5 Differential Equations

Transcription

1 Differenial Equaions. Slope Fields and Euler s Mehod. Growh and Deca. Separaion of Variables. The Logisic Equaion Sailing (Eercise 7, p. 9) Cooe Populaion (Eample, p. 7) Elk Populaion (Eample 6, p. 99) Foresr (Eercise 60, p. ) Paul s Noes Chaper Overview Chaper combines echnological and concepual opics wih a subsanial number of real-life applicaions. While presening he maerial in his chaper, be aware ha he process of differenial equaions can be difficul for sudens. The chaper begins b aking sudens hrough he process of solving differenial equaions for boh general and paricular soluions. An emphasis is placed on he use of slope fields as soluions, paricularl for differenial equaions ha canno be easil solved. From his poin, sudens are read o appl he concep of differenial equaions. There are plen of opporuniies for boh ou, as he eacher, and our sudens o eplore a number of real-world phenomena. Tr o selec he applicaions ou use based on he ineress of our sudens. Some sudens ma enjo he growh and deca principles of financial mahemaics, while ohers ma prefer he scienific applicaions. The use of he logisic equaion is an imporan concep o cover for sudens in BC Calculus. The use of his differenial equaion presens sudens wih a more accurae depicion of populaion models, paricularl when here are consrains wihin he model, such as he maimum capaci of a zoo. Radioacive Deca (Eample, p. 79) Clockwise from op lef, Web Picure Blog/Shuersock.com; Infiniumgu/Shuersock.com; Sephen Aaron Rees/Shuersock.com; Foo/Gallo Images/Ge Images; Derek R. Audee/Shuersock.com An insrucional video from Paul, including ke poins and conceps covered in he chaper, is available a LarsonCalculusforAP.com. 67 Chaper Planning Guide Chaper Resources Ineracive Technolog Complee Soluions Guide Manual Available Noe Taking on Insrucor Guide AIECompanion Sie Tes Teacher s Bank Resource Guide PowerLecure Videos wih EamView Te Cognero Specific Publisher DVDs Tesing Sofware 67

2 6 Chaper Differenial Equaions Paul s Noes Secion Overview An insrucional video from Paul, including eaching sraegies for he secion, is available a LarsonCalculusforAP.com. Essenial Quesion How do ou approimae he paricular soluion of a differenial equaion? Tell sudens ha he will learn how o answer his quesion b using slope fields and Euler s Mehod. Lesson Moivaor An inroducion o slope fields will allow sudens o skech paricular soluions of differenial equaions. Similarl, Euler s Mehod allows sudens o approimae paricular soluions of differenial equaions numericall. Era Eample Deermine wheher each funcion is a soluion of he differenial equaion + = 0. a. = cos Soluion b. = e No a soluion c. = C sin Soluion. Slope Fields and Euler s Mehod Use iniial condiions o find paricular soluions of differenial equaions. Use slope fields o approimae soluions of differenial equaions. Use Euler s Mehod o approimae soluions of differenial equaions. General and Paricular Soluions In his e, ou will learn ha phsical phenomena can be described b differenial equaions. Recall ha a differenial equaion in and is an equaion ha involves,, and derivaives of. For eample, = 0 Differenial equaion is a differenial equaion. In he ne secion, ou will see ha problems involving radioacive deca, populaion growh, and Newon s Law of Cooling can be formulaed in erms of differenial equaions. A funcion = f () is called a soluion of a differenial equaion if he equaion is saisfied when and is derivaives are replaced b f () and is derivaives. For eample, differeniaion and subsiuion would show ha = e is a soluion of he differenial equaion + = 0. I can be shown ha ever soluion of his differenial equaion is of he form = Ce General soluion of + = 0 where C is an real number. This soluion is called he general soluion. Some differenial equaions have singular soluions ha canno be wrien as special cases of he general soluion. Such soluions, however, are no considered in his e. The order of a differenial equaion is deermined b he highes-order derivaive in he equaion. For insance, = is a firs-order differenial equaion. In Secion., Eample 9, ou saw ha he second-order differenial equaion s () = has he general soluion s() = 6 + C + C General soluion of s () = which conains wo arbirar consans. I can be shown ha a differenial equaion of order n has a general soluion wih n arbirar consans. Verifing Soluions Deermine wheher each funcion is a soluion of he differenial equaion = 0. a. = sin b. = e c. = Ce Soluion a. Because = sin, = cos, and = sin, i follows ha = sin sin = sin 0. So, = sin is no a soluion. b. Because = e, = e, and = e, i follows ha = e e = 0. So, = e is a soluion. c. Because = Ce, = Ce, and = Ce, i follows ha = Ce Ce = 0. So, = Ce is a soluion for an value of C. 6

3 Secion. Slope Fields and Euler s Mehod 69 Geomericall, he general soluion of a firs-order differenial equaion represens a famil of curves known as soluion curves, one for each value assigned o he arbirar consan. For insance, ou can verif ha ever funcion of he form = C is a soluion of he differenial equaion General soluion of + = 0 + = 0. Figure. shows four of he soluion curves corresponding o differen values of C. As discussed in Secion., paricular soluions of a differenial equaion are obained from iniial condiions ha give he values of he dependen variable or one of is derivaives for paricular values of he independen variable. The erm iniial condiion sems from he fac ha, ofen in problems involving ime, he value of he dependen variable or one of is derivaives is known a he iniial ime = 0. For insance, he second-order differenial equaion s () = having he general soluion s() = 6 + C + C General soluion of s () = migh have he following iniial condiions. s(0) = 0, s (0) = 6 Iniial condiions In his case, he iniial condiions ield he paricular soluion s() = Paricular soluion Finding a Paricular Soluion See LarsonCalculusforAP.com for an ineracive version of his pe of eample. For he differenial equaion = 0 verif ha = C is a soluion. Then find he paricular soluion deermined b he iniial condiion = when =. Soluion You know ha = C is a soluion because =C and = (C ) (C ) = 0. Furhermore, he iniial condiion = when = ields = C General soluion = C( ) Subsiue iniial condiion. = C Solve for C. 7 and ou can conclude ha he paricular soluion is =. Paricular soluion 7 Tr checking his soluion b subsiuing for and in he original differenial equaion. Noe ha o deermine a paricular soluion, he number of iniial condiions mus mach he number of consans in he general soluion. C = C = C = C = General soluion: = C C = C = C = C = Soluion curves for + = 0 Figure. Paul s Noes Era Eample For he differenial equaion + = 0, verif ha = Ce is a soluion. Then find he paricular soluion deermined b he iniial condiion = when = 0. + = Ce + (Ce ) = e = 0; 69

4 70 Chaper Differenial Equaions Paul s Noes Era Eample Skech a slope field for he differenial equaion = + for he poins (, ), (, ), and (0, ). Slope Fields Solving a differenial equaion analicall can be difficul or even impossible. However, here is a graphical approach ou can use o learn a lo abou he soluion of a differenial equaion. Consider a differenial equaion of he form =F(, ) Differenial equaion where F(, ) is some epression in and. A each poin (, ) in he -plane where F is defined, he differenial equaion deermines he slope =F(, ) of he soluion a ha poin. If ou draw shor line segmens wih slope F(, ) a seleced poins (, ) in he domain of F, hen hese line segmens form a slope field, or a direcion field, for he differenial equaion =F(, ). Each line segmen has he same slope as he soluion curve hrough ha poin. A slope field shows he general shape of all he soluions and can be helpful in geing a visual perspecive of he direcions of he soluions of a differenial equaion. Insigh The abili o idenif, inerpre, and draw a slope field is esed on boh he AP Calculus AB and BC Eams. Skeching a Slope Field Skech a slope field for he differenial equaion = for he poins (, ), (0, ), and (, ). Soluion The slope of he soluion curve a an poin (, ) is Era Eample Mach each slope field wih is differenial equaion. a. F(, ) =. Slope a (, ) So, he slope a each poin can be found as shown. Slope a (, ): = = Slope a (0, ): = 0 = Slope a (, ): = = 0 Draw shor line segmens a he hree poins wih heir respecive slopes, as shown in Figure.. Figure. Idenifing Slope Fields for Differenial Equaions Mach each slope field wih is differenial equaion. a. b. c. b. i. = + ii. = iii. = Soluion a. You can see ha he slope a an poin along he -ais is 0. The onl equaion ha saisfies his condiion is =. So, he graph maches equaion (ii). b. You can see ha he slope a he poin (, ) is 0. The onl equaion ha saisfies his condiion is = +. So, he graph maches equaion (i). c. c. You can see ha he slope a an poin along he -ais is 0. The onl equaion ha saisfies his condiion is =. So, he graph maches equaion (iii). i. = + ii. = iii. = (a) ii (b) iii (c) i Teaching Sraegies Poin ou ha paern recogniion is ver helpful when maching a slope field wih is differenial equaion. For insance, in Eample par (a), ou can see ha a an given value of, he slope is he same for all values of. Similarl, in Eample par (c), a an given value of, he slope is he same for all values of. From his, ou can conclude ha he slope in par (a) is a funcion sricl in erms of and he slope in par (c) is a funcion sricl in erms of. 70

5 Secion. Slope Fields and Euler s Mehod 7 A soluion curve of a differenial equaion =F(, ) is simpl a curve in he -plane whose angen line a each poin (, ) has slope equal o F(, ). This is illusraed in Eample. Skeching a Soluion Using a Slope Field Skech a slope field for he differenial equaion = +. Use he slope field o skech he soluion ha passes hrough he poin (, ). Soluion Make a able showing he slopes a several poins. The able shown is a small sample. The slopes a man oher poins should be calculaed o ge a represenaive slope field. 0 0 = + Paul s Noes Era Eample Skech a slope field for he differenial equaion =. Use he slope field o skech he soluion ha passes hrough he poin (, ). Ne, draw line segmens a he poins wih heir respecive slopes, as shown in Figure.. Slope field for = + Paricular soluion for = + passing hrough (, ) Figure. Figure. Afer he slope field is drawn, sar a he iniial poin (, ) and move o he righ in he direcion of he line segmen. Coninue o draw he soluion curve so ha i moves parallel o he nearb line segmens. Do he same o he lef of (, ). The resuling soluion is shown in Figure.. In Eample, noe ha he slope field shows ha increases o infini as increases. Technolog Drawing a slope field b hand is edious. In pracice, slope fields are usuall drawn using a graphing uili. If ou have access o a graphing uili ha can graph slope fields, r graphing he slope field for he differenial equaion in Eample. One eample of a slope field drawn b a graphing uili is shown a he righ. Common Errors When skeching a paricular soluion, such as in Eample, sudens ma no follow he slope field correcl bu sill pass hrough he given poin. Be sure o check ha our sudens undersand how o follow a slope field. Teaching Sraegies When skeching slope fields, such as in Eample, sudens should no be overl concerned wih properl drawing each line segmen eacl o scale. I is more imporan o focus on he general characerisics of each segmen, such as seepness, wheher he slope is posiive or negaive, and, in some cases, wheher he segmen is horizonal or verical. Generaed b Maple 7

6 7 Chaper Differenial Equaions Paul s Noes Teaching Sraegies I ma be beneficial o sudens o poin ou ha because h is he change in and F is he slope, hf is he change in. So, o find he value of i, ou simpl add he change in o i. Era Eample 6 Use Euler s Mehod o approimae he paricular soluion of he differenial equaion = passing hrough he poin (0, ). Use five seps of h = 0.. n 0 n n n n n.6.0 Lesson Closer When will he slope field for d d = have horizonal line segmens? When will i have verical line segmens? Sample answer: I will have horizonal line segmens when = 0, = 0, or = ; I will have horizonal line segmens when =. Euler s Mehod Euler s Mehod is a numerical approach o approimaing he paricular soluion of he differenial equaion =F(, ) ha passes hrough he poin ( 0, 0 ). From he given informaion, ou know ha he graph of he soluion passes hrough he poin ( 0, 0 ) and has a slope of F( 0, 0 ) a his poin. This gives ou a saring poin for approimaing he soluion. From his saring poin, ou can proceed in he direcion indicaed b he slope. Using a small sep h, move along he angen line unil ou arrive a he poin (, ), where = 0 + h and = 0 + hf( 0, 0 ) as shown in Figure.. Then, using (, ) as a new saring poin, ou can repea he process o obain a second poin (, ). The values of i and i are shown below. = 0 + h = 0 + hf( 0, 0 ) = + h = + hf(, ) n = n + h n = n + hf( n, n ) When using his mehod, noe ha ou can obain beer approimaions of he eac soluion b choosing smaller and smaller sep sizes. Approimaing a Soluion Using Euler s Mehod Use Euler s Mehod o approimae he paricular soluion of he differenial equaion = passing hrough he poin (0, ). Use a sep of h = 0.. Soluion Using h = 0., 0 = 0, 0 =, and F(, ) =, ou have 0 = 0, = 0., = 0., = 0. and he firs hree approimaions are = 0 + hf( 0, 0 ) = + (0.)(0 ) = 0.9 = + hf(, ) = (0.)(0. 0.9) = 0. = + hf(, ) = 0. + (0.)(0. 0.) = 0.7. The firs en approimaions are shown in he able. You can plo hese values o see a graph of he approimae soluion, as shown in Figure.6. n n n For he differenial equaion in Eample 6, ou can verif he eac soluion o be he equaion = + e. Figure.6 compares his eac soluion wih he approimae soluion obained in Eample 6. 0 Insigh Euler s Mehod is esed onl on he AP Calculus BC Eam. 0 Figure Eac soluion curve Euler approimaion h Slope F( 0, 0 ) 0. Figure h (, ) hf( 0, 0 ) (, ) Eac soluion Approimae soluion

7 See margin. Secion. Slope Fields and Euler s Mehod 7. Eercises See CalcCha.com for uorial help and worked-ou soluions o odd-numbered eercises. Verifing a Soluion In Eercises, verif Finding a Paricular Soluion In Eercises he soluion of he differenial equaion. and 6, some of he curves corresponding o differen values of C in he general soluion Soluion Differenial Equaion of he differenial equaion are shown in. = Ce he graph. Find he paricular soluion ha Proof = passes hrough he poin shown on he graph.. = e Proof + = e. + = C =. = Ce 6. = C Proof + = 0 = e = 0 = d. ln = d = Proof (, ). = C sin C cos Proof + = 0 (0, ) 6. = C e cos + C e sin + + = 0 Proof 7. = cos ln sec + an + = an Proof. = (e + e ) Proof + = e Verifing a Paricular Soluion In Eercises 9, verif he paricular soluion of he differenial equaion. Differenial Equaion Soluion and Iniial Condiion 9. = sin cos cos + = sin Proof ( π ) = 0 0. = 6 sin + = 6 cos Proof (0) =. = e 6 = Proof (0) =. = e cos = sin Proof ( π ) = Deermining a Soluion In Eercises, deermine wheher he funcion is a soluion of he differenial equaion () 6 = 0.. = cos No a soluion. = sin Soluion. = e Soluion 6. = ln No a soluion 7. = C e + C e + C sin + C cos Soluion. = e sin Soluion Deermining a Soluion In Eercises 9, deermine wheher he funcion is a soluion of he differenial equaion = e. 9. See margin. 9. = 0. = cos. = e. = ( + e ). = ln. = e Graphs of Paricular Soluions In Eercises 7 and, he general soluion of he differenial equaion is given. Use a graphing uili o graph he paricular soluions for he given values of C. 7. See margin. 7. = 0. + = 0 = C + = C C = 0, C =±, C =± C = 0, C =, C = 9. = Ce 6 Finding a Paricular Soluion In Eercises 9, verif ha he general soluion saisfies he differenial equaion. Then find he paricular soluion ha saisfies he iniial condiion(s). + 6 = 0 = when = 0 = e = C + = 0 = when = + =. = C sin + C cos + 9 = 0 = when = π 6 = when = π 6. = C + C ln + = 0 = 0 when = = when = = sin cos = ln Assignmen Guide Es.,, 9,,, 7 odd,, 9,, 7 odd,, 0 6, 60, 6, 69 77,,,, 7, No a soluion 0. No a soluion. Soluion. Soluion. No a soluion. Soluion 7.. C = 0 C = C = C = C = 7

8 . = +. = e ( ). = + C 6. = + C 7. = ln( + ) + C. = ln( + e ) + C 9. = ln + C 0. = sin + C. = cos + C. = an + C. = ( 6) + ( 6) + C. = 6 ( + ) + C. = e + C 6. = 0e + C d d Undef d d d d d d 0 7 Chaper Differenial Equaions. See margin.. = C + C. = e (C + C ) + = = 0 = 0 when = = when = 0 = when = = 0 when = Finding a General Soluion In Eercises 6, use inegraion o find a general soluion of he differenial equaion.. d d = 6. d d = 0 7. d d = d. + d = e + e 9. d d =. d = sin d. d d = 6. d d = e 0. d = cos d d. d = an d. d = + 6. d d = e Slope Field In Eercises 7 0, a differenial equaion and is slope field are given. Complee he able b deermining he slopes (if possible) in he slope field a he given poins See margin. 7. d d = d π = cos d d d d d = d π = an d 6 6. See margin. (a) (c) Maching In Eercises, mach he differenial equaion wih is slope field. [The slope fields are labeled (a), (b), (c), and (d).] Slope Field In Eercises and 6, use he slope field for he differenial equaion o skech he graph of he soluion ha passes hrough he given poin. Then discuss he graph of he soluion as. To prin an enlarged cop of he graph, go o MahGraphs.com.. = 6. See margin., > 0; (, 0) 6. =, > 0; (0, ) 6 (b) (d). = sin b. = cos c. =e d. = a 6 Slope Field In Eercises 7 60, (a) skech he slope field for he differenial equaion, (b) use he slope field o skech he soluion ha passes hrough he given poin, and (c) discuss he graph of he soluion as and. To prin a se of coordinae aes for our graph, go o MahGraphs.com. 7. =, (, ). =, (, ) 9. =, (, ) 60. = +, (0, ) See margin. 0 d d 0. (, 0) 6. (0, ) 7. (a) and (b) (, ) 6 6 d d 0 As,. As,. (c) As,. As, See Addiional Answers beginning on page AA.

9 Secion. Slope Fields and Euler s Mehod 7 Slope Field In Eercises 6 6, use a compuer algebra ssem o (a) graph he slope field for he differenial equaion and (b) graph he soluion saisfing he specified iniial condiion See margin. 6. d d = 0. d 6. = 0.0(0 ) d (0) = (0) = 6. d d = 0.( ) 6. d d = e sin π (0) = (0) = Euler s Mehod In Eercises 6 70, use Euler s Mehod o make a able of values for he approimae soluion of he differenial equaion wih he specified iniial value. Use n seps of size h See margin. 6. = +, (0) =, n = 0, h = = +, (0) =, n = 0, h = =, (0) =, n = 0, h = =0.( ), (0) =, n =, h = = e, (0) =, n = 0, h = = cos + sin, (0) =, n = 0, h = 0. Euler s Mehod In Eercises 7 7, complee he able using he eac soluion of he differenial equaion and wo approimaions obained using Euler s Mehod o approimae he paricular soluion of he differenial equaion. Use h = 0. and h = 0., and compue each approimaion o four decimal places See margin () (eac) () (h = 0.) () (h = 0.) Differenial Iniial Eac Equaion Condiion Soluion 7. d = d (0, ) = e 7. d d = (0, ) = + 7. d d = + cos (0, 0) = (sin cos + e ) 7. Euler s Mehod Compare he values of he approimaions in Eercises 7 7 wih he values given b he eac soluion. How does he error change as h increases? As h increases (from 0. o 0.), he error increases. 7. A slope field for he differenial equaion =F(, ) consiss of small line segmens a various poins (, ) in he plane. The line segmen equals he slope =F(, ) of he soluion a he poin (, ). 7. Temperaure A ime = 0 minues, he emperaure of an objec is 0 F. The emperaure of he objec is changing a he rae given b he differenial equaion d d = ( 7). See margin. (a) Use a graphing uili and Euler s Mehod o approimae he paricular soluions of his differenial equaion a =,, and. Use a sep size of h = 0.. (A graphing uili program for Euler s Mehod is available a LarsonCalculusforAP.com.) (b) Compare our resuls wih he eac soluion = 7 + 6e. (c) Repea pars (a) and (b) using a sep size of h = 0.0. Compare he resuls. 76. HOW DO YOU SEE IT? The graph shows a soluion of one of he following differenial equaions. Deermine he correc equaion. Eplain our reasoning. See margin. (a) = (b) = (c) = (d) = See margin. WRITING ABOUT CONCEPTS 77. General and Paricular Soluions In our own words, describe he difference beween a general soluion of a differenial equaion and a paricular soluion. 7. Slope Field Eplain how o inerpre a slope field. 79. Euler s Mehod Describe how o use Euler s Mehod o approimae a paricular soluion of a differenial equaion. 0. Finding Values I is known ha = Ce k is a soluion of he differenial equaion = Is i possible o deermine C or k from he informaion given? If so, find is value.. Find he Error Describe and correc he error in deermining wheher = e is a soluion of + = 0. + = 6e ( e ) + (e ) = 6e + e + e = 6e + 6e = 0 So, = e is a soluion. should be 6e ; No a soluion 79. Begin wih a poin ( 0, 0 ) ha saisfies he iniial condiion ( 0 ) = 0. Then, using a small sep size h, calculae he poin (, ) = ( 0 + h, 0 + hf( 0, 0 )). Coninue generaing he sequence of poins ( n + h, n + hf( n, n )) or ( n+, n+ ). 0. k = 0.07; C canno be deermined. 6. (a) and (b) (a) and (b) 6. (a) and (b) 6. (a) and (b) 7. (a) () =.7 () = () = 6.9 (b) () =. () = 97.0 () = 7.79 (c) Euler s Mehod: () =.9 () = () = 6.6 Eac soluion: () =. () = 97.0 () = 7.79 The approimaions are beer using h = When = 0, = 0. Therefore, (d) is no possible. When > 0 and > 0, < 0 (decreasing funcion). Therefore, (c) is he equaion. 77. The general soluion is a famil of curves ha saisfies he differenial equaion. A paricular soluion is one member of he famil ha saisfies given condiions See Addiional Answers beginning on page AA. 7

10 76 Chaper Differenial Equaions. should be + ; + = + ; No a soluion. False; = is a soluion of = 0, bu = + is no a soluion. 6. False; The slope field could represen man differen differenial equaions, such as = (a) (b) = + π + ; 0.. Find he Error Describe and correc he error in deermining wheher = + + ( ) is a soluion of + = ( + ). See margin. + = ( + + ) + ( + + ) = = + + So, = + + is no a soluion. True or False? In Eercises 6, deermine wheher he saemen is rue or false. If i is false, eplain wh or give an eample ha shows i is false.. If = f () is a soluion of a firs-order differenial equaion, hen = f () + C is also a soluion. See margin.. The general soluion of a differenial equaion is =.9 + C + C. To find a paricular soluion, ou mus be given wo iniial condiions. True. Slope fields represen he general soluions of differenial equaions. True 6. A slope field shows ha he slope a he poin (, ) is 6. This slope field represens he famil of soluions for he differenial equaion = +. See margin. 7. Errors and Euler s Mehod The eac soluion of he differenial equaion d d = where (0) =, is = e. (a) Use a graphing uili o complee he able, where is he eac value of he soluion, is he approimae soluion using Euler s Mehod wih h = 0., is he approimae soluion using Euler s Mehod wih h = 0., e is he absolue error, e is he absolue error, and r is he raio e e See margin.. Think Abou I I is known ha = e k is a soluion of he differenial equaion 6 = 0. Find he values of k. k =± 9. Think Abou I I is known ha = A sin ω is a soluion of he differenial equaion + 6 = 0. Find he values of ω. ω =± Calculus AP Eam Preparaion Quesions 90. Muliple Choice Which differenial equaion corresponds o he slope field below? D (A) d d = ln (B) d d = (C) d d = ln (D) d d = ln 9. Muliple Choice Consider he differenial equaion = 6. Le = f () be he paricular soluion of he differenial equaion wih f (0) =. If Euler s Mehod, saring a = 0 wih hree seps of equal size, is used o approimae f (0.6), wha is he resuling approimaion? B (A). (B).96 (C). (D).6 9. Free Response Consider he differenial equaion d = ( )cos. d Le = f () be he paricular soluion of he differenial equaion wih he iniial condiion f (0) =. The funcion f is defined for all real numbers. See margin. (a) A porion of he slope field of he differenial equaion is shown below. Skech he soluion curve hrough he poin (0, ). e e r (b) Wha can ou conclude abou he raio r as h changes? (c) Predic he absolue error when h = 0.0. (b) Wrie an equaion of he angen line o he soluion curve in par (a) a he poin (π, ). Use he equaion o approimae f (.) See Addiional Answers beginning on page AA.

11 Secion. Growh and Deca 77. Growh and Deca Use separaion of variables o solve a simple differenial equaion. Use eponenial funcions o model growh and deca in applied problems. Differenial Equaions In Secion., ou learned o analze he soluions visuall of differenial equaions using slope fields and o approimae soluions numericall using Euler s Mehod. Analicall, ou have learned o solve onl wo pes of differenial equaions hose of he forms =f () and =f (). In his secion, ou will learn how o solve a more general pe of differenial equaion. The sraeg is o rewrie he equaion so ha each variable occurs on onl one side of he equaion. This sraeg is called separaion of variables. (You will sud his sraeg in deail in Secion..) Paul s Noes Secion Overview An insrucional video from Paul, including eaching sraegies for he secion, is available a LarsonCalculusforAP.com. = Solving a Differenial Equaion Original equaion = Mulipl boh sides b. d = d Inegrae wih respec o. d = d d = d = + C Appl Power Rule. = C Rewrie, leing C = C. So, he general soluion is = C. When ou inegrae boh sides of he equaion in Eample, ou do no need o add a consan of inegraion o boh sides. When ou do, ou sill obain he same resul. d = d + C = + C = + (C C ) = + C Some people prefer o use Leibniz noaion and differenials when appling separaion of variables. The soluion o Eample is shown below using his noaion. d d = d = d d = d = + C = C Eploraion Connecing Conceps You can use implici differeniaion o check he soluion o Eample. Eploraion In Eample, he general soluion of he differenial equaion is = C. Use a graphing uili o skech he paricular soluions for C =±, C =±, and C = 0. Describe he soluions graphicall. Is he following saemen rue of each soluion? The slope of he graph a he poin (, ) is equal o wice he raio of and. Eplain our reasoning. Are all curves for which his saemen is rue represened b he general soluion? Essenial Quesion How are differenial equaions used in applicaion problems, such as he eponenial growh and deca model? Tell sudens ha he will learn how o answer his quesion hrough an inroducion o separaion of variables and b eploring differen real-life scenarios in which his echnique can be applied. Lesson Moivaor This lesson will inroduce sudens o he concep of solving differenial equaions b separaion of variables, which leads o he eponenial growh and deca model. B appling hese conceps, sudens will be able o solve man real-life problems modeled b differenial equaions. Connecing Conceps Encouraging sudens o use differeniaion o check heir answers o differenial equaions will no onl help o build heir confidence, i will also reinforce he connecion beween differeniaion and is inverse process, inegraion. Era Eample Solve he differenial equaion. =( + ) = Ce The are hperbolas; Yes, = ; Yes 77

12 7 Chaper Differenial Equaions Paul s Noes Era Eample The rae of change of is proporional o. When = 0, =, and when =, =. Wha is he value of when =? Abou 07 Growh and Deca Models In man applicaions, he rae of change of a variable is proporional o he value of. When is a funcion of ime, he proporion can be wrien as shown. Rae of change of is proporional o. d d = k The general soluion of his differenial equaion is given in he ne heorem. THEOREM. Eponenial Growh and Deca Model If is a differeniable funcion of such ha > 0 and = k for some consan k, hen = Ce k where C is he iniial value of, and k is he proporionali consan. Eponenial growh occurs when k > 0, and eponenial deca occurs when k < 0. Proof = k Wrie original equaion. = k Separae variables. d = k d Inegrae wih respec o. d = k d d = d Teaching Sraegies Sudens ofen sruggle wih he las sep in he proof of he eponenial growh and deca model. I ma be beneficial o ake he ime o poin ou ha because C is a consan, C = e C is also a consan. ln = k + C = e k+c = e k e C Find aniderivaive of each side. Eponeniae each side. Proper of eponens = Ce k Le C = e C. So, all soluions of = k are of he form = Ce k. Remember ha ou can differeniae he funcion = Ce k wih respec o o verif ha = k. Using an Eponenial Growh Model The rae of change of is proporional o. When = 0, =, and when =, =. Wha is he value of when =? Soluion Because = k, ou know ha and are relaed b he equaion = Ce k. You can find he values of he consans C and k b appling he iniial condiions. = Ce 0 C = When = 0, =. 7 6 (,.67) (, ) = e 0.66 = e k k = ln 0.66 When =, =. So, he model is = e When =, he value of is e 0.66().67. (See Figure.7.) Using logarihmic properies, he value of k in Eample can also be wrien as ln. So, he model becomes = e (ln ), which can be rewrien as = ( ). (0, ) If he rae of change of is proporional o, hen follows an eponenial model. Figure.7 7

13 Secion. Growh and Deca 79 Technolog Mos graphing uiliies have curve-fiing capabiliies ha can be used o find models ha represen daa. Use he eponenial regression feaure of a graphing uili and he informaion in Eample o find a model for he daa. How does our model compare wih he given model? Radioacive deca is measured in erms of half-life he number of ears required for half of he aoms in a sample of radioacive maerial o deca. The rae of deca is proporional o he amoun presen. The half-lives of some common radioacive isoopes are lised below. Uranium ( U),70,000,000 ears Pluonium ( 9 Pu),00 ears Carbon ( C) 7 ears Radium ( 6 Ra) 99 ears Einseinium ( Es) 76 das Radon ( Rn). das Nobelium ( 7 No) seconds Paul s Noes Era Eample How long will i ake for grams of radium o deca o gram? Abou 7 ears Radioacive Deca Ten grams of he pluonium isoope 9 Pu were released in a nuclear acciden. How long will i ake for he 0 grams o deca o gram? Soluion Le represen he mass (in grams) of he pluonium. Because he rae of deca is proporional o, ou know ha = Ce k, where is he ime in ears. To find he values of he consans C and k, appl he iniial condiions. Using he fac ha = 0 when = 0, ou can wrie 0 = Ce k(0) 0 = Ce 0 which implies ha C = 0. Ne, using he fac ha he half-life of 9 Pu is,00 ears, ou have = 0 = when =,00, so ou can wrie = 0e k(,00) = e,00k,00 ln = k k. The Fukushima Daiichi nuclear disaser occurred afer an earhquake and sunami. Several of he reacors a he plan eperienced full meldowns. So, he model is = 0e Half-life model To find he ime i would ake for 0 grams o deca o gram, ou can solve for in he equaion = 0e The soluion is approimael 0,09 ears. From Eample, noice ha in an eponenial growh or deca problem, i is eas o solve for C when ou are given he value of a = 0. The ne eample demonsraes a procedure for solving for C and k when ou do no know he value of a = 0. Connecing Noaions The eponenial deca model in Eample could also be wrien as = 0( ),00. This model is much easier o derive, bu for some applicaions i is no as convenien o use. Connecing Noaions In Eample, have sudens derive he model = 0( ),00, and ask hem which form he prefer. Sudens will likel sh awa from he form = 0( ),00 because i is a bi more obscure. Foo/Gallo Images/Ge Images 79

14 0 Chaper Differenial Equaions Populaion Growh Paul s Noes Era Eample An eperimenal insec populaion increases according o he law of eponenial growh. There were 0 insecs afer he hird da of he eperimen and 00 insecs afer he fourh da. Approimael how man insecs were in he original populaion? 6 insecs See LarsonCalculusforAP.com for an ineracive version of his pe of eample. An eperimenal populaion of frui flies increases according o he law of eponenial growh. There were 00 flies afer he second da of he eperimen and 00 flies afer he fourh da. Approimael how man flies were in he original populaion? Soluion Le = Ce k be he number of flies a ime, where is measured in das. Noe ha is coninuous, whereas he number of flies is discree. Because = 00 when = and = 00 when =, ou can wrie 00 = Ce k and 00 = Ce k. From he firs equaion, ou know ha C = 00e k. Subsiuing his value ino he second equaion produces he following. 00 = 00e k e k 00 = 00e k = e k Era Eample Si monhs afer i sops adverising, a manufacuring compan noices ha is sales have dropped from 00,000 unis per monh o 60,000 unis per monh. The sales follow an eponenial paern of decline. Wha will he sales be afer anoher monhs? Abou 0,000 unis ln = k ln = k 0.9 k So, he eponenial growh model is = Ce 0.9. To solve for C, reappl he condiion = 00 when = and obain 00 = Ce 0.9() C = 00e.096 C. So, he original populaion (when = 0) consised of approimael = C = flies, as shown in Figure.. Number of frui flies Figure. = e 0.9 (, 00) (0, ) Time (in das) (, 00) Declining Sales Four monhs afer i sops adverising, a manufacuring compan noices ha is sales have dropped from 00,000 unis per monh o 0,000 unis per monh. The sales follow an eponenial paern of decline. Wha will he sales be afer anoher monhs? Soluion Use he eponenial deca model = Ce k, where is measured in monhs. From he iniial condiion ( = 0), ou know ha C = 00,000. Moreover, because = 0,000 when =, ou have 0,000 = 00,000e k 0. = e k ln(0.) = k 0.0 k. So, afer more monhs ( = 6), ou can epec he monhl sales rae o be Unis sold (in housands) (0, 00,000) (, 0,000) (6, 7,00) = 00,000e 0.0 = 00,000e 0.0(6) 7,00 unis. 6 7 Time (in monhs) See Figure.9. Figure.9 0

15 Secion. Growh and Deca In Eamples hrough, ou did no acuall have o solve he differenial equaion = k. (This was done once in he proof of Theorem..) The ne eample demonsraes a problem whose soluion involves he separaion of variables echnique. The eample concerns Newon s Law of Cooling, which saes ha he rae of change in he emperaure of an objec is proporional o he difference beween he objec s emperaure and he emperaure of he surrounding medium. Newon s Law of Cooling Le represen he emperaure (in F) of an objec in a room whose emperaure is kep a a consan 60. The objec cools from 00 o 90 in 0 minues. How much longer will i ake for he emperaure of he objec o decrease o 0? Soluion From Newon s Law of Cooling, ou know ha he rae of change in is proporional o he difference beween and 60. This can be wrien as =k( 60), To solve his differenial equaion, use separaion of variables, as shown. Paul s Noes Era Eample 6 You are making sew. When ou ake i off he sove, he emperaure of he sew is 0 F. The room emperaure is 7 F. Afer minues, he sew has cooled o F. How long will i ake o cool he sew o a serving emperaure of 9 F? Abou 0 minues ( d = k( 60) d d = k d 60) 60 d = k d ln 60 = k + C Differenial equaion Separae variables. Inegrae each side. Find aniderivaive of each side. Because > 60, 60 = 60, and ou can omi he absolue value signs. Using eponenial noaion, ou have 60 = e k+c = 60 + Ce k. C = e C Using = 00 when = 0, ou obain 00 = 60 + Ce k(0) = 60 + C which implies ha C = 0. Because = 90 when = 0, 90 = e k(0) 0 = 0e 0k k = 0 ln. So, k and he model is = e When = 0, ou obain 0 = e = 0e = e ln = Cooling model.09 minues. So, i will require abou.09 more minues for he objec o cool o a emperaure of 0. (See Figure.0.) Temperaure (in F) 0 0 (0, 00) 00 0 (0, 90) (.09, 0) 60 0 = e Time (in minues) Figure.0 Teaching Sraegies While problems such as Eample 6 are no ver common on he AP Eam, i is beneficial o emphasize o sudens ha he soluion o Eample 6 closel resembles he proof of he eponenial growh and deca model. Lesson Closer. Le d = p, where p is a d nonzero consan. Deermine wheher each funcion is a possible soluion of he differenial equaion. Eplain our reasoning. a. = e p No; Sample answer: I canno be wrien in he form = Ce p. b. = e p Yes; Sample answer: I is of he form = Ce p. c. = e p+6 Yes; Sample answer: I can be wrien as = e 6 e p, which is of he form = Ce p. d. = p + No; Sample answer: I canno be wrien in he form = Ce p. e. = p + No; Sample answer: I canno be wrien in he form = Ce p.. The rae of growh of a populaion of rabbis is modeled b he differenial equaion =. The original populaion (when = 0) consised of 0 rabbis. Approimael how man rabbis were here afer hree ears? Abou 60 rabbis

16 Chaper Differenial Equaions Assignmen Guide Es.,,, 9,,,, 9 odd,,, 6, = + + C. = + C. = Ce. = 6 Ce. = C = C 7. = Ce ( ). = Ce 9. = C( + ) 0. = 00 Ce. dq d = k ; Q = k + C. dp = k( ) d. (a). (a) P = k ( ) + C 9 (0, 0) (b) = 6 6e Eercises See CalcCha.com for uorial help and worked-ou soluions o odd-numbered eercises.. d d = +. d d = +. = Solving a Differenial Equaion In Eercises 0, solve he differenial equaion. 0. See margin. d. d = d. d = 6 6. = 7. =. = ( + ) 9. ( + ) = = 00. See margin. Wriing and Solving a Differenial Equaion In Eercises and, wrie and solve he differenial equaion ha models he verbal saemen.. The rae of change of Q wih respec o is inversel proporional o he square of.. The rae of change of P wih respec o is proporional o.. See margin. Slope Field In Eercises and, a differenial equaion, a poin, and a slope field are given. (a) Skech wo approimae soluions of he differenial equaion on he slope field, one of which passes hrough he given poin. (b) Use inegraion o find he paricular soluion of he differenial equaion and use a graphing uili o graph he soluion. Compare he resul wih he skech in par (a). To prin an enlarged cop of he graph, go o MahGraphs.com.. d = (6 ), d 9. d d = d (0, 0). d =, ( 0, ) Finding a Paricular Soluion In Eercises, find he funcion = f () passing hrough he poin (0, 0) wih he given firs derivaive. Use a graphing uili o graph he soluion.. See margin. d 6. d = 9 7. d d = d. d = Wriing and Solving a Differenial Equaion In Eercises 9 and 0, wrie and solve he differenial equaion ha models he verbal saemen. Evaluae he soluion a he specified value of he independen variable. 9. The rae of change of N is proporional o N. When = 0, N = 0, and when =, N = 00. Wha is he value of N when =? 9 0. The rae of change of P is proporional o P. When = 0, P = 000, and when =, P = 70. Wha is he value of P when =? (0, ) Finding an Eponenial Funcion In Eercises, find he eponenial funcion = Ce k ha passes hrough he wo given poins. (, ) (, ) (, ) 6.. (0, ), ), ) (, ) 6. See margin. WRITING ABOUT CONCEPTS. Describing Values Describe wha he values of C and k represen in he eponenial growh and deca model, = Ce k. 6. Eponenial Growh and Deca Give he differenial equaion ha models eponenial growh and deca. Increasing Funcion In Eercises 7 and, deermine he quadrans in which he soluion of he differenial equaion is an increasing funcion. Eplain. (Do no solve he differenial equaion.) 7. See margin. 7. d d = = e [( )ln( )] e 0.0 See margin. d. d = = e [( )ln( )] e 0.9 See margin. ) ) (0, ) (b) = e. = (0, 0) 6. = (0, 0) 7. = 0e 6 (0, 0) 0 (0, ). = ( ) e [( )ln( )] 6.7e 0.9. = Ce (ln 0) 0.000e.06 0,. See Addiional Answers beginning on page AA.

17 Radioacive Deca In Eercises 9 6, complee he able for he radioacive isoope See margin. Amoun Amoun Half-life Iniial afer afer Isoope (in ears) Quani 000 Years 0,000 Years 9. 6 Ra 99 0 g 0. 6 Ra 99. g. 6 Ra g. C 7 g. C 7 g. C 7.6 g. 9 Pu,00. g 6. 9 Pu,00 0. g 7. Radioacive Deca Radioacive radium has a half-life of approimael 99 ears. Wha percen of a given amoun remains afer 00 ears? 9.76%. Carbon Daing Carbon- daing assumes ha he carbon dioide on Earh oda has he same radioacive conen as i did cenuries ago. If his is rue, he amoun of C absorbed b a ree ha grew several cenuries ago should be he same as he amoun of C absorbed b a ree growing oda. A piece of ancien charcoal conains onl % as much of he radioacive carbon as a piece of modern charcoal. How long ago was he ree burned o make he ancien charcoal? (The half-life of C is 7 ears.),6. ears Compound Ineres In Eercises 9, complee he able for a savings accoun in which ineres is compounded coninuousl. 9. See margin. Iniial Annual Time o Amoun afer Invesmen Rae Double 0 Years 9. $000 % 0. $,000 %. $70 7 r. $,00 0 r. $00 $9.. $6000 $90.9 Compound Ineres In Eercises, find he principal P ha mus be invesed a rae r, compounded monhl, so ha $,000,000 will be available for reiremen in ears.. See margin.. r = 7 %, = 0 6. r = 6%, = 0 7. r = %, =. r = 9%, = Compound Ineres In Eercises 9 and 0, find he ime necessar for $000 o double when i is invesed a a rae of r compounded (a) annuall, (b) monhl, (c) dail, and (d) coninuousl See margin. 9. r = 7% 0. r =.% 6. (a) baceria (b) = 6 e[( )ln( )].6e 0. (c) 7 baceria (d).9 hours Secion. Growh and Deca Populaion In Eercises, he populaion (in millions) of a counr in 0 and he epeced annual growh rae k of he populaion are given. (Source: U.S. Census Bureau, Inernaional Daa Base). See margin. (a) Find he eponenial growh model P = Ce k for he populaion b leing = 0 correspond o 00. (b) Use he model o predic he populaion of he counr in 0. (c) Discuss he relaionship beween he sign of k and he change in populaion for he counr. Counr 0 Populaion k. Lavia Egp Uganda Hungar Modeling Daa One hundred baceria are sared in a culure and he number N of baceria is couned each hour for hours. The resuls are shown in he able, where is he ime in hours. See margin. 0 N (a) Use he regression capabiliies of a graphing uili o find an eponenial model for he daa. (b) Use he model o esimae he ime required for he populaion o quadruple in size. 6. Baceria Growh The number of baceria in a culure is increasing according o he law of eponenial growh. There are baceria in he culure afer hours and 0 baceria afer hours. See margin. (a) Find he iniial populaion. (b) Wrie an eponenial growh model for he baceria populaion. Le represen he ime in hours. (c) Use he model o deermine he number of baceria afer hours. (d) Afer how man hours will he baceria coun be,000? 7. Learning Curve The managemen a a cerain facor has found ha a worker can produce a mos 0 unis in a da. The learning curve for he number of unis N produced per da afer a new emploee has worked das is N = 0( e k ). Afer 0 das on he job, a paricular worker produces 9 unis. See margin. (a) Find he learning curve for his worker. (b) How man das should pass before his worker is producing unis per da? 7. (a) N 0( e 0.00 ) (b) 6 das 9. Amoun afer 000 r:.96 g Amoun afer 0,000 r: 0.6 g 0. Iniial quani:. g Amoun afer 0,000 r: 0.0 g. Iniial quani: 7.6 g Amoun afer 000 r:.9 g. Iniial quani: 0.09 g Amoun afer 000 r:.9 g. Amoun afer 000 r:. g Amoun afer 0,000 r:.9 g. Iniial quani:.06 g Amoun afer 0,000 r: 0. g. Iniial quani:.6 g Amoun afer 0,000 r:.6 g 6. Iniial quani: 0. g Amoun afer 000 r: 0. g 9. Time o double:.7 r Amoun afer 0 r: $0. 0. Time o double:.6 r Amoun afer 0 r: $,9.. Annual rae:.9% Amoun afer 0 r: $.67. Annual rae:.7% Amoun afer 0 r: $7,67.. Annual rae: 9.0% Time o double: 7.0 r. Annual rae: % Time o double: 7. r. $,7. 6. $9, $6,77.7. $06,7. 9. (a) 0. r (b) 9.9 r (c) 9.90 r (d) 9.90 r 0. (a).9 r (b).6 r (c).60 r (d).60 r. (a) P =.7e (b).07 million (c) Because k < 0, he populaion is increasing.. (a) P = 0.e 0.0 (b) 0.9 million (c) Because k > 0, he populaion is increasing.. (a) P =.6e 0.0 (b).0 million (c) Because k > 0, he populaion is increasing.. (a) P = 0e 0.00 (b) 9.70 million (c) Because k < 0, he populaion is decreasing.. (a) N = 00.96(.) (b) 6. h

18 . (a) Boh funcions represen eponenial growh because he graphs are increasing. (b) g has a greaer k value because is graph is increasing a a greaer rae han he graph of f. 9. (a) Because he populaion increases b a consan each monh, he rae of change from monh o monh will alwas be he same. So, he slope is consan, and he model is linear. (b) Alhough he percenage increase is consan each monh, he rae of growh is no consan. The rae of change of is d d = r, which is an eponenial model. 6. 9% F Chaper Differenial Equaions. HOW DO YOU SEE IT? The funcions f and g are boh of he form = Ce k. See margin. (a) Do he funcions f and g represen eponenial growh 6 g or eponenial deca? Eplain. (b) Assume boh funcions have he f same value of C. 6 Which funcion has a greaer value of k? Eplain. 9. Insec Populaion (a) Suppose an insec populaion increases b a consan number each monh. Eplain wh he number of insecs can be represened b a linear funcion. (b) Suppose an insec populaion increases b a consan percenage each monh. Eplain wh he number of insecs can be represened b an eponenial funcion. 60. FORESTRY See margin. The value of a rac of imber is V() = 00,000e 0., where is he ime in ears, wih = 0 corresponding o 00. If mone earns ineres coninuousl a 0%, hen he presen value of he imber a an ime is A() = V()e 0.0. Find he ear in which he imber should be harvesed o maimize he presen value funcion Sound Inensi The level of sound β (in decibels) wih an inensi of I is β(i) = 0 log 0( I I 0 ) where I 0 is an inensi of 0 6 wa per square cenimeer, corresponding roughl o he faines sound ha can be heard. Deermine β(i) for he following. (a) I = 0 wa per square cenimeer (whisper) 0 db (b) I = 0 9 wa per square cenimeer (bus sree corner) 70 db 9 db (c) I = 0 6. wa per square cenimeer (air hammer) (d) I = 0 wa per square cenimeer (hreshold of pain) 0 db 6 6. See margin. 6. Noise Level Wih he insallaion of noise suppression maerials, he noise level in an audiorium was reduced from 9 o 0 decibels. Use he funcion in Eercise 6 o find he percen decrease in he inensi level of he noise as a resul of he insallaion of hese maerials. 6. Newon s Law of Cooling When an objec is removed from a furnace and placed in an environmen wih a consan emperaure of 0 F, is core emperaure is 00 F. One hour afer i is removed, he core emperaure is 0 F. Find he core emperaure hours afer he objec is removed from he furnace. 6. Newon s Law of Cooling A conainer of ho liquid is placed in a freezer ha is kep a a consan emperaure of 0 F. The iniial emperaure of he liquid is 60 F. Afer minues, he liquid s emperaure is 60 F. How much longer will i ake for is emperaure o decrease o 0 F?. minues longer Calculus AP Eam Preparaion Quesions 6. Muliple Choice Which saemen corresponds o he differenial equaion dp d = k ()? A (A) The rae of change of P is inversel proporional o boh and. (B) The rae of change of P is proporional o and inversel proporional o. (C) The rae of change of P is proporional o boh and. (D) The rae of change of P is proporional o and inversel proporional o. 66. Muliple Choice Which of he following is he soluion of he differenial equaion d d = where () =? A (A) = 6 for > (B) = 6 for > (C) = for > (D) = for 67. Muliple Choice Le = f () be a soluion of he differenial equaion d d = k, where k is a consan. Values of f for seleced values of are given in he able below. Which of he following is an epression for f ()? D 0 f () 0 (A) + (B) + (C) e ( )ln + (D) e ( )ln Sephen Aaron Rees/Shuersock.com

19 Secion. Separaion of Variables. Separaion of Variables Recognize and solve differenial equaions ha can be solved b separaion of variables. Use differenial equaions o model and solve applied problems. Separaion of Variables Consider a differenial equaion ha can be wrien in he form M() + N() d d = 0 where M is a coninuous funcion of alone and N is a coninuous funcion of alone. As ou saw in Secion., for his pe of equaion, all erms can be colleced wih d and all erms wih d, and a soluion can be obained b inegraion. Such equaions are said o be separable, and he soluion procedure is called separaion of variables. Below are some eamples of differenial equaions ha are separable. Original Differenial Equaion Rewrien wih Variables Separaed d d = ( + + ) d = d + d d = 0 d = d (sin ) =cos d = co d e + = e + d = d Separaion of Variables See LarsonCalculusforAP.com for an ineracive version of his pe of eample. Find he general soluion of ( + ) d d =. Soluion To begin, noe ha = 0 is a soluion. To find oher soluions, assume ha 0 and separae variables as shown. ( + ) d = d Differenial form d = + d Now, inegrae o obain d = + d ln = ln( + ) + C ln = ln + + C = e C + =±e C +. Separae variables. Inegrae. Because = 0 is also a soluion, ou can wrie he general soluion as = C +. General soluion Insigh Separable differenial equaions appear on boh he AP Calculus AB and BC Eams. Connecing Conceps Be sure o check our soluions hroughou his chaper. In Eample, ou can check he soluion = C + b differeniaing and subsiuing ino he original equaion. ( + ) d d = ( + ) C + =? (C + ) C + = C + So, he soluion checks. Paul s Noes Secion Overview An insrucional video from Paul, including eaching sraegies for he secion, is available a LarsonCalculusforAP.com. Essenial Quesion How do ou solve separable differenial equaions? Tell sudens ha he will learn how o answer his quesion b using separaion of variables. Lesson Moivaor An undersanding of separaion of variables will allow sudens o find he general and paricular soluions for various differenial equaions. This lesson also eplores a number of real-life applicaions ha involve he use of separaion of variables. Connecing Conceps Using differeniaion o check heir answers allows sudens o see how he consan C fis ino he general soluion. Era Eample Find he general soluion of ( cos ) d = sec. d = an + C

20 6 Chaper Differenial Equaions Paul s Noes Era Eample Given he iniial condiion () =, find he paricular soluion of he equaion = 0. = ln + Era Eample Find he equaion of he curve ha passes hrough he poin (, ) and has a slope of a an poin (, ). = + In some cases, i is no feasible o wrie he general soluion in he eplici form = f (). The ne eample illusraes such a soluion. Implici differeniaion can be used o verif his soluion. Finding a Paricular Soluion Given he iniial condiion (0) =, find he paricular soluion of he equaion d + e ( ) d = 0. Soluion Noe ha = 0 is a soluion of he differenial equaion bu his soluion does no saisf he iniial condiion. So, ou can assume ha 0. To separae variables, ou mus rid he firs erm of and he second erm of e. So, ou should mulipl b e and obain he following. d + e ( ) d = 0 e ( ) d = d ( ) d = e d ln = e + C From he iniial condiion (0) =, ou have 0 = + C which implies ha C =. So, he paricular soluion has he implici form ln = e + ln + e =. You can check his b differeniaing and rewriing o ge he original equaion. Teaching Sraegies Sudens ofen have difficul deermining a which sep o subsiue he iniial condiion ino he general soluion. Assure sudens ha he can do his a an poin afer he have found he general soluion. As long as heir algebra is correc, he should arrive a he same paricular soluion. For insance, in Eample, subsiuing he iniial condiion (, ) ino he equaion ln = + C resuls in a consan of C = ln +. Using he eponenial properies, ou can wrie ln = + ln + as Finding a Paricular Soluion Curve Find he equaion of he curve ha passes hrough he poin (, ) and has a slope of a an poin (, ). Soluion Because he slope of he curve is, ou have d d = wih he iniial condiion () =. Separaing variables and inegraing produces d = d, 0 ln = + C = e ( )+C = Ce. Because = when =, i follows ha = Ce and C = e. So, he equaion of he specified curve is = (e)e = e ( ), > 0. Because he soluion is no defined a = 0 and he iniial condiion is given a =, is resriced o posiive values. See Figure (, ) = e = e ( )/ 6 0 Figure. = e ( ). 6

21 Secion. Separaion of Variables 7 Applicaions Wildlife Populaion The rae of change of he number of cooes N() in a populaion is direcl proporional o 60 N(), where is he ime in ears. When = 0, he populaion is 00, and when =, he populaion has increased o 00. Find he populaion when =. Soluion Because he rae of change of he populaion is proporional o 60 N(), or 60 N, ou can wrie he differenial equaion dn = k(60 N). d You can solve his equaion using separaion of variables. dn = k(60 N) d Differenial form dn = k d Separae variables. 60 N ln 60 N = k + C ln Inegrae. 60 N = k C 60 N = e k C Assume N < 60. N = 60 Ce k General soluion Using N = 00 when = 0, ou can conclude ha C = 0, which produces N = 60 0e k. Then, using N = 00 when =, i follows ha 00 = 60 0e k e k = k So, he model for he cooe populaion is N = 60 0e 0.6. Model for populaion When =, ou can approimae he populaion o be N = 60 0e 0.6() cooes. The model for he populaion is shown in Figure.. Noe ha N = 60 is he horizonal asmpoe of he graph and is he carring capaci of he model. You will learn more abou carring capaci in Secion.. N Algebra Review For help wih he algebra in solving for C in Eample, see Eample in he Chaper Algebra Review on page A. For help wih he algebra in solving for k in Eample, see Eample (a) in he Chaper Algebra Review on page A. Paul s Noes Common Errors Sudens ofen do no ake he ime o properl read applicaion problems. For insance, in Eample, sudens migh glance pas he phrase direcl proporional o, resuling in he use of he differenial equaion dn = 60 N insead of he d correc differenial equaion dn = k(60 N). d Era Eample The rae of change in he number of foes N() in a populaion is direcl proporional o 0 N(), where is he ime in ears. When = 0, he populaion is 0, and when =, he populaion has increased o 00. Find he populaion when =. Abou foes Number of cooes (0, 00) Figure. Derek R. Audee/Shuersock.com (, 00) (, ) N = 60 0e Time (in ears) 7

22 Chaper Differenial Equaions Paul s Noes Era Eample A new produc is inroduced hrough an adverising campaign o a populaion of. million poenial cusomers. The rae a which he populaion hears abou he produc is assumed o be proporional o he number of people who are no e aware of he produc. B he end of 6 monhs, 7,000 people have heard of he produc. How man will have heard of i b he end of ear? Abou 660,000 people Modeling Adverising Awareness A new cereal produc is inroduced hrough an adverising campaign o a populaion of million poenial cusomers. The rae a which he populaion hears abou he produc is assumed o be proporional o he number of people who are no e aware of he produc. B he end of ear, half of he populaion has heard of he produc. How man will have heard of i b he end of ears? Soluion Le be he number of people (in millions) a ime who have heard of he produc. This means ha ( ) is he number of people (in millions) who have no heard of i, and d d is he rae a which he populaion hears abou he produc. From he given assumpion, ou can wrie he differenial equaion as shown. Rae of change of d = k( ) d is proporional o he difference beween and. You can solve his equaion using separaion of variables. d = k( ) d Differenial form d = k d Separae variables. ln = k + C Inegrae. ln = k C Mulipl each side b. = e k C Assume <. = Ce k General soluion To solve for he consans C and k, use he iniial condiions. Tha is, because = 0 when = 0, ou can deermine ha C =. Similarl, because = 0. when =, i follows ha 0. = e k, which implies ha k = ln So, he paricular soluion is = e Paricular soluion This model is shown in Figure.. Using he model, ou can deermine ha he number of people who have heard of he produc afer ears is = e 0.69() 0.7 or 70,000 people. Algebra Review For help wih he algebra in solving for C in Eample, see Eample in he Chaper Algebra Review on page A. For help wih he algebra in solving for k in Eample, see Eample (b) in he Chaper Algebra Review on page A. Poenial cusomers (in millions) (0, 0) Figure. (, 0.0) (, 0.7) = e 0.69 Time (in ears)

23 Secion. Separaion of Variables 9 Modeling a Chemical Reacion During a chemical reacion, subsance A is convered ino subsance B a a rae ha is proporional o he square of he amoun of A. When = 0, 60 grams of A is presen, and afer hour ( = ), onl 0 grams of A remains unconvered. How much of A is presen afer hours? Soluion Le be he amoun of unconvered subsance A a an ime. From he given assumpion abou he conversion rae, ou can wrie he differenial equaion as shown. Rae of change of d d = k is proporional o he square of. You can solve his equaion using separaion of variables. d = k d d = k d Differenial form Separae variables. Paul s Noes Era Eample 6 During a chemical reacion, subsance A is convered ino subsance B a a rae ha is proporional o he square of he amoun of A. When = 0, 0 grams of A is presen, and afer hour ( = ), onl grams of A remain. How much of A is presen afer hours? Abou 0. grams = k + C Inegrae. = General soluion k + C To solve for he consans C and k, use he iniial condiions. Tha is, because = 60 when = 0, ou can deermine ha C = 60. Similarl, because = 0 when =, i follows ha 0 = k ( 60) which implies ha k =. So, he paricular soluion is = Subsiue for k and C. ( ) ( 60) = Paricular soluion Using he model, ou can deermine ha he unconvered amoun of subsance A afer hours is = 60 () +. grams. In Figure., noe ha he chemical conversion is occurring rapidl during he firs hour. Then, as more and more of subsance A is convered, he conversion rae slows down. Amoun (in grams) 60 (0, 60) = (, 0) 0 (,.) Figure. Time (in hours) Eploraion In Eample 6, he rae of conversion was assumed o be proporional o he square of he unconvered amoun. How does he resul change when he conversion is proporional o he unconvered amoun? Eploraion If he rae of conversion is proporional o he unconvered amoun, hen d = k, which leads o d he model = Ce k. Using he iniial condiions given in Eample 6 gives C = 60 and k = ln 6. Subsiuing = ino = 60e ( ln 6) gives =.67 grams. 9

24 Paul s Noes Era Eample 7 A populaion of 0 bobcas has been inroduced ino a naional park. The fores service esimaes ha he maimum populaion he park can susain is 0 bobcas. Afer ears, he populaion is esimaed o be bobcas. According o a Gomperz growh model, how man bobcas will here be ears afer heir inroducion? Abou 7 bobcas Lesson Closer. Lis he seps necessar o find he paricular soluion of a separable differenial equaion. Sample answer: Collec all erms wih d and all erms wih d, inegrae each side of he equaion, solve for, and subsiue he iniial condiion(s) ino he general soluion o find missing consan(s).. Find he general soluion of he equaion d d = e. = ln e + C 90 Chaper Differenial Equaions The ne eample describes a growh model called a Gomperz growh model. This model assumes ha he rae of change of is proporional o he produc of and he naural log of L, where L is he populaion limi. Modeling Populaion Growh A populaion of 0 wolves has been inroduced ino a naional park. The fores service esimaes ha he maimum populaion he park can susain is 00 wolves. Afer ears, he populaion is esimaed o be 0 wolves. According o a Gomperz growh model, how man wolves will here be 0 ears afer heir inroducion? Soluion Le be he number of wolves a an ime. From he given assumpion abou he rae of growh of he populaion, ou can wrie he differenial equaion as shown. Rae of change of is proporional o d 00 = k ln d he produc of and he naural log of he raio of 00 and. Using separaion of variables or a compuer algebra ssem, ou can find he general soluion o be = 00e Ce k. General soluion To solve for he consans C and k, use he iniial condiions. Tha is, because = 0 when = 0, ou can deermine ha C = ln Similarl, because = 0 when =, i follows ha 0 = 00e.06e k which implies ha k 0.9. So, he paricular soluion is = 00e.06e 0.9. Paricular soluion Using he model, ou can esimae he wolf populaion afer 0 ears o be = 00e.06e 0.9(0) 00 wolves. In Figure., noe ha afer 0 ears he populaion has reached abou half of he esimaed maimum populaion. Tr checking he growh model o see ha i ields = 0 when = 0 and = 0 when =. Number of wolves Figure. = 00e.06e 0.9 (, 0) (0, 0) (0, 00) 6 0 Time (in ears) Algebra Review For help wih he algebra in solving for C in Eample 7, see Eample in he Chaper Algebra Review on page A. For help wih he algebra in solving for k in Eample 7, see Eample (c) in he Chaper Algebra Review on page A. Technolog If ou have access o a compuer algebra ssem, r using i o find he general soluion and he paricular soluion o Eample 7. 90

25 Secion. Separaion of Variables 9. Eercises See CalcCha.com for uorial help and worked-ou soluions o odd-numbered eercises.. dr ds = 0.7r. d d = Finding a General Soluion Using Separaion of Variables In Eercises, find he general soluion of he differenial equaion.. See margin. dr. ds = 0.7s. d d =. d d = 6. d d = 6 7. ( + ) =. = 9. = sin 0. = cos π. =. 6 =. ln = 0. 7e = 0 Finding a Paricular Soluion Using Separaion of Variables In Eercises, find he paricular soluion ha saisfies he iniial condiion. Differenial Equaion. See margin. Iniial Condiion. e = 0 (0) = =0 () = 9 7. ( + ) + =0 ( ) =. ln = 0 () = 9. ( + ) ( + ) = 0 (0) = 0. = 0 (0) =. du dv = uv sin v u(0) =. dr ds = er s r(0) = 0. dp kp d = 0 P(0) = P 0. dt + k(t 70) d = 0 T(0) = 0 Finding a Paricular Soluion In Eercises, find an equaion of he graph ha passes hrough he poin and has he given slope.. See margin.. (0, ), = 7. (9, ), = 6. (, ), = 9 6. (, ), = Using Slope In Eercises 9 and 0, find all funcions f having he indicaed proper. 9. The angen o he graph of f a he poin (, ) inersecs he -ais a ( +, 0). = Ce 0. All angens o he graph of f pass hrough he origin. = C Slope Field In Eercises, skech a few soluions of he differenial equaion on he slope field and hen find he general soluion analicall. To prin an enlarged cop of he graph, go o MahGraphs.com.. d d =. d d = d. d =. See margin. d. = 0.( ) d Euler s Mehod In Eercises, (a) use Euler s Mehod wih a sep size of h = 0. o approimae he paricular soluion of he iniial value problem a he given -value, (b) find he eac soluion of he differenial equaion analicall, and (c) compare he soluions a he given -value.. See margin. Differenial Equaion Iniial Condiion -Value. d = 6 d (0, ) = 6. d d + 6 = 0 (0, ) = 7. d + = d (, ) =. d d = ( + ) (, 0) =. 9. Radioacive Deca The rae of decomposiion of radioacive radium is proporional o he amoun presen a an ime. The half-life of radioacive radium is 99 ears. Wha percen of a presen amoun will remain afer 0 ears? 97.9% of he original amoun 0. Chemical Reacion In a chemical reacion, a cerain compound changes ino anoher compound a a rae proporional o he unchanged amoun. There is 0 grams of he original compound iniiall and grams afer hour. When will 7 percen of he compound be changed? 0. hours Assignmen Guide Es.,,, 7, 0,, 7, 6,,,, 7,,, 60, 6 6, 6, r = Ce 0.7s. r = 0.7s + C. = C. = C. + = C = C 7. = C( + ). = C 9. = C cos 0. = 6 sin π + C π. = + C. = 6 + C. = Ce (ln ). 6 = 7e + C. = e = 7. = e ( +). = (ln ) + 9. = + 0. =. u = e ( cos v ). r = ln ( + e s). P = P 0 e k. T = 70( + e k ). = = 7. =. =. Graphs will var.. Graphs will var. = + Ce. Graphs will var. = + Ce ( ). (a) = 0.60 (b) = e (c) = (a) = 0.6 (b) = 9 + (c) = (a).0 (b) = + (c) =. (a).770 (b) = an( ) (c).0096 = + C. Graphs will var. + = C 9

26 . (a) d = k( ) d (b) a (c) Proof. (a) d = k( ) d (b) b (c) Proof. (a) d = k( ) d (b) c (c) Proof. (a) d d = k (b) d (c) Proof. (a) w = 00 0e k (b) w = 00 0e w = 00 0e w = 00 0e (c). r;.6 r;.0 r (d) 00 lb N = + e 0.. ds = ks(l S); d 0L S = 0 + (L 0)e Lk. = Chaper Differenial Equaions Slope Field In Eercises, (a) wrie a differenial equaion for he saemen, (b) mach he differenial equaion wih a possible slope field, and (c) verif our resul b using a graphing uili o graph a slope field for he differenial equaion. [The slope fields are labeled (a), (b), (c), and (d).] To prin an enlarged cop of he graph, go o MahGraphs.com.. See margin. (a) (b) (c) 9 9 (d)... The rae of change of wih respec o is proporional o he difference beween and.. The rae of change of wih respec o is proporional o he difference beween and.. The rae of change of wih respec o is proporional o he produc of and he difference beween and.. The rae of change of wih respec o is proporional o.. Weigh Gain A calf ha weighs 60 pounds a birh gains weigh a he rae dw = k(00 w) d where w is weigh in pounds and is ime in ears. (a) Solve he differenial equaion. (b) Use a graphing uili o graph he paricular soluions for k = 0., 0.9, and. (c) The animal is sold when is weigh reaches 00 pounds. Find he ime of sale for each of he models in par (b). (d) Wha is he maimum weigh of he animal for each of he models in par (b)? 6. Weigh Gain A calf ha weighs w 0 pounds a birh gains weigh a he rae dw d = 00 w See margin. where w is weigh in pounds and is ime in ears. Solve he differenial equaion. w = 00 (00 w 0 )e 9 7. Biolog A an ime, he rae of growh of he populaion N of deer in a sae park is proporional o he produc of N and L N, where L = 00 is he maimum number of deer he park can susain. When = 0, N = 00, and when =, N = 00. Wrie N as a funcion of. See margin.. Sales Growh The rae of change in sales S (in housands of unis) of a new produc is proporional o he produc of S and L S, where L (in housands of unis) is he esimaed maimum level of sales. When = 0, S = 0. Wrie and solve he differenial equaion for his sales model. See margin. Adverising Awareness In Eercises 9 and 0, use he adverising awareness model described in Eample o find he number of people (in millions) aware of he produc as a funcion of ime (in ears). 9. = 0 when = 0; = 0.7 when = 0. = 0 when = 0; = 0.9 when = e.6 e. Chemical Reacion In Eercises and, use he chemical reacion model given in Eample 6 o find he amoun as a funcion of, and use a graphing uili o graph he funcion.. See margin.. = grams when = 0; = grams when =. = 7 grams when = 0; = grams when = Using a Gomperz Growh Model In Eercises and, use he Gomperz growh model described on page 90 o find he growh funcion, and skech is graph.. See margin.. L = 00; = 00 when = 0; = 0 when =. L = 000; = 00 when = 0; = 6 when = 6. See margin.. Biolog A populaion of eigh beavers has been inroduced ino a new welands area. Biologiss esimae ha he maimum populaion he welands can susain is 60 beavers. Afer ears, he populaion is beavers. According o a Gomperz growh model, how man beavers will be presen in he welands afer 0 ears? 6. Biolog A populaion of 0 rabbis has been inroduced ino a new region. I is esimaed ha he maimum populaion he region can susain is 00 rabbis. Afer ear, he populaion is esimaed o be 90 rabbis. According o a Gomperz growh model, how man rabbis will be presen afer ears? 7. Chemical Miure A 00-gallon ank is full of a soluion con aining pounds of a concenrae. Saring a ime = 0, disilled waer is admied o he ank a he rae of gallons per minue, and he well-sirred soluion is wihdrawn a he same rae. See margin. (a) Find he amoun Q of he concenrae in he soluion as a funcion of. (Hin: Q +Q 0 = 0) (b) Find he ime when he amoun of concenrae in he ank reaches pounds = = = 00e.609e = 000e.06e beavers 6. rabbis 7. (a) Q = e ( 0) (b) 0. min 9

27 See margin.. Chemical Miure A 00-gallon ank is half full of disilled waer. A ime = 0, a soluion conaining 0. pound of concenrae per gallon eners he ank a he rae of gallons per minue, and he well-sirred miure is wihdrawn a he same rae. Find he amoun Q of concenrae in he ank afer 0 minues. ( Hin: Q + Q 0 = ) 9. Chemical Reacion In a chemical reacion, a compound changes ino anoher compound a a rae proporional o he unchanged amoun, according o he model d d = k. (a) Solve he differenial equaion. = Ce k (b) The iniial amoun of he original compound is 0 grams, and he amoun remaining afer hour is 6 grams. When will 7% of he compound have been changed? Abou 6. h 60. Snow Removal The rae of change in he number of miles s of road cleared per hour b a snowplow is inversel proporional o he deph h of snow. Tha is, ds dh = k h. ln(h ) s =, h ln Find s as a funcion of h given ha s = miles when h = inches and s = miles when h = 6 inches ( h ). 6. Chemisr A we owel hung from a clohesline o dr loses moisure hrough evaporaion a a rae proporional o is moisure conen. Afer hour, he owel has los 0% of is original moisure conen. Afer how long will i have los 0%? Abou. h 6. Biolog Le and be he sizes of wo inernal organs of a paricular mammal a ime. Empirical daa indicae ha he relaive growh raes of hese wo organs are equal, and can be modeled b d d = d d. = C Use his differenial equaion o wrie as a funcion of. 6. Invesmen A large corporaion sars a ime = 0 o inves par of is receips a a rae of P dollars per ear in a fund for fuure corporae epansion. The fund earns r percen ineres per ear compounded coninuousl. The rae of growh of he amoun A in he fund is given b da d = ra + P where A = 0 when = 0. Solve his differenial equaion for A as a funcion of. A = P r (er ) Web Picure Blog/Shuersock.com. lb gal Secion. Separaion of Variables 9 Invesmen In Eercises 6 66, use he resul of Eercise See margin. 6. Find A for P = $7,000, r = %, and = 0 ears. 6. The corporaion needs $60,000,000 in ears and he fund earns 7 % ineres compounded coninuousl. Find P. 66. The corporaion needs $,000,000 and i can inves $,000 per ear in a fund earning % ineres compounded coninuousl. Find. 67. Using a Gomperz Growh Model Use he Gomperz growh model described in Eample 7. See margin. (a) Use a graphing uili o graph he slope field for he growh model when k = 0.0 and L = 000. (b) Describe he behavior of he graph as. (c) Solve he growh model for L = 000, 0 = 00, and k = 0.0. (d) Graph he equaion ou found in par (c). Deermine he concavi of he graph. WRITING ABOUT CONCEPTS 6. Separaion of Variables In our own words, describe how o recognize and solve differenial equaions ha can be solved b separaion of variables. Answers will var. Separaion of Variables In Eercises 69 7, deermine wheher he differenial equaion is separable. If he equaion is separable, rewrie i in he form N() d = M() d. (Do no solve he differenial equaion.) See margin. 69. ( + ) d + d = = 7. + = 7. = + 7. SAILING Ignoring resisance, a sailboa saring from res acceleraes (dv d) a a rae proporional o he difference beween he velociies of he wind and he boa. (a) The wind is blowing a 0 knos, and afer half-hour, he boa is moving a 0 knos. Wrie he veloci v as a funcion of ime. v = 0( e.6 ) (b) Use he resul of par (a) o wrie he disance raveled b he boa as a funcion of ime. s 0 +.(e.6 ) 6. $,, $,9, r 67. (a) (b) As, L. (c) = 000e.0e 0.0 (d) The graph is concave upward on (0,.7) and concave downward on (.7, ). 69. Separable; + d = d 70. Separable; d = d 7. No separable 7. Separable; d = ( + ) d 9

28 9 Chaper Differenial Equaions 7. For graph (a), he -inercep is (0, 6), so C =. For graph (b), he -inercep is (0, ), so C =. For graph (c), he -inercep is (0, ), so C =. For graph (d), he -inercep is (0, ), so C = False; = is separable, bu = 0 is no a soluion. 7. HOW DO YOU SEE IT? Recall from Eample ha he general soluion of ( + ) d d = is = C +. The graphs below show he paricular soluions for C = 0.,,, and. Mach he value of C wih each graph. Eplain our reasoning. See margin. 9 7 True or False? In Eercises 7 and 76, deermine wheher he saemen is rue or false. If i is false, eplain wh or give an eample ha shows i is false. 7. The funcion = 0 is alwas a soluion of a differenial equaion ha can be solved b separaion of variables. 76. The differenial equaion See margin. = + can be wrien in separaed variables form. True a b c d Calculus AP Eam Preparaion Quesions 77. Muliple Choice If d d = + +, hen = A (A) an[ln( + ) + C]. (B) an[ln( + )] + C. (C) ln( + ) + C. (D) an ( ln + + C ). 7. Muliple Choice If d d = sec and = when = 0, hen = B (A) e an +. (B) e an. (C) e an +. (D) an Free Response Consider he differenial equaion d d =. (a) Find he paricular soluion = f () o he differenial equaion wih he iniial condiion f () = 0. = e + + (b) For he paricular soluion = f () described in par (a), find lim f (). e + Orhogonal Trajecories A common problem in elecrosaics, hermodnamics, and hdrodnamics involves finding a famil of curves, each of which is orhogonal o all members of a given famil of curves. For eample, he figure shows a famil of circles + = C, each of which inersecs he lines in he famil = K a righ angles. Two such families of curves are said o be muuall orhogonal, and each curve in one of he families is called an orhogonal rajecor of he oher famil. In elecrosaics, lines of force are orhogonal o he equipoenial curves. In hermodnamics, he flow of hea across a plane surface is orhogonal o he isohermal curves. In hdrodnamics, he flow (sream) lines are orhogonal rajecories of he veloci poenial curves. In (a) (f ), find he orhogonal rajecories of each famil. Use a graphing uili o graph several members of each famil. (a) = Ce (b) = C (c) = C (d) = C (e) = C Each line = K is an orhogonal rajecor of he famil of circles. (f ) = C A worked-ou soluion o he Secion Projec can be found in he Teacher s Resource Manual. 9

29 Secion. The Logisic Equaion 9. The Logisic Equaion Solve and analze logisic differenial equaions. Use logisic differenial equaions o model and solve applied problems. Logisic Differenial Equaion In Secion., he eponenial growh model was derived from he fac ha he rae of change of a variable is proporional o he value of. You observed ha he differenial equaion d d = k has he general soluion = Ce k. Eponenial growh is unlimied, bu when describing a populaion, here ofen eiss some upper limi L pas which growh canno occur. This upper limi L is called he carring capaci, which is he maimum populaion () ha can be susained or suppored as ime increases. A model ha is ofen used o describe his pe of growh is he logisic differenial equaion d d = k ( L) Logisic differenial equaion where k and L are posiive consans. A populaion ha saisfies his equaion does no grow wihou bound, bu approaches he carring capaci L as increases. From he equaion, ou can see ha if is beween 0 and he carring capaci L, hen d d > 0, and he populaion increases. If is greaer han L, hen d d < 0, and he populaion decreases. The general soluion of he logisic differenial equaion is derived in he ne eample. Deriving he General Soluion Solve he logisic differenial equaion d d = k ( L). Soluion Begin b separaing variables. d d = k ( L) Wrie differenial equaion. d = k d Separae variables. ( L) ( L) d = k d Inegrae each side. ( + L ) d = k d Rewrie lef side using parial fracions. ln ln L = k + C Find aniderivaive of each side. ln L = k C Mulipl each side b and simplif. L = e k C Eponeniae each side. L = e C e k Proper of eponens L = be k Le ±e C = b. L Solving his equaion for produces he general soluion = + be. k Insigh Logisic differenial equaions appear onl on he AP Calculus BC Eam. On he free-response secion, ou ma be asked o solve a logisic differenial equaion b separaing variables. You ma also be asked o find he carring capaci and he inflecion poin (if an), and hen eplain wha he represen in he cone of he problem. Remark A review of he mehod of parial fracions is given in Secion 7.. Paul s Noes Secion Overview An insrucional video from Paul, including eaching sraegies for he secion, is available a LarsonCalculusforAP.com. Essenial Quesion How do ou solve logisic differenial equaions? Tell sudens ha he will learn how o answer his quesion b using separaion of variables and observing he behavior of he funcion as ime goes on o infini. Lesson Moivaor An undersanding of logisic differenial equaions will allow sudens o solve real-life problems involving a carring capaci. Era Eample Solve he logisic differenial equaion d d = k ( 0). = 0 + be k Teaching Sraegies Sudens ma need o see he inermediae seps below o undersand how L = be k can be wrien as L = + be. k L = be k L = be k L = + be k L = + be k = L + be k 9

30 Paul s Noes Era Eample Verif ha he equaion = saisfies he logisic + e differenial equaion, and find he iniial condiion. saisfies he differenial equaion = ( ) ; (0) = 96 Chaper Differenial Equaions From Eample, ou can conclude ha all soluions of he logisic differenial equaion are of he general form L = + be. k The graph of he funcion is called he logisic curve, as shown in Figure.6. In he ne eample, ou will verif a paricular soluion of a logisic differenial equaion and find he iniial condiion. L = L Logisic curve Noe ha as, L. Figure.6 Eploraion Use a graphing uili o invesigae he effecs of he values of L, b, and k on he graph of L = + be. k Include some eamples o suppor our resuls. Verifing a Paricular Soluion Verif ha he equaion = + e saisfies he logisic differenial equaion, and find he iniial condiion. Soluion Comparing he given equaion wih he general form derived in Eample, ou know ha L =, b =, and k =. You can verif ha saisfies he logisic differenial equaion as follows. = ( + e ) Rewrie using negaive eponen. =( )( + e ) ( 6e ) Appl Power Rule. = ( )( e + e + e ) Rewrie. = ( e + e ) Rewrie using = + e. = ( + e ) Rewrie fracion using long division. = ( ( + e )) Mulipl fracion b. = ( ) Rewrie using = + e. So, saisfies he logisic differenial equaion = ( ). The iniial condiion can be found b leing = 0 in he given equaion. = + e (0) = So, he iniial condiion is (0) =. 96 Eploraion L The graph of = + be k has horizonal asmpoes a = 0 and = L, and is range is (0, L). So, a change in L changes he horizonal asmpoe = L and he range accordingl. If L < 0, hen he graph is refleced abou he -ais. The value of b affecs he -inercep, which is he poin ( L 0,. As posiive values of b + b) ge larger, he -inercep approaches (0, 0). As posiive values of b ge smaller, he -inercep approaches (0, L). When b < 0, he graph has a verical asmpoe. The value of k affecs he seepness of he graph. As k ges larger, he graph increases more quickl. As k ges smaller, he graph increases more slowl. When k < 0, he graph is refleced abou he -ais.

31 Secion. The Logisic Equaion 97 Verifing he Upper Limi Verif ha he upper limi of = is. + e Algebraic Soluion To verif he upper limi, find he limi of as. lim = lim + e lim = lim ( + e ) = + 0 = So, he upper limi of is, which is also he carring capaci L =. Numerical Soluion X Y= Deermining he Poin of Inflecion Skech a graph of = +. Calculae in erms of and. Then deermine he poin of inflecion. Soluion From Eample, ou know ha = ( ). Le =. + e Y As increases, he values of appear o reach an upper limi of. So, he upper limi of appears o be, which is also he carring capaci L =. Paul s Noes Teaching Sraegies Do no dismiss Eamples and. Alhough hese eamples are no widel esed on he AP Eam, he are imporan for a solid concepual undersanding of he meaning of poins of inflecion and limis a infini in his cone. Era Eample Verif ha he upper limi of = + e is. lim = lim + e lim = lim ( + e ) = + 0 = Now calculae in erms of and. = ( ) + ( ) = ( ) ( ) Differeniae using Produc Rule. Facor and simplif. When < <, < 0 and he graph of is concave downward. When 0 < <, > 0 and he graph of is concave upward. So, a poin of inflecion mus occur a =. The corresponding -value is = + e + e = e = = ln. The poin of inflecion is ( ln, ), as shown in Figure.7. In Eample, he poin of inflecion occurs a = L. This is rue for an logisic growh curve for which he soluion sars below he carring capaci L. (See Eercise.) Concave upward 6 Figure.7 Concave downward ( ) Poin of ln, inflecion Era Eample Skech a graph of = + e. Calculae in erms of and. Then deermine he poin of inflecion = ( ) ; = ( ) ; (ln, ) Teaching Sraegies Alhough he poin of inflecion occurs a = L for an logisic growh curve for which he soluion sars below he carring capaci L, i is he corresponding -value ha is of ineres in real-life applicaions. This is because his -value represens he ime a which he rae of change of he populaion begins o decrease. 97

32 9 Chaper Differenial Equaions Paul s Noes Era Eample Graph a slope field for he logisic differenial equaion =0. ( 00). Then graph soluion curves for he iniial condiions (0) = 00, (0) = 00, and (0) = 00. Graphing a Slope Field and Soluion Curves Graph a slope field for he logisic differenial equaion =0.0 ( 00). Then graph soluion curves for he iniial condiions (0) = 00, (0) = 00, and (0) = 00. Soluion You can use a graphing uili o graph he slope field shown in Figure.. The soluion curves for he iniial condiions (0) = 00, (0) = 00, and (0) = 00 are shown in Figures (0, 00) 600 (0, 00) (0, 00) (0, 00) Slope field for Paricular soluion for =0.0 ( 00) =0.0 ( 00) and iniial condiion (0) = 00 Figure. Figure.9 (0, 00) (0, 00) Paricular soluion for Paricular soluion for =0.0 ( 00) ( 00) and iniial condiion (0) = 00 and iniial condiion (0) = 00 Figure.0 Figure. Noe ha as increases wihou bound, he soluion curves in Figures.9. all end o he same limi, which is he carring capaci of 00. 9

33 Secion. The Logisic Equaion 99 Applicaion Solving a Logisic Differenial Equaion A sae game commission releases 0 elk ino a game refuge. Afer ears, he elk populaion is 0. The commission believes ha he environmen can suppor no more han 000 elk. The growh rae of he elk populaion p is dp d = kp ( 000) p, 0 p 000 where is he number of ears. a. Wrie a model for he elk populaion in erms of. b. Graph he slope field for he differenial equaion and he soluion ha passes hrough he poin (0, 0). c. Use he model o esimae he elk populaion afer ears. d. Find he limi of he model as. Soluion a. You know ha L = 000. So, he soluion of he equaion is of he form p = be. k Because p(0) = 0, ou can solve for b as follows. 0 = be k(0) 0 = b = 99 + b Then, because p = 0 when =, ou can solve for k = + 99e k() k 0.9 So, a model for he elk populaion is 000 p = + 99e. 0.9 b. Using a graphing uili, ou can graph he slope field of dp d = 0.9p ( 000) p and he soluion ha passes hrough (0, 0), as shown in Figure.. c. To esimae he elk populaion afer ears, subsiue for in he model. 000 p = + 99e 0.9() Subsiue for. 000 = + 99e.9 Simplif. 66 Simplif. d. As increases wihou bound, he denominaor of e 0.9 ges closer and closer o. So, 000 lim 0.9 = e Infiniumgu/Shuersock.com Slope field for dp d = 0.9p ( 000) p 0 and he soluion passing hrough (0, 0) Figure. Paul s Noes Era Eample 6 A ime = 0, a bacerial culure weighs grams. Afer hour, he culure weighs grams. The maimum weigh of he culure is 0 grams. The growh rae of he weigh w of he culure is dw d = kw ( w 0), w 0, where is he ime in hours. a. Wrie a model for he weigh of he culure in erms of. 0 w = + e.096 b. Graph he slope field for he differenial equaion and he soluion ha passes hrough he poin (0, ) (0, ) w 6 0 c. Use he model o esimae he weigh of he culure afer hours. Abou 7.7 grams d. Find he limi of he model as. 0 Lesson Closer The populaion P of a species saisfies he logisic differenial equaion dp d = P ( 000) P, where he iniial populaion is 00. Wha is lim P()?,000 99

34 00 Chaper Differenial Equaions Assignmen Guide Es. 9,, 7, 0,, 6,,, 9. (a) 0.7 (b) 00 (c) 70 (d).9 r (e) dp d = 0.7P [ 00] P 0. (a) 0. (b) 000 (c) (d).7 r (e) dp d = 0.P [ 000] P. (a) 0. (b) 6000 (c). (d) 0.6 r (e) dp d = 0.P [ 6000] P. (a) 0. (b) 000 (c) (d) 0.0 r (e) dp d = 0.P [ 000] P. (a) (b) 00 (c) P (d) 0. (a) 0. (b) 0 (c) P Eercises See CalcCha.com for uorial help and worked-ou soluions o odd-numbered eercises. Maching In Eercises, mach he logisic equaion wih is graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) (c) (d) = d + e. = + e. = b +. = e + e Verifing a Paricular Soluion In Eercises, verif ha he equaion saisfies he logisic differenial equaion d d = k ( L). Then find he iniial condiion.. = + e 6. = 0 (0) = (0) = + e 7. = + 6e. = (0) = 7 (0) = 7 + e Using a Logisic Equaion In Eercises 9, he logisic equaion models he growh of a populaion. Use he equaion o (a) find he value of k, (b) find he carring capaci, (c) find he iniial populaion, (d) deermine when he populaion will reach 0% of is carring capaci, and (e) wrie a logisic differenial equaion ha has he soluion P(). 9. See margin P() = + 9e 0. P() = e P() = + 999e 0.. P() = + e 0. a c Using a Logisic Differenial Equaion In Eercises 6, he logisic differenial equaion models he growh rae of a populaion. Use he equaion o (a) find he value of k, (b) find he carring capaci, (c) use a compuer algebra ssem o graph a slope field, and (d) deermine he value of P a which he populaion growh rae is he greaes. 6. See margin.. dp d = P ( 00) P. dp d = 0.P ( 0) P. dp = 0.P 0.000P d 6. dp = 0.P 0.000P d Solving a Logisic Differenial Equaion In Eercises 7 0, find he logisic equaion ha saisfies he iniial condiion. Then use he logisic equaion o find when = and = See margin. Logisic Differenial Equaion 7. d d = ( 6). d d =. ( 0) 9. d d = 0 0. d d = Iniial Condiion (0, ) (0, 7) (0, ) (0, ) WRITING ABOUT CONCEPTS. Describing a Value Describe wha he value of L represens in he logisic differenial equaion d d = k ( L). L is he carring capaci.. Deermining Values I is known ha L = + be k is a soluion of he logisic differenial equaion d d = 0.7 ( 00). L represens he value ha approaches as approaches infini. Is i possible o deermine L, k, and b from he informaion given? If so, find heir values. If no, which value(s) canno be deermined and wha informaion do ou need o deermine he value(s)? See margin.. Separaion of Variables Is he logisic differenial equaion separable? Eplain. See margin (d). (a) 0. (b) 0 (c) P (d) (a) 0. (b) 600 (c) P (d) = 6 + e ;.6; = ; e.; 0 9. = + e 0.; 9.; = + e ( 0); 9.6; 0.0. No, i is no possible o deermine b. However, L = 00 and k = 0.7. You need an iniial condiion o deermine b.. Yes; I can be wrien as d k ( L) = d. 00

35 See margin.. HOW DO YOU SEE IT? The growh of a populaion is modeled b a logisic equaion as shown in he graph below. Wha happens o he rae of growh as he populaion increases? Wha do ou hink causes his o occur in real-life siuaions, such as animal or human populaions? Answers will var. Sample answer: There migh be limis on available food or space.. Endangered Species A conservaion organizaion releases Florida panhers ino a game preserve. Afer ears, here are 9 panhers in he preserve. The Florida preserve has a carring capaci of 00 panhers. (a) Wrie a logisic equaion ha models he populaion of panhers in he preserve. See margin. (b) Find he populaion afer ears. 70 panhers (c) When will he populaion reach 00? 7.7 r (d) Wrie a logisic differenial equaion ha models he growh rae of he panher populaion. Then repea par (b) using Euler s Mehod wih a sep size of h =. Compare he approimaion wih he eac answer. See margin. (e) Afer how man ears is he panher populaion growing mos rapidl? Eplain. See margin. 6. Baceria Growh A ime = 0, a bacerial culure weighs gram. Two hours laer, he culure weighs grams. The maimum weigh of he culure is 0 grams. (a) Wrie a logisic equaion ha models he weigh of he bacerial culure. See margin. (b) Find he culure s weigh afer hours.. g (c) When will he culure s weigh reach grams? 6.60 hours (d) Wrie a logisic differenial equaion ha models he growh rae of he culure s weigh. Then repea par (b) using Euler s Mehod wih a sep size of h =. Compare he approimaion wih he eac answer. See margin. (e) Afer how man hours is he culure s weigh increasing mos rapidl? Eplain. See margin. 7. Finding a Derivaive Show ha if = + be k hen d d = k( ). Proof Secion. The Logisic Equaion 0. Poin of Inflecion For an logisic growh curve, show ha he poin of inflecion occurs a = L when he soluion sars below he carring capaci L. Proof True or False? In Eercises 9 and 0, deermine wheher he saemen is rue or false. If i is false, eplain wh or give an eample ha shows i is false. 9. For he logisic differenial equaion False; d d d = k ( L) < 0 and he populaion d decreases o approach L. if > L, hen d d > 0 and he populaion increases. 0. For he logisic differenial equaion d d = k ( L) if 0 < < L, hen d d > 0 and he populaion increases. True Calculus AP Eam Preparaion Quesions. Free Response A populaion is modeled b a funcion () ha saisfies he logisic differenial equaion d d = ( ). (a) Wha is lim ()? (b) For wha value of is he populaion growing he fases? = 0 See margin. (c) Find he paricular soluion saisfing (0) =.. Free Response Consider he differenial equaion d d = 0.9 ( 00). See margin. Le = f () be he paricular soluion of he differenial equaion wih f (0) = 0. (a) A slope field for his differenial equaion is given below. Skech possible soluion curves hrough he poins (0, 0) and (, 00) (b) Use Euler s Mehod, saring a = 0 wih wo seps of equal size, o approimae f (). (c) Wha is he range of f for 0? 00. (a) P = + 7e 0.60 (d) dp d = 0.60P ( 00) P ; 69. panhers; This approimaion is close o he eac answer. (e) The populaion is growing mos rapidl when P = L = 00 = 00, corresponding o 7.7 ears (a) = + 9e (d) d d = ( ln 9 ) ( 0) ;.7g; This approimaion is quie a bi less han he eac answer. (e) The weigh is increasing mos rapidl when = L = 0 = 0, corresponding o.7 hours.. (c) = + e. (a) (0, 0) (b) 09.6 (c) (00, 0) (, 00) 6 0 0

36 0 Chaper Differenial Equaions Assignmen Guide Es., 7 odd, 0,, 6 0 even,,,, 6,,,, 7. = C. = + C. = sin + C 6. = cos + C 7. = e + C. = e + C d d d d. (a) and (b) (0, ). (a) and (b) (, ) Review Eercises See CalcCha.com for uorial help and worked-ou soluions o odd-numbered eercises.. Deermining a Soluion Deermine wheher he funcion = is a soluion of he differenial equaion + = 0. Soluion. Deermining a Soluion Deermine wheher he funcion = sin is a soluion of he differenial equaion = 0. No a soluion Finding a General Soluion In Eercises, use inegraion o find a general soluion of he differenial equaion.. See margin.. d d = + 7. d = cos d 7. d d = e. d d = d 6. d = sin. d d = e Slope Field In Eercises 9 and 0, a differenial equaion and is slope field are given. Complee he able b deermining he slopes (if possible) in he slope field a he given poins See margin. 9. d d = d d d 0. d = sin π Slope Field In Eercises and, (a) skech he slope field for he differenial equaion, and (b) use he slope field o skech he soluion ha passes hrough he given poin. To prin a se of coordinae aes for our graph, go o MahGraphs.com.. See margin.. =, (0, ). = +, (, ) Euler s Mehod In Eercises and, use Euler s Mehod o make a able of values for he approimae soluion of he differenial equaion wih he specified iniial value. Use n seps of size h.. See margin.. =, (0) =, n = 0, h = 0.0. =, (0) =, n = 0, h = 0. 0 Solving a Differenial Equaion In Eercises 0, solve he differenial equaion. 0. See margin.. d = 6. d 7. d = ( + ). d d d = + d d = 0 9. ( + ) = 0 0. ( + ) = 0. See margin. Wriing and Solving a Differenial Equaion In Eercises and, wrie and solve he differenial equaion ha models he verbal saemen.. The rae of change of wih respec o is inversel proporional o he cube of.. The rae of change of wih respec o is proporional o 0. Finding an Eponenial Funcion In Eercises 6, find he eponenial funcion = Ce k ha passes hrough he wo poins. 6. See margin.... (, ) ) 0, ), ) (0, ), 6) ) ) (, ) 6. (, ) (, ) 7. See margin. 7. Air Pressure Under ideal condiions, air pressure decreases coninuousl wih he heigh above sea level a a rae proporional o he pressure a ha heigh. The baromeer reads 0 inches a sea level and inches a,000 fee. Find he baromeric pressure a,000 fee.. Radioacive Deca Radioacive radium has a halflife of approimael 99 ears. The iniial quani is grams. How much remains afer 70 ears? 9. Populaion Growh A populaion grows coninuousl a a rae of.%. How long will i ake he populaion o double? Abou 7. r. = + + C 6. = + Ce 7. = + C. = ( + C) 9. = Ce ( + ) 0. = Ce. d d = k k ; = + C. d = k(0 ); d = 0k k + C. e[( )ln(0 )] e0.79. = e [ ( )ln(0)] e = 9 0 e[( )ln(0 )] 9 0 e = e [ ( )ln ] 6.96e Abou 7.79 in.. Abou 0. g 0. See Addiional Answers beginning on page AA.

37 Review Eercises 0 0. Compound Ineres Find he balance in an accoun when $000 is deposied for ears a an ineres rae of % compounded coninuousl. $77.. Sales The sales S (in housands of unis) of a new produc afer i has been on he marke for ears is given b S = Ce k. See margin. (a) Find S as a funcion of when 000 unis have been sold afer ear and he sauraion poin for he marke is 0,000 unis (ha is, lim S = 0). (b) How man unis will have been sold afer ears?. Sales The sales S (in housands of unis) of a new produc afer i has been on he marke for ears is given b S = ( e k ). (a) Find S as a funcion of when 000 unis have been sold afer ear. S = ( e 0.7 ) (b) How man unis will saurae his marke?,000 unis (c) How man unis will have been sold afer ears?, unis Finding a General Soluion Using Separaion of Variables In Eercises 6, find he general soluion of he differenial equaion. 6. See margin.. d d =. d d =. 6 = 0 6. e sin = 0 Finding a Paricular Soluion Using Separaion of Variables In Eercises 7 0, find he paricular soluion ha saisfies he iniial condiion See margin. Differenial Equaion Iniial Condiion 7. = 0 () =. e = 0 (0) = 9. ( + ) ( + ) = 0 (0) = 0. cos = 0 (0) = Slope Field In Eercises and, skech a few soluions of he differenial equaion on he slope field and hen find he general soluion analicall. To prin an enlarged cop of he graph, go o MahGraphs.com.. d d =. d d = 9. ds = k(l S) d S = L( e k ) 0. (a) S = 00( e 0. ) (b) S = 00( e 0.0 ). dp = kp(l P) dn CL P = e Lkn + C. See margin.. (a) P = P Using a Logisic Equaion In Eercises and, he logisic equaion models he growh of a populaion. Use he equaion o (a) find he value of k, (b) find he carring capaci, (c) find he iniial populaion, (d) deermine when he populaion will reach 0% of is carring capaci, and (e) wrie a logisic differenial equaion ha has he soluion P().. See margin. 0. P() = + e. P() = e 0. Solving a Logisic Differenial Equaion In Eercises and 6, find he logisic equaion ha passes hrough he given poin. 6. See margin.. d d = ( 0) (0, ) (0, ) 6. d d =.76 ( ) See margin. 7. Environmen A conservaion deparmen releases 00 brook rou ino a lake. I is esimaed ha he carring capaci of he lake for he species is 0,00. Afer he firs ear, here are 000 brook rou in he lake. (a) Wrie a logisic equaion ha models he number of brook rou in he lake. (b) Find he number of brook rou in he lake afer ears. (c) When will he number of brook rou reach 0,000?. Environmen Wrie a logisic differenial equaion ha models he growh rae of he brook rou populaion in Eercise 7. Then repea par (b) using Euler s Mehod wih a sep size of h =. Compare he approimaion wih he eac answer. See margin. 9. Sales Growh The rae of change in sales S (in housands of unis) of a new produc is proporional o L S a an ime (in ears), where L is he esimaed maimum level of sales (in housands of unis). When = 0, s = 0. Wrie and solve he differenial equaion for his sales model. See margin. 0. Sales Growh Use he resul of Eercise 9 o wrie S as a funcion of for (a) L = 00 and S = when =, and (b) L = 00 and S = 0 when =. See margin. Learning Theor In Eercises and, assume ha he rae of change in he proporion P of correc responses afer n rials is proporional o he produc of P and L P, where L is he limiing proporion of correc responses.. See margin.. Wrie and solve he differenial equaion for his learning heor model.. Use he soluion of Eercise o wrie P as a funcion of n, and hen use a graphing uili o graph he soluion. (a) L =.00 (b) L = e 0.7n P = 0.0 when n = 0 P = 0. when n = 0 P = 0. when n = P = 0.60 when n = 0 (b) P = P + e 0.7n 0 0. (a) S 0e.79 (b) 0,96 unis. = + C. = + C. = Ce 6. = ln cos + C 7. = 6. = e + 9. = + 0. = sin +. Graphs will var. + = C. Graphs will var. = + Ce. (a) 0. (b) 0 (c) 0 (d) 6. r (e) dp d = 0.P ( 0) P. (a) 0. (b) 00 (c) 0 (d) 7.9 r (e) dp d = 0.P ( 00) P 0. = + 9e 6. = + e.76 0,00 7. (a) P = + 6e 0. (b) 7, rou (c).9 r. dp d = 0.P ( 0,00) P ; 6,70 rou; This approimaion is close o he eac answer. 0

38 See margin. 0 Chaper Differenial Equaions AP Eam Pracice Quesions for Chaper Wha You Need o Know... You ma have o skech a slope field for a given differenial equaion a a specified number of poins. You ma have o skech a soluion curve on a given porion of a slope field. When his occurs, make sure ha our curve follows he slope field appropriael, and passes hrough he indicaed poin. Be prepared o solve differenial equaions compleel using separaion of variables wih a given iniial condiion. The final answer should be of he form = f (). On he AP Calculus BC Eam, be prepared o do a leas wo ieraions of Euler s Mehod, showing all of our work wihou using a calculaor. Pracice Quesions Secion, Par A, Muliple Choice, No Technolog. Which of he following is a slope field for d d =? (A) (B) (C) (D). A populaion P grows according o he equaion dp d = kp, where k is a consan and is measured in ears. If he populaion riples ever ears, wha is he value of k? B (A) ln (B) ln (C) ln (D). Le = f () be a soluion of he differenial equaion = k, where k is a consan. If f (0) = and f (6) =, which of he following is an epression for f ()? A (A) e ( 6)ln( ) (B) e ( 6)ln (C) + (D) + B. If d d = and ( ) =, find (). (A) e (B) (C) (D). Which of he following is he soluion of he differenial equaion d d = wih he iniial condiion () =? B (A) = (B) = (C) = (D) = 6. Which of he following differenial equaions produces he slope field shown below? A (A) d (B) d (C) d (D) d d = 0 ( ) d = ( ) d = ( ) d = ( 6) Secion, Par B, Muliple Choice, Technolog Permied 7. Consider he differenial equaion =0.( )( + ) wih an iniial value of (0) =. Using Euler s Mehod wih a sep of h =, wha is he approimae value of ()? A (A) 6. (B) 6. (C).7 (D).70 C 0

39 AP Eam Pracice Quesions for Chaper 0 Secion, Par A, Free Response, Technolog Permied. A an ime 0, in hours, he rae of growh of a populaion of baceria is given b d d = 0.. Iniiall, here are 00 baceria. See margin. (a) Solve for, he number of baceria presen, a an ime 0. (b) Wrie and evaluae an epression o find he average number of baceria in he populaion for 0 0. (c) Wrie an epression ha gives he average rae of baceria growh over he firs 0 hours of growh. Indicae unis of measure. Secion, Par B, Free Response, No Technolog 9. Le = f () be a paricular soluion of he differenial equaion d d = (b) While he slope field in par (a) is drawn onl a nine poins, i is defined a ever poin in he -plane. Describe all poins in he -plane for which he slopes are posiive. (c) Find he paricular soluion in he form of = f () o he given differenial equaion wih he iniial condiion f (0) =.. Consider he differenial equaion =() wih a paricular soluion in he form of = f () ha saisfies he iniial condiion f () =. See margin. (a) Use Euler s Mehod, saring a = wih wo seps of equal size, o approimae (.). Show he work ha leads o our answer. (b) Find he paricular soluion of he given differenial equaion ha passes hrough (, ) and sae is domain.. Consider he differenial equaion d d =. See margin. (a) Le = f () be he funcion ha saisfies he differenial equaion wih iniial condiion f () =. Use Euler s Mehod, saring a = wih a sep size of 0., o approimae f (.). Show he work ha leads o our answer. (b) Find d d. Deermine wheher he approimaion found in par (a) is less han or greaer han f (.). Jusif our answer. (c) Find he paricular soluion of he given differenial equaion ha passes hrough (, ). wih f () =. See margin.. A an ime 0, he rae of he spread of an epidemic (a) Find d a he poin (, ). is modeled b a funcion ha saisfies he differenial d equaion (b) Wrie an equaion for he line angen o he graph d of f a (, ) and use i o approimae f (.). Is he d = 0 ( 000) approimaion for f (.) greaer han or less han f (.)? Eplain our reasoning. In an isolaed own of 000 inhabians, 00 people (c) Find he soluion of he given differenial equaion have a disease a he beginning of he week. See margin. ha saisfies he iniial condiion f () =. (a) Is he disease spreading faser when 00 people 0. Consider he differenial equaion d have he disease or when 00 people have he d = ( ). See margin. disease? Eplain our reasoning. (a) On he aes provided, skech a slope field for (b) Wrie a model for he populaion = f () a an he given differenial equaion a he nine poins ime 0. indicaed.. (a) (c) Wha is lim ()?. Consider he differenial equaion d d =. (a) On he aes provided, skech a slope field for he given differenial equaion a he welve poins indicaed. See margin. (b) Find d d in erms of and. (c) Find he paricular soluion of he given differenial equaion ha saisfies he iniial condiion (0) =.. (a) = 00e 0. 0 (b) 00e 0 0. d 0 = 0(e ) 0 (c) 00e 0 0. d baceria 0 per hour 9. (a) (b) = + ; f (.).0;.0 > f (.) because f (.) < 0. (c) f () = ln + 0. (a) (b) d d > 0 when < (c) f () = + e ( ). (a) =.,. f (.). (b) f () = + Domain: (, ). (a) =., =. f (.). (b) d d = + ; On he inerval [,.], is posiive. Because d d =, d and have he same d sign on [,.]. Because = f () =, d and are d boh posiive on [,.]. Therefore, d > 0 and he d approimaion found in par (a) is less han f (.). (c) f () = e (a) I is spreading faser when 00 people have he disease; Because when = 00, d d > d, when = 00. d (b) f () = e 0. (c) 000 (b) d d = (c) = + 0

40 See margin. 06 Chaper Differenial Equaions Sample Rubric Eercise..(a).(b).(a).(b).(c).(d).(e) Toal Poins 06 Poins for wriing logisic equaion Toal possible poins for finding he limi for saing wha he value of he limi represens for finding he number of infeced sudens for skeching he soluion Toal possible poins for finding when he virus is spreading mos rapidl for finding he number of sudens infeced for jusifing answer Toal possible poins for finding he rae of change for inerpreing answer Toal possible poins for finding he values of S for jusifing answer Toal possible poins for finding he number of infeced sudens for comparing resuls Poins 00. S = + 79e 0.. (a) 00; I represens he maimum number of sudens on campus ha can be infeced wih he virus. (b) Abou 6 sudens. (a) Performance Task Spread of an Influenza Virus Throughou hisor, influenza viruses have caused pandemics or global epidemics. The influenza pandemic of 9 99 occurred in hree waves. The firs wave occurred in he lae spring and summer of 9, he second wave occurred in he fall of 9, and he final wave occurred in he spring of 99. B he ime i ended, approimael. million people across he world had died as a resul of he pandemic, wih 67,000 deahs in he Unied Saes alone. Eercises In Eercises, use he following informaion. On a small college campus wih 00 sudens, en sudens reurn from spring break wih a conagious flu virus. The rae a which he virus spreads hrough he campus is given b he differenial equaion ds d = 0.S ( 00) S where S is he number of infeced sudens and is he number of das.. See margin.. Wriing a Logisic Equaion Wrie a logisic equaion ha models he number of sudens infeced wih he virus afer das.. Using a Logisic Equaion Use he equaion ou wroe in Eercise. (a) Wha is lim S()? Wha does his value represen? (b) How man sudens are infeced wih he virus on Da?. Using a Logisic Differenial Equaion The slope field for he given differenial equaion is shown. To prin an enlarged cop of he graph, go o MahGraphs.com (b) Da 0; 00 sudens; The poin of inflecion occurs when S = 00, so (a) Skech he soluion ha saisfies he iniial condiion. (b) On wha da is he virus spreading mos rapidl? How man sudens are infeced wih he virus a his poin? Jusif our answer. (c) Wha is he rae of change of he number of sudens infeced wih he virus when he virus is spreading mos rapidl? Inerpre our answer. (d) Wha are all values of S for which he rae of change of he number of people infeced is decreasing? Jusif our answer. (e) Repea Eercise (b) using Euler s Mehod wih a sep size of h =. Compare our resuls. Evere Hisorical/Shuersock.com 6 0 (c) 6.; A da 0, he rae of change of he spreading virus is abou 7 sudens per da. (d) 00 < S < 00; To deermine where he rae of change is decreasing, ou need o deermine where he derivaive of ds d is negaive. The criical value of d S d is 00, and using he Firs Derivaive Tes, d S d is negaive on he inerval (00, 00). (e) S 9 when = ; The value found b using Euler s Mehod is less han he value found in Eercise (b).