MATH 201 Final, Fall 2008

1 FINAL EXAMINATION MATH 201 January 24, 2008; 11:30 A.M.-1:30 P.M. Name: Signature: Circle Your Section Number: 17. 18 19 20 24 25 Instructions: • There are two types of questions: PART I consists of six work-out problems. Give detailed solutions. PART II consists of eight multiple-choice questions each with exactly one correct answer. Circle the appropriate answer. Grading policy: • 10 points for each problem of PART I. • 5 points for each problem of PART II. • 0 point for no, wrong, or more than one answer of PART II. GRADE OF PART I /60 GRADE OF PART II /40 TOTAL GRADE /100 2 Part I (1). Use Lagrange Multipliers to find the absolute maximum and minimum values for the function f(x, y, z) = x − y + z on the unit sphere x2 + y2 + z2 − 1 = 0. 3 Part I (2). Use cylindrical coordinates to find the volume of the solid bounded above by the surface z = e √x2 + y2 , below by the xy−plane, and laterally by the cylinder x2 + y2 = 1. 4 Part I (3). Find the absolute maximum and minimum values of the function f(x, y) = x2 +3y −3xy over the triangular region R bounded by the x−axis, the y−axis, and the line x + y = 6. 5 Part I (4). Evaluate the integral � � R 2y + x y − 2x dA, where R is the trapezoid with vertices (−1, 0), (−2, 0), (0, 4), and (0, 2), by using the substitution u = y − 2x and v = 2y + x. 6 7 Part I (5). Integrate the function f(x, y) = √x2 + y2 over the parametric curve C : −→r (t) = (cos t + t sin t)−→i + (sin t − t cos t)−→j , 0 ≤ t ≤ √ 3. 8 Part I (6). Sketch the region of integration of the integral � 4 0 � 2 √y x3 cos(xy) dx dy, and evaluate the integral by reversing its order of integration. 9 Part II (1). If f(x, y) satisfies Laplace’s equation fxx + fyy = 0, then the value of the integral � C fydx − fxdy over all simple closed curves C to which Green’s theorem applies is (a) 0. (b) 1. (c) −1. (d) π. (e) −π. Part II (2). The mass of the annular metal plate bounded by the circles r = 1 and r = 2 and whose density function is δ(r, θ) = cos2 θ is (a) π/4. (b) π/2. (c) 3π/4. (d) 3π/2. (e) None of the above. 10 Part II (3). The volume of the solid lying below the half-cone z = √x2 + y2 and inside the sphere x2 + y2 + z2 = 8 is given by the triple integral (a) � 2π 0 � π π/4 � 2 √ 0 2 ρ2 sin φ dρ dφ dθ. (b) � 2π 0 � π/4 π/3 � 2 √ 0 2 ρ sin2 φ dρ dφ dθ. (c) � 2π 0 � π/2 π/4 � 2 0 ρ2 sin φ dρ dφ dθ. (d) � π 0 � π π/4 � 2 0 ρ2 sin φ dρ dφ dθ. (e) None of the above. Part II (4). If an = � 1/√n; 1 ≤ n ≤ 100 (−1)n/n; n > 100, then the series �∞ n=1 an is (a) convergent. (b) absolutely convergent. (c) conditionally convergent. (d) divergent. (e) None of the above. 11 Part II (5). The power series ∞ � k=2 (−1)k (2x − 1)k k6k has interval of absolute convergence (a) −5/2 < x ≤ 7/2. (b) −5/2 < x < 7/2. (c) −5/2 ≤ x < 7/2. (d) −5/2 ≤ x ≤ 7/2. (e) None of the above. Part II (6). The directional derivative of the function f(x, y, z) = x2 + 4y2 − 9z2 at the point P(3, 0, −4) in the direction from P to the origin is (a) 52. (b) −52. (c) 54. (d) −54. (e) None of the above. 12 Part II (7) The function f(x, y) = x2 − 4xy + y3 + 4y (a) has local minimum at (4, 2) and local maximum at (4/3, 2/3). (b) has local minimum at (4, 2) and saddle point at (4/3, 2/3). (c) has local maximum at (4, 2) and saddle point at (4/3, 2/3). (d) has saddle point at (4, 2) and local minimum at (4/3, 2/3). (e) None of the above. Part II (8) An equation of the tangent plane to the surface 2x2−y+5z2 = 0 at the point (1, 7, 1) is (a) 3x − y + z = −3. (b) x + y + 2z = 10. (c) 10x − y + 4z = 7. (d) 4x − y + 10z = 7. (e) None of the above.