt: c, or ny constnt c R Similrly:, n 3 3 Tis seems to suest te more enerl ormul: n n n, or n N, wic is esy to prove i we use te lebric ormul : n n n3 n n b b b b b n n b or te binomil ormul : n n n n! were n nk k k k k k! n k! Formul is clle te Power Rule or erivtives or s well n N We cn sow tt te power rule ols or n R First, ceck tt ols or n,, wic suests tt it ols or n Z proo will be iven in te net section Also ceck tt ols or n / n n / 3 Te ollowin enerl rules cn lso esily be veriie: 3 4 c c ', or ny constnt c R, n : ' ' n 5 ' ' by recllin tt n tt Formuls 3 to 5 llow us to eten ormuls n to oter epressions see emples 5 n 6 in te tetbook 6
7 Let us turn now to te erivtive o n eponentil unction, : R, ce First, u te lebric property:, we in tt :, tereore, we nee to etermine ' te enerl erivtive ' ' 5 ' ' ' ' Tere re mny possible equivlent einitions o te number e, wic is te bsis o te nturl loritmic unction suc s: 6 e n n Here, we eine e s te irrtionl number e,3 n suc tt e 7 note tt inormlly 7 is equivlent wit 6 7 mens tt e is te prticulr eponentil unction te unction e is see Fiure below or wic ' te slope o te tnent line t oriin or Fiure : Deinition o te number e U 5, tis prouces te useul ormul: 8 e e Emple : Do emple 8 n 9 in te tetbook Homework: Problems,3,4,6,9,,,3,5,6,8,,3,7,33,36,38,47,5,5,6,67,68,7,7,75,77 rom Eercise Set 3 7
8 8 Te Prouct Rule n te Quotient Rule: Tese two enerl ormuls or erivtives te prouct rule n te quotient rule will llow us to clculte erivtives o combintions o unctions lernt in Te enerl ormuls re: ' ' te prouct rule n ' ' te quotient rule Memorize tese ormuls correctly py specil ttention to te in ormul Proo o : ' ' Proo o : ' ' Note tt or we nee to keep trck o wic epression unction is te numertor n wic is te enomintor, wile or te orer o te terms o te prouct is not importnt Emple : Do Emples, 3, 4 n 5 in section 3 in te tetbook Memorize te ormuls sown t te en o te section 3 Homework: Problems,3,6,5,6,,3,5,7,9,3,33,35,43,46,49,54 rom Eercise Set 3 in te tetbook
9 3 Derivtives o trionometric unctions Gol: Derive ormuls or ' cos cos ' tn tn ' cot cot ' Beore pplyin ormul in to erive tese, note te rps o n o its erivtive built by mesurin te slope o te tnent lines to te rp o t ierent points below: Fiure : Te rps o n o its erivtive ' Tereore oo uess or U ormul in : ' is ' cos Let us veriy tis lebriclly: cos cos cos cos Tereore, in orer to in ', we nee to prove ormul or wic we uesse s bein in cos n or For provin tt, consier te sector o unit circle sown in Fiure below: Fiure : A sector o n unit circle: note tt BC, OC cos, AO n AD tn Also, we ve tt: Are OBC Aresector OAB AreOAD cos tn cos cos 9
10 by u te squeeze teorem Tereore: Tereore, we ve sown tt: ' 3 u u rom te it lws is n importnt result wic we will use to in, n n importnt it o its own Note tt implies more enerlly tt u, were u is ny epression wic pproces Emple : See Emple 5 in section 33 in te tetbook cos In orer to in, note tt : cos cos cos Tereore: cos 4 wic implies te more enerl result: cos u 5 u u Returnin now to inin ormul or ', substitute n 4 in to in tt 6 cos, s uesse compre wit te rp in Fiure Similrly: cos cos cos cos cos cos cos cos ter u in n 4 Tereore: 7 cos 6 n 7 re importnt ormuls or te erivtives o n cos wic we will use oten
11 By u now te quotient rule ormul in, we cn esily erive tt eercise: 8 tn n tt: 9 cos cot A ormuls 6, 7, 8 n 9 to te tble o erivtives n memorize tis tble Emple : Emple in section 33 in te tetbook n eercises,,, 39 n 46 rom eercise set 33 Homework: eercises 7,9,,3,5,5,7,4,4,44,49 n 5 rom eercise set 33 4 Te Cin Rule: Gol: Derive n use rule or compose unction o te orm: u Motivtion: Until now, we know ow to clculte te erivtives o most o te simple lebric, trionometric n eponentil unctions However, wt i we nee to clculte te erivtive o compose unctions suc s: or Te cin rule erive in tis section will provie enerl ormul to clculte suc erivtives n oters Note tt: u u u u u u u u u u u u Formul is te cin rule, written sortly s: u u u n wic res : Te erivtive o unction pplie to noter unction is te u erivtive o te outer unction, evlute t te inner unction times te erivtive o te inner unction Emple : Do Emple in section 34: Clculte ' or Do Emple, tt is clculte ' or U, we cn esily erive: ln e ln ln e ln ' e ln ln n or ' e Tereore: 3 ln, erive in, n in reement wit tis Tis is te enerliztion o ormul or
12 U te cin rule, we esily enerlize ll ierentition ormul erive so r in te tble o ierentition, by tin n etr u or ec, or emple: n n n or n=,,3 becomes: 4 n u n n u u', or n=,,3, n so on or te oters Eercise: Generlize s suc ll ormuls in te tble o erivtives to obtin te erivtives o compose unctions Emple : Do Emples 3,4 n 8 in section 34 in te tetbook Homework: eercises,,3,4,7,9,,,8,39,4,47,5,63, 65 n 73 rom eercise set 34 in te tetbook 5 Implicit Dierentition: In sections to 4 we lernt ow to clculte te erivtive o unction y y, wic is iven eplicitly in terms o noter vrible, suc s: y or y 3, or, in enerl: y Mny unctions, owever, re eine by n implicit reltion between n y o te orm: E, y In tis cse we cn still clculte y' by tkin te erivtive o n oten u cin rule Tis meto is clle implicit ierentition Emple : Emple on pe Consier te implicit eqution: 3 y 5 were we consier tt y y Clculte y' b Fin te eqution o te tnent line to te circle 3 y 5 t te point 3,4 c Look lso t solution in te tetbook in wic y' is oun by irst solvin or y eplicitly
13 Note tt in mny cses or n eqution o te orm, y cnnot be oun or it cnnot be esily oun eplicitly Tereore te meto outline in points n b bove is te only meto we cn use to in y' Emple : Emple on pe Consier te implicit eqution : y 6y Fin y' by implicit ierentition ; b Fin te eqution o te tnent line to tis curve te olium o Descrtes t 3,3 c t wt points in te irst qurnt, y wit n y on tis curve is te tnent line orizontl? Emple 3: Consier y 7y Clculte ' y n ' y Importnt Note: Te simplicity o te process bove ies some iiculties, tou Tt is, in n implicit ormul o te type, y my not be eine or it my not be unction o t, or ' my not eist t However, i etermines y s unction o tt is, i y cn be oun uniquely rom, n i tis is ierentible t, ten te erivtive y oun s in te process bove by implicit ierentition ives us te correct vlue o ' 3
14 Emple 4: Consier 6 y By implicit ierentition, we in y' Tis is correct y Wt is, owever, y '? Note tt 6 oes not necessrily ives y s unction y=, unless etr conitions re speciie suc s y Besies, even i we ssume tt 6 ives unction y= t some, tis unction my not be ierentible t! In te problems in our tetbook, tou, we ssume tt n implicit eqution etermines ierentible unction t ny were we wnt to clculte erivtives, suc tt implicit ierentition ives te correct erivtive Emple 5: Emple 4 on pe 3: Fin 4 4 y '' i y 6 m m r m n n Emple 6: Do problem 48 rom eercise set 35 to sow tt 7 y ' y ' y r y n ierentition r u implicit From now on, we consier 7 correct n we use it 7 cn lso be sown or ny rel power r s well Homework: Problems,,3,5,6,,6,,8,33,35,36,43 n 65 rom Eercise Set 35 6 Derivtives o loritmic unctions Loritmic Dierentition In tis section we will complete te tble o erivtives wit ormuls or te erivtives o te loritmic lo n ln unctions: We will lso lern tecnique clle loritmic ierentition wic llows us to clculte reltively quickly te erivtives o complicte epressions, typiclly involvin proucts, rctions n / or powers n eponents I Te erivtives o lo n ln : First we sow tt: lo ln note tt in we nee : n Wy? 4
15 y To sow, recll tt y lo u Use implicit ierentition in n recll tt see 4: ln u' y ln y' y' rom, wic sows y ln ln Consier now e in to in tt: 3 ln u Tereore: Formuls n 3 ive te erivtives o loritmic unctions n tey cn be enerlize or composite unctions s: 4 u lo u' n 5 u ln u ln u' u Te simplicity o ormuls 3 n 5 mkes te nturl loritm n te nturl eponentil very useul in Clculus n in oter sciences A ormuls, 3, 4 n 5 to te tble o erivtives n memorize ll ormuls in te tble o erivtives Emple pe 8: Fin ' Emple : pe 9: Fin ' y i ln 3 y y i y ln Emple 3: pe 9: Fin y ' i y ln Emple 4: pe 9: Fin ' y i y lo II Loritmic ierentition: To clculte reltively quickly te erivtives o more complicte unctions involvin proucts, quotients, powers n/or eponents, we cn pply te nturl loritm to bot sies o te eqution irst Tis tecnique is clle loritmic ierentition Note tt tere re some cses o te orm y, y'? clculte witout loritmic ierentition, wen y cnnot be Emple 7: pe 9: Clculte y ' i y 3/ Emple 8 problem 43 rom Eercise set 6: Clculte y ' i y Homework: problems,5,6,,3,5,,4,6,3,33,39,4,44,46 n 49 rom eercise set 6 5
16 9 Relte rtes : In mny prcticl problems, quntities cne usully wit time I we know te rte o cne o one quntity: t n reltion y y or E, y, ten we cn in y by tkin te erivtive wit respect to t o t y or Te rtes erivtives n re relte, n tereore tese rtes o cne re clle relte rtes t t Emple : pe 43: 3 Air is bein pumpe into spericl blloon so tt its volume increses t rte o cm / sec Consierin tt tis process continues ineinitely, ow st is te rius o te blloon incre wen te imeter o te blloon is 5 cm? Wen solvin relte rtes problems, it is useul to ollow te ollowin uielines: Step : Trnslte ll iven inormtion in terms o mtemticl quntities: Let t be time Drw, i possible, irm wic is vli t ALL times On tis iure lbel or in te quntities o interest t, y t, n oter iven quntities constnts, etc Write ormul or te iven Epress te require rte o cne t erivtives y in enerl t in terms o Step : Fin te reltion between y n : 3 E, y vli t ALL times Step 3: Dierentite 3 n solve or y t Emple pe 45: A ler t lon rests inst verticl wll I te bottom o te ler slies wy rom te wll t rte o t/sec, ow st is te top o te ler sliin own te wll wen te bottom o te ler is 6 t rom te wll? Emple 3 pe 45 : 6
17 Emple 4 pe 47: Homework: problems,5,7,,,,5,6,,3,6,37,44,45,46 rom problem set 39 Cpter Review In tis cpter, we : Lerne te einition o te erivtive o unction t point: ', n its interprettion n use in terms o te slope o te tnent line m, instntneous velocity v n instntneous rte o cne o T y unction: Section Lerne te einition o ', its interprettion s unction, n te notion o ierentible unctions Cses o non-ierentibility n te rpicl menin o ierentible n non-ierentible unctions Section Derive n lerne memorize ormuls or te erivtives o, e,, polynomils,, c,,,, trionometric n loritmic unctions n ornize tese in tble o erivtives Sections,, 3 n 4 n 6; Lerne ow to clculte erivtives o unctions eine implicitly u implicit ierentition Section 5; Lerne ow to solve relte rtes problems n ow to use loritmic ierentition Section 9 n 6 Lern te min results einitions, teorems n ormuls rom tese sections best write review uie to inclue tese, ten solve te review problems rom pes 63 to 67 in te tetbook, incluin te Concept Ceck, True/ Flse n Eercises types o problems 7
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