MATH 201 Final, Fall 2009

Math 201 — Fall 2009–10 Calculus and Analytic Geometry III, sections 1–8, 24–26 Final Exam, February 1 — Duration: 2.5 hours GRADES: 1 (15 pts) 2 (12 pts) 3 (10 pts) 4 (12 pts) 5 (14 pts) 6 (12 pts) 7 (15 pts) 8 (15 pts) 9 (25 pts) 10 (28 pts) 11 (14 pts) 12 (18 pts) 13 (10 pts) Total/200 YOUR NAME: YOUR AUB ID#: PLEASE CIRCLE YOUR SECTION: Section 1 Section 2 Section 3 Section 4 Lecture MWF 3 Lecture MWF 3 Lecture MWF 3 Lecture MWF 3 Professor Makdisi Professor Makdisi Professor Makdisi Professor Makdisi Recitation F 11 Recitation F 2 Recitation F 4 Recitation F 9 Ms. Nassif Ms. Nassif Ms. Nassif Ms. Nassif Section 5 Section 6 Section 7 Section 8 Lecture MWF 10 Lecture MWF 10 Lecture MWF 10 Lecture MWF 10 Professor Raji Professor Raji Professor Raji Professor Raji Recitation T 11 Recitation T 3:30 Recitation T 8 Recitation T 2 Professor Raji Ms. Itani Ms. Itani Ms. Itani Section 24 Section 25 Section 26 Lecture MWF 2 Lecture MWF 2 Lecture MWF 2 Professor Tlas Professor Tlas Professor Tlas Recitation F 11 Recitation F 12 Recitation F 3 Dr. Yamani Dr. Yamani Professor Tlas INSTRUCTIONS: 1. Write your NAME and AUB ID number, and circle your SECTION above. 2. Solve the problems inside the booklet. Explain your steps precisely and clearly to ensure full credit. Partial solutions will receive partial credit. 3. You may use the back of each page for scratchwork OR for solutions. There are five blank sheets at the end, for extra scratchwork or solutions. If you need to continue a solution on another page, INDICATE CLEARLY WHERE THE GRADER SHOULD CONTINUE READING. 4. Closed book and notes. NO CALCULATORS ALLOWED. Turn OFF and put away any cell phones. GOOD LUCK! i An overview of the exam problems. Take a minute to look at all the questions, THEN solve each problem on its corresponding page INSIDE the booklet. 1. (5 pts each part, 15 pts total) Determine whether each of the following series converges or diverges: (a) ∞ � n=1 √ n2 − 1 √ n5 + 1 (b) ∞ � n=1 tan−1 n (c) ∞ � n=1 (n + 1)(0.5)n 2. (12 pts) Find the interval of convergence of the following series. Remember to test the endpoints. ∞ � n=1 (x − 5)n √n 3n 3. (5 pts each part, 10 pts total) a) Find the first four nonzero terms of the Taylor series for ex and cos 3x centered at a = 0. b) Compute the limit: lim x→0 ex − 1 − x 1 − cos 3x . 4. (6 pts each part, 12 pts total) a) Find the second-order Taylor polynomial of the function f(x) = √x, centered at x = 100. b) Using your answer P2 above, give an approximate value for √ 99 and estimate the error. 5. (7 pts each part, 14 pts total) We are given a function f(x, y) satisfying: ⃗∇f ��� (1,2,3)= (4, −5, 6), ⃗∇f ��� (2,4,6)= (8, −9, 10), ⃗∇f ��� (2,0,1)= (2, 0, 2). a) Which of the following differences do you expect to be largest, and why? f(1.1, 2.1, 3.1) − f(1, 2, 3), f(2.1, 4.1, 6.1) − f(2, 4, 6), f(2.1, 0.1, 1.1) − f(2, 0, 1). b) Let g(r, s) = f(r − s, r + s, 2rs). Find ∂g ∂s ��� (r,s)=(3,1). 6. (4 pts each part, 12 pts total) Some level curves of a function f(x, y) are drawn below. The rest of problem 6 is on page iii. ii 6, continued. a) Give two unit vectors ⃗uA, ⃗uB such that ⃗uA points in the same direction as ⃗∇f ��� A, and similarly ⃗uB points in the same direction as ⃗∇f ��� B. b) Based on the figure, what is a reasonable approximation to the directional deriva- tives D⃗uAf ��� A, D⃗uBf ��� B? c) Deduce a reasonable approximation to each of the vectors ⃗∇f ��� A and ⃗∇f ��� B. 7. (15 pts) Find and classify the critical points of the function f(x, y) = x4 12 − 3xy + 3y2 2 . 8. (15 pts total) Let f(x, y, z) = 8x + y + z, and consider the surface S given by xyz = 1 in the first octant only: S = {(x, y, z) | xyz = 1, x, y, z > 0}. a) (12 pts) Use Lagrange multipliers to find the minimum value of f on the surface S. b) (3 pts) Show that f does not have a maximum value on S. (Why are the values of f not bounded above on S?) 9. (5 pts each part, 25 pts total) Consider the integral I = � 1 x=0 � x y=0 (x + y) dydx. a) Draw the region of integration. b) Evaluate I. c) Set up but do not evaluate I in the order dxdy. d) Set up but do not evaluate I in polar coordinates, in the order drdθ. e) Set up but do not evaluate I in polar coordinates, in the order dθdr. 10. (7 pts each part, 28 pts total) In this problem, D is the solid ball with center (0, 0, 1) and radius 1. The equations for the boundary of D in various coordinate systems are: x2 + y2 + (z − 1)2 = 1, r2 + (z − 1)2 = 1, ρ = 2 cos ϕ. The density of D is δ(x, y, z) = � x2 + y2 + z2. We wish to compute the total mass of D. The rest of problem 10 is on page iv. iii Problem 10, continued. a) Set up but do not evaluate an integral giving the total mass of D in spherical coordinates. b) Evaluate the integral from part a). c) Do the same as part a) (without evaluating!) in cylindrical coordinates. d) Do the same as part a) (no evaluating!) in rectangular (i.e., xyz) coordinates. 11. (7 pts each part, 14 pts total) Consider the two vector fields in the plane ⃗F = (0, x), ⃗G = (0, y). a) One of the two vector fields is conservative. Explain why, and find a potential function for the conservative field. b) Let C be the cardioid, given in polar coordinates by r = 1 + cos θ. We orient C counterclockwise. Find the work integrals � C ⃗F · d⃗r and � C ⃗G · d⃗r. 12. (6 pts each part, 18 pts total) The curves C1, C2, and C3 are shown in the figure below. They all go from (2, 0) to (−2, 0). a) Compute the work integral � C1 ⃗F · d⃗r, where ⃗F = (−y, x) x2 + y2 . b) Do the same for the integral along C2. (Hint: what is the curl of ⃗F?) c) Do the same for C3, and use this to show that ⃗F is not conservative. 13. (10 pts) Given the vector field in space ⃗F = (x, y, z). Let S be the upper hemisphere of center (0, 0, 0) and radius 2, oriented with a normal vector that points away from the origin. Compute the flux �� S ⃗F · ⃗n dσ of ⃗F across S. iv 1. (5 pts each part, 15 pts total) Determine whether each of the following series converges or diverges: (a) ∞ � n=1 √ n2 − 1 √ n5 + 1 (b) ∞ � n=1 tan−1 n (c) ∞ � n=1 (n + 1)(0.5)n 1 2. (12 pts) Find the interval of convergence of the following series. Remember to test the endpoints. ∞ � n=1 (x − 5)n √n 3n 2 3. (5 pts each part, 10 pts total) a) Find the first four nonzero terms of the Taylor series for ex and cos 3x centered at a = 0. b) Compute the limit: lim x→0 ex − 1 − x 1 − cos 3x . 3 4. (6 pts each part, 12 pts total) a) Find the second-order Taylor polynomial of the function f(x) = √x, centered at x = 100. b) Using your answer P2 above, give an approximate value for √ 99 and estimate the error. 4 5. (7 pts each part, 14 pts total) We are given a function f(x, y) satisfying: ⃗∇f ��� (1,2,3)= (4, −5, 6), ⃗∇f ��� (2,4,6)= (8, −9, 10), ⃗∇f ��� (2,0,1)= (2, 0, 2). a) Which of the following differences do you expect to be largest, and why? f(1.1, 2.1, 3.1) − f(1, 2, 3), f(2.1, 4.1, 6.1) − f(2, 4, 6), f(2.1, 0.1, 1.1) − f(2, 0, 1). b) Let g(r, s) = f(r − s, r + s, 2rs). Find ∂g ∂s ��� (r,s)=(3,1). 5 6. (4 pts each part, 12 pts total) Some level curves of a function f(x, y) are drawn below. a) Give two unit vectors ⃗uA, ⃗uB such that ⃗uA points in the same direction as ⃗∇f ��� A, and similarly ⃗uB points in the same direction as ⃗∇f ��� B. b) Based on the figure, what is a reasonable approximation to the directional deriva- tives D⃗uAf ��� A, D⃗uBf ��� B? c) Deduce a reasonable approximation to each of the vectors ⃗∇f ��� A and ⃗∇f ��� B. 6 7. (15 pts) Find and classify the critical points of the function f(x, y) = x4 12 − 3xy + 3y2 2 . 7 8. (15 pts total) Let f(x, y, z) = 8x + y + z, and consider the surface S given by xyz = 1 in the first octant only: S = {(x, y, z) | xyz = 1, x, y, z > 0}. a) (12 pts) Use Lagrange multipliers to find the minimum value of f on the surface S. b) (3 pts) Show that f does not have a maximum value on S. (Why are the values of f not bounded above on S?) 8 9. (5 pts each part, 25 pts total) Consider the integral I = � 1 x=0 � x y=0 (x + y) dydx. a) Draw the region of integration. b) Evaluate I. c) Set up but do not evaluate I in the order dxdy. d) Set up but do not evaluate I in polar coordinates, in the order drdθ. e) Set up but do not evaluate I in polar coordinates, in the order dθdr. 9 10. (7 pts each part, 28 pts total) In this problem, D is the solid ball with center (0, 0, 1) and radius 1. The equations for the boundary of D in various coordinate systems are: x2 + y2 + (z − 1)2 = 1, r2 + (z − 1)2 = 1, ρ = 2 cos ϕ. The density of D is δ(x, y, z) = � x2 + y2 + z2. We wish to compute the total mass of D. a) Set up but do not evaluate an integral giving the total mass of D in spherical coordinates. b) Evaluate the integral from part a). c) Do the same as part a) (without evaluating!) in cylindrical coordinates. d) Do the same as part a) (no evaluating!) in rectangular (i.e., xyz) coordinates. 10 11. (7 pts each part, 14 pts total) Consider the two vector fields in the plane ⃗F = (0, x), ⃗G = (0, y). a) One of the two vector fields is conservative. Explain why, and find a potential function for the conservative field. b) Let C be the cardioid, given in polar coordinates by r = 1 + cos θ. We orient C counterclockwise. Find the work integrals � C ⃗F · d⃗r and � C ⃗G · d⃗r. 11 12. (6 pts each part, 18 pts total) The curves C1, C2, and C3 are shown in the figure below. They all go from (2, 0) to (−2, 0). a) Compute the work integral � C1 ⃗F · d⃗r, where ⃗F = (−y, x) x2 + y2 . b) Do the same for the integral along C2. (Hint: what is the curl of ⃗F?) c) Do the same for C3, and use this to show that ⃗F is not conservative. 12 13. (10 pts) Given the vector field in space ⃗F = (x, y, z). Let S be the upper hemisphere of center (0, 0, 0) and radius 2, oriented with a normal vector that points away from the origin. Compute the flux �� S ⃗F · ⃗n dσ of ⃗F across S. 13 Blank sheet. 14 Blank sheet. 15 Blank sheet. 16 Blank sheet. 17 Blank sheet. 18