MATH 201 Exam 2, Fall 2014

Math 201 — Fall 2009–10 Calculus and Analytic Geometry III, sections 1–8, 24–26 Quiz 2, December 2 — Duration: 1 hour GRADES: 1 (/15) 2 (/15) 3 (/15) 4 (/16) 5 (/19) 6 (/20) TOTAL/100 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 three extra 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. Let the function f(x) be given by f(x) =    0, when 0 ≤ x < π x − π, when π ≤ x < 2π and f(x) is periodic with period 2π. a) (5 pts) Sketch the graph of f(x) for x ∈ [−2π, 4π]. b) (10 pts) The Fourier series of f(x) is � n≥0 an cos nx+� n≥1 bn sin nx. Find ONLY the coefficients bn. 2. a) (6 pts) Plot the polar graph of the curve C : r = 1 + sin θ. Also draw the line L : y = 4/9 on your graph. b) (3 pts) Convert the equation of L to polar coordinates. c) (6 pts) Find the (r, θ)-coordinates of the two points of intersection on L ∩ C. 3. Consider the following moving point in space: P(t) = (3t, √ 6 et, 1 2e2t). a) (5 pts) Find the velocity and the speed of P(t) at the instant t = 0. b) (5 pts) What is the arclength of the curve given by P(t) for 0 ≤ t ≤ ln 5? Simplify your answer. c) (5 pts) Suppose we have a function f(x, y, z) with the property ⃗∇f|(3, √ 6 e, 1 2 e2) = (e2, − √ 6 e, 5). Find d dtf(P(t)) at the instant when the point P(t) passes through (3, √ 6 e, 1 2e2). 4. a) (8 pts) Show that lim (x,y)→(0,0) x2y x4 + y2 does not exist. b) (8 pts) Show that lim (x,y)→(0,0) x2y x2 + y2 does exist (hint: the limit is equal to 0). 5. Consider the function f(x, y, z) = zex3y. a) (6 pts) Find the gradient of f(x, y, z) at P0(1, 1, −1). b) (7 pts) Find the equation of the tangent plane to the surface f(x, y, z) = −e at P0. c) (6 pts) Determine the direction in which f(x, y, z) increases most rapidly when the point (x, y, z) moves away from P0. Your answer should be a unit vector. 6. Given a function f(x, y) satisfying f(1, 2) = 4, ⃗∇f ��� (1,2)= (3, 4). a) (6 pts) Approximately how much is f(1.03, 1.99)? b) (7 pts) Find a direction ⃗u in which the directional derivative D⃗uf ��� (1,2)= 0. Your answer ⃗u should be a unit vector. c) (7 pts) Let S be the graph of f. In other words, S = {(x, y, z) ∈ R3 | z = f(x, y)}. Find the equation of the tangent plane to S at the point P0(1, 2, 4) ∈ S. (Be careful.) ii 1. Let the function f(x) be given by f(x) =    0, when 0 ≤ x < π x − π, when π ≤ x < 2π and f(x) is periodic with period 2π. a) (5 pts) Sketch the graph of f(x) for x ∈ [−2π, 4π]. b) (10 pts) The Fourier series of f(x) is � n≥0 an cos nx+� n≥1 bn sin nx. Find ONLY the coefficients bn. 1 2. a) (6 pts) Plot the polar graph of the curve C : r = 1 + sin θ. Also draw the line L : y = 4/9 on your graph. b) (3 pts) Convert the equation of L to polar coordinates. c) (6 pts) Find the (r, θ)-coordinates of the two points of intersection on L ∩ C. 2 3. Consider the following moving point in space: P(t) = (3t, √ 6 et, 1 2e2t). a) (5 pts) Find the velocity and the speed of P(t) at the instant t = 0. b) (5 pts) What is the arclength of the curve given by P(t) for 0 ≤ t ≤ ln 5? Simplify your answer. c) (5 pts) Suppose we have a function f(x, y, z) with the property ⃗∇f|(3, √ 6 e, 1 2 e2) = (e2, − √ 6 e, 5). Find d dtf(P(t)) at the instant when the point P(t) passes through (3, √ 6 e, 1 2e2). 3 4. a) (8 pts) Show that lim (x,y)→(0,0) x2y x4 + y2 does not exist. b) (8 pts) Show that lim (x,y)→(0,0) x2y x2 + y2 does exist (hint: the limit is equal to 0). 4 5. Consider the function f(x, y, z) = zex3y. a) (6 pts) Find the gradient of f(x, y, z) at P0(1, 1, −1). b) (7 pts) Find the equation of the tangent plane to the surface f(x, y, z) = −e at P0. c) (6 pts) Determine the direction in which f(x, y, z) increases most rapidly when the point (x, y, z) moves away from P0. Your answer should be a unit vector. 5 6. Given a function f(x, y) satisfying f(1, 2) = 4, ⃗∇f ��� (1,2)= (3, 4). a) (6 pts) Approximately how much is f(1.03, 1.99)? b) (7 pts) Find a direction ⃗u in which the directional derivative D⃗uf ��� (1,2)= 0. Your answer ⃗u should be a unit vector. c) (7 pts) Let S be the graph of f. In other words, S = {(x, y, z) ∈ R3 | z = f(x, y)}. Find the equation of the tangent plane to S at the point P0(1, 2, 4) ∈ S. (Be careful.) 6 Blank sheet. 7 Blank sheet. 8 Blank sheet. 9