MATH 201 Final, Fall 2005

Met To Se facan Out American University of Beirut Math 201, Fall 2005-06, sections 5-8 Final exam, February 1, 2006, 2.5 hours Remarks and instructions: Remember to write your name, AUB ID number, and SECTION on your exam booklet. The sections are: Section 5, Tu 11:00, with Ms. Jaber — Section 6, Tu 12:30, with Ms. Jaber — Section 7, Tu 2:00, with Ms. Jaber — Section 8, Tu 3:30, with Prof. Makdisi. The exam is open book and notes. Calculators are not allowed. Please make it clear in your exam booklet which problem you are solving on each page. Remember to justify your work carefully. The problems are listed in the order of the material in the book, not in order of increasing difficulty. Take a few minutes to look over the exam to decide which problems you wish to work on first. Remember to budget your time wisely. The total number of points on the exam is 157. Good luck! Question 1 (15 pts = 3 pts for each part). Which of the following series converge or diverge, and why? veel = ninn ; —. 4 +sin(n?) [nr a (a) tai (b) du ayyCO) aan (d) ds [FI (e) sia Ea Question 2 (12 pts = 6 pts for each part). O.4 (a) Express | In(1 + 2°) dz as a series. z=0 (b) Find a specific partial sum s,y of the above series such that |sv — L| < 107}, You do not have to check the hypotheses of the theorem that you use to estimate the error. Question 3 (12 pts = 6 pts for each part). (a) Find the third-order Taylor approximation Ps to f(z) =e” at the center 2 = 1. Your answer should have the form P3(2) = co +e1(@ — 1) + e2(2- 1)? +.¢3(a — 1)8 for specific numbers Cdye ey C3 (b) As usual, we write f(x) = P3(xr)+R3(2), where R3 is the error in the Taylor approximation. We assume that 0.9 < « < 1.1. Find an explicit constant A for which |R3(x)| < A. (You do not need to simplify your expression for A). Question 4 (12 pts). Find ail the critical points of f(z2,y) = et + Qxry + y?, and clagsify each critical point as a local maximum, a local minimum, or a saddle point. Queskul, 7 S33 rinale Question 5 (12 pts total). We wish to minimize and maximize the function f(2,y,z) = cy’=* on the triangle given by the part of the plane x 4 x+y +z = 2 lying in the first octant. (a) (2 pts) Use common sense to explain why the minimum value of side f on the triangle is 0, which is attained on the edges of the triangle. wen (b) (10 pts) Use Lagrange multipliers to find the maximum value of tein, s< f on the triangle. Note that by part (a) you can assume that 2, y, 2 #0. Question 6 (15 pts = 5 pts for each part). (a) Given the function f(c,y) = Vy + wy. Find Vf and Vf a2)" x (b) Use the linear approximation (ie., the increment theorem) to obtain an approximate value of f (1.02, 2.01). : 5 . Jad, (c) Given a moving point P(t) = (t2 + 4t + 4,¢? — 2), find the value of qe] at the instant when the moving point passes through (1, 2). ot ¢ 4 Bessive ; Question 7 (20 pts total). Let D be the solid upside-down pyramid with vertices O(0,0,0), A(1, 0,1), B(O, 1,1), and C(0,0, 1). Note that the equation of the plane OAB is z = x+y. The density of D is given by é(z,y,z) = x. (a) (9 pts for this part) Using wyz-coordinates, set up (5 pts) and evaluate (4 pts) an integral in the order ff, ty (ie., dy dz dz) that computes the total mass of D. (b) (5 pts) Set up but do not evaluate the same integral in the order fof J’. ME Sa Sy Jz (c) (6 pts) Set up but do not evaluate the same integral in cylindrical coordinates (any order is fine, but I recommend to £. fi.) Question 8 (16 pts = 8 pts for each part). Let D be the part of the solid sphere x? + y? + z? < 2 which is below the plane z= 1 and above the cone z = —\/a? + y?. (a) Set up but do not evaluate an integral for the volume of D in cylindrical coordinates. (b) Set up but do not evaluate an integral for the volume of D in spherical coordinates. Remark: for both (a) and (b), you will need to split your integral into two parts. Question 9 (12 pts = 6 pts for each part). {a) Find a potential function f(x, y,z) for the conservative vector field F = (y+ 1,2 + yz?, y2z + 27), (b) Show that the vector field G = (Qez,z? + y223, 27 + 2?y3) is not conservative (i.e., does not have a potential function). & ct . ~ 2 Question 10 (16 pts total). Given the vector field F = (0,32) 3 lem and the closed curve C given by parts of the parabola y = x? and 22 the line y = 2. We orient C counterclockwise and use an outward- ~ & pointing normal. 272 CQ ou 3 cf (a) (7 pts) Directly calculate the work (i.e., circulation) integral eh : F.Tds, by parametrizing C’. * C ; a LG (b) (7 pts) Use Green’s theorem to re-caleulate | F-Tds. \ JIC (c} (2 pts) Use Green’s theorem to calculate the flux integral F. fds. [ QQ vec on NY Question 11 (15 pts). Given the vector field F = (0,y,0) and the surface § shaped like a cup, given by the part of the paraboloid y = 2? + 27 — 3 cut off by the zz-plane. We orient S using the normal vector shown, which points generally away from the origin. Set up but do not evaluate the surface flux integral // F - ido in terms of an explicit integral that involves only x s and 2.