Math 116 Final Exam December 12, 2014

Transcription

1 Math 6 Fial Exam December 2, 24 Name: EXAM SOLUTIONS Istructor: Sectio:. Do ot ope this exam util you are told to do so. 2. This exam has 4 pages icludig this cover. There are 2 problems. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur to a problem o which you are stuck. 3. Do ot separate the pages of this exam. If they do become separated, write your ame o every page ad poit this out to your istructor whe you had i the exam. 4. Please read the istructios for each idividual problem carefully. Oe of the skills beig tested o this exam is your ability to iterpret mathematical questios, so istructors will ot aswer questios about exam problems durig the exam. 5. Show a appropriate amout of work (icludig appropriate explaatio) for each problem, so that graders ca see ot oly your aswer but how you obtaied it. Iclude uits i your aswer where that is appropriate. 6. You may use ay calculator except a TI-92 (or other calculator with a full alphaumeric keypad). However, you must show work for ay calculatio which we have leared how to do i this course. You are also allowed two sides of a 3 5 ote card. 7. If you use graphs or tables to fid a aswer, be sure to iclude a explaatio ad sketch of the graph, ad to write out the etries of the table that you use. 8. Tur off all cell phoes ad pagers, ad remove all headphoes. 9. O the last page of this exam you will fid a page cotaiig formulas for some commo Taylor series.. You must use the methods leared i this course to solve all problems. Problem Poits Score Total

2 Math 6 / Fial (December 2, 24) page 2. [ poits] You are stadig o Loely Hill while Frakli s robot army surrouds you. You have a Electro-Magetic Pulse (EMP) device that deactivates all of the robots withi a oe mile radius of the peak of Loely Hill as show i the shaded regio below. The poit P is oe mile from Loely Hill for your referece. Let R(x) give the desity, i robots per square mile, of robots x miles from the peak of Loely Hill. A table of values for R(x) is give below. x R(x) P Peak a. [3 poits] Write a itegral which gives the exact umber of robots that are deactivated by the EMP. Your aswer may cotai the fuctio R(x). Do ot evaluate your itegral. 2π xr(x)dx b. [5 poits] Use the Trapezoidal method with as may subdivisios as possible to estimate the total umber of robots that are deactivated by the EMP. Write out all of the terms for your estimate. LEF T (4) = 2π.25(()5 + (.25) + (.5)4 + (.75)35) RIGHT (4) = 2π.25((.25) + (.5)4 + (.75)35 + ()5) T RAP (4) = RIGHT (4) + LEF T (4)

3 Math 6 / Fial (December 2, 24) page 3 c. [3 poits] You are very happy that the EMP worked. Let H(t) give the level of happiess you feel t secods after Frakli s robots are deactivated. Determie the value of a so that H(t) = 3e t + at 2 4t 4 is a solutio to the differetial equatio dh dt = H + 2t2. H (t) = 3e t + 2at 4 3e t + 2at 4 = (3e t + at 2 4t 4) + 2t 2 2at = at 2 4t + 2t 2 2a = at 4 + 2t a(2 t) = 2(2 t) a = 2 This implies that at 2 + t 2 = ad 2at = 4t, so a = 2.

4 Math 6 / Fial (December 2, 24) page 4 2. [6 poits] Let f(x) = xe x2. a. [4 poits] Fid the Taylor series of f(x) cetered at x =. Be sure to iclude the first 3 ozero terms ad the geeral term. We ca use the Taylor series of e y to fid the Taylor series for e x2 by substitutig y = x 2. ( x 2 ) e x2 =! = = + ( x 2 ) + ( x2 ) 2 2! + + ( x2 )! + Therefore the Taylor series of xe x2 ( ) x 2+ xe x2 =! = is = x x 3 + x5 2! + + ( ) x 2+ +! b. [2 poits] Fid f (5) (). We kow that f (5) () 5! will appear as the coefficiet of the degree 5 term of the Taylor series. Usig part (a), we see that the degree 5 term has coefficiet 7!. Therefore f (5) () = 5! = 259, 459, 2 7! 3. [3 poits] Determie the exact value of the ifiite series 2! + 4 2! 8 3! + + ( ) 2! + Notice that this is the Taylor series for e y applied to y = 2. Therefore, the series has exact value e 2.

5 Math 6 / Fial (December 2, 24) page 5 4. [8 poits] Frakli s robots start buildig more robots to replace their deactivated comrades. The iitial umber of robots i Frakli s army is 8. Each miute, the umber of robots icreases by 5%. At the ed of each miute, you fire a EMP which immediately deactivates 5 robots. a. [3 poits] Let R deote the umber of active robots i Frakli s army immediately after the EMP is fired for the -th time. Fid R ad R 2. R = (.5)8 5 R 2 = (.5)((.5)8 5) 5 b. [4 poits] Fid a closed form expressio for R (i.e. evaluate ay sums ad solve ay recursio). R = 8(.5) 5(.5) i i= = 8(.5) 5(.5 ).5 c. [ poit] Fid lim R. No justificatio is ecessary. lim R =

6 Math 6 / Fial (December 2, 24) page 6 5. [ poits] Frakli s robot army is surroudig you! a. [6 poits] Cosider the polar curves r = cos(θ) r = si(θ) + 2 Frakli s robot army occupies the shaded regio betwee these two curves. Write a expressio ivolvig itegrals that gives the area occupied by Frakli s robot army. Do ot evaluate ay itegrals. Area = 2 2π (si(θ) + 2) 2 dθ 2 π (cos(θ)) 2 dθ b. [5 poits] Your fried, Kazilla, pours her magic potio o the groud. Suddely, a flock of wild chickes surrouds you. The chickes occupy the shaded regio eclosed withi the polar curve r = + 2 cos(θ) as show below. Write a expressio ivolvig itegrals that gives the perimeter of the regio occupied by the flock of wild chickes. Do ot evaluate ay itegrals. We use the arc legth formula: Arc Legth = b a (r(θ)) 2 + (r (θ)) 2 dθ Note that r (θ) = 2 si(θ). Also, the shaded regio of lies betwee θ = 2π/3 ad θ = 4π/3 (you ca see this by settig r(θ) =, ad testig that r(π) =, so it lies o the boudary of the shaded regio.) Arc Legth = 4π/3 2π/3 ( + 2 cos(θ)) 2 + ( 2 si(θ)) 2 dθ

7 Math 6 / Fial (December 2, 24) page 7 6. [8 poits] Suppose that f(x), g(x), h(x) ad k(x) are all positive, differetiable fuctios. Suppose that for all < x <, ad that < f(x) < x < g(x) < x 2 < h(x) < x 2 < k(x) < x for x >. Determie whether the followig statemets are always, sometimes or ever true by circlig the appropriate aswer. No justificatio is ecessary. a. [2 poits] g(x)dx coverges. Always Sometimes Never b. [2 poits] f(x)dx diverges. Always Sometimes Never c. [2 poits] h() diverges. Always Sometimes Never d. [2 poits] k() coverges. Always Sometimes Never

8 Math 6 / Fial (December 2, 24) page 8 7. [4 poits] Chickes cotiue to appear aroud you, ad Frakli s army is hesitat to advace. a. [6 poits] Let F (t) give the total umber of chickes that have arrived after t secods. You observe that F (t) obeys the followig differetial equatio df dt = e F t 2. If there are iitially 2 chickes, fid a formula (i terms of t) for F (t). e F df = t 2 dt e F = t3 3 + C F (t) = l( t3 3 + C) Sice F () = 2, we see that so C = e 2, ad 2 = l(c) F (t) = l( t3 3 + e2 ) b. [4 poits] A large, familiar-lookig chicke steps forward from the flock ad clucks, Koo Koo Katcha!. This large chicke waddles towards Frakli followig the parametric equatios x(t) = si(πt) + y(t) = l(t + ) π where t is the time, i secods, after the chicke steps forward from the flock ad both x ad y are measured i feet. Fid the chicke s speed secods after it steps forward. Iclude uits. Now we plug these ito the speed formula x (t) = cos(πt) y (t) = t + Speed = (x (t)) 2 + (y (t)) 2 ) whe t =. Speed = (cos(π)) 2 + ( )2 = 22

9 Math 6 / Fial (December 2, 24) page 9 c. [4 poits] Frakli says, BEEP BOOP BEEP. YOU RE RIGHT, WHAT HAVE I BECOME? A sigle robot tear falls from Frakli s robot eye. Cosider the regio i the xy-plae bouded by y = si(x), x = π, x = 2π, ad the x-axis. The volume of Frakli s tear is x + 2 give by rotatig this regio aroud the x-axis. Write a itegral givig the volume of Frakli s tear. Do ot evaluate this itegral. 2π π π ( ) si(x) 2 dx x + 2

10 Math 6 / Fial (December 2, 24) page 8. [7 poits] Cosider the power series (x + 2) 3. a. [2 poits] At which x-value is the iterval of covergece of this power series cetered? This power series is cetered o x = 2. (x + 2) b. [5 poits] The radius of covergece for the power series 3 is 3. iterval of covergece for this power series. Thoroughly justify your aswer. Fid the Sice the radius of covergece for this power series is 3 ad it is cetered o x = 2, the iterval of covergece cotais the ope iterval ( 2 3, 2 + 3) = ( 5, ). Now we oly eed to check the edpoits x = 5 ad x =. ( + 2) For x = : 3 = diverges by the p-test with p = (this is the harmoic series). ( 5 + 2) ( ) For x = 5: 3 = which coverges by the alteratig series test. Therefore, the iterval of covergece for this power series is [ 5, ). 9. [5 poits] Fid the radius of covergece for the power series Let the -th term be deoted by a a + (2( + ))!x 2(+) (!) 2 = a (( + )!) 2 (2)!x 2 = (2 + 2)(2 + )x 2 (( + ) 2 (2)! (!) 2 x2 Therefore, we ca use the ratio test: a + lim = lim (2 + 2)(2 + )x 2 a (( + ) 2 = 4x2. So this series coverges for x with 4x 2 <, or rather with x 2 < 4 radius of covergece is /2. which implies that the

11 Math 6 / Fial (December 2, 24) page. [ poits] Determie whether the followig series coverge or diverge. Justify your aswers. ( ) l() a. [5 poits] =2 Note that this series is alteratig ad that the absolute values of the terms l() form a decreasig sequece that coverges to as approaches. Therefore, this series coverges by the alteratig series test. b. [5 poits] This ca be doe with a compariso or limit compariso test. For compariso: > ( ) 3 = 3 3 By the p-test with p = /2 <, we have that the series also diverges. diverges. By the compariso test,

12 Math 6 / Fial (December 2, 24) page 2. [9 poits] Circle all true statemets. a. [3 poits] The itegral si(x) dx I. coverges by the compariso test because < x ad Cdx coverges. si(x) C for some costat C for II. diverges by the compariso test because diverges. si(x) x for < x ad x dx III. diverges because lim x si(x). IV. coverges by the alteratig series test because the values of si(x) oscillate betwee ad. b. [3 poits] The series = e 2! e 2 I. coverges because lim! =. II. coverges because factorials grow faster tha expoetial fuctios. III. diverges by the ratio test. IV. diverges by the compariso test because e2! diverges. e for =,, 2, 3,... ad = e c. [3 poits] The differetial equatio dy = t(y 2)(l(y)) defied for t > ad y > has dt I. a ustable equilibrium solutio at t =. II. a stable equilibrium solutio at y = 2. III. a stable equilibrium solutio at y =. IV. a ustable equilibrium solutio at y = 2.

13 Math 6 / Fial (December 2, 24) page 3 2. [8 poits] Frakli, your friedly ew eighbor, is buildig a large chicke sactuary. You decide to help Frakli build a special chicke coop with volume (i cubic km) give by the itegral x cos(x 2 ) dx. This itegral is difficult to evaluate precisely, so you decide to use the methods you ve leared this semester to help out Frakli. Your fried ad presidet-elect, Kazilla, stops by to give you a had. She suggests fidig the 4th degree Taylor polyomial, P 4 (x), for the fuctio cos(x 2 ) ear x =. a. [4 poits] Fid P 4 (x). We ca use the Taylor series expasio for cos(x 2 ) so Therefore cos(y) = cos(x 2 ) = = ( ) y 2 (2)! ( ) (x 2 ) 2 = (2)! = y2 2! + + ( ) y 2 + (2)! = (x2 ) 2 2! P 4 = ( x4 2! ) = x4 2! + + ( ) (x 2 ) 2 (2)! + b. [4 poits] Substitute P 4 (x) for cos(x 2 ) i the itegral ad compute the resultig itegral by had, showig all of your work. x P 4 (x)dx = = = x4 x x 4 x = x=

14 Math 6 / Fial (December 2, 24) page 4 Kow Taylor series (all aroud x = ): si(x) = = ( ) x 2+ (2 + )! = x x3 3! + + ( ) x 2+ + for all values of x (2 + )! ( ) x 2 cos(x) = (2)! = = x2 2! + + ( ) x 2 + for all values of x (2)! e x = = x! = + x + x2 2! + + x + for all values of x! ( ) + x l( + x) = = x x2 2 + x3 3 + ( )+ x + for < x ( + x) p = + px + p(p ) 2! x 2 + p(p )(p 2) 3! x 3 + for < x < x = x = + x + x 2 + x x + for < x < =