Chapter 2 The Derivative Applied Calculus 97

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1 Caper Te Derivaive Applie Calculus 97 Secion 3: Power an Sum Rules for Derivaives In e ne few secions, we ll ge e erivaive rules a will le us fin formulas for erivaives wen our funcion comes o us as a formula. Tese are very algebraic secion, an you soul ge los of pracice. As we learn new rules, we will look a some basic applicaions. Builing Blocks Tese are e simples rules rules for e basic funcions. We won prove ese rules; we ll jus use em. Bu firs, le s look a a few so a we can see ey make sense. Eample 1 Fin e erivaive of y = f ( ) = m + b Tis is a linear funcion, so is grap is is own angen line! Te slope of e angen line, e erivaive, is e slope of e line: f '( ) = m Eample Rule: Te erivaive of a linear funcion is is slope =135 Fin e erivaive of ( ). f Tink abou is one grapically, oo. Te grap of f() is a orizonal line. So is slope is zero. f ' = ( ) 0 Eample 3 Rule: Te erivaive of a consan is zero f = Fin e erivaive of ( ) Tis quesion is callenging using limis, as you saw in e previous secion. We will sow you e long way o o i, en give you a soran rule o bypass all is. f ( + ) f ( ) Recall e formal efiniion of e erivaive: f '( ) = lim. 0 Using our funcion f ( ) =, f ( + ) = ( + ) = + +. Ten f f '( ) = lim 0 + = lim 0 ( + ) f ( ) = lim 0 ( + ) = lim 0 = lim 0 f = From all a, we fin e ( ) + + ( + ) = Tis caper is (c) 013. I was remie by Davi Lippman from Sana Calaway's remi of Conemporary Calculus by Dale Hoffman. I is license uner e Creaive Commons Aribuion license.

2 Caper Te Derivaive Applie Calculus 98 Luckily, ere is a any rule we use o skip using e limi: Eample 4 n 1 Power Rule: Te erivaive f ( ) = is ( ) g = 4. Fin e erivaive of ( ) 3 f n = n 3 Using e power rule, we know a if f ( ) =, en f ( ) = 3. Noice a g is 4 imes e funcion f. Tink abou wa is cange means o e grap of g i s now 4 imes as all as g 4 f f e grap of f. If we fin e slope of a secan line, i will be = = 4 ; eac slope will be 4 imes e slope of e secan line on e f grap. Tis propery will ol for e slopes of angen lines, oo: 3 3 ( 4 ) = 4 ( ) = 4 3 = 1 Rule: Consans come along for e rie; ( kf ) = kf ' Here are all e basic rules in one place.

3 Caper Te Derivaive Applie Calculus 99 Derivaive Rules: Builing Blocks In wa follows, f an g are iffereniable funcions of. (a) Consan Muliple Rule: ( kf ) = kf ' (b) Sum (or Difference) Rule: ( f + g) = f ' + g' (or ( f g) = f ' g' = n n n 1 (c) Power Rule: ( ) Special cases: ( k) = 0 ( ) = 1 = e () Eponenial Funcions: ( ) e (because (because ( a ) = ln a a 0 k = k ) 1 = ) ) ln = 1 (e) Naural Logarim: ( ) Te sum, ifference, an consan muliple rule combine wi e power rule allow us o easily fin e erivaive of any polynomial. Eample 5 p = Fin e erivaive of ( ) 1003 = = ( ) = = ( 17 ) + ( 13 ) ( 1.8) + ( 1003) 10 8 ( ) + 13 ( ) 1.8 ( ) + ( 1003) 9 7 ( ) + 13( 8 ) 1.8( 1)

4 Caper Te Derivaive Applie Calculus 100 You on ave o sow every single sep. Do be careful wen you re firs working wi e rules, bu prey soon you ll be able o jus wrie own e erivaive irecly: Eample Fin ( ) Wriing ou e rules, we' wrie = 17() 33(1) + 0 = 34 ( ) 33 Once you're familiar wi e rules, you can, in your ea, muliply e imes e 17 an e 33 imes 1, an jus wrie ( ) = Te power rule works even if e power is negaive or a fracion. In orer o apply i, firs ranslae all roos an basic raional epressions ino eponens: Eample 7 Fin e erivaive of y = 4 3 5e 4 + Firs sep ranslae ino eponens: 4 y 3 5e 3 1/ 4 = + = 4 + 5e 4 Now you can ake e erivaive: e = 3 1/ / 5 3 = e = 4 ( + 5e ) ( ) ( ) 1/ e. If ere is a reason o, you can rewrie e answer wi raicals an posiive eponens: / e = + + 5e 5 Be careful wen fining e erivaives wi negaive eponens. We can immeiaely apply ese rules o solve e problem we sare e caper wi - fining a angen line.

5 Caper Te Derivaive Applie Calculus 101 Eample 8 Fin e equaion of e line angen o g( ) = 10 wen =. Te slope of e angen line is e value of e erivaive. We can compue g ( ) =. To fin e slope of e angen line wen = 3, evaluae e erivaive a a poin. g ( ) = () = 4. Te slope of e angen line is -4. To fin e equaion of e angen line, we also nee a poin on e angen line. Since e angen line ouces e original funcion a =, we can fin e poin by evaluaing e original funcion: g (3) = 10 = 6. Te angen line mus pass roug e poin (, 6). Using e poin-slope equaion of a line, e angen line will ave equaion y 6 = 4( ). Simplifying o slope-inercep form, e equaion is y = Graping, we can verify is line is inee angen o e curve. We can also use ese rules o elp us fin e erivaives we nee o inerpre e beavior of a funcion. Eample 9 In a memory eperimen, a researcer asks e subjec o memorize as many wors from a lis as possible in 10 secons. Recall is ese, en e subjec is given 10 more secons o suy, an so on. Suppose e number of wors remembere afer secons of suying coul be /5 moele by W () = 4. Fin an inerpre W (0). 8 W () /5 3/5 = =, so 8 ( ) 3/5 W (0) = Since W is measure in wors, an is in secons, W' as unis wors per secon. W (0) 0.65 means a afer 0 secons of suying, e subjec is learning abou 0.7 more wors for eac aiional secon of suying.

6 Caper Te Derivaive Applie Calculus 10 Business an Economics Ne we will elve more eeply ino some business applicaions. To o a, we firs nee o review some erminology. Suppose you are proucing an selling some iem. Te profi you make is e amoun of money you ake in minus wa you ave o pay o prouce e iems. Bo of ese quaniies epen on ow many you make an sell. (So we ave funcions ere.) Here is a lis of efiniions for some of e erminology, ogeer wi eir meaning in algebraic erms an in grapical erms. Your cos is e money you ave o spen o prouce your iems. Te Fie Cos (FC) is e amoun of money you ave o spen regarless of ow many iems you prouce. FC can inclue ings like ren, purcase coss of macinery, an salaries for office saff. You ave o pay e fie coss even if you on prouce anying. Te Toal Variable Cos (TVC) for q iems is e amoun of money you spen o acually prouce em. TVC inclues ings like e maerials you use, e elecriciy o run e macinery, gasoline for your elivery vans, maybe e wages of your proucion workers. Tese coss will vary accoring o ow many iems you prouce. Te Toal Cos (TC, or someimes jus C) for q iems is e oal cos of proucing em. I s e sum of e fie cos an e oal variable cos for proucing q iems. Te Marginal Cos (MC) a q iems is e cos of proucing e ne iem. Really, i s MC(q) = TC(q + 1) TC(q). In many cases, oug, i s easier o approimae is ifference using calculus (see Eample below). An some sources efine e marginal cos irecly as e erivaive, MC(q) = TC'(q). In is course, we will use bo of ese efiniions as if ey were inercangeable. Te unis on marginal cos is cos per iem. Wy is i OK a are ere wo efiniions for Marginal Cos (an Marginal Revenue, an Marginal Profi)? We ave been using slopes of secan lines over iny inervals o approimae erivaives. In is eample, we ll urn a aroun we ll use e erivaive o approimae e slope of e secan line. Noice a e cos of e ne iem efiniion is acually e slope of a secan line, over an inerval of 1 uni: C ( ) ( ) ( q + 1) 1 MC q = C q = 1 So is is approimaely e same as e erivaive of e cos funcion a q: MC ( q) = C' ( q) In pracice, ese wo numbers are so close a ere s no pracical reason o make a isincion. For our purposes, e marginal cos is e erivaive is e cos of e ne iem.

7 Caper Te Derivaive Applie Calculus 103 Eample 10 Te able sows e oal cos (TC) of proucing q iems. a) Wa is e fie cos? b) Wen 00 iems are mae, wa is e oal variable cos? Te average variable cos? c) Wen 00 iems are mae, esimae e marginal cos. Iems, q Toal Cos, TC 0 $0, $35, $45, $53,000 a) Te fie cos is $0,000, e cos even wen no iems are mae. b) Wen 00 iems are mae, e oal cos is $45,000. Subracing e fie cos, e oal variable cos is $45,000 - $0,000 = $5,000. Te average variable cos is e oal variable cos ivie by e number of iems, so we woul ivie e $5,000 oal variable cos by e 00 iems mae. $5,000 00= $15. On average, eac iem a a variable cos of $15. c) We nee o esimae e value of e erivaive, or e slope of e angen line a q = ,000 35,000 Fining e secan line from q=100 o q=00 gives a slope of = 100. Fining ,000 45,000 e secan line from q=00 o q=300 gives a slope of = 80. We coul esimae e angen slope by averaging ese secan slopes, giving us an esimae of $90/iem. Tis ells us a afer 00 iems ave been mae, i will cos abou $90 o make one more iem. Eample 11 Te cos o prouce iems is unre ollars. (a) Wa is e cos for proucing 100 iems? 101 iems? Wa is cos of e 101 s iem? (b) For C() =, calculae C'() an evaluae C' a = 100. How oes C '(100) compare wi e las answer in par (a)? (a) Pu C() = = 1/ unre ollars, e cos for iems. Ten C (100) = $1000 an C(101) = $ , so i coss $4.99 for a 101 s iem. Using is efiniion, e marginal cos is $4.99. (b) 1 1 1/ = = so C ( ) C 1 1 (100) = 100 = 0 unre ollars = $5.00. Noe ow close ese answers are! Tis sows (again) wy i s OK a we use bo efiniions for marginal cos.

8 Caper Te Derivaive Applie Calculus 104 Deman is e funcional relaionsip beween e price p an e quaniy q a can be sol (a is emane). Depening on your siuaion, you mig ink of p as a funcion of q, or of q as a funcion of p. Your revenue is e amoun of money you acually ake in from selling your proucs. Revenue is price quaniy. Te Toal Revenue (TR, or jus R) for q iems is e oal amoun of money you ake in for selling q iems. Te Marginal Revenue (MR) a q iems is e cos of proucing e ne iem, MR(q) = TR(q + 1) TR(q). Jus as wi marginal cos, we will use bo is efiniion an e erivaive efiniion MR(q) = TR (q). Your profi is wa s lef over from oal revenue afer coss ave been subrace. Te Profi (P) for q iems is TR(q) TC(q), e ifference beween oal revenue an oal coss Te average profi for q iems is P/q. Te marginal profi a q iems is P(q + 1) P(q), or P ( q) Grapical Inerpreaions of e Basic Business Ma Terms Illusraion/Eample: Here are e graps of TR an TC for proucing an selling a cerain iem. Te orizonal ais is e number of iems, in ousans. Te verical ais is e number of ollars, also in ousans. Firs, noice ow o fin e fie cos an variable cos from e grap ere. FC is e y- inercep of e TC grap. (FC = TC(0).) Te grap of TVC woul ave e same sape as e grap of TC, sife own. (TVC = TC FC.)

9 Caper Te Derivaive Applie Calculus 105 MC(q) = TC(q + 1) TC(q), bu a s impossible o rea on is grap. How coul you isinguis beween TC(40) an TC(403)? On is grap, a inerval is oo small o see, an our bes guess a e secan line is acually e angen line o e TC curve a a poin. (Tis is e reason we wan o ave e erivaive efiniion any.) MC(q) is e slope of e angen line o e TC curve a (q, TC(q)). MR(q) is e slope of e angen line o e TR curve a (q, TR(q)). Profi is e isance beween e TR an TC curve. If you eperimen wi your clear plasic ruler, you ll see a e bigges profi occurs eacly wen e angen lines o e TR an TC curves are parallel. Tis is e rule profi is maimize wen MR = MC. wic we'll eplore laer in e caper. Eample 1 Te eman, D, for a prouc a a price of p ollars is given by marginal revenue wen e price is $10. D( p) = p. Fin e Firs we nee o form a revenue equaion. Since Revenue = Price Quaniy, an e eman equaion sows e quaniy of prouc a can be sol, we ave R( p) = D( p) p = p p = 00 p 0. p 3 ( ) Now we can fin marginal revenue by fining e erivaive R ( p) = p = p ( ) ( ) A a price of $10, R (10) = ( 10) = 140. ollars of Revenue Noice e unis for R' are, so R (10) = 140 means a wen e price is ollar of price $10, e revenue will increase by $140 for eac ollar e price was increase..3 Eercises 1. Fill in e values in e able for ( 3 f ( ) ), ( f ( ) + g ( ) ), an ( g( ) f ( ) ) 3. f() f '() g() g '() ( 3 f ( ) ) ( f ( ) + g ( ) ) ( 3 g( ) f ( ) )

10 Caper Te Derivaive Applie Calculus 106. Fin (a) D( 1 ) (b) 3. Fin (a) D( 9 ) (b) /3 ( 7 ) (c) D( 1 e 3 ) () (c) D( 1 4 ) () D( π ) In problems 4 8, (a) calculae f '(1) an (b) eermine wen f '() = f() = f() = f() = f() = f() = Were o f() = an g() = 3 1 ave orizonal angen lines? 10. I akes T() = ours o weave small rugs. Wa is e marginal proucion ime o weave a rug? (Be sure o inclue e unis wi your answer.) 11. I coss C() = ollars o prouce golf balls. Wa is e marginal proucion cos o make a golf ball? Wa is e marginal proucion cos wen = 5? wen = 100? (Inclue unis.) 1. An arrow so sraig up from groun level wi an iniial velociy of 18 fee per secon will be a eig () = fee a secons. (a) Deermine e velociy of e arrow wen = 0, 1 an secons. (b) Wa is e velociy of e arrow, v(), a any ime? (c) A wa ime will e velociy of e arrow be 0? () Wa is e greaes eig e arrow reaces? (e) How long will e arrow be alof? (f) Use e answer for e velociy in par (b) o eermine e acceleraion, a() = v '(), a any ime. 13. If an arrow is so sraig up from groun level on e moon wi an iniial velociy of 18 fee per secon, is eig will be () = fee a secons. Do pars (a) (e) of problem 40 using is new equaion for. 14. f() = 3 + A + B + C wi consans A, B an C. Can you fin coniions on e consans A, B an C wic will guaranee a e grap of y = f() as wo isinc "verices"? (Here a "vere" means a place were e curve canges from increasing o ecreasing or from ecreasing o increasing.)