The tangent line and the velocity problems. The derivative at a point and rates of change. , such as the graph shown below:

Transcription

1 Capter 3: Derivatives In tis capter we will cover: 3 Te tanent line an te velocity problems Te erivative at a point an rates o cane 3 Te erivative as a unction Dierentiability 3 Derivatives o constant, power, polynomials an eponential unctions 3 Te Prouct Rule an te Quotient Rule 33 Derivatives o trionometric unctions 34 Cain Rule Te erivative o eneral eponential unctions 35 Implicit ierentiation 36 Derivatives o loaritmic unctions Loaritmic ierentiation 39 Relate rates 3 Capter Review 3 Te tanent line an te velocity problems Te erivative at a point an rates o cane A Te tanent problem: Consier te rap o a unction, suc as te rap sown below: Fiure : Te rap o a eneric unction an calculation o te tanent line

2 How can we eine an in te tanent line to tis rap at a point on te curve P a, a? Te einition o te tanent line: We remember tat to in tanent line at te point P, we consier a eneric point Q, on te rap, an calculate irst te slope o te secant line PQ: m PQ a, as sown in te iure a Ten te tanent line at P a, a is te line wic passes trou P an wic as te slope: m T a mpq, provie tat tis it eists QP a a Eample : Calculate te equation o te tanent line to te rap o at, P, P, an P,4 Note: We sometimes reer to te slope o te tanent line to te rap o at a point P a, a as te slope o te unction at P a, a Tis is because i we zoom in enou near te point P, ten te unction i smoot at te point will appear to be a line It is useul to see tat ormula can also be written as: m T a a, by makin in te substitution a Eample : Let unction Calculate te equation o te tanent line at te point P, on te rap o tis B Te velocity problem: We ave seen in tat or an object wic moves alon a strait line wit te position iven by s t, we ave tat te averae velocity o te object over a time interval t,t is iven by: 3 4 v t t t t cane in position ivie by te cane in time over tis time interval an, Av t t eine te instantaneous velocity or instantaneous spee at v a v a, a Av a Eample 3: Do problem 3 rom te eercise set 7 a t a as were t is an arbitrary time t a

3 Equations, an 4, wic are very similar, all ive te instantaneous rate o cane o te unction at P a, a Tis quantity is calle te erivative o te unction at a an is enote by ' a Deinition : Te erivative o a unction at a point in its omain a D is iven by: 5 ' a a a a a a, i tis it eists Optional Note: In eneral, or a unction y, te ierence quotient: y 5 represents te averae relative rate o cane o tis unction, wen canes by rom to a Te relative instantaneous rate o cane o te quantity y at a is ' a, wic is a a also te slope o te tanent line to te rap o at a Eample 4: Do eample 6 on pae 48 En Optional Eample 5: a Fin te erivative o te unction 4 9 at a point a b Fin an equation o te tanent line to te parabola 4 9 at P 3,6 Eample 5: Calculate,, - an or tese points an interpret tese as te instantaneous rate o cane o at Note tat wen te erivative is lare in manitue at a, ten te y values o te unction near a cane rapily, an wen te erivative is close to, ten te curve is relatively lat near a, an te unction canes very slowly near a 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 3 Te erivative as a unction Dierentiability Note tat in we calculate te erivative o a unction at a speciic point a in its omain It is oten best to calculate i possible ' at any point in its omain Tereore, eine: ' wen tis it eists Note: t We can try to use also ' as a eneralization o 5 rom altou te t t ormula above is more natural, an tereore almost always use Given any number, te epression sown in eines a new unction ', calle te erivative o Eample : I at Tis can be interprete as te slope o te tanent line to te rap o at Te omain o smaller tan te omain o is te set D ' eists 3, calculate ' Ten raw te raps o an ' on te same aes an conirm tat ' ives te slope o te tanent line to te rap o at eac point Eample : For omains, calculate ' Fin te omains o ' an o Grap bot unctions on teir respective Deinition : a A unction is ierentiable at a i ' a eists b A unction Eample 3: corresponin interval is ierentiable on, b or, aor a, or, a i ' eists at eac point o te ' Were is ierentiable an were it is not ierentiable? Question: In eneral, ow can a unction ail to be ierentiable? First, let us establis te ollowin important result: 4

5 Teorem : Consier a unction : D R an a D I is ierentiable at a, ten is continuous at a a an take te it o tis equality, rememberin tat a Proo: Consier a a a En o proo a a ' a eists Tereore, one way in wic a unction is not ierentiable at a is wen te unction is iscontinuous at a To unerstan best ow a unction may ail to be ierentiable at a, eine: a ' l a a an 3 a a a ' r a a a a Since ierentiability at a means tat a ' a eists, ten we can establis: a a Teorem : Consier a unction : D R an a D is ierentiable at a i an only i: ' l a an ' r a eists an i ' l a ' r a Tereore, a unction can ail to be ierentiable i one o te ollowin takes place: is iscontinuous at a tink also o te etreme case in wic as te same slope on eiter, or sie o te point but it is iscontinuous at a, suc as :, wic is iscontinuous at, or Sow tat tis unction is not ierentiable at, by calculatin ' an ' ' l a or it oes not eist or ' a or it oes not eist Note tat an ininite semi tanent r line means tat te semi tanent line is vertical ' l a an ' r a bot eist but ' l a ' r a tis case correspons to a corner in te rap o See also Fiure 7 on pae 59 in te tetbook or rapical illustration o tese cases 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 3 Derivatives o constant, power, polynomials an eponential unctions From now on an or a ew sections we inten to establis ormulas or erivatives o elementary unctions alebraic an trionometric, or transcenental unctions an or combinations o tese Tis will allow us to quickly calculate te erivatives o tese unctions, an tese calculations will be use in many applications as it ives te relative instantaneous rate o cane o tese epressions Finally, we will collect all tese ormulas in a table o erivatives, to oranize an elp us wit memorizin tese In tis section, we start wit erivin ormulas or te erivatives o constant, power, polynomial an eponential unction U ormula rom, it is easy to see tat: c, or any constant c R Similarly:, an 3 3 Tis seems to suest te more eneral ormula: n n n, or n N, wic is easy to prove i we use te alebraic ormula : n n n3 n n a ba a b a b ab b n n a b or te binomial ormula : a n n n n! were n nk k a k k k k! n k! Formula is calle te Power Rule or erivatives or as well n N We can sow tat te power rule ols or n R First, ceck tat ols or n,, wic suests tat it ols or n Z a proo will be iven in te net section Also ceck tat ols or n / an n / 3 Te ollowin eneral rules can also easily be veriie: 3 4 c c ', or any constant c R, an : ' ' an 5 ' ' by recallin tat an tat Formulas 3 to 5 allow us to eten ormulas an to oter epressions see eamples 5 an 6 in te tetbook 6

7 Let us turn now to te erivative o an eponential unction a, : R, ce a First, u te alebraic property: a a a, we in tat :, tereore, we nee to etermine ' te eneral erivative ' a ' 5 ' a ' a ' ' Tere are many possible equivalent einitions o te number e, wic is te basis o te natural loaritmic unction suc as: 6 e n n Here, we eine e as te irrational number e,3 n suc tat e 7 note tat inormally 7 is equivalent wit 6 7 means tat e is te particular eponential unction te unction e is see Fiure below a or wic ' te slope o te tanent line at oriin or Fiure : Deinition o te number e U 5, tis prouces te useul ormula: 8 e e Eample : Do eample 8 an 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 3 Te Prouct Rule an te Quotient Rule: Tese two eneral ormulas or erivatives te prouct rule an te quotient rule will allow us to calculate erivatives o combinations o unctions learnt in Te eneral ormulas are: ' ' te prouct rule an ' ' te quotient rule Memorize tese ormulas correctly pay special attention to te in ormula Proo o : ' ' Proo o : ' ' Note tat or we nee to keep track o wic epression unction is te numerator an wic is te enominator, wile or te orer o te terms o te prouct is not important Eample : Do Eamples, 3, 4 an 5 in section 3 in te tetbook Memorize te ormulas sown at 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 33 Derivatives o trionometric unctions Goal: Derive ormulas or ' cos cos ' tan tan ' cot cot ' Beore applyin ormula in to erive tese, note te raps o an o its erivative built by measurin te slope o te tanent lines to te rap o at ierent points below: Fiure : Te raps o an o its erivative ' Tereore a oo uess or U ormula in : ' is ' cos Let us veriy tis alebraically: cos cos cos cos Tereore, in orer to in ', we nee to prove a ormula or wic we uesse as bein in cos an or For provin tat, consier te sector o a unit circle sown in Fiure below: Fiure : A sector o an unit circle: note tat BC, OC cos, AO an AD tan Also, we ave tat: Area OBC Areasector OAB AreaOAD cos tan cos cos 9

10 by u te squeeze teorem Tereore: Tereore, we ave sown tat: ' 3 u u rom te it laws, an an important it o its own Note tat implies more enerally tat u, were u is any epression wic approaces Eample : See Eample 5 in section 33 in te tetbook is an important result wic we will use to in cos In orer to in, note tat : cos cos cos Tereore: cos 4 wic implies te more eneral result: cos u 5 u u Returnin now to inin a ormula or ', substitute an 4 in to in tat 6 cos, as uesse compare wit te rap in Fiure Similarly: cos cos cos cos cos cos cos cos ater u aain an 4 Tereore: 7 cos 6 an 7 are important ormulas or te erivatives o an cos wic we will use oten

11 By u now te quotient rule ormula in, we can easily erive tat eercise: 8 tan an tat: 9 cos cot A ormulas 6, 7, 8 an 9 to te table o erivatives an memorize tis table Eample : Eample in section 33 in te tetbook an eercises,,, 39 an 46 rom eercise set 33 Homework: eercises 7,9,,3,5,5,7,4,4,44,49 an 5 rom eercise set Te Cain Rule: Goal: Derive an use a rule or a compose unction o te orm: u Motivation: Until now, we know ow to calculate te erivatives o most o te simple alebraic, trionometric an eponential unctions However, wat i we nee to calculate te erivative o compose unctions suc as: or Te cain rule erive in tis section will provie a eneral ormula to calculate suc erivatives an oters Note tat: u u u u u u u u u u u u Formula is te cain rule, written sortly as: u u u an wic reas : Te erivative o a unction applie to anoter unction is te u erivative o te outer unction, evaluate at te inner unction times te erivative o te inner unction Eample : Do Eample in section 34: Calculate ' or Do Eample, tat is calculate ' or U, we can easily erive: an or ' ln a a e ln a ln a e ln a ' e ln a a ln a Tereore: 3 wit tis a a lna Tis is te eneralization o ormula or e, erive in, an in areement

12 U te cain rule, we easily eneralize all ierentiation ormula erive so ar in te table o ierentiation, by tain an etra u or eac, or eample: n n n or n=,,3 becomes: 4 n u n n u u', or n=,,3, an so on or te oters Eercise: Generalize as suc all ormulas in te table o erivatives to obtain te erivatives o compose unctions Eample : Do Eamples 3,4 an 8 in section 34 in te tetbook Homework: eercises,,3,4,7,9,,,8,39,4,47,5,63, 65 an 73 rom eercise set 34 in te tetbook 35 Implicit Dierentiation: In sections to 4 we learnt ow to calculate te erivative o a unction y y, wic is iven eplicitly in terms o anoter variable, suc as: y or y 3, or, in eneral: y Many unctions, owever, are eine by an implicit relation between an y o te orm: E, y In tis case we can still calculate y' by takin te erivative o an oten u cain rule Tis meto is calle implicit ierentiation Eample : Eample on pae Consier te implicit equation: 3 y 5 were we consier tat y y a Calculate y' b Fin te equation o te tanent line to te circle 3 y 5 at te point 3,4 c Look also at solution in te tetbook in wic y' is oun by irst solvin or y eplicitly

13 Note tat in many cases or an equation o te orm, y cannot be oun or it cannot be easily oun eplicitly Tereore te meto outline in points a an b above is te only meto we can use to in y' Eample : Eample on pae Consier te implicit equation : y 6y a Fin y' by implicit ierentiation ; b Fin te equation o te tanent line to tis curve te olium o Descartes at 3,3 c at wat points in te irst quarant, y wit an y on tis curve is te tanent line orizontal? Eample 3: Consier y 7y Calculate ' y an ' y Important Note: Te simplicity o te process above ies some iiculties, tou Tat is, in an implicit ormula o te type, y may not be eine or it may not be a unction o at, or ' may not eist at However, i etermines y as a unction o tat is, i y can be oun uniquely rom, an i tis is ierentiable at, ten te erivative y oun as in te process above by implicit ierentiation ives us te correct value o ' 3

14 Eample 4: Consier 6 y By implicit ierentiation, we in y' Tis is correct y Wat is, owever, y '? Note tat 6 oes not necessarily ives y as a unction y=, unless etra conitions are speciie suc as y Besies, even i we assume tat 6 ives a unction y= at some, tis unction may not be ierentiable at! In te problems in our tetbook, tou, we assume tat an implicit equation etermines a ierentiable unction at any were we want to calculate erivatives, suc tat implicit ierentiation ives te correct erivative Eample 5: Eample 4 on pae 3: Fin 4 4 y '' i y 6 m m r m n n Eample 6: Do problem 48 rom eercise set 35 to sow tat 7 y ' y ' y r y n ierentiation r u implicit From now on, we consier 7 correct an we use it 7 can also be sown or any real power r as well Homework: Problems,,3,5,6,,6,,8,33,35,36,43 an 65 rom Eercise Set Derivatives o loaritmic unctions Loaritmic Dierentiation In tis section we will complete te table o erivatives wit ormulas or te erivatives o te loaritmic lo an ln unctions: a We will also learn a tecnique calle loaritmic ierentiation wic allows us to calculate relatively quickly te erivatives o complicate epressions, typically involvin proucts, ractions an / or powers an eponents I Te erivatives o lo an ln a : First we sow tat: lo a ln a note tat in we nee : a an Wy? 4

15 y To sow, recall tat y lo a a a u Use implicit ierentiation in an recall tat see 4: a ln a u' a y ln a y' y' rom, wic sows y a ln a ln a Consier now a e in to in tat: 3 ln u Tereore: Formulas an 3 ive te erivatives o loaritmic unctions an tey can be eneralize or composite unctions as: 4 u lo a u' an 5 u ln a u ln u' u Te simplicity o ormulas 3 an 5 makes te natural loaritm an te natural eponential very useul in Calculus an in oter sciences A ormulas, 3, 4 an 5 to te table o erivatives an memorize all ormulas in te table o erivatives Eample pae 8: Fin ' Eample : pae 9: Fin ' y i ln 3 y y i y ln Eample 3: pae 9: Fin y ' i y ln Eample 4: pae 9: Fin ' y i y lo II Loaritmic ierentiation: To calculate relatively quickly te erivatives o more complicate unctions involvin proucts, quotients, powers an/or eponents, we can apply te natural loaritm to bot sies o te equation irst Tis tecnique is calle loaritmic ierentiation Note tat tere are some cases o te orm y, y'?, wen y cannot be calculate witout loaritmic ierentiation Eample 7: pae 9: Calculate y ' i y 3/ Eample 8 proble m 43 rom Eercise set 6: Calculate y ' i y Homework: problems,5,6,,3,5,,4,6,3,33,39,4,44,46 an 49 rom eercise set 6 5

16 39 Relate rates : In many practical problems, quantities cane usually wit time I we know te rate o cane o one quantity: t an a relation y y or E, y, ten we can in y by takin te erivative wit respect to t o t y or Te rates erivatives an are relate, an tereore tese rates o cane are calle relate rates t t Eample : pae 43: 3 Air is bein pumpe into a sperical balloon so tat its volume increases at a rate o cm / sec Consierin tat tis process continues ineinitely, ow ast is te raius o te balloon increa wen te iameter o te balloon is 5 cm? Wen solvin relate rates problems, it is useul to ollow te ollowin uielines: Step : Translate all iven inormation in terms o matematical quantities: Let t be time Draw, i possible, a iaram wic is vali at ALL times On tis iure label or in te quantities o interest t, y t, an oter iven quantities constants, etc Write a ormula or te iven Epress te require rate o cane t erivatives y in eneral t in terms o Step : Fin te relation between y an : 3 E, y vali at ALL times Step 3: Dierentiate 3 an solve or y t Eample pae 45: A laer t lon rests aainst a vertical wall I te bottom o te laer slies away rom te wall at a rate o t/sec, ow ast is te top o te laer sliin own te wall wen te bottom o te laer is 6 t rom te wall? Eample 3 pae 45 : 6

17 Eample 4 pae 47: Homework: problems,5,7,,,,5,6,,3,6,37,44,45,46 rom problem set 39 3 Capter Review In tis capter, we : Learne te einition o te erivative o a unction at a point: ' a, an its interpretation an use in terms o te slope o te tanent line m, instantaneous velocity va an instantaneous rate o cane o a T y unction: Section Learne te einition o ', its interpretation as a unction, an te notion o ierentiable unctions Cases o non-ierentiability an te rapical meanin o ierentiable an non-ierentiable unctions Section Derive an learne memorize ormulas or te erivatives o a, e a,, polynomials,, c,,,, trionometric an loaritmic unctions an oranize tese in a table o erivatives Sections,, 3 an 4 an 6; Learne ow to calculate erivatives o unctions eine implicitly u implicit ierentiation Section 5; Learne ow to solve relate rates problems an ow to use loaritmic ierentiation Section 9 an 6 Learn te main results einitions, teorems an ormulas rom tese sections best write a review uie to inclue tese, ten solve te review problems rom paes 63 to 67 in te tetbook, incluin te Concept Ceck, True/ False an Eercises types o problems 7