Math 116 Second Midterm March 21, 2016

Transcription

1 Mah 6 Second Miderm March, 06 UMID: EXAM SOLUTIONS Iniials: Insrucor: Secion:. Do no open his exam unil you are old o do so.. Do no wrie your name anywhere on his exam. 3. This exam has pages including his cover. There are 9 problems. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem on which you are suck. 4. Do no separae he pages of his exam. If hey do become separaed, wrie your UMID on every page and poin his ou o your insrucor when you hand in he exam. 5. Please read he insrucions for each individual problem carefully. One of he skills being esed on his exam is your abiliy o inerpre mahemaical quesions, so insrucors will no answer quesions abou exam problems during he exam. 6. Show an appropriae amoun of work (including appropriae explanaion) for each problem so ha graders can see no only your answer, bu also how you obained i. Include unis in your answer where ha is appropriae. 7. You may use a TI-84, TI-89, TI-Nspire or oher approved calculaor. However, you mus show work for any calculaion which we have learned how o do in his course. You are also allowed wo sides of a 3 5 noe card. 8. If you use graphs or ables o find an answer, be sure o include an explanaion and skech of he graph, and o wrie ou he enries of he able ha you use. 9. Turn off all cell phones, pagers, and smarwaches, and remove all headphones. Problem Poins Score Toal 00

2 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page. [4 poins] Deermine if he following inegrals converge or diverge. If he inegral converges, circle he word converges and give he exac value (i.e. no decimal approximaions). If he inegral diverges, circle diverges. In eiher case, you mus give full evidence supporing your answer, showing all your work and indicaing any heorems abou improper inegrals you use. Any direc evaluaion of inegrals mus be done wihou using a calculaor. a. [7 poins] x, where a > 0 is a consan e ax + Converges Diverges x b = lim e ax + b = lim b ( e ab + a x ab + = lim e ax + b a+ ) ea+ a = ea+ a = ae a+ ae u du = In he second equaliy, we used he subsiuion u = ax +. b. [7 poins] x + sin x x Converges Diverges

3 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 3 Since sin x for any x, The improper inegral x + sin x x x x = x x ( ) diverges because he improper inegral ( x x ) x diverges (p = ) while he improper inegral x converges (p = > ). So using ( ), he inegral in quesion diverges by he comparison es. Alernaively, we can use he inequaliy x + sin x x x x x which is valid for all x, he p-es and he comparison es.

4 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 4. [3 poins] Leia and Han are imprisoned in a cell whose door is made ou of seel and has a hickness of 3 fee. Luke uses his lighsaber o cu hrough he door in he shape of he curve given by he polar coordinaes equaion r = cos ( θ + π 4 ) where r is measured in fee. a. [6 poins] Wrie an expression involving inegrals for he volume of he piece ha Luke cus ou of he door. ( π 5 3 )) cos ( θ + π dθ f 3 4

5 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 5 b. [7 poins] Sill considering he polar curve r = cos ( θ + π 4 ) graphed in he xy-plane, wrie an explici expression involving inegrals for he lengh of he par of he curve ha lies o he righ of he y-axis. π π ( ( 5 )) 3 + cos ( θ + π + 4 ) 0 sin ( θ + π 4 ( 3 + cos ( θ + π 4 ) ) ) dθ f Alernaively: where and dθ dy dθ π π ( ) ( ) dy + dθ f dθ dθ = ( 5 sin θ)(3 + cos(θ + π/4)) + (5 cos θ) sin(θ + π/4) [3 + cos(θ + π/4)] = (5 cos θ)(3 + cos(θ + π/4)) + (5 sin θ) sin(θ + π/4) [3 + cos(θ + π/4)]

6 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 6 3. [3 poins] O-guk s playful son, O-ghan, is running on he xy-plane. His posiion seconds afer he begins running is x = y = sin() +. Assume x and y are in meers. a. [3 poins] Does O-ghan pass hough he origin? Briefly jusify. x = 0 when = 0 so when =. For his value of, y = sin() + 0. So he didn pass hrough he origin. b. [4 poins] How fas is O-ghan running a = 5? Give your answer in exac form (i.e. no decimal approximaions). Include unis. ( 5 ) + (cos(5)) m s c. [6 poins] Find an equaion, in xy-coordinaes, of he angen line o his pah a =. The slope of he angen line is given by m = dy/d /d = cos() = cos() The equaion of he angen line is y sin() = cos()(x 0) or equivalenly y = cos()x + sin()

7 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 7 4. [5 poins] Drake is running for presiden. Suppose F () is he fracion of he oal populaion of he counry who suppors him monhs afer he announces he is running. Drake gains supporers a a seady rae of % of he oal populaion of he counry per monh, bu he also seadily loses 3% of his supporers per monh. Wrie a differenial equaion ha models F (). df d = F 5. [6 poins] Adele is also running for presiden. Suppose P (), he oal number of supporers she has in millions days afer she announces, is modeled by he differenial equaion wih k > 0. dp d = kp (00 P ) a. [4 poins] Find he equilibrium soluions o his differenial equaion and indicae sabiliies for each. Make sure your answer is clear. The equilibrium P = 0 is unsable and he equilibrium P = 00 is sable. b. [ poins] If Adele sars wih one million supporers, wha is he maximum number of supporers she can ge in he long run? You do no need o show your work. 00, 000, 000 supporers

8 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 8 6. [5 poins] In he following quesions, circle he correc answer. You do no need o show any work, bu make sure your answer is clear. No poins will be given for unclear answers. a. [3 poins] The value of A for which he funcion y = e x +A 3x solves he equaion y + 8y = xy is b. [3 poins] The funcion g is posiive, decreasing and differeniable. The soluion curves of he differenial equaion y = e x g(y) are concave up concave down changing concaviy c. [3 poins] Suppose ha h(x) is an increasing differeniable funcion wih h(0) = 0 and lim h(x) = 5. The value of he inegral (h(x)) 4 h (x) x 0 diverges is 5 4 is is is 0 d. [3 poins] Suppose a is a consan, and he funcion h saisfies 0 x. The inegral 0 (h(x)) converges always never someimes x /a h(x) x a for e. [3 poins] The funcion f saisfies inegral g(x) converges x x 3 f(x) x for x and f(x) = g(x ). always never someimes The

9 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 9 7. [9 poins] The graph of a slope field corresponding o a differenial equaion is shown below. y x a. [3 poins] On he slope field, carefully skech a soluion curve passing hrough he poin (, ) wih domain 0 x 5. b. [4 poins] The slope field picured above is he slope field for one of he following differenial equaions. Which one? Circle your answer. You do no need o show your work. dy dy = (x )(y )(y 3) dy = (x )(y ) (y 3) = (x + )(y + )(y + 3) dy = (x )(y ) (y 3) c. [ poins] If we use Euler s mehod saring a he poin (, ) and use x = 0., would we ge an overesimae or an underesimae for he value of y(.5)? Circle your answer. You do no need o show your work. overesimae underesimae

10 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 0 8. [3 poins] Brianne is hiking, and he emperaure of he air in C afer she s raveled x km is a soluion o he differenial equaion y + y sin x = 0 a. [7 poins] Find he general soluion of he differenial equaion. Wriing he equaion as dy = y sin x and separaing he variables we ge y dy = sin x ln y = cos x + C y = Ae cos x b. [ poins] If he emperaure was 0 C a he beginning of he hike, find T (x), he emperaure of he air in C afer she s raveled x km. Show your work. From (a), T (x) = Ae cos x. Since T (0) = 0 we ge A = 0 e. Thus, T (x) = 0 e ecos x c. [4 poins] Brianne raveled 7 km on he hike. Using he informaion given in (b), find he coldes air emperaure she encounered on he hike. Give an exac answer (i.e. no decimal approximaions). We wan o find he minimum of he funcion T (x) = 0 e ecos x over he inerval [0, 7]. The criical poins are 0, π, π. Checking he oupus of T a hose poins and he endpoin 7 we find ha he minimum is T (π) = 0. e

11 Mah 6 / Exam (March, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 9. [ poins] a. [6 poins] Show ha he following inegral diverges. Give full evidence supporing your answer, showing all your work and indicaing any heorems abou improper inegrals you use. cos( ) d cos( ) cos() because he funcion F (x) = cos x is decreasing in he inerval [0, ]. Therefore, cos( ) cos() The improper inegral cos() d = cos() d diverges by he p-es since p =. So he inegral cos( ) diverges by he comparison es (noice ha cos() > 0). d b. [6 poins] Find he limi lim x x cos( ) x d Noice ha by (a), his is. We use L Hopial s rule along wih he nd Fundamenal Theorem in he numeraor: lim x x cos( ) x d = lim x cos( x ) x x ( ) = lim cos = cos(0) = x x