UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 This problem se concenraes on pracice wih aniderivaives. You will ge los of pracice finding simple aniderivaives as well as finding aniderivaives graphically using slope fields. You will also ge some pracice wih problems involving accumulaed change (i.e. he area under curves) and definie inegrals.. (4.3-33) The Saue of Libery is 92 meers high, including he 46 meer pedesal upon which i sands. How far from he base should an individual sand o ensure ha he view angle, θ, is maximized? (Hin: See he exbook for a picure.) 2. (4.3-35) An oil spill has fouled 2 miles of Pacific shoreline. The oil company responsible has been given foureen days o clean up he shoreline, afer which a fine will be levied in he amoun of $, /day. The local cleanup crew can scrub five miles of beach per day a a cos of $5/day. Addiional crews can be brough in a a cos of $8,, plus $8/day for each crew. Deermine how many addiional crews should be brough in o minimize he oal cos o he company and how much he cleanup will cos. 3. (4.3-33) Consider a cylindrical cell wih radius r and heigh r/2. Assume ha he cell gains energy a a rae proporional o is surface area (i.e., nuriens diffusing in from ouside of he cell) and ha he cell loses energy a a rae proporional o is volume (i.e., all pars of he cell are using energy). If he cell is rying o maximize is ne gain of energy, deermine he opimal value of r. Noe: Your final expression will depend on your proporionaliy consans. 4. (5.) Find he general aniderivaive of he funcions (a) (5.-2) f(x) = 4 (b) (5.-8) f() = 4 + 4 2 (c) (5.2-) f(x) = 2x (d) (5.-4) f(x) = 4 sin(5x) (e) (5.-6) f(x) = 4e x (f) (5.-22) f(u) = 6u + 3 cos u 5. Find he aniderivaives of he following funcions (a) f(x) = 2x + 4 (b) p(z) = 5 4z (c) q(x) = x 2 + (d) f() = 3 3 2 + 2 (e) f(x) = x 3 cos x (f) q(c) = sin c cos c (g) g(u) = e u u (for x > ) (h) f(x) = 2x 4 (for x > 2) 6. Find he aniderivaives of he following funcions. These are rickier and may require some reversing of he chain rule. (a) f(x) = 2xe x2 (b) g(x) = 3x 2 sin(x 3 ) (c) f() = 2 + (d) p(u) = sin( u) (e) f(x) = sin x (cos x) 3 (f) h(z) = z 2 (z 3 ) 3 2
UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 (g) f(x) = x sin(x 2 x) 2 sin(x2 x) (h) f(x) = x x 2 + 7. Skech he slope field for he following differenial equaions and hus skech he aniderivaive wih he given iniial value. (a) df dx = ex2 where f() = (b) df dx = ex x where f() = (c) df dx = x +e where f() = x (d) df dx = 2 + sin x where f(2) = (e) df dx = x sin x where f() = 8. A rocke sars from a saionary posiion and acceleraes a a rae of 2m/s 2 for 6 seconds and hen ravels a a consan speed (i.e. is acceleraion is zero) for anoher 6 seconds. How far has he rocke ravelled (assuming i is always moving in a sraigh line). Soluion: The velociy of he rocke is given by { 2 if 6, v() = 2 if 6 2. The graph of he funcion is v() 2 6 2 We are asked o find he disance he rocke has ravelled which is he accumulaed change in he velociy of he 2 seconds he rocke has been airborne. Thus we need o find he area under he graph. We have spli his ino a riangle and a recangle, for which he areas are 36 and 72 respecively. Thus he oal disance ravelled is 8 meers.
UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 9. Rainwaer fills a ank a a rae of 4 gallons per hour. If i rains for 6 hours how much rain does he ank collec? Soluion: Rain is being colleced for 6 hours and every hour he ank collecs 4 gallons so he oal rain colleced is 4 6 = 24 gallons.. A hydroelecric dam produces 2 Megawas of power for every million lires of waer which flows hrough i. Over a period of 4 hours, you measure r() million lires per hour flowing hrough he urbines. If r() = 4 (x 2) 2 + 5, how many Megawas were produced in hose 4 hours? Soluion: The graph of he rae of flow is r() 5 4 I is a semicircle wih radius 2 cenred a (2, 5). Thus we spli he area under he graph ino a recangle wih area 2 and a semicircle wih area 2π. So he oal amoun of waer ha flowed hrough in he 4 hours is 2 + 2π million lires. Thus he oal number of Megawas produced is wice his number, i.e. 4 + 4π megawas.. A car sars a a saionary posiion and acceleraes a.5m/s 2 for 2 seconds and hen deceleraes a.5m/s 2 unil coming o a sandsill. How far did he car ravel?
UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 Soluion: The car sars from a saionary posiion so if v() is he velociy of he car a ime hen v() =. Thus he velociy of he car is given by {.5 if 2, v() = 3.5( 2) if 2. Thus he graph of he velociy is v() 3 2 8 The car comes o a sandsill when he velociy is zero again so we need o solve he equaion 3.5( 2) = 3.5 + = 4 =.5 = 8. We can divide he area under he graph ino wo riangles, he lef wih area.5 2 3 = 3 and he riangle on he righ wih area.5 6 3 = 9 so he oal disance ravelled is 3+9 = 2 meers. 2. Rain adds 5 lires of waer per minue o your rain ank. You check he rain ank and find ha i currenly has 2 lires of waer in i. Afer 2 minues you noice ha he ank has had a hole in he boom his whole ime! Waer is able o flow ou of he hole a a rae of lires per minue. You immediaely begin repairing he hole. During his ime, waer is able o drain ou a a rae of 3 lires per minue, where is he ime elapsed since you begun he repair. I akes you 3 minues. How much waer is in he ank, hour afer you checked he level of he ank?
UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 Soluion: Les say ha you sar fixing he hole a =. Tha means you firs checked he ank level a = 2 and one hour afer his would be = 4. The rae of waer coming ino he ank due o rain is consan and is given by I() = 5. The rae of waer flowing ou of he ank due o he hole is given by if 2 O() = 3 if 3 if 3 4. The oal waer flowing in is herefore T () = I() O() which is 5 if 2 T () = 5 + 3 if 3 5 if 3 4. A graph of his is given below. T () 5 2 5 3 4 5 Thus we can spli he area under he graph up ino four regions. Each of hese areas is easy o calculae and remembering ha he area under he graph is negaive, he oal area is 75 2 + 75 2 + 5 = 5 This represens he oal change in he amoun of waer in he ank in lires. Since we sared wih 2 lires here mus now be 5 lires lef. 3. Evaluae he definie inegrals (hin: i migh be bes o do quesions and 2 firs) (a) (b) (c) 2 2x + 4 dx 5 4z dz x 2 + dx
UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) 2π π π 3 2 3 π π π π/2 2 5 4 3 3 2 + 2 d x 3 cos x dx sin c cos c dc e u u du 2x 4 dx 2xe x2 dx 3x 2 sin(x 3 ) dx 2 + d sin( u) du sin x (cos x) 3 dx z 2 (z 3 ) 3 2 dz x sin(x 2 x) 2 sin(x2 x) dx x x2 + dx