MATH 201 Exam, Spring 2008

Final Exam.; Math. 201; Jan. 31, 1996; 11:00 A.M.-1:00 P.M. Name: Signature: Student number: Section number (Encircle): 3 5 6 1. Instructions: • No calculators are allowed. • Make sure that you have exactly 20 questions. • Each question has only one correct answer. • Make sure to circle your answers appropiately below. 2. Grading policy: • 5 points for each correct answer. • -1 point (penalty) for each wrong answer. • 0 point for zero or more than one answer. Circle the appropiate answers: 1. a b c d 11. a b c d 2. a b c d 12. a b c d 3. a b c d 13. a b c d 4. a b c d 14. a b c d 5. a b c d 15. a b c d 6. a b c d 16. a b c d 7. a b c d 17. a b c d 8. a b c d 18. a b c d 9. a b c d 19. a b c d 10. a b c d 20. a b c d (0) 1 1. The area of the region lying inside the circle r = −3 cos θ and outside the cardioid r = 1 − cos θ is (a) 3(π + 3 √ 2 − 1). (b) π. (c) 2π. (d) 2(π + 2 √ 3 − 2). 2. The points of intersection of the curves r = 2 and r = 4 cos 2θ are (a) (2, 0), (2, π), (2, ±π/2). (b) (2, ±π/6), ±π/3, (2, ±2π/3), (2, ±5π/6). (c) (2, ±π/12), ±5π/12, (2, ±13π/12), (2, ±17π/12). (d) (2, ±π/4), (2, ±5π/4). 3. The surface x2/16 − y2/25 = z/4 is (a) a circular paraboloid. (b) a hyperbolic paraboloid. (c) an elliptic cone. (d) a one-sheeted hyperboloid. 4. If f(x, y) = x2y3/[x8 + y4] for (x, y) ̸= (0, 0) and f(0, 0) = 0, then (a) lim(x,y)→(0,0) f(x, y) = 1/2. (b) f is discontinuous at (0,0). (c) lim(x,y)→(0,0) f(x, y) = 0. (d) f is continuous at (0,0). (1) 2 5. An estimate of the error obtained by taking the first four terms of the series ∞ � n=1 (−1)n+1(0.01)n/n3 is (a) 1.56 × 10−10. (b) 8 × 10−11. (c) 1.56 × 10−12. (d) 8 × 10−13. 6. The interval of convergence of the power series ∞ � n=0 2n(x − 3)n/n2 is (a) 5/2 < x < 7/2. (b) 1 ≤ x ≤ 5. (c) 5/2 ≤ x ≤ 7/2. (d) 1 < x < 5. 7. The Maclaurian series of the integral � x 0 sint4 t2 dt is (a) �∞ n=0(−1)nx8n+1/(8n + 1)[(2n + 1)!]. (b) �∞ n=0(−1)nx8n+3/(8n + 3)[(2n + 1)!]. (c) �∞ n=0(−1)nx8n+1/(8n + 3)[(2n + 1)!]. (d) �∞ n=0(−1)nx8n+3/(8n + 3)[(2n + 3)!]. (2) 3 8. If an = ( 3 5)n + 4n n! and bn = n(e 1 n − 1), then (a) the sequences {an} and {bn} diverge. (b) the sequences {an} and {bn} converge. (c) the sequence {an} diverges and {bn} converges. (d) the sequence {an} converges and {bn} diverges. 9. If an = 2n+1 n2(n+1)2 and bn = 1 n(ln n)ln2, then (a) the series �∞ n=1 an and �∞ n=1 bn diverge. (b) the series �∞ n=1 an and �∞ n=1 bn converge. (c) the series �∞ n=1 an diverges and �∞ n=1 bn converges. (d) the series �∞ n=1 an converges and �∞ n=1 bn diverges. 10. If an = (1 − 1 n)n and bn = cos nπ √n , then (a) the series �∞ n=1 an diverges and �∞ n=1 bn converges. (b) the series �∞ n=1 an and �∞ n=1 bn converge. (c) the series �∞ n=1 an and �∞ n=1 bn diverge. (d) the series �∞ n=1 an converges and �∞ n=1 bn diverges. 11. The Taylor series of ln x at x = 3 is (a) ln 3 + �∞ n=1(−1)n(x − 3)n/n3n. (b) ln 3 + �∞ n=1(−1)n+1(x − 3)n/n!3n. (c) ln 3 + �∞ n=1(−1)n+1(x − 3)n/n3n. (d) ln 3 + �∞ n=1(−1)n(x − 3)n/n!3n. (3) 4 12. If z = f(x, y) where x = r cos θ and x = rsinθ, then (a) f 2 x + f 2 y = r2f 2 r + f 2 θ . (b) f 2 x + f 2 y = f 2 r + r−2f 2 θ . (c) f 2 x + f 2 y = f 2 r + f 2 θ . (d) f 2 x + f 2 y = f 2 r − f 2 θ . 13. An equation of the tangent plane to the surface x2+2xy+2y2+z = 12 at the point P(2, −3, 2)is (a) −8x + 18y − z = −72. (b) 2x + 8y − z = −22. (c) 2x + 18y − z = −52. (d) −8x + 8y − z = −42. 14. The tangent line to the curve of intersection of the paraboloid x2 + y2−z = 0 and the ellipsoid 2x2+3y2+z2−9 = 0 through the point P(1, −1, 2) has parametric equations: (a) x = 1 + 7t, y = −1 + 6t, z = 2 + 2t. (b) x = 1 + 5t, y = −1 − 7t, z = 2 + t. (c) x = 1 − 3t, y = −1 + 9t, z = 2 + 3t. (d) x = 1 + 4t, y = −1 − 7t, z = 2 − t. (4) 5 15. The rate of change of the temperature function T(x, y) = 4x2 − y2 + 16z2 at the point P(−1, 2, 1) is zero in the direction of the vector: (a) < 3, 4, −2 >. (b) < 1, −1, 2 >. (c) < 1, −2, 0 >. (d) < 0, 8, −1 >. 16. The function f(x, y) = 3x2 − 4xy + 5y2 − 7 admits -7 as a (a) local maximum value. (b) an absolute minimum value. (c) a value at a saddle point. (d) an absolute maximum value. 17. The value of the double integral � 2 0 � 1 y/2 yex3dxdy is (a) (e − 1)/4. (b) 5(e − 1)/4. (c) 2(e − 1)/3. (d) (e − 1)/3. (5) 6 18. The average value of the function e−(x2+y2) over the first quadrant of the disc x2 + y2 ≤ 1 is (a) (e − 1)/2. (b) e − 1. (c) 1 − e−1. (d) (1 − e−1)/2. 19. The volume of the solid bounded by the paraboloids z = 2x2 + y2 and z = 12 − x2 − 2y2 is (a) 20π. (b) 24π. (c) 18π. (d) 22π. 20. The solid whose volume is given by the triple integral in spherical coordinates as � 2π 0 � π/6 0 � 2cosφ 0 ρ2sinφdρdφdθ is bounded by (a) a cone and a cylinder. (b) a sphere and a cylinder. (c) a cone and a sphere. (d) a sphere and a paraboloid. (6) 7