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.
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.
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 = 17 + 13 1.8 + 10 8 Fin e erivaive of ( ) 1003 = = 17 10 8 ( 17 + 13 1.8 + 1003) = 17 10 = 170 10 8 ( 17 ) + ( 13 ) ( 1.8) + ( 1003) 10 8 ( ) + 13 ( ) 1.8 ( ) + ( 1003) 9 7 ( ) + 13( 8 ) 1.8( 1) + 0 9 + 104 7 1.8
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 6 17 33 + 1 Fin ( ) Wriing ou e rules, we' wrie 17 33 + 1 = 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 ( 17 33 + 1) = 34 33 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: 4 3 + 5e = 3 1/ 4 4 1 1/ 5 3 = 3 4 4 + 5 e = 4 ( + 5e ) ( ) ( ) 1/ 5 + 16 + 5e. If ere is a reason o, you can rewrie e answer wi raicals an posiive eponens: 3 3 16 1/ 5 + 16 + 5e = + + 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.
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 = 4 +14. 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 () 4 5 5 3/5 3/5 = =, so 8 ( ) 3/5 W (0) = 0 0.65 5 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.
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 1 = 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.
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 $0,000 100 $35,000 00 $45,000 300 $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 = 00. 45,000 35,000 Fining e secan line from q=100 o q=00 gives a slope of = 100. Fining 00 100 53,000 45,000 e secan line from q=00 o q=300 gives a slope of = 80. We coul esimae 300 00 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) = $1004.99, 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.
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.)
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) 00 0. = 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 = 00 0. p p = 00 p 0. p 3 ( ) Now we can fin marginal revenue by fining e erivaive R ( p) = 00 1 0. 3p = 00 0.6 p ( ) ( ) A a price of $10, R (10) = 00 0.6( 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 ( ) ) 0 3 4 3 1 1 1 0 4 3 1
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 '() = 0. 4. f() = 5 + 13 5. f() = 5 40 + 73 6. f() = 3 + 9 + 6 7. f() = 3 + 3 + 3 1 8. f() = 3 + + 1 9. Were o f() = 10 + 3 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 () = 16 + 18 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 () =.65 + 18 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.)