Chaper The Derivaive Business Calculus 1 Chaper : The Derivaive Precalculus Iea: Slope an Rae of Change The slope of a line measures how fas a line rises or falls as we move from lef o righ along he line. I measures he rae of change of he y-coorinae wih respec o changes in he x-coorinae. If he line represens he isance ravele over ime, for example, hen is slope represens he velociy. In Figure 1, you can remin yourself of how we calculae slope using wo poins on he line: m = { slope from P o Q } = rise run = y y 1 x x 1 = y x Figure 1 We woul like o be able o ge ha same sor of informaion (how fas he curve rises or falls, velociy from isance) even if he graph is no a sraigh line. Bu wha happens if we ry o fin he slope of a curve, as in Figure? We nee wo poins in orer o eermine he slope of a line. How can we fin a slope of a curve, a jus one poin? Figure The answer, as suggese in Figure is o fin he slope of he angen line o he curve a ha poin. Mos of us have an inuiive iea of wha a angen line is. Unforunaely, angen line is har o efine precisely. Definiion: A secan line is a line beween wo poins on a curve. Can -quie-o-i-ye Definiion: A angen line is a line a one poin on a curve. ha oes is bes o be he curve a ha poin? I urns ou ha he easies way o efine he angen line is o efine is slope.
Chaper The Derivaive Business Calculus Secion 1: Insananeous Rae of Change an Tangen Lines Insananeous Velociy Suppose we rop a omao from he op of a 100 foo builing an ime is fall. Figure Some quesions are easy o answer irecly from he able: (a) How long i i ake for he omao n o rop 100 fee? (b) How far i he omao fall uring he firs secon? (c) How far i he omao fall uring he las secon? () How far i he omao fall beween =.5 an = 1? (.5 secons) (100 84 = 16 fee) (64 0 = 64 fee) (96 84 = 1 fee) Some oher quesions require a lile calculaion: (e) Wha was he average velociy of he omao uring is fall? Average velociy = isance fallen oal ime = posiion ime = 100 f.5 s = 40 f/s. (f) Wha was he average velociy beween =1 an = secons? Average velociy = posiion ime = 6 f 84 f s 1 s = 48 f 1 s = 48 f/s. Some quesions are more ifficul. (g) How fas was he omao falling 1 secon afer i was roppe? This quesion is significanly ifferen from he previous wo quesions abou average velociy. Here we wan he insananeous velociy, he velociy a an insan in ime. Unforunaely he omao is no equippe wih a speeomeer so we will have o give an approximae answer. One crue approximaion of he insananeous velociy afer 1 secon is simply he average velociy uring he enire fall, 40 f/s. Bu he omao fell slowly a he beginning an rapily near he en so he " 40 f/s" esimae may or may no be a goo answer.
Chaper The Derivaive Business Calculus We can ge a beer approximaion of he insananeous velociy a =1 by calculaing he average velociies over a 1 fee shor ime inerval near = 1. The average velociy beween = 0.5 an = 1 is = 4 f/s, an he 0.5 s 0 fee average velociy beween = 1 an = 1.5 is = 40 f/s so we can be reasonably sure ha he.5 s insananeous velociy is beween 4 f/s an 40 f/s. In general, he shorer he ime inerval over which we calculae he average velociy, he beer he average velociy posiion will approximae he insananeous velociy. The average velociy over a ime inerval is, which is he ime slope of he secan line hrough wo poins on he graph of heigh versus ime (Fig. 4). The insananeous velociy a a paricular ime an heigh is he slope of he angen line o he graph a he poin given by ha ime an heigh. Figure 4 Average velociy = posiion ime = slope of he secan line hrough poins. Insananeous velociy = slope of he line angen o he graph.
Chaper The Derivaive Business Calculus 4 Tangen Lines Do his! The graph below is he graph of x y f. We wan o fin he slope of he angen line a he poin (1, ). Firs, raw he secan line beween (1, ) an (, 1) an compue is slope. Now raw he secan line beween (1, ) an (1.5, 1) an compue is slope. Compare he wo lines you have rawn. Which woul be a beer approximaion of he angen line o he curve a (1, )? Now raw he secan line beween (1, ) an (1., 1.5) an compue is slope. Is his line an even beer approximaion of he angen line? Now raw your bes guess for he angen line an measure is slope. Do you see a paern in he slopes? Figure 5 You shoul have noice ha as he inerval go smaller an smaller, he secan line go closer o he angen line an is slope go closer o he slope of he angen line. Tha s goo news we know how o fin he slope of a secan line. Example: Now le's look a he problem of fining he slope of he line L (Figure 6) which is angen o f(x) = x a he poin (,4). We coul esimae he slope of L from he graph, bu we won'. Insea, we will use he iea ha secan lines over iny inervals approximae he angen line.
Chaper The Derivaive Business Calculus 5 Figure 6 Figure 7 We can see ha he line hrough (,4) an (,9) on he graph of f is an approximaion of he slope of he angen line, an we can calculae ha slope exacly: m = y/ x = (9 4)/( ) = 5. Bu m = 5 is only an esimae of he slope of he angen line an no a very goo esimae. I's oo big. We can ge a beer esimae by picking a secon poin on he graph of f which is closer o (,4) he poin (,4) is fixe an i mus be one of he poins we use. From Figure 7, we can see ha he slope of he line hrough he poins (,4) an (.5,6.5) is a beer approximaion of he slope of he angen line a (,4): m = y/ x = (6.5 4)/(.5 ) =.5/.5 = 4.5, a beer esimae, bu sill an approximaion. We can coninue picking poins closer an closer o (,4) on he graph of f, an hen calculaing he slopes of he lines hrough each of hese poins an he poin (,4): Poins o he lef of (,4) Poins o he righ of (,4) x y = x slope of line hrough x y = x slope of line hrough (x,y) an (,4) (x,y) an (,4) 1.5.5.5 9 5 1.9.61.9.5 6.5 4.5 1.99.9601.99.01 4.0401 4.01 The only hing special abou he x values we picke is ha hey are numbers which are close, an very close, o x =. Someone else migh have picke oher nearby values for x. As he poins we pick ge closer an closer o he poin (,4) on he graph of y = x, he slopes of he lines hrough he poins an (,4) are beer approximaions of he slope of he angen line, an hese slopes are geing closer an closer o 4. We can bypass much of he calculaing by no picking he poins one a a ime: le's look a a general poin near (,4). Define x = + h so h is he incremen from o x (Fig. 8). If h is small, hen x = + h is close o an
Chaper The Derivaive Business Calculus 6 he poin (+h, f(+h) ) = (+h, (+h) ) is close o (,4). The slope m of he line hrough he poins (,4) an (+h, (+h) ) is a goo approximaion of he slope of he angen line a he poin (,4): Figure 8 m = y x = (+h) 4 (+h) = {4 + 4h + h } 4 h = 4h + h h = h(4 + h) h = 4 + h. If h is very small, hen m = 4 + h is a very goo approximaion o he slope of he angen line, an m = 4 + h is very close o he value 4. The value m = 4 + h is he slope of he secan line hrough he wo poins (,4) an ( +h, (+h) ). As h ges smaller an smaller, his slope approaches he slope of he angen line o he graph of f a (,4). In some applicaions, we nee o know where he graph of a funcion f(x) has horizonal angen lines (slopes = 0). In Fig., he slopes of he angen lines o graph of y = f(x) are 0 when x = or x 4.5. Example: A righ is he graph of y = g(x). A wha values of x oes he graph of y = g(x) in Fig. 9 have horizonal angen lines? Figure 9 Soluion: The angen lines o he graph of g are horizonal (slope = 0) when x 1, 1,.5, an 5.
Chaper The Derivaive Business Calculus 7 Secion : The Derivaive Definiion of he Derivaive The angen line problem an he insananeous velociy problem are he same problem. In each problem we wane o know how rapily somehing was changing a an insan in ime, an he answer urne ou o be fining he slope of a angen line, which we approximae wih he slope of a secan line. This iea is he key o efining he slope of a curve. The Derivaive: The erivaive of a funcion f a a poin (x, f(x)) is he insananeous rae of change. The erivaive is he slope of he angen line o he graph of f a he poin (x, f(x)). The erivaive is he slope of he curve f(x) a he poin (x, f(x)). A funcion is calle iffereniable a (x, f(x)) if is erivaive exiss a (x, f(x)). Noaion for he Derivaive: The erivaive of y = f(x) wih respec o x is wrien as x f ' (rea alou as f prime of x ), or ' y f (rea alou as ee why ee ex ), or y ( y prime ) or The noaion ha resembles a fracion is calle Leibniz noaion. I isplays no only he name of he funcion (f or y), bu also he name of he variable (in his case, x). I looks like a fracion because he y erivaive is a slope. In fac, his is simply wrien in Roman leers insea of Greek leers. x Verb forms: We fin he erivaive of a funcion, or ake he erivaive of a funcion, or iffereniae a funcion. y We use an aapaion of he noaion o mean fin he erivaive of f(x): Formal Algebraic Definiion: f x f x h f x f f '( x) lim (*) h0 h Pracical Definiion: The erivaive can be approximae by looking a an average rae of change, or he slope of a secan line, over a very iny inerval. The inier he inerval, he closer his is o he rue insananeous rae of change, slope of he angen line, or slope of he curve. Looking Ahea: We will have mehos for compuing exac values of erivaives from formulas soon. If he funcion is given o you as a able or graph, you will sill nee o approximae his way. (* informaion abou lim is in Chaper 5: Opional Topics)
Chaper The Derivaive Business Calculus 8 This is he founaion for he res of his chaper. I s remarkable ha such a simple iea (he slope of a angen line) an such a simple efiniion (for he erivaive f ' ) will lea o so many imporan ieas an applicaions. The Derivaive as a Funcion We now know how o fin (or a leas approximae) he erivaive of a funcion for any x-value; his means we can hink of he erivaive as a funcion, oo. The inpus are he same x s; he oupu is he value of he erivaive a Example: Fig. 10 is he graph of a funcion showing values of f ' x : y f x. We can use he informaion in he graph o fill in a able Figure 10 A various values of x, raw your bes guess a he angen line an measure is slope. You migh have o exen your lines so you can rea some poins. In general, your esimae of he slope will be beer if you choose poins ha are easy o rea an far away from each oher. Here are my esimaes for a few values of x (pars of he angen lines I use are shown): x y f x f ' x = he esimae SLOPE of he angen line o he curve a he poin x, y. 0 0 1 1 1 0 0 1 1 0 4 1 1 5 0.5 We can esimae he values of f (x) a some non-ineger values of x, oo: f (.5) 0.5 an f (1.) 0..
Chaper The Derivaive Business Calculus 9 We can even hink abou enire inervals. For example, if 0 < x < 1, hen f(x) is increasing, all he slopes are posiive, an so f (x) is posiive. The values of f (x) efiniely epen on he values of x, an f (x) is a funcion of x. We can use he resuls in he able o help skech he graph of f (x) in Fig. 11. Figure 11 Example: Fig. 1 is he graph of he heigh h() of a rocke a ime. Skech he graph of he velociy of he rocke a ime. (Velociy is he erivaive of he heigh funcion, so i is he slope of he angen o he graph of posiion or heigh.) Figure 1
Chaper The Derivaive Business Calculus 10 Soluion: The lower graph in Fig. 1 shows he velociy of he rocke. This is v() = h (). Figure 1 Secion : Raes in Real Life So far we have emphasize he erivaive as he slope of he line angen o a graph. Tha inerpreaion is very visual an useful when examining he graph of a funcion, an we will coninue o use i. Derivaives, however, are use in a wie variey of fiels an applicaions, an some of hese fiels use oher inerpreaions. The following are a few inerpreaions of he erivaive ha are commonly use. General Graphical Physical Rae of Change: f '(x) is he rae of change of he funcion a x. If he unis for x are years an he unis for f people f(x) are people, hen he unis for are, a rae of change in populaion. year Slope: f '(x) is he slope of he line angen o he graph of f a he poin ( x, f(x) ). Velociy: If f(x) is he posiion of an objec a ime x, hen f '(x) is he velociy of he objec a ime x. If he unis for x are hours an f(x) is isance measure in miles, hen he unis for f '(x) = f miles are hour miles per hour, which is a measure of velociy.,
Chaper The Derivaive Business Calculus 11 Acceleraion If f(x) is he velociy of an objec a ime x, hen f '(x) is he acceleraion of he objec a ime x. If he unis are for x are hours an f(x) has he unis miles, hen he unis for he acceleraion f '(x) = hour f are miles/hour = miles, miles per hour per hour. hour hour Business Marginal Cos If f(x) is he oal cos of x objecs, hen f '(x) is he marginal cos, a a proucion level of x. This marginal cos is approximaely he aiional cos of making one more objec once we have alreay mae x objecs. If he unis for x are bicycles an he unis for f(x) are ollars, hen he unis for f '(x) = f are ollars, he cos per bicycle. bicycle Marginal Profi If f(x) is he oal profi from proucing an selling x objecs, hen f '(x) is he marginal profi, he profi o be mae from proucing an selling one more objec. If he unis for x are bicycles an he unis for f(x) are ollars, hen he unis for f '(x) = f are ollars, ollars per bicycle, which is he bicycle profi per bicycle. In business conexs, he wor "marginal" usually means he erivaive or rae of change of some quaniy. One of he srenghs of calculus is ha i provies a uniy an economy of ieas among iverse applicaions. The vocabulary an problems may be ifferen, bu he ieas an even he noaions of calculus are sill useful. Business an Economics Terms Suppose you are proucing an selling some iem. The profi you make is he amoun of money you ake in minus wha you have o pay o prouce he iems. Boh of hese quaniies epen on how many you make an sell. (So we have funcions here.) Here is a lis of efiniions for some of he erminology, ogeher wih heir meaning in algebraic erms an in graphical erms. Your cos is he money you have o spen o prouce your iems. The Fixe Cos (FC) is he amoun of money you have o spen regarless of how many iems you prouce. FC can inclue hings like ren, purchase coss of machinery, an salaries for office saff. You have o pay he fixe coss even if you on prouce anyhing.
Chaper The Derivaive Business Calculus 1 The Toal Variable Cos (TVC) for q iems is he amoun of money you spen o acually prouce hem. TVC inclues hings like he maerials you use, he elecriciy o run he machinery, gasoline for your elivery vans, maybe he wages of your proucion workers. These coss will vary accoring o how many iems you prouce. The Toal Cos (TC) for q iems is he oal cos of proucing hem. I s he sum of he fixe cos an he oal variable cos for proucing q iems. The Average Cos (AC) for q iems is he oal cos ivie by q, or TC/q. You can also alk abou he average fixe cos, FC/q, or he average variable cos, TVC/q. The Marginal Cos (MC) a q iems is he cos of proucing he nex iem. Really, i s MC(q) = TC(q + 1) TC(q). In many cases, hough, i s easier o approximae his ifference using calculus (see Example below). An some sources efine he marginal cos irecly as he erivaive, MC(q) = TC'(q). In his course, we will use boh of hese efiniions as if hey were inerchangeable. Deman is he funcional relaionship beween he price p an he quaniy q ha can be sol (ha is emane). Depening on your siuaion, you migh hink of p as a funcion of q, or of q as a funcion of p. Your revenue is he amoun of money you acually ake in from selling your proucs. Revenue is price quaniy. The Toal Revenue (TR) for q iems is he oal amoun of money you ake in for selling q iems. The Average Revenue (AR) for q iems is he oal revenue ivie by q, or TR/q. The Marginal Revenue (MR) a q iems is he cos of proucing he nex iem, MR(q) = TR(q + 1) TR(q). Jus as wih marginal cos, we will use boh his efiniion an he erivaive efiniion MR(q) = TR (q). Your profi is wha s lef over from oal revenue afer coss have been subrace. The Profi (π) for q iems is TR(q) TC(q).
Chaper The Derivaive Business Calculus 1 The average profi for q iems is π/q. The marginal profi a q iems is π(q + 1) π(q), or ' q Example: Why is i OK ha are here wo efiniions for Marginal Cos (an Marginal Revenue, an Marginal Profi)? We have been using slopes of secan lines over iny inervals o approximae erivaives. In his example, we ll urn ha aroun we ll use he erivaive o approximae he slope of he secan line. Noice ha he cos of he nex iem efiniion is acually he slope of a secan line, over an inerval of 1 uni: MC q Cq 1 C q 1 1 1 1 So his is approximaely he same as he erivaive of he cos funcion a q: MC q C' q In pracice, hese wo numbers are so close ha here s no pracical reason o make a isincion. For our purposes, he marginal cos is he erivaive is he cos of he nex iem. Graphical Inerpreaions of he Basic Business Mah Terms Illusraion/Example: Here are he graphs of TR an TC for proucing an selling a cerain iem. The horizonal axis is he number of iems, in housans. The verical axis is he number of ollars, also in housans. Figure 14
Chaper The Derivaive Business Calculus 14 Firs, noice how o fin he fixe cos an variable cos from he graph here. FC is he y-inercep of he TC graph. (FC = TC(0).) The graph of TVC woul have he same shape as he graph of TC, shife own. (TVC = TC FC.) We alreay know ha we can fin average raes of change by fining slopes of secan lines. AC, AR, MC, an MR are all raes of change, an we can fin hem wih slopes, oo. AC(q) is he slope of a iagonal line, from (0, 0) o (q, TC(q)). AR(q) is he slope of he line from (0, 0) o (q, TR(q)). MC(q) = TC(q + 1) TC(q), bu ha s impossible o rea on his graph. How coul you isinguish beween TC(40) an TC(40)? On his graph, ha inerval is oo small o see, an our bes guess a he secan line is acually he angen line o he TC curve a ha poin. (This is he reason we wan o have he erivaive efiniion hany.) MC(q) is he slope of he angen line o he TC curve a (q, TC(q)). In a similar way, MR(q) is he slope of he angen line o he TR curve a (q, TR(q)). Profi is he isance beween he TR an TC curve. If you experimen wih your clear plasic ruler, you ll see ha he bigges profi occurs exacly when he angen lines o he TR an TC curves are parallel. This is he rule profi is maximize when MR = MC. Raes in Real Life Example: You can esimae a ree s age in years by muliplying is iameer (measure in inches) by is growh facor (a number ha epens on he species). Accoring o he Missouri Deparmen of Conservaion, he Growh facor for a coonwoo ree is. a. Suppose you fin a coonwoo ree in Missouri ha is 6 inches in iameer. How ol woul you esimae i o be? b. Wha are he unis of he growh facor? c. Is his growh facor a erivaive? Soluion: a. The coonwoo ree shoul be abou 6 = 1 years ol. b. The unis of he growh facor are years per inch (because when we muliply he growh facor by inches, we ge years). c. Yes, he growh facor is a erivaive. I has fracional unis (years per inch), so i represens a rae. In his case, i s he erivaive of he funcion ha gives he age of a ree as a funcion of is iameer. The funcion is linear, so he erivaive in his case is he consan slope, years per inch.
Chaper The Derivaive Business Calculus 15 Example: The lengh of ay (ha is, ayligh) in Seale is a funcion of he ay of he year. For example, on Augus 1 h, 01, here were abou 14 hours 4 minues of ayligh. In Seale, Augus is he summer, approaching he auumnal equinox. The ays are ecreasing in lengh by abou hree minues per ay. So he erivaive of his funcion is abou minues per ay. On January 15, 01, which is winerime in Seale, here were abou 8 hours 5 minues of ayligh, an he erivaive was abou (posiive) minues per ay; he lengh of he ay was increasing by abou minues a ay. Secion 4: Derivaives of Formulas In his secion, we ll ge he erivaive rules ha will le us fin formulas for erivaives when our funcion comes o us as a formula. This is a very algebraic secion, an you shoul ge los of pracice. When you ell someone you have suie calculus, his is he one skill hey will expec you o have. There s no a lo of eep meaning here hese are sricly algebraic rules. Builing Blocks These are he simples rules rules for he basic funcions. We won prove hese rules; we ll jus use hem. Bu firs, le s look a a few so ha we can see hey make sense. Example: Fin he erivaive of y f x mx b Soluion: This is a linear funcion, so is graph is is own angen line! The slope of he angen line, he erivaive, is he slope of he line: f ' x m Rule: The erivaive of a linear funcion is is slope Example: Fin he erivaive of f x 15. Soluion: Think abou his one graphically, oo. The graph of f(x) is a horizonal line. So is slope is zero. f ' x 0 Rule: The erivaive of a consan is zero Example: I will jus ell you ha he erivaive of f x x is ' x x g x 4x. Wha will is erivaive be? funcion f. Now hink abou he Soluion: Think abou wha his change means o he graph of g i s now 4 imes as all as he graph of f. If g 4f f we fin he slope of a secan line, i will be 4 ; each slope will be 4 imes he slope of he x x x secan line on he f graph. This propery will hol for he slopes of angen lines, oo:
Chaper The Derivaive Business Calculus 16 4x 4 x 4 x 1x Rule: Consans come along for he rie. OK, enough of ha. Here are he basic rules, all in one place. Derivaive Rules: Builing Blocks In wha follows, f an g are iffereniable funcions of x. (a) Consan Muliple Rule: kf kf ' (b) Sum (or Difference) Rule: f g f ' g' (or f g f ' g' nx n n1 (c) Power Rule: Special cases: k 0 x x 1 e x x () Exponenial Funcions: e (because (because x x a ln a a ln 1 x (e) Naural Logarihm: x 0 k kx ) 1 x x ) ) The sum, ifference, an consan muliple rule combine wih he power rule allow us o easily fin he erivaive of any polynomial. 10 8 Example: Fin he erivaive of p x 17x 1x 1.8x 100
Chaper The Derivaive Business Calculus 17 Soluion: 17x 17 17 10 170x 1x 10 8 17x 1x 1.8 x 100 10 8 x 1 x 1.8 x 100 9 7 x 18 x 1.81 0 9 10 104x 8 7 1.8x 100 1.8 No, you on have o show every single sep. Do be careful when you re firs working wih he rules, bu prey soon you ll be able o jus wrie own he erivaive irecly: Example: 17x x 1 4x. The power rule works even if he power is negaive or a fracion. In orer o apply i, firs ranslae all roos an oneovers ino exponens: Example: Fin he erivaive of y 4 5e 4 Soluion: Firs sep ranslae ino exponens: 4 y 5e 1/ 4 4 5e 4 Now you can ake he erivaive: 4 5e 4 1 1/ 4 1/ 4 5 5 4 5 1/ e 16 5e. 4 5e Be careful when fining he erivaives wih negaive exponens. Example : The cos o prouce x iems is x hunre ollars. (a) Wha is he cos for 100 iems? 101 iems? Wha is cos of he 101 s iem? (b) For f(x) = las answer in par (a)? x, calculae f '(x) an evaluae f ' a x = 100. How oes f '(100) compare wih he
Chaper The Derivaive Business Calculus 18 Soluion: (a) Pu f(x) = x = x 1/ hunre ollars, he cos for x iems. Then f(100) = $1000 an f(101) = $1004.99, so i coss $4.99 for ha 101 s iem. Using his efiniion, he marginal cos is $4.99. (b) f '(x) = 1 x 1/ 1 1 = so f '(100) = x 100 = 1 hunre ollars = $5.00. Noe how close 0 hese answers are! This shows (again) why i s OK ha we use boh efiniions for marginal cos. Prouc an Quoien Rules The basic rules will le us ackle simple funcions. Bu wha happens if we nee he erivaive of a combinaion of hese funcions? Example: Fin he erivaive of g x 4x 11 x Soluion: This funcion is no a simple sum or ifference of polynomials. I s a prouc of polynomials. We can simply muliply i ou o fin is erivaive: g x 4x 11 x 4x g ' x 16x 11 6x 4 11x 1x 5 7 5 Example: Fin he erivaive of f x 4x x 1.5x 11 x 7.5x 10x Soluion: This funcion is no a simple sum or ifference of polynomials. I s a prouc of polynomials. We coul simply muliply i ou o fin is erivaive as before who wans o voluneer? Noboy? We ll nee a rule for fining he erivaive of a prouc so we on have o muliply everyhing ou. Is he rule wha we hope i is, ha we can jus ake he erivaives of he facors an muliply hem? Unforunaely, no ha won give he righ answer. Example: Fin he erivaive of g x 4x 11 x 11 6x Soluion: We alreay worke ou he erivaive. I s g' x 16x iffereniaing he facors an muliplying hem? We ge from he correc answer.. Wha if we ry 1x 1 1x, which is oally ifferen The rules for fining erivaives of proucs an quoiens are a lile complicae, bu hey save us he much more complicae algebra we migh face if we were o ry o muliply hings ou. They also le us eal wih proucs where he facors are no polynomials. We can use hese rules, ogeher wih he basic rules, o fin erivaives of many complicae looking funcions.
Chaper The Derivaive Business Calculus 19 Derivaive Rules: Prouc an Quoien Rules In wha follows, f an g are iffereniable funcions of x. (f) Prouc Rule: fg f ' g fg' The erivaive of he firs facor imes he secon lef alone, plus he firs lef alone imes he erivaive of he secon. The prouc rule can exen o a prouc of several funcions; he paern coninues ake he erivaive of each facor in urn, muliplie by all he oher facors lef alone, an a hem up. (g) Quoien Rule: f f ' g g g fg' The numeraor of he resul resembles he prouc rule, bu here is a minus insea of a plus; he minus sign goes wih he g. The enominaor is simply he square of he original enominaor no erivaives here. Example: Fin he erivaive of F e ln Soluion: This is a prouc, so we nee o use he prouc rule. I like o pu own empy parenheses o remin myself of he paern; ha way I on forge anyhing. F ' Then I fill in he parenheses he firs se ges he erivaive of ges e lef alone, an he fourh ges he erivaive of F ' ln. e ln e 1 Noice ha his was one we couln have one by muliplying ou. Example: Fin he erivaive of 4 x 4 y 16x x e, he secon ges ln lef alone, he hir Soluion: This is a quoien, so we nee o use he quoien rule. Again, I fin i helpful o pu own he empy parenheses as a emplae: y ' Then I fill in all he pieces:
Chaper The Derivaive Business Calculus 0 y' x 4 x 4x ln 44 16x x 4 48x 16x Now for gooness sakes on ry o simplify ha! Remember ha simple epens on wha you will o nex; in his case, we were aske o fin he erivaive, an we ve one ha. Please STOP! Chain Rule There is one more ype of complicae funcion ha we will wan o know how o iffereniae: composiion. The Chain Rule will le us fin he erivaive of a composiion. (This is he las erivaive rule we will learn!) y 4x 15x. Example: Fin he erivaive of Soluion: This is no a simple polynomial, so we can use he basic builing block rules ye. I is a prouc, so we coul wrie i as y 4x 15x 4x 15x4 x 15x an use he prouc rule. Or we coul muliply i ou an simply iffereniae he resuling polynomial. I ll o i he secon way: y 4x y' 64x 5 15x 480x 16x 6 450x 10x 4 5x y 4x 15x Example: Fin he erivaive of 0 Soluion: We coul wrie i as a prouc wih 0 facors an use he prouc rule, or we coul muliply i ou. Bu I on wan o o ha, o you? We nee an easier way, a rule ha will hanle a composiion like his. The Chain Rule is a lile complicae, bu i saves us he much more complicae algebra of muliplying somehing like his ou. I will also hanle composiions where i wouln be possible o muliply i ou. The Chain Rule is he mos common place for suens o make misakes. Par of he reason is ha he noaion akes a lile geing use o. An par of he reason is ha suens ofen forge o use i when hey shoul. When shoul you use he Chain Rule? Almos every ime you ake a erivaive.
Chaper The Derivaive Business Calculus 1 Derivaive Rules: Chain Rule In wha follows, f an g are iffereniable funcions wih y f u an u gx (h) Chain Rule (Leibniz noaion): y y u u Noice ha he u s seem o cancel. This is one avanage of he Leibniz noaion; i can remin you of how he chain rule chains ogeher. (h) Chain Rule (using prime noaion): f ' x f ' u g' x f ' g x g' x (h) Chain Rule (in wors): The erivaive of a composiion is) he erivaive of he ousie TIMES he erivaive of wha s insie. I recie he version in wors each ime I ake a erivaive, especially if he funcion is complicae. y 4x 15x. Example: Fin he erivaive of Soluion: This is he same one we i before by muliplying ou. This ime, le s use he Chain Rule: The insie funcion is wha appears insie he parenheses: 4x 15x. The ousie funcion is he firs hing we fin as we come in from he ousie i s he square funcion, somehing. We wan he erivaive of he ousie ( somehing) TIMES he erivaive of wha s insie (which is 1x 15 ): y y' 4x 15x 4x 15x 1x 15 (By he way, if you muliply his ou, you ge he same answer we go before. Hurray! Algebra works!) y 4x 15x Example: Fin he erivaive of 0 Soluion: Now we have a way o hanle his one. I s he erivaive of he ousie TIMES he erivaive of wha s insie. y 4x y' 0 0 15x 19 4x 15x 1x 15 Example : Differeniae x 5 e. Soluion: This isn a simple exponenial funcion; i s a composiion. Typical calculaor or compuer synax can help you see wha he insie funcion is here. On a TI calculaor, for example, when you push he x e key, i opens up parenheses: e ^ ( This ells you ha he insie of he exponenial funcion is he exponen. Here, he
Chaper The Derivaive Business Calculus insie is he exponen x 5. Now we can use he Chain Rule: We wan he erivaive of he ousie TIMES he erivaive of wha s insie. The ousie is he e o he funcion, so is erivaive is he same hing. The erivaive of wha s insie is x. So x 5 x 5 e e x Example: The able gives values for f, f ', g an g ' a a number of poins. Use hese values o eermine ( f g )(x) an ( f g ) '(x) a x = 1 an 0. x f(x) g(x) f '(x) g '(x) ( f g )(x) ( f g ) '(x) 1 1 0 0 1 1 1 1 0 1 1 0 1 0 1 Soluion: ( f g )( 1) = f( g( 1) ) = f( ) = 0 an ( f g )(0) = f( g(0) ) = f( 1 ) = 1. ( f g ) '( 1) = f '( g( 1) ). g '( 1 ) = f '( ). (0) = ()(0) = 0 an ( f g ) '( 0 ) = f '( g( 0 ) ). g '( 0 ) = f '( 1 ). ( ) = ( 1)() =. I ll le you o he res. Derivaives of Complicae Funcions You re now reay o ake he erivaive of some mighy complicae funcions. Bu how o you ell wha rule applies firs? Come in from he ousie wha o you encouner firs? Tha s he firs rule you nee. Use he Prouc, Quoien, an Chain Rules o peel off he layers, one a a ime, unil you re all he way insie. x Example: Fin e ln5x 7 Soluion: Coming in from he ousie, I see ha his is a prouc of wo (complicae) funcions. So I ll nee he Prouc Rule firs. I ll fill in he pieces I know, an hen I can figure he res as separae seps an subsiue in a he en: x x e ln5x 7 e ln5x 7 x e ln5x 7 Now as separae seps, I ll fin x x e e (using he Chain Rule) an 1 5x 7 ln5x 7 5 Finally, o subsiue hese in heir places: (also using he Chain Rule).
Chaper The Derivaive Business Calculus 5 7 5 1 7 5 ln 7 5 ln x e x e x e x x x (An please on ry o simplify ha!) Example: Differeniae 4 1 e z Soluion: Don panic! As you come in from he ousie, wha s he firs hing you encouner? I s ha 4 h power. Tha ells you ha his is a composiion, a (complicae) funcion raise o he 4 h power. Sep One: Use he Chain Rule. The erivaive of he ousie TIMES he erivaive of wha s insie. 1 1 4 1 4 e e e z Now we re one sep insie, an we can concenrae on jus he 1 e par. Now, as you come in from he ousie, he firs hing you encouner is a quoien his is he quoien of wo (complicae) funcions. Sep Two: Use he Quoien Rule: 1 1 1 9 1 e e e e Now we ve gone one more sep insie, an we can concenrae on jus he 1 e par. Now we have a prouc. Sep Three: Use he Prouc Rule: 1 1 1 e e e An now we re all he way in no more erivaives o ake. Sep Four: Now i s jus a quesion of subsiuing back be careful now! 1 1 1 e e e, so 1 1 1 1 9 1 e e e e e, so
Chaper The Derivaive Business Calculus 4 z Phew! e 4 1 e 1 9 e 1 e 1 e 1 e 1 4 Wha if he Derivaive Doesn Exis? A funcion is calle iffereniable a a poin if is erivaive exiss a ha poin.. We ve been acing as if erivaives exis everywhere for every funcion. This is rue for mos of he funcions ha you will run ino in his class. Bu here are some common places where he erivaive oesn exis. Remember ha he erivaive is he slope of he angen line o he curve. Tha s wha o hink abou. Where can a slope no exis? If he angen line is verical, he erivaive will no exis. Example: This is he graph of f(x) = x = x 1/. Noice ha he angen line o his curve a x = 0 is verical. So is slope oes no exis, an so he erivaive oes no exis a x = 0. Figure 15 Where can a angen line no exis? If here is a corner in he graph, he erivaive will no exis a ha poin because here is no well-efine angen line (a eeering angen, if you will.) Or if here is a jump in he graph, he angen line will be ifferen on eiher sie an he erivaive can exis. Example: This is he graph of he Greaes Ineger Funcion, a basic sep funcion. There is no single angen line a x = 1; he angen lines on eiher sie are ifferen. So he erivaive oes no exis a x = 1.
Chaper The Derivaive Business Calculus 5 Figure 16 Secion 5: Secon Derivaive an Concaviy Secon Derivaive an Concaviy Graphically, a funcion is concave up if is graph is curve wih he opening upwar (Fig. 17a). Similarly, a funcion is concave own if is graph opens ownwar (Fig. 17b). Figure 17 For example, An Epiemic: Suppose an epiemic has sare, an you, as a member of congress, mus ecie wheher he curren mehos are effecively fighing he sprea of he isease or wheher more rasic measures an more money are neee. In Fig. 18, f(x) is he number of people who have he isease a ime x, an wo ifferen siuaions are shown. In boh (a) an (b), he number of people wih he isease, f(now), an he rae a which new people are geing sick, f '(now), are he same. The ifference in he wo siuaions is he concaviy of f, an ha ifference in concaviy migh have a big effec on your ecision. Figure 18
Chaper The Derivaive Business Calculus 6 In (a), f is concave own a "now", he slopes are ecreasing, an i looks as if i s ailing off. We can say f is increasing a a ecreasing rae. I appears ha he curren mehos are saring o bring he epiemic uner conrol. In (b), f is concave up, he slopes are increasing, an i looks as if i will keep increasing faser an faser. I appears ha he epiemic is sill ou of conrol. The ifferences beween he graphs come from wheher he erivaive is increasing or ecreasing. The erivaive of a funcion f is a funcion ha gives informaion abou he slope of f. The erivaive ells us if he original funcion is increasing or ecreasing. Because f is a funcion, we can ake is erivaive. This secon erivaive also gives us informaion abou our original funcion f. The secon erivaive gives us a mahemaical way o ell how he graph of a funcion is curve. The secon erivaive ells us if he original funcion is concave up or own. Secon Derivaive Le y f x The secon erivaive of f is he erivaive of x y ' f '. Using prime noaion, his is x Using Leibniz noaion, he secon erivaive is wrien erivaive of f. If f '' or y ''. You can rea his alou as y ouble prime. ' x is posiive on an inerval, he graph of f x f ' y or f. This is rea alou as he secon y is concave up on ha inerval. We can say ha f is increasing (or ecreasing) a an increasing rae. If f ' x y f x is concave up on ha inerval. We can say ha f ' is negaive on an inerval, he graph of is increasing (or ecreasing) a a ecreasing rae. 7 ' for f x x Example: Fin f ' x Soluion: Firs, we nee o fin he firs erivaive: 6 x 1x f ' Then we ake he erivaive of ha funcion: f '' x 6 5 f ' x 1x 16x
Chaper The Derivaive Business Calculus 7 If f(x) represens he posiion of a paricle a ime x, hen v(x) = f '(x) will represen he velociy (rae of change of he posiion) of he paricle an a(x) = v '(x) = f ''(x) will represen he acceleraion (he rae of change of he velociy) of he paricle. Example : The heigh (fee) of a paricle a ime secons is 4 + 8. Fin he heigh, velociy an acceleraion of he paricle when = 0, 1, an secons. Soluion: f() = 4 + 8 so f(0) = 0 fee, f(1) = 5 fee, an f() = 8 fee. The velociy is v() = f '() = 8 + 8 so v(0) = 8 f/s, v(1) = f/s, an v() = 4 f/s. A each of hese imes he velociy is posiive an he paricle is moving upwar, increasing in heigh. The acceleraion is a() = 6 8 so a(0) = 8 f/s, a(1) = f/s an a() = 4 f/s. Inflecion Poins Definiion: An inflecion poin is a poin on he graph of a funcion where he concaviy of he funcion changes, from concave up o own or from concave own o up. Example: Which of he labele poins in Fig. 19 are inflecion poins? Figure 19 Soluion: The concaviy changes a poins b an g. A poins a an h, he graph is concave up on eiher sie, so he concaviy oes no change. A poins c an f, he graph is concave own on eiher sie. An a poin e, even hough he graph looks srange here, he graph is concave own on boh sies he concaviy oes no change. Inflecion poins happen when he concaviy changes. Because we know he connecion beween he concaviy of a funcion an he sign of is secon erivaive, we can use his o fin inflecion poins. Working Definiion: An inflecion poin is a poin on he graph where he secon erivaive changes sign.
Chaper The Derivaive Business Calculus 8 In orer for he secon erivaive o change signs, i mus eiher be zero or be unefine. So o fin he inflecion poins of a funcion we only nee o check he poins where f ''(x) is 0 or unefine. Noe ha i is no enough for he secon erivaive o be zero or unefine. We sill nee o check ha he sign of f changes sign. The funcions in he nex example illusrae wha can happen. Example: Le f(x) = x, g(x) = x 4 an h(x) = x 1/ (Fig. 0). For which of hese funcions is he poin (0,0) an inflecion poin? Figure 0 Soluion: Graphically, i is clear ha he concaviy of f(x) = x an h(x) = x 1/ changes a (0,0), so (0,0) is an inflecion poin for f an h. The funcion g(x) = x 4 is concave up everywhere so (0,0) is no an inflecion poin of g. We can also compue he secon erivaives an check he sign change. If f(x) = x, hen f '(x) = x an f ''(x) = 6x. The only poin a which f ''(x) = 0 or is unefine (f ' is no iffereniable) is a x = 0. If x < 0, hen f ''(x) < 0 so f is concave own. If x > 0, hen f ''(x) > 0 so f is concave up. A x = 0 he concaviy changes so he poin (0,f(0)) = (0,0) is an inflecion poin of x. If g(x) = x 4, hen g '(x) = 4x an g ''(x) = 1x. The only poin a which g ''(x) = 0 or is unefine is a x = 0. If x < 0, hen g ''(x) > 0 so g is concave up. If x > 0, hen g ''(x) > 0 so g is also concave up. A x = 0 he concaviy oes no change so he poin (0, g(0)) = (0,0) is no an inflecion poin of x 4. Keep his example in min!.
Chaper The Derivaive Business Calculus 9 If h(x) = x 1/, hen h '(x) = 1 x / an h ''(x) = 9 x 5/. h'' is no efine if x = 0, bu h ''(negaive number) > 0 an h ''(posiive number) < 0 so h changes concaviy a (0,0) an (0,0) is an inflecion poin of h. Example: Skech he graph of a funcion wih f() =, f '() = 1, an an inflecion poin a (,). Soluion: Two soluions are given in Fig. 1. Secion 6: Opimizaion Figure 1 In heory an applicaions, we ofen wan o maximize or minimize some quaniy. An engineer may wan o maximize he spee of a new compuer or minimize he hea prouce by an appliance. A manufacurer may wan o maximize profis an marke share or minimize wase. A suen may wan o maximize a grae in calculus or minimize he hours of suy neee o earn a paricular grae. Wihou calculus, we only know how o fin he opimum poins in a few specific examples (for example, we know how o fin he verex of a parabola). Bu wha if we nee o opimize an unfamiliar funcion? The bes way we have wihou calculus is o examine he graph of he funcion, perhaps using echnology. Bu our view epens on he viewing winow we choose we migh miss somehing imporan. In aiion, we ll probably only ge an approximaion his way. (In some cases, ha will be goo enough.) Calculus provies ways of rasically narrowing he number of poins we nee o examine o fin he exac locaions of maximums an minimums, while a he same ime ensuring ha we haven misse anyhing imporan. Local Maxima an Minima Before we examine how calculus can help us fin maximums an minimums, we nee o efine he conceps we will evelop an use.
Chaper The Derivaive Business Calculus 0 Definiions: f has a local maximum a a if f(a) f(x) for all x near a f has a local minimum a a if f(a) f(x) for all x near a f has a local exreme a a if f(a) is a local maximum or minimum. The plurals of hese are maxima an minima. We ofen simply say max or min; i saves a lo of syllables. Some books say relaive insea of local. The process of fining maxima or minima is calle opimizaion. A poin is a local max (or min) if i is higher (lower) han all he nearby poins. These poins come from he shape of he graph. Definiions: f has a global maximum a a if f(a) f(x) for all x in he omain of f. f has a global minimum a a if f(a) f(x) for all x in he omain of f. f has a global exreme a a if f(a) is a global maximum or minimum. Some books say absolue insea of global A poin is a global max (or min) if i is higher (lower) han every poin on he graph. These poins come from he shape of he graph an he winow hrough which we view he graph. The local an global exremes of he funcion in Fig. are labele. You shoul noice ha every global exreme is also a local exreme, bu here are local exremes ha are no global exremes. Figure
Chaper The Derivaive Business Calculus 1 If h(x) is he heigh of he earh above sea level a he locaion x, hen he global maximum of h is h(summi of M. Everes) = 9,08 fee. The local maximum of h for he Unie Saes is h(summi of M. McKinley) = 0,0 fee. The local minimum of h for he Unie Saes is h(deah Valley) = 8 fee. Example: The able shows he annual calculus enrollmens a a large universiy. Which years ha local maximum or minimum calculus enrollmens? Wha were he global maximum an minimum enrollmens in calculus? year 1980 81 8 8 84 85 86 87 88 89 90 enrollmen 157 14 178 16 189 1450 15 158 1567 1545 1571 Soluion: There were local maxima in 198 an 1987; he global maximum was 158 suens in 1987. There were local minima in 198 an 1989; he global minimum was 16 suens in 198. I choose no o hink of 1980 as a local minimum or 1990 as a local maximum. However, some books woul inclue he enpoins. Fining Maxima an Minima of a Funcion Wha mus he angen line look like a a local max or min? Look a hese wo graphs again you ll see ha a all he exreme poins, he angen line is horizonal (so f = 0). There is one cusp in he blue graph he angen line if verical here (so f is unefine). Tha gives us he clue how o fin exreme values. Definiion: A criical number for a funcion f is a value x = a in he omain of f where eiher f (a) = 0 or f (a) is unefine. Definiion: A criical poin for a funcion f is a poin (a, f(a)) where a is a criical number of f. Useful Fac: A local max or min of f can only occur a a criical poin.
Chaper The Derivaive Business Calculus Example: Fin he criical poins of f(x) = x 6x + 9x +. Soluion: A criical number of f can occur only where f '(x) = 0 or where f oes no exis. f '(x) = x 1x + 9 = (x 4x + ) = (x 1)(x ) so f '(x) = 0 only a x = 1 an x =. There are no places where f is unefine. The criical numbers are x = 1 an x=. So he criical poins are (1, 6) an (, ). These are he only possible locaions of local exremes of f. We haven iscusse ye how o ell wheher eiher of hese poins is acually a local exreme of f, or which kin i migh be. Bu we can be cerain ha no oher poin is a local exreme. The graph of f (Fig. ) shows ha (1, f(1) ) = (1, 6) is a local maximum an (, f() ) = (, ) is a local minimum. This funcion oes no have a global maximum or minimum. Figure
Chaper The Derivaive Business Calculus Example : Fin all local exremes of f(x) = x. Soluion: f(x) = x is iffereniable for all x, an f '(x) = x. The only place where f '(x) = 0 is a x = 0, so he only caniae is he criical poin (0,0). Bu if x > 0 hen f(x) = x > 0 = f(0), so f(0) is no a local maximum. Similarly, if x < 0 hen f(x) = x < 0 = f(0) so f(0) is no a local minimum. The criical poin (0,0) is he only caniae o be a local exreme of f, an his caniae i no urn ou o be a local exreme of f. The funcion f(x) = x oes no have any local exremes. (Fig. 4) Figure 4 Remember his example! I is no enough o fin he criical poins -- we can only say ha f migh have a local exreme a he criical poins. Firs an Secon Derivaive Tess Is ha criical poin a Maximum or Minimum (or Neiher)? Once we have foun he criical poins of f, we sill have he problem of eermining wheher hese poins are maxima, minima or neiher. All of he graphs in Fig. 5 have a criical poin a (, ). I is clear from he graphs ha he poin (,) is a local maximum in (a) an (), (,) is a local minimum in (b) an (e), an (,) is no a local exreme in (c) an (f).
Chaper The Derivaive Business Calculus 4 Figure 5 The criical numbers only give he possible locaions of exremes, an some criical numbers are no he locaions of exremes. The criical numbers are he caniaes for he locaions of maxima an minima. f ' an Exreme Values of f Four possible shapes of graphs are shown here in each graph, he poin marke by an arrow is a criical poin, where f (x) = 0. Wha happens o he erivaive near he criical poin? Figure 6 A a local max, such as in he graph on he lef, he funcion increases on he lef of he local max, hen ecreases on he righ. The erivaive is firs posiive, hen negaive a a local max. A a local min, he funcion ecreases o he lef an increases o he righ, so he erivaive is firs negaive, hen posiive. When here isn a local exreme, he funcion coninues o increase (or ecrease) righ pas he criical poin he erivaive oesn change sign.
Chaper The Derivaive Business Calculus 5 The Firs Derivaive Tes for Exremes: Fin he criical poins of f. For each criical number c, examine he sign of f o he lef an o he righ of c. Wha happens o he sign as you move from lef o righ? If f '(x) changes from posiive o negaive a x = c, hen f has a local maximum a (c, f(c)). If f '(x) changes from negaive o posiive a x = c, hen f has a local minimum a (c, f(c)). If f '(x) oes no change sign a x = c, hen (c, f(c)) is neiher a local max nor a local min. Example: Fin he criical poins of f(x) = x 6x + 9x + an classify hem as local max, local min, or neiher.. Soluion: We alreay foun he criical poins; hey are (1, 6) an (, ). Now we can use he firs erivaive es o classify each. Recall ha f (x) = f '(x) = x 1x + 9 = (x 4x + ) = (x 1)(x ). The facore form is easies o work wih here, so le s use ha. (1, 6) You coul choose a number slighly less han 1 o plug ino he formula for f perhaps use x = 0, or x = 0.9. Then you coul examine is sign. Bu I on care abou he numerical value, all I m inerese in is is sign. An for ha, you on have o o any plugging in: If x is a lile less han 1, hen x 1 is negaive, an x is negaive. So f = (x 1)(x ) will be pos(neg)(neg) = posiive. For x a lile more han 1, you can evaluae f a a number more han 1 (bu less han, you on wan o go pas he nex criical poin!) perhaps x =. Or you can make a quick sign argumen like wha I i above. For x a lile more han 1, f = (x 1)(x ) will be pos(pos)(neg) = negaive. f changes from posiive o negaive, so here is a local max a (1, 6) (, ) f changes from negaive o posiive, so here is a local min a (, ). This confirms wha we saw before in he graph.
Chaper The Derivaive Business Calculus 6 Figure 7 f '' an Exreme Values of f The concaviy of a funcion can also help us eermine wheher a criical poin is a maximum or minimum or neiher. For example, if a poin is a he boom of a concave up funcion (Fig. 8), hen he poin is a minimum. Figure 8 The Secon Derivaive Tes for Exremes: Fin all criical poins of f. For hose criical poins where f (c) = 0, fin f (c). (a) If f ''(c) < 0 hen f is concave own an has a local maximum a x = c. (b) If f ''(c) > 0 hen f is concave up an has a local minimum a x = c. (c) If f ''(c) = 0 hen f may have a local maximum, a minimum or neiher a x = c.
Chaper The Derivaive Business Calculus 7 Figure 9 The caroon faces can help you remember he Secon Derivaive Tes. Example: f(x) = x 15x + 4x 7 has criical numbers x = 1 an 4. Use he Secon Derivaive Tes for Exremes o eermine wheher f(1) an f(4) are maximums or minimums or neiher. Soluion: We nee o fin he secon erivaive: f f f x x 15x ' x 6x 0 '' x 1x 0 4x 7 x 4 Then we jus nee o evaluae f a each criical number: x = 1: f '' 1 11 0 0; here is a local maximum a x = 1. x = 4: '' 4 14 0 0 f ; here is a local minimum a x = 4. Many suens like he Secon Derivaive Tes. The Secon Derivaive Tes is ofen easier o use han he Firs Derivaive Tes. You only have o fin he sign of one number for each criical number raher han wo. An if your funcion is a polynomial, is secon erivaive will probably be a simpler funcion han he erivaive. Bu if you neee a prouc rule, quoien rule, or chain rule o fin he firs erivaive, fining he secon erivaive can be a lo of work. An, even if he secon erivaive is easy, he Secon Derivaive Tes oesn always give an answer. The Firs Derivaive Tes will always give you an answer. Use whichever es you wan o. Bu remember you have o o some es o be sure ha your criical poin acually is a local max or min. Global Maxima an Minima In applicaions, we ofen wan o fin he global exreme; knowing ha a criical poin is a local exreme is no enough.
Chaper The Derivaive Business Calculus 8 For example, if we wan o make he greaes profi, we wan o make he absoluely greaes profi of all. How o we fin global max an min? There are jus a few aiional hings o hink abou. Enpoin Exremes The local exremes of a funcion occur a criical poins hese are poins in he funcion ha we can fin by hinking abou he shape (an using he erivaive o help us). Bu if we re looking a a funcion on a close inerval, he enpoins coul be exremes. These enpoin exremes are no relae o he shape of he funcion; hey have o o wih he inerval, he winow hrough which we re viewing he funcion. Figure 0 In Fig. 0, i appears ha here are hree criical poins one local min, one local max, an one ha is neiher one. Bu he global max, he highes poin of all, is a he lef enpoin. The global min, he lowes poin of all, is a he righ enpoin. How o we ecie if enpoins are global max or min? I s easier han you expece simply plug in he enpoins, along wih all he criical numbers, an compare y-values. Example : Fin he global max an min of f(x) = x x 9x + 5 for x 6. Soluion: f '(x) = x 6x 9 = (x + 1)(x ). We nee o fin criical poins, an we nee o check he enpoins. (i) f '(x) = (x + 1)(x ) = 0 when x = 1 an x =. (ii) f is a polynomial so f is efine everywhere. (iii) The enpoins of he inerval are x = an x = 6. Now we simply compare he values of f a hese 4 values of x: f( ) =, f( 1) = 10, f() =, an f(6) = 59.
Chaper The Derivaive Business Calculus 9 The global minimum of f on [, 6] is, when x =, an he global maximum of f on [, 6] is 59, when x = 6. If here s only one criical poin If he funcion has only one criical poin an i s a local max (or min), hen i mus be he global max (or min). To see his, hink abou he geomery. Look a he graph on he lef here is a local max, an he graph goes own on eiher sie of he criical poin. Suppose here was some oher poin ha was higher hen he graph woul have o urn aroun. Bu ha urning poin woul have shown up as anoher criical poin. If here s only one criical poin, hen he graph can never urn back aroun. When in oub, graph i an look. Figure 1 If you are rying o fin a global max or min on an open inerval (or he whole real line), an here is more han one criical poin, hen you nee o look a he graph o ecie wheher here is a global max or min. Be sure ha all your criical poins show in your graph, an ha you go a lile beyon ha will ell you wha you wan o know. Example: Fin he global max an min of f(x) = x 6x + 9x +. Soluion: We have previously foun ha (1, 6) is a local max an (, ) is a local min. This is no a close inerval, an here are wo criical poins, so we mus urn o he graph of he funcion o fin global max an min. The graph of f (Fig. ) shows ha poins o he lef of x = 4 have y-values greaer han 6, so (1, 6) is no a global max. Likewise, if x is negaive, y is less han, so (, ) is no a global min. There are no enpoins, so we ve exhause all he possibiliies. This funcion oes no have a global maximum or minimum.
Chaper The Derivaive Business Calculus 40 Figure To fin Global Exremes: The only places where a funcion can have a global exreme are criical poins or enpoins. (a) If he funcion has only one criical poin, an i s a local exreme, hen i is also he global exreme. (b) If here are enpoins, fin he global exremes by comparing y-values a all he criical poins an a he enpoins. (c) When in oub, graph he funcion o be sure. Secion 7: Applie Opimizaion We have use erivaives o help fin he maximums an minimums of some funcions given by equaions, bu i is very unlikely ha someone will simply han you a funcion an ask you o fin is exreme values. More ypically, someone will escribe a problem an ask your help in maximizing or minimizing somehing: "Wha is he larges volume package which he pos office will ake?"; "Wha is he quickes way o ge from here o here?"; or "Wha is he leas expensive way o accomplish some ask?" In his secion, we ll iscuss how o fin hese exreme values using calculus. Max/Min Applicaions Example: The manager of a garen sore wans o buil a 600 square foo recangular enclosure on he sore s parking lo in orer o isplay some equipmen. Three sies of he enclosure will be buil of rewoo
Chaper The Derivaive Business Calculus 41 fencing, a a cos of $7 per running foo. The fourh sie will be buil of cemen blocks, a a cos of $14 per running foo. Fin he imensions of he leas cosly such enclosure. The process of fining maxima or minima is calle opimizaion. The funcion we re opimizing is calle he objecive funcion. The objecive funcion can be recognize by is proximiy o es wors (greaes, leas, highes, farhes, mos, ) Look a he garen sore example; he cos funcion is he objecive funcion. In many cases, here are wo (or more) variables in he problem. In he garen sore example again, he lengh an wih of he enclosure are boh unknown. If here is an equaion ha relaes he variables we can solve for one of hem in erms of he ohers, an wrie he objecive funcion as a funcion of jus one variable. Equaions ha relae he variables in his way are calle consrain equaions. The consrain equaions are always equaions, so hey will have equals signs. For he garen sore, he fixe area relaes he lengh an wih of he enclosure. This will give us our consrain equaion. Once we have a funcion of jus one variable, we can apply he calculus echniques we ve jus learne o fin maxima or minima. Max-Min Sory Problem Technique: (a) Translae he English saemen of he problem line by line ino a picure (if ha applies) an ino mah. This is ofen he hares sep! (b) Ienify he objecive funcion. Look for es wors. (b1) If you seem o have wo or more variables, fin he consrain equaion. Think abou he English meaning of he wor consrain, an remember ha he consrain equaion will have an = sign. (b) Solve he consrain equaion for one variable an subsiue ino he objecive funcion. Now you have an equaion of one variable. (c) Use calculus o fin he opimum values. (Take erivaive, fin criical poins, es. Don forge o check he enpoins!) () Look back a he quesion o make sure you answere wha was aske. Translae your number answer back ino English. Example: The manager of a garen sore wans o buil a 600 square foo recangular enclosure on he sore s parking lo in orer o isplay some equipmen. Three sies of he enclosure will be buil of rewoo fencing, a a cos of $7 per running foo. The fourh sie will be buil of cemen blocks, a a cos of $14 per running foo. Fin he imensions of he leas cosly such enclosure.
Chaper The Derivaive Business Calculus 4 Soluion: Firs, ranslae line by line ino mah an a picure: Tex The manager of a garen sore wans o buil a 600 square foo recangular enclosure on he sore s parking lo in orer o isplay some equipmen. Three sies of he enclosure will be buil of rewoo fencing, a a cos of $7 per running foo. The fourh sie will be buil of cemen blocks, a a cos of $14 per running foo. Fin he imensions of he leas cosly such enclosure. Translaion Le x an y be he imensions of he enclosure, wih y being he lengh of he sie mae of blocks. Then: Area = A = xy = 600 x + y coss $7 per foo y coss $14 per foo So Cos = C = 7(x + y) + 14y = 14x + 1y Fin x an y so ha C is minimize. Figure The objecive funcion is he cos funcion, an we wan o minimize i. As i sans, hough, i has wo variables, so we nee o use he consrain equaion. The consrain equaion is he fixe area A = xy = 600. Solve A for x o ge x 600 y, an hen subsiue ino C: 600 8400 C 14 1y 1y. y y Now we have a funcion of jus one variable, so we can whack i wih calculus (fin he criical poins, ec.) 8400 C ' 1 y C is unefine for y = 0, an C = 0 when y = 0 or y = 0.
Chaper The Derivaive Business Calculus 4 Of hese hree criical numbers, only y = 0 makes sense (is in he omain of he acual funcion) remember ha y is a lengh, so i can be negaive. An y = 0 woul mean here was no enclosure a all, so i couln have an area of 600 square fee. Tes y = 0: (I chose he secon erivaive es) 16800 C '' 0 y, so his is a local minimum. Since his is he only criical poin in he omain, his mus be he global minimum. When y = 0, x = 0. The imensions of he enclosure ha minimize he cos are 0 fee 0 fee. Marginal Revenue = Marginal Cos You ve probably hear before ha profi is maximize when marginal cos an marginal revenue are equal. Now you can see why people say ha! (Even hough i s no compleely rue.) General Example: Suppose we wan o maximize profi. Now we know wha o o fin he profi funcion, fin is criical poins, es hem, ec. Bu remember ha Profi = Revenue Cos. So Profi = Revenue Cos. Tha is, he erivaive of he profi funcion is MR MC. Now le s fin he criical poins hose will be where Profi = 0 or is unefine. Profi = 0 when MR MC = 0, or where MR = MC. Tha s where he saying comes from! Here s a more accurae way o express his: Profi has criical poins when Marginal Revenue an Marginal Cos are equal. In all he cases we ll see in his class, Profi will be very well behave, an we won have o worry abou looking for criical poins where Profi is unefine. Bu remember ha no all criical poins are local max! The places where MR = MC coul represen local max, local min, or neiher one. Example: A company sells q ribbon winers per year a $p per ribbon winer. The eman funcion for ribbon winers is given by: p 00 0. 0q The ribbon winers cos $0 apiece o manufacure, plus here are fixe coss of $9000 per year. Fin he quaniy where profi is maximize.
Chaper The Derivaive Business Calculus 44 Soluion: We wan o maximize profi, bu here isn a formula for profi showing ye. So le s make one. We can fin a funcion for Revenue = pq using he eman funcion for p. R q 00 0.0qq 00q 0.0q We can also fin a funcion for Cos, using he variable cos of $0 per ribbon winer, plus he fixe cos: C q 9000 0q Puing hem ogeher, we ge a funcion for Profi: q Rq Cq 00q 0.0q 9000 0q 0.0q 70q 9000 Now we have wo choices. We can fin he criical poins of Profi by aking he erivaive of π(q) irecly, or we can fin MR an MC an se hem equal. (Naurally, you ll ge he same answer eiher way.). I ll use MR = MC his ime. MR 00 0.04q MC 0 00 0.04q 0 70 0.04q q 6750 The only criical poin is a q = 6750. Now we nee o be sure his is a local max an no a local min. In his case, I ll look o he graph of π(q) i s a ownwar opening parabola, so his mus be a local max. An since i s he only criical poin, i mus also be he global max. Profi is maximize when hey sell 6750 ribbon winers. Average Cos = Marginal Cos Average cos is minimize when average cos = marginal cos is anoher saying ha isn quie rue; in his case, he correc saemen is: Average Cos has criical poins when Average Cos an Marginal Cos are equal. Le s look a a geomeric argumen here:
Chaper The Derivaive Business Calculus 45 Figure 4 Remember ha he average cos is he slope of he iagonal line, he line from he origin o he poin on he oal cos curve. If you move your clear plasic ruler aroun, you ll see (an feel) ha he slope of he iagonal line is smalles when he iagonal line jus ouches he cos curve when he iagonal line is acually a angen line when he average cos is equal o he marginal cos. Example: The cos in ollars o prouce q jars of gourme salsa is given by Cq 160 q.1q. quaniy where he average cos is minimum. Soluion: q C 160 Aq. 1q. We coul fin he criical poins by fining ' q q MC q. 160.1q.q q cos o marginal cos; I ll o he laer his ime. q 160.1q q 1600 q q 40 The criical poin of average cos is when q = 40. Fin he. So I wan o solve: A, or by seing average Noice ha we sill have o confirm ha he criical poin is a minimum. For his, we can use he firs or secon erivaive es on A q. A' A'' q 160.1 q 0 q q 0 The secon erivaive is posiive for all posiive q, so ha means his is a local min. Average cos is minimize when hey prouce 40 jars of salsa; a ha quaniy, he average cos is $10 per jar. (Mighy expensive salsa.)
Chaper The Derivaive Business Calculus 46 Secion 8: Oher Applicaions Tangen Line Approximaion Back when we firs hough abou he erivaive, we use he slope of secan lines over iny inervals o approximae he erivaive: f ' a y x f x f x a a Now ha we have oher ways o fin erivaives, we can exploi his approximaion o go he oher way. Solve he expression above for f(x), an you ll ge he angen line approximaion: The Tangen Line Approximaion (TLA) To approximae he value of f(x) using TLA, fin some a where 1. a an x are close, an. You know he exac values of boh f (a) an f (a). Then f x f a f ' a x a Anoher way o look a he same formula: y f ' ax How close is close? I epens on he shape of he graph of f. In general, he closer he beer. Figure 5 Example: Suppose we know ha g(0) = 5 an g (0) = 1.4. Using jus his informaion, we can approximae he values of g a some nearby poins: g() 5 + (1.4)( 0) = 9.
Chaper The Derivaive Business Calculus 47 g(18) 5 + (1.4)(18 0) =. Noe ha we on know if hese approximaions are close bu hey re he bes we can o wih he limie informaion we have o sar wih. Noe also ha 18 an are sor of close o 0, so we can hope hese approximaions are prey goo. We feel more confien using his informaion o approximae g(0.00). We feel very unsure using his informaion o approximae g(55). Elasiciy We know ha eman funcions are ecreasing, so when he price increases, he quaniy emane goes own. Bu wha abou revenue = price quaniy? Will revenue go own because he eman roppe so much? Or will revenue increase because eman in rop very much? Elasiciy of eman is a measure of how eman reacs o price changes. I s normalize ha means he paricular prices an quaniies on maer, so we can compare onions an cars. The formula for elasiciy of eman involves a erivaive, which is why we re iscussing i here. Elasiciy of Deman Given a eman funcion ha gives q in erms of p, The elasiciy of eman is E p q q p (Noe ha since eman is a ecreasing funcion of p, he erivaive is negaive. Tha s why we have he absolue values so E will always be posiive.) If E < 1, we say eman is inelasic. In his case, raising prices increases revenue. If E > 1, we say eman is elasic. In his case, raising prices ecreases revenue. If E = 1, we say eman is uniary. E = 1 a criical poins of he revenue funcion. Example: A company sells q ribbon winers per year a $p per ribbon winer. The eman funcion for ribbon winers is given by: p 00 0. 0q Fin he elasiciy of eman when he price is $70 apiece. Will an increase in price lea o an increase in revenue?
Chaper The Derivaive Business Calculus 48 q Soluion: Firs, we nee o solve he eman equaion so i gives q in erms of p, so ha we can fin : p p 00 0.0q, so q 15000 50p. q We nee o fin q when p = 70: q = 11500. We also nee 50 p p q 70 E. q p 11500 Now compue 50 0. E < 1, so eman is inelasic. Increasing he price woul lea o an increase in revenue; i seems ha he company shoul increase is price. The eman for proucs ha people have o buy, such as onions, ens o be inelasic. Even if he price goes up, people sill have o buy abou he same amoun of onions, an revenue will no go own. The eman for proucs ha people can o wihou, or pu off buying, such as cars, ens o be elasic. If he price goes up, people will jus no buy cars righ now, an revenue will rop. Chaper Exercises 1. Wha is he slope of he line hrough (,9) an (x, y) for y = x an x =.97? x =.001? x = +h? Wha happens o his las slope when h is very small (close o 0)?. Wha is he slope of he line hrough (,4) an (x, y) for y = x an x = 1.98? x =.0? x = +h? Wha happens o his las slope when h is very small (close o 0)?. Fig. 6 shows he emperaure uring a ay in Ames. (a) Wha was he average change in emperaure from 9 am o 1 pm? (b) Esimae how fas he emperaure was rising a 10 am an a 7 pm? Figure 6 4. Fig. 7 shows he isance of a car from a measuring posiion locae on he ege of a sraigh roa. (a) Wha was he average velociy of he car from = 0 o = 0 secons?
Chaper The Derivaive Business Calculus 49 (b) Wha was he average velociy of he car from = 10 o = 0 secons? (c) Abou how fas was he car raveling a = 10 secons? a = 0 s? a = 0 s? () Wha oes he horizonal par of he graph beween = 15 an = 0 secons mean? (e) Wha oes he negaive velociy a = 5 represen? Figure 7 5. Fig. 8 shows he isance of a car from a measuring posiion locae on he ege of a sraigh roa. (a) Wha was he average velociy of he car from = 0 o = 0 secons? (b) Wha was he average velociy from = 10 o = 0 secons? (c) Abou how fas was he car raveling a = 10 secons? a = 0 s? a = 0 s? Figure 8 6. Use he funcion in Fig. 9 o fill in he able an hen graph m(x). Figure 9
Chaper The Derivaive Business Calculus 50 x y = f(x) m(x) = he esimae slope of he angen line o y=f(x) a he poin (x,y) 0 0.5 1.0 1.5.0.5.0.5 4.0 7. The graph of y = f(x) is given in Fig. 40. Se up a able of values for x an m(x) (he slope of he line angen o he graph of y=f(x) a he poin (x,y) ), an hen graph he funcion m(x). Figure 40 8. (a) A wha values of x oes he graph of f in Fig. 41 have a horizonal angen line? (b) A wha value(s) of x is he value of f he larges? smalles? (c) Skech he graph of m(x) = he slope of he line angen o he graph of f a he poin (x,y). Figure 41 9. (a) A wha values of x oes he graph of g in Fig. 4 have a horizonal angen line? (b) A wha value(s) of x is he value of g he larges? smalles? (c) A wha value(s) of x is he slope of g he larges? smalles?
Chaper The Derivaive Business Calculus 51 Figure 4 10. Mach he siuaion escripions wih he corresponing ime velociy graphs in Fig. 16. (a) A car quickly leaving from a sop sign. (b) A car seaely leaving from a sop sign. (c) A suen bouncing on a rampoline. () A ball hrown sraigh up. (e) A suen confienly sriing across campus o ake a calculus es. (f) An unprepare suen walking across campus o ake a calculus es. Figure 4 11. Fig. 44 shows he emperaure uring a summer ay in Chicago. Skech he graph of he rae a which he emperaure is changing. (This is jus he graph of he slopes of he lines which are angen o he emperaure graph.)
Chaper The Derivaive Business Calculus 5 Figure 44 1. Fig. 45 shows six graphs, hree of which are erivaives of he oher hree. Mach he funcions wih heir erivaives. Figure 45 1. Mach he graphs of he hree funcions in Fig. 46 wih he graphs of heir erivaives. Figure 46 14: Fill in he values in he able for f x, f x g x, an gx f x x f(x) f '(x) g(x) g '(x) f x f x g x gx f x 0 4 1 1 1 0 4 1.
Chaper The Derivaive Business Calculus 5 15: Use he values in he able o fill in he res of he able. x f(x) f '(x) g(x) g '(x) f x gx 0 4 1 1 1 0 4 1 f g x x g f x x 16. Use he informaion in Fig. 47 o plo he values of he funcions f + g, f. g an f/g an heir erivaives a x = 1,, an. 17. Calculae x 5 x 7 Figure 47 by (a) using he prouc rule an (b) expaning he prouc an hen iffereniaing. Verify ha boh mehos give he same resul. 18. If he prouc of f an g is a consan ( f(x) g(x) = k for all x), hen how are an g g x x relae? 19. If he quoien of f an g is a consan ( f g x x f f x x k for all x), hen how are g. f ' an f. g ' relae? In problems 0 5, (a) calculae f '(1) an (b) eermine when f '(x) = 0. 0. f(x) = x 5x + 1 1. f(x) = 5x 40x + 7. f(x) = x + 9x + 6. f(x) = x + x + x 1
Chaper The Derivaive Business Calculus 54 4. f(x) = x + x + x 1 7x 5. f(x) = x + 4 6.Deermine x 1 7x an 7. Fin (a) x x x e an (b) e x 8. Fin (a) e, (b) e 5 ( 5 + 1 ). 9: Where o f(x) = x 10x + an g(x) = x 1x have horizonal angen lines?. 0. f(x) = x + A x + B x + C wih consans A, B an C. Can you fin coniions on he consans A, B an C which will guaranee ha he graph of y = f(x) has wo isinc "verices"? (Here a "verex" means a place where he curve changes from increasing o ecreasing or from ecreasing o increasing.)
Chaper The Derivaive Business Calculus 55 1. An arrow sho sraigh up from groun level wih an iniial velociy of 18 fee per secon will be a heigh h(x) = 16x + 18x fee afer x secons. (Fig.48) (a) Deermine he velociy of he arrow when x = 0, 1 an secons. (b) Wha is he velociy of he arrow, v(x), a any ime x? (c) A wha ime x will he velociy of he arrow be 0? () Wha is he greaes heigh he arrow reaches? (e) How long will he arrow be alof? (f) Use he answer for he velociy in par (b) o eermine he acceleraion, a(x) = v '(x), a any ime x. Figure 48. If an arrow is sho sraigh up from groun level on he moon wih an iniial velociy of 18 fee per secon, is heigh will be h(x) =.65x + 18x fee a x secons. Do pars (a) (f) of problem 1 using his new equaion for h. In problems - 8, iffereniae each funcion an fin he equaion of he angen line a x = a. y x x ; a = 0. 1 4. y 1 ; a = 1 x 4 5. 1 y x ; a = 0.5 x 5 y e 6. ; a = 0 7. 8. y y x x e e ; a = 0 x x e e ; a = 0. 9. A manufacurer has eermine ha an employee wih ays of proucion experience will be able o prouce approximaely P() = + 15( 1 e 0. ) iems per ay.
Chaper The Derivaive Business Calculus 56 (a) Approximaely how many iems will a beginning employee be able o prouce each ay? (b) How many iems will an experience employee be able o prouce each ay? (c) Wha is he marginal proucion rae of an employee wih 5 ays of experience? (Wha are he unis of your answer, an wha oes his answer mean?) 40. The air pressure P(h), in pouns per square inch, a an aliue of h fee above sea level is approximaely P(h) = 14.7 e 0.000085h. (a) Wha is he air pressure a sea level? Wha is he air pressure a an aliue of 0,000 fee? (b) A wha aliue is he air pressure 10 pouns per square inch? (c) If you are in a balloon which is 000 fee above he Pacific Ocean an is rising a 500 fee per minue, how fas is he air pressure on he balloon changing? () If he emperaure of he gas in he balloon remaine consan uring his ascen, wha woul happen o he volume of he balloon? For problems 41-5, calculae 41. y Ax B 4. y Ax Bx C 4. 44. 45. 46. 47. 48. y y A Bx A Bx Bx y Ae Bx y xe Ax Ax e e 1 y Ax B y y'. The leers A D represen consans. 49. Ax B y Cx D 50. y ln Ax B 51. 5. y 1 ln Ax B Ax B y ln Ax B
Chaper The Derivaive Business Calculus 57 1 5. y ln Ax B 54. For y = Ax + Bx + C, (a) fin y ', (b) fin he value(s) of x so ha y ' = 0, an (c) fin y ". (You shoul recognize he par (b) answer from inermeiae algebra. Wha is i?) 55. y = Ax(B x) = ABx Ax, (a) fin y ', (b) fin he value(s) of x so ha y ' = 0, an (c) fin y ". 56. y = Ax + Bx + C., (a) fin y ', (b) fin he value(s) of x so ha y ' = 0, an (c) fin y ". In problems 57 an 58, each quoaion is a saemen abou a quaniiy of somehing changing over ime. Le f() represen he quaniy a ime. For each quoaion, ell wha f represens an wheher he firs an secon erivaives of f are posiive or negaive. 57. (a) "Unemploymen rose again, bu he rae of increase is smaller han las monh." (b) "Our profis ecline again, bu a a slower rae han las monh." (c) "The populaion is sill rising an a a faser rae han las year." 58. (a) "The chil's emperaure is sill rising, bu slower han i was a few hours ago." (b) "The number of whales is ecreasing, bu a a slower rae han las year." (c) "The number of people wih he flu is rising an a a faser rae han las monh." 59. On which inervals is he funcion in Fig. 49 (a) concave up? (b) concave own? Figure 49 60. On which inervals is he funcion in Fig. 50 (a) concave up? (b) concave own? Figure 50 61. Skech he graphs of funcions which are efine an concave up everywhere an which have
Chaper The Derivaive Business Calculus 58 (a) no roos. (b) exacly 1 roo. (c) exacly roos. () exacly roos. In problems 6 65, a funcion an values of x so ha f '(x) = 0 are given. Use he Secon Derivaive Tes o eermine wheher each poin (x, f(x)) is a local maximum, a local minimum or neiher 6. f(x) = x 15x + 6, x = 0, 5. 6. g(x) = x x 9x + 7, x = 1,. 64. h(x) = x 4 8x, x =, 0,. 65. f(x) = x. ln(x), x = 1/e. 66. Which of he labele poins in Fig. 51 are inflecion poins? Figure 51 67. Which of he labele poins in Fig. 5 are inflecion poins? Figure 5 68. How many inflecion poins can a (a) quaraic polynomial have? (b) cubic polynomial have? (c) polynomial of egree n have? 69. Fill in he able wih "+", " ", or "0" for he funcion in Fig. 5. x 0 1 f x f ' x f '' x
Chaper The Derivaive Business Calculus 59 Figure 5 70. Fill in he able wih "+", " ", or "0" for he funcion in Fig. 54 x 0 1 f x f ' x f '' x Figure 54 In problems 71 76, fin he erivaive an secon erivaive of each funcion. 71. f(x) = 7x + 5x 7. f(x) = (x 8) 5 7. f(x) = (6x x ) 10 74. f(x) = x. (x + 7) 5 75. f(x) = (x + ) 6 76. f(x) = x + 6x 1 77. f(x) = lnx 4 78. Fin he equaion of he line angen o f(x) = e x a he poin (, e ). Where will his angen line inersec he x axis? Where will he angen line o f(x) = e x a he poin (p, e p ) inersec he x axis? 79. Fin all exremes of f(x) = x 1x + 7 an use he Firs Derivaive Tes o eermine if hey are maximums, minimums or neiher. In problems 80 87, fin all of he criical poins an local maximums an minimums of each funcion. 80. f(x) = x + 8x + 7 81. f(x) = x 1x + 7 8. f(x) = x 6x + 5 8. f(x) = (x 1) (x ) 84. f(x) = ln( x 6x + 11 )
Chaper The Derivaive Business Calculus 60 85. f(x) = x 96x + 4 In problems 86 9, fin all criical poins an global exremes of each funcion on he given inervals. 86. f(x) = x 6x + 5 on he enire real number line. 87. f(x) = x on he enire real number line. 88. f(x) = x x + 5 on he enire real number line. f x x e 89. x on he enire real number line. 90. f(x) = x 6x + 5 on [, 5]. 91. f(x) = x on [, 1]. 9. f(x) = x x + 5 on [, 1]. x 9. f x x e on [ 1, ]. 94.Fin all of he criical poins of he funcion in Fig. 55 an ienify hem as local max, local min, or neiher. Fin he global max an min on he inerval. Figure 55 95. Fin all of he criical poins of he funcion in Fig. 56 an ienify hem as local max, local min, or neiher. Fin he global max an min on he inerval. Figure 56 96. Suppose f(1) = 5 an f '(1) = 0. Wha can we conclue abou he poin (1,5) if (a) f '(x) < 0 for x < 1, an f '(x) > 0 for x > 1? (b) f '(x) < 0 for x < 1, an f '(x) < 0 for x > 1? (c) f '(x) > 0 for x < 1, an f '(x) < 0 for x > 1? () f '(x) > 0 for x < 1, an f '(x) > 0 for x > 1?
Chaper The Derivaive Business Calculus 61 97. Wha will he n erivaive of a quaraic polynomial be? The r erivaive? The 4 h erivaive? 98. Wha will he r erivaive of a cubic polynomial be? The 4 h erivaive? 99. Wha can you say abou he n h an (n+1) s erivaives of a polynomial of egree n? 100. Skech he graph of a coninuous funcion f so ha (a) f(1) =, f '(1) = 0, an he poin (1,) is a local maximum of f. (b) f() = 1, f '() = 0, an he poin (,1) is a local minimum of f. (c) f(5) = 4, f '(5) = 0, an he poin (5,4) is no a local minimum or maximum of f. 101. Define A(x) o be he area boune beween he x axis, he graph of f, an a verical line a x ( 0 x 10 ). See Fig. 57. (a) A wha value of x is A(x) minimum? (b) A wha value of x is A(x) maximum? Figure 57 10. Define S(x) o be he slope of he line hrough he poins (0,0) an ( x, f(x) ) ( 0 x 10 ). See Fig. 58. (a) A wha value of x is S(x) minimum? (b) A wha value of x is S(x) maximum? Figure 58 10. (a) You have 00 fee of fencing o enclose a recangular vegeable garen. Wha shoul he imensions of your garen be in orer o enclose he larges area? (b) Show ha if you have P fee of fencing available, he garen of greaes area is a square.
Chaper The Derivaive Business Calculus 6 (c) Wha are he imensions of he larges recangular garen you can enclose wih P fee of fencing if one ege of he garen borers a sraigh river an oes no nee o be fence? () Jus hinking calculus will no help wih his one: Wha o you hink is he shape of he larges garen which can be enclose wih P fee of fencing if we o no require he garen o be recangular? Wha o you hink is he shape of he larges garen which can be enclose wih P fee of fencing if one ege of he garen borers a river an oes no nee o be fence? 104. (a) You have 00 fee of fencing available o consruc a recangular pen wih a fence ivier own he mile (see Fig. 59). Wha imensions of he pen enclose he larges oal area? (b) If you nee iviers, wha imensions of he pen enclose he larges area? (c) Wha are he imensions in pars (a) an (b) if one ege of he pen borers on a river an oes no require any fencing? Figure 59 105. You have 10 fee of fencing o consruc a pen wih 4 equal size salls. If he pen is recangular an shape like he one in Fig. 60, wha are he imensions of he pen of larges area an wha is ha area? Figure 60 106. Suppose you ecie o fence he recangular garen in he corner of your yar. Then wo sies of he garen are boune by he yar fence which is alreay here, so you only nee o use he 80 fee of fencing o enclose he oher wo sies. Wha are he imensions of he new garen of larges area? Wha are he imensions of he recangular garen of larges area in he corner of he yar if you have F fee of new fencing available? 107. (a) You have been aske o bi on he consrucion of a square-boome box wih no op which will hol 100 cubic inches of waer. If he boom an sies are mae from he same maerial, wha are he imensions of he box which uses he leas maerial? (Assume ha no maerial is wase.) (b) Suppose he box in par (a) uses ifferen maerials for he boom an he sies. If he boom maerial coss 5 per square inch an he sie maerial coss per square inch, wha are he imensions of he leas expensive box which will hol 100 cubic inches of waer?
Chaper The Derivaive Business Calculus 6 108. Problem 107 is a "classic" problem which has many variaions. We coul require ha he box be wice as long as i is wie, or ha he box have a op, or ha he ens cos a ifferen amoun han he fron an back, or even ha i coss some amoun of money o wel each inch of ege. Wrie an solve a variaion of Problem 107. 109. U.S. posal regulaions sae ha he sum of he lengh an girh (isance aroun) of a package mus be no more han 108 inches. (Fig. 61) (a) Fin he imensions of he accepable box wih a square en which has he larges volume. (b) Fin he imensions of he accepable box which has he larges volume if is en is a recangle wice as long as i is wie. Figure 61 110. D. Simonon claims ha he "prouciviy levels" of people in ifferen fiels can be escribe as a funcion of heir "career age" by p() = e a e b where a an b are consans which epen on he fiel of work, an career age is approximaely 0 less han he acual age of he iniviual. (a) Base on his moel, a wha ages o mahemaicians (a=.0, b=.05), geologiss (a=.0, b=.04), an hisorians (a=.0, b=.0) reach heir maximum prouciviy? (b) Simonon says "Wih a lile calculus we can show ha he curve ( p() ) maximizes a = Use calculus o show ha Simonon is correc. 1 b a ln( b a )." Noe: Moels of his ype have uses for escribing he behavior of groups, bu i is angerous an usually invali o apply group escripions or comparisons o iniviuals in he group. (Scienific Genius, by Dean Simonon, Cambrige Universiy Press, 1988, pp. 69 7) 111. You own a small airplane which hols a maximum of 0 passengers. I coss you $100 per fligh from S. Thomas o S. Croix for gas an wages plus an aiional $6 per passenger for he exra gas require by he exra weigh. The charge per passenger is $0 each if 10 people charer your plane (10 is he minimum number you will fly), an his charge is reuce by $1 per passenger for each passenger over 10 who goes (ha is, if 11 go hey each pay $9, if 1 go hey each pay $8, ec.). Wha number of passengers on a fligh will maximize your profis?