MAT 127 Final ExamDecember 13, 2010 8:15-10:45amName:first name last nameID:Section: L01 L02 L03 (circle yours)MWF 9:35-10:30am TuTh 5:20-6:40pm TuTh 2:20-3:40pmDO NOT OPEN THIS EXAM YETInstructions(1) Fill in your name and Stony Brook ID number and circle your lecture number at the top ofthis cover sheet.(2) This exam is closed-book and closed-notes; no calculators, no phones.(3) Please write legibly to receive credit. Circle or box your final answers. If your solution toa problem does not fit on the page on which the problem is stated, please indicate on thatpage where in the e xam to find (the rest of) your solution.(4) You may continue your solutions on additional sheets of paper provided by the proctors. Ifyou do so, please write your name and ID number at the top of each of them and staple themto the back of the exam (stapler available); otherwise, these sheets may get lost.(5) Anything handed in will be graded; incorrect statements will be penalized even if they arein addition to complete and correct solutions. If you do not want something graded, pleaseerase it or cross it out.(6) Leave your answers in exact form (e.g.√2, not ≈ 1.4) and simplify them as much as possible(e.g. 1/2, not 2/4) to receive full credit.(7) Show your work; correct answers only will receive only partial credit (unless noted otherwise).(8) Be care ful to avoid making grievous errors that are subject to heavy penalties.(9) If you need more blank paper, ask a proctor.Out of fairness to others, please stop working and close the exam as soon as the time iscalled. A significant number of points will be taken off your exam score if you continue workingafter the time is calle d. You will be given a two-minute warning before the end.Some Taylor Series11 − x=∞Xn=0xnif |x| < 1 cos x =∞Xn=0(−1)nx2n(2n)!ex=∞Xn=0xnn!sin x =∞Xn=0(−1)nx2n+1(2n+1)!to receive full credit, justify any other power series expansion you use1 (10pts)2 (10pts)3ab+c (20pts)4 (20pts)5 (20pts)Subtotal (80pts)6abc+d (10pts)7 (20pts)8 (10pts)9 (10pts)10A/B (20pts)Subtotal (70pts)Total (150pts)Problem 1 (10pts)Answer Only. Determine whether each of the following sequences or series converges or not. Ineach case, clearly circle either YES or NO, but not both. Each correct answer is worth 2 points.You may use the blank space between the questions to figure out the answer, but no partial creditwill be awarded and no justification is exp e cted for your answers on this problem.(a) the sequence an= 1 +cos3nnYES NO(b) the sequence an= n2(1 − e1/n) YES NO(c) the series∞Xn=1n + (−1)nn2+ 1YES NO(d) the series∞Xn=1(−1)nn2n + 1YES NO(e) the series∞Xn=12n√3n+ 5nYES NOProblem 2 (10pts)Answer Only. P ut your answer to each question in the corresponding box in the simplest possibleform. No credit will be awarded if the answer i n the box is wrong; partial credit may be awardedif the answer in the box is correct, but not in the simplest possible form.(a; 5pts) Write the number 1.109 = 1.1090909 . . . as a simple fraction(b; 5pts) Find the li mit of the sequence recursively defined bya1= 4, an+1= 4 −3ann ≥ 1.Assume that this sequence c onverges.Problem 3 (20pts)Find Taylor series expansions of the following functions around the given point. In each case,determine the radius of convergence of the resulting power series and its interval of convergence.(a; 10pts) f(x) = x2+ 2x around x = −2(b; 10pts) f(x) =x4 + x2around x = 0(c; bonus 10pts)1f(x) =15 − 12x2+ 4x4around x = 01this part is relatively hard and subject to harsh grading; do an d doub le-check the rest of the exam firstProblem 4 (20pts)(a; 8pts) Find the radius and interval of convergence of the power seriesf(x) =∞Xn=1xn√n.(b; 4pts) Find limx−→0f(x) −xx2(c; 8pts) Find the Taylor series expansion for the function g = g(x) given byg(x) =Zx0f(u) − uu2duaround x = 0. What are the radius and interval of convergence of this power series?Problem 5 (20pts)Show that the following series are convergent and find their sums.(a; 10pts)∞Xn=0(−1)nπ2n9n(2n)!(b; 10pts)∞Xn=1n2n3nProblem 6 (10pts)All questions in this problem refer to the series∞Xn=1(−1)n−1(n+2) · n! · 2n(a; 3pts) Explain why this series converges.(b; 4pts) What is the minimal numb er of terms required to approximate the sum of this series witherror le ss than 1/1000? Justify your answer.(c; 3pts) Based on your answer in part (b), estimate the sum of this series with error less than1/1000; leave your answer as a simple fraction p/q for some integers p and q with no commonfactor. Is your estimate an under- or over-estimate for the sum? Explain why. (If you do not knowhow to do (b), take the answer to (b) to be 2).(d; bonus 8pts)2Find the sum of the infinite series exactly.2this part is relatively hard and subject to harsh grading; do an d doub le-check the rest of the exam firstProblem 7 (20pts)Find the general real solution to each of the following differential equations.(a; 6pts) y′′= 0, y = y(x)(b; 7pts) y′′+ 4y′+ 5y = 0, y = y(x)(c; 7pts) y′′+ 4y′− 5y = 0, y = y(x)Problem 8 (10pts)Consider the four differential equations for y = y(x):(a) y′= x(x2− 1) (b) y′= x(y2− 1) (c) y′= y(x2− 1) (d) y′= y(y2− 1).Each of the two diagrams below shows the direction field for one of these equations:Iyx11IIyx11Each of the two diagrams below shows three solution curves for one of these equations:IIIyx11IVyx11(so ALL three curves in diagram III are solution curves for either (a), or (b), or (c), or (d); same (?)for ALL three curves)Match each of the diagrams to the corresponding differential equation (the match is one-to-one):diagram I II III IVequationAnswer Only: no explanation is required.do not write below this line or your work on this problem will be voidgrader’s use onlycorrect − repeats 0- 1 2 3 4points 0 2 5 9 10Problem 9 (10pts)Answer Only. A two-species interaction is modeled by the following system of differential equa-tions(dxdt= x −110x2−140xydydt=12y −1100xy(x, y) = (x(t), y(t)),where t denotes time.(a; 2pts) Which of the following best describes the interaction modeled by this system?(i) predator-prey (ii) competition for same resources (iii) cooperation for mutual benefitCircle your answer above.(b; 8pts) This system has 3 equilibrium (constant) solutions; find all of them and explain theirsignificance relative to the interaction the system is modeling. Put one equilibrium solution in eachbox below and use the space to the right of the box to