Math 1b. Calculus, Series, and Differential Equations. Final Exam Solutions

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1 Mah b. Calculus, Series, and Differenial Equaions. Final Exam Soluions Spring 6. (9 poins) Evaluae he following inegrals. 5x + 7 (a) (x + )(x + ) dx. (b) (c) x arcan x dx x(ln x) dx Soluion. (a) Using parial fracions, we can deermine ha (b) Using inegraion b pars, we can deermine ha 5x + 7 dx = ln x + + ln x + + C. (x + )(x + ) (c) Using inegraion b subsiuion, we can deermine ha x arcan x dx = π. x(ln x) dx = ln().. ( poins) Suppose ha ou wish o model a populaion wih a differenial equaion of he form dp/ = f(p ), where P () is he populaion a ime. Experimens have been performed on he populaion ha give he following informaion: The populaion a P = remains consan. A populaion of < P < will decrease. A populaion of P = does no change. A populaion of < P < increases. A populaion of P > will decrease. Which of he following differenial equaions bes models his populaion? Circle he correc answer. (a) dp (b) dp (c) dp = (P )(P ) = P ( P )(P ) = P (P )( P )

2 (d) dp (e) dp = P ( P )( P ) = ( P )(P ) Soluion. (b) or (c).. (7 poins) A bag of sand originall weighing lb is lifed a a consan rae. As i rises, sand leaks ou a a consan rae. The sand is half gone b he ime he bag has been lifed 8 f. (a) How man pounds of sand leak ou of he bag per foo as he bag is lifed? (b) How much work was done in lifing he bag 8 f? To receive full credi for our work, indicae clearl wha our variable is and wha ou are considering o be zero. Soluion. (a) lb/f (b) 8 x dx = 9 f-lb. graphs_final_b.nb. (8 poins) Le R be he region bounded b he curves = f(x) and = g(x) shown in he graph below. H-, L f HxL - - x ghxl - - H,-L (a) Wrie a definie inegral ha will give he area of he region R. (b) Wrie a definie inegral ha will give he volume of he solid generaed when he region R is revolved abou he horizonal line =. (c) If he base of a solid V is he region R and he cross-secions of he solid perpendicular o he x-axis are squares, wrie a definie inegral ha will give he he volume of V. Soluion. (a) (b) f(x) g(x) dx π( g(x)) π( f(x)) dx

3 (c) (f(x) g(x)) dx 5. (8 poins) A resor own is laid ou along he seashore in he shape of a semicircle of radius miles wih he diameer graphs_final_b.nb of he semicircle bordering he ocean. People wan o live close o he cener of he own (indicaed b he solid black disk). The densi of he populaion (individuals per square mile) a a disance of x miles from he cener of own is given b ρ(x) = 6 x. Resor Ocean (a) Wrie a Riemann sum ha approximaes he oal populaion of he resor. (b) Use our answer from par (a) o wrie a definie inegral ha represens he oal populaion of he own and evaluae he inegral. Soluion. n (a) πx i ρ(x i ) x. (b) i= πxρ(x) dx = π x(6 x) dx = 9π. 6. (5 poins) In an isolaed region of he Canadian Norhwes Terriories, a populaion of arcic wolves and a populaion of silver foxes compee for resources. The wo species have a common, limied food suppl, which consiss mainl of mice. If x() = he populaion of arcic wolves (in housands) () = he populaion of silver foxes (in housands), he ineracion of he wo species can be modeled b he following ssem of differenial equaions, dx d = x x x = x. (a) Find he nullclines of he ssem for x and. (Axes are provided on he nex page.) (b) Find all of he equilibrium poins for x and. (c) Wha happens o he arcic wolf populaion in he absence of silver foxes? Wha happens o he silver fox populaion in he absence of arcic wolves? (d) Skech and label he nullclines for x and. Be sure o indicae he direcion of he soluion on he nullclines and in he regions bounded b he nullclines. (e) Skech he soluion rajecor wih he iniial condiions x() =.5 and () =.5, indicaing he direcion of our soluion curve. Soluion.

4 (a) The x-nullclines of he ssem occur when dx/ = x x x = x( x ) = or are a x = and = x +. The -nullclines of he ssem occur when d/ = / x/ = (/ x/) = or are a = and = x/ + /. (b) The equilibrium poins for x and occur a (, ), (, /), (, ), and (/, /). (c) The arcic wolf populaion grows logisicall wih a carring capaci of in he absence of silver foxes. The silver fox populaion grows logisicall wih a carring capaci of / in he absence of arcic wolves. (d) and (e) 7. (8 poins) A dosage d of a drug is given dail a =,,,,... das. The drug decas exponeniall a a rae r in he blood sream. Thus, he amoun in he bloodsream afer n + doses is d + de r + de r + + de nr (a) Find he level of he drug afer an infinie number of doses. Tha is, find d + de r + de r + + de nr + (b) If r =., wha dosage is needed o mainain a drug level of? Soluion. (a) d + de r + de r + + de nr + = (b) If = d e., hen d = e.. d e r

5 8. ( poins) The following polnomials are second-degree Talor polnomials for funcions whose graphs are given below. Mach each Talor polnomial wih he appropriae graph. (a) T (x) = (x ) (x ) (b) T (x) = + (x ) 5 (x ) Uniled- (c) T (x) = (x ) + 9(x ) Uniled- (d) T (x) = + 8 (x + ) (x + ) x x Uniled- graphs_final_b.nb (i) - (ii) x x (iii) - (iv) - Soluion. (i) (c), (ii) (d), (iii) (b), (iv) (a). 9. (9 poins) Find a power series represenaion a x = for each of he following funcions. x (a) x (b) x cos x 5

6 (c) x sin Soluion. x (a) x = x + x + x + x + x 5 + (b) x cos x = x x! + x6! x8 6! + x 8! (c) x sin = = x x ( 6! ! 7! 9! ( 8! ! 7! 9! ) ) = x5 5 x9 9! + x 5! 7 7 7! + 9!. (8 poins) Find he inerval of convergence for he power series n= (x + 5) n n n. If he inerval of convergence is finie, make sure ha ou deermine he convergence a each endpoin and jusif our conclusions for he convergence or divergence a he endpoins. Soluion. Using he Raio Tes, a n+ a n = (x + 5) n+ n+ n + n n (x + 5) n = ( ) n x + 5, n + we can deermine ha lim n a n+ /a n = x + 5 /. This limi is less han one on he inerval x + 5 < or 8 < x <, and he series converges on his inerval. We mus consider he endpoins separael. A x =, he series n= is a divergen p-series. A x = 8, he series n= n n n = n n= ( ) n n n = n= ( ) n n converges b he Alernaing Series Tes. Thus, he power series converges on he inerval 8 x <.. (6 poins) Mach each slope field wih one of he following differenial equaions. (a) d = (b) d = (c) d = (d) d = (e) d = cos (f) d = ( ) 6

7 graphs_final_b.nb graphs_final_b.nb (g) d = cos (h) d = ( ) graphs_final_b.nb - graphs_final_b.nb - (i) (ii) (iii) - (iv) - Soluion. (i) (e), (ii) (f), (iii) (a), (iv) (g).. (6 poins) Mach each of he following graphs of versus o he differenial equaion for which i could be a soluion. (a) = (b) 5 + = (c) + = (d) + + = 7

8 graphs_final_b.nb graphs_final_b.nb graphs_final_b.nb (i) graphs_final_b.nb (ii) (iii) (iv) Soluion. (i) (c), (ii) (b), (iii) (a), (iv) (d).. (8 poins) A new 5-gallon juice dispenser in he cafeeria is filled wih a frui drink ha is 8% orange juice and % pineapple juice. Ever hour, gallons of juice is consumed. Bu due o an error, he dispenser is coninuousl replenished ever hour wih an orange-pineapple mixure ha is % orange and 6% pineapple. Assume ha he dispensed juice is alwas well-mixed. (a) Wrie down a differenial equaion for P (), he amoun of pineapple juice in he dispenser a ime. Be sure o include our iniial condiion. (b) B solving he differenial equaion, find he amoun of pineapple juice in he conainer afer hours. Soluion. (a) dp P = rae in rae ou = 6 5 = 6 P. The iniial condiion is P () = (b) The differenial equaion ma be solved as a firs-order linear differenial equaion or b using separaion of variables, P () = 9 6e /. The amoun of pineapple juice in he conainer afer hours is P () = 9 6e /. 8