MAT 127 Final ExamDecember 14, 20098:15-10:45amName: ID:Section: L01 L02 L03 L04 (circle yours)MWF 9:35-10:30am MW 5:20-6:45pm TuTh 2:20-3:40pm TuTh 5:20-6:40pmDO NOT OPEN THIS EXAM YETInstructions(1) This exam is closed-book and closed-notes; no calculators, no phones.(2) Please write legibly. Circle or box your final answers.(3) Show your work. Correct answers only will receive only partial credit.(4) Simplify your answers as much as possible.(5) Leave your answers in exact form (e.g.√2, not ≈ 1.4).(6) If you need more blank paper, ask a proctor.(7) Please write your name and ID number on any additional sheets you’d like to be gradedand staple them to the back of the exam (stapler provided); indicate in the exam that thesolution continues on the attached sheets.(8) Anything handed in will be graded; incorrect statements will be penalized even if theyare in addition to complete and correct solutions. If you do not want something graded,please erase it or cross it out.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 called. 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)2ab+c (20pts)3 (20pts)4 (20pts)Subtotal (70pts)5 (20pts)6 (20pts)7 (20pts)8A/B (20pts)Subtotal (80pts)Total (150pts)Problem 1 (10pts)Determine whether each of the following sequences or series converges or not. In each case, clearlycircle either YES or NO, but not both. Each correct answer is worth 2 points. You may use theblank space between the questions to figure out the answer, but no partial credit will be awardedand no justification is expected for your answers on this problem.(a) the sequence an= 1 − (−1)nYES NO(b) the sequence an= 1 +cos(n)n2YES NO(c) the series∞Xn=1cos(1/n) YES NO(d) the series∞Xn=1(−1)n√nYES NO(e) the series∞Xn=1n + sin nn2YES NO3Problem 2 (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) = x3around x = 2(b; 10pts) f(x) =x1 − 4x2around x = 0(c; bonus 10pts)1f(x) =16 − 5x + x2around x = 01this part is relatively hard and subject to harsh grading; do and double-check the rest of the exam first4Problem 3 (20pts)(a; 8pts) Find the radius and interval of convergence of the power seriesf(x) =∞Xn=1√nxn.(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?5Problem 4 (20pts)Show that the following series are convergent and find their sums.(a; 10pts)∞Xn=02n(ln 3)nn!(b; 10pts)∞Xn=1(−1)nn2n6Problem 5 (20pts)Explain why each of the following series converges. Then estimate its sum to within 1/18 using theminimal possible number of terms, justifying your estimate; leave your answer as a simple fractionp/q for some integers p and q with no common factor. Is your estimate an under- or over-estimatefor the sum? Explain why.(a; 10pts)∞Xn=1(−1)n−1n2(b; 10pts)∞Xn=11n37Problem 6 (20pts)Find the general real solution to each of the following differential equations.(a; 6pts) 9y′′+ 4y = 0, y = y(x)(b; 7pts) 9y′′− 12y′+ 4y = 0, y = y(x)(c; 7pts) 9y′′− 12y′= 0, y = y(x)8Problem 7 (20pts)Consider the four differential equations for y = y(x):(a) y′= y − 1, (b) y′= y2− 1, (c) y′= x(y2− 1) (d) y′= x + y − 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 IVequationExplain your reasoning below.9Problem 8A (20pts)Only the higher of your scores on Problems 8A and 8B will count toward the total for the examA tank contains 100 liters of salt solution with 500 grams of salt dissolved in it. A salt solutioncontaining 2g of salt per liter enters the tank at a rate of 5 liters per minute. The solution is keptthoroughly mixed and drains at a rate of 5L/min (so the volume in the tank stays constant). Lety(t) be the amount of salt in the tank, measured in grams, after t minutes.(a; 8pts) Explain (based on the above information) why the function y =y(t) solves the initial-valueproblemy′= 10 −y20, y = y(t), y(0) = 500.(b; 8pts) Find the solution y =y(t) to the initial-value problem stated in (a).(c; 4pts) How long will it take for the amount of salt in the tank to reach 300 grams?10Problem 8B (20pts)Only the higher of your scores on Problems 8A and 8B will count toward the total for the exam(a; 8pts) Show that the orthogonal trajectories to the family of curves xy = k are described by thedifferential equationy′=xy, y = y(x).(b; 6pts) Find the general solution to the differential equation stated i n (a).(c; 6pts) Sketch at least 3 representatives of the original family of curves and at least 3 orthogonaltrajectories on the same diagram; indicate clearly which is which.11you can write on this