Name:Lab Hour:MATH 215 – Fall 2001FINAL EXAMThis exam contains 9 problems on 11 sheets of paper. The problems are worth a total of200 points, with each problem weighted as indicated. Do all of your work in this booklet andWRITE YOUR ANSWER IN THE CORRESPONDING BOX. SHOW YOUR WORK!!Problem Possible points SCORE1 152 203 254 205 206 407 208 209 20TOTAL 2001Problem 1.- (15 points) Use Green’s theorem to computeIC(x3− y3)dx + (x3+ y3)dywhere C is the oriented boundary of the region between the circles x2+ y2= 2 and x2+ y2= 4.Answer:2Problem 2. (20 points) Find the absolute maximum and absolute minimum values of thefunction f(x, y) = (x−1)2+(y−1)2on the rectangular domain D = {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ 2}and the points where these values are attained.Answer: Max Value: Pt(s). attained:Min Value: Pt(s). attained:3Problem 3. (25 points)Find the volume of the solid that lies within the sphere x2+ y2+ z2= 4, above the xy-planeand below the cone z =px2+ y2.Final Answer:4Problem 4.- (4x5=20 points) Please write the correct answers in the appropriate boxes. NOPARTIAL CREDIT.(I) Suppose that f(x, y, z) is a smooth function and~F (x, y, z) a smooth vector field in space.Next to each of the following expressions, write one of: “(s)calar”, “(v)ector” or “(m)eaningless”,according to whether the expression is scalar-valued, vector-valued or makes no sense:(a) the directional derivative of f in the direction (1, 1, 1)(b) curl (grad f)(c) div (grad~F )(d) grad (div~F )(e) curl (curl~F )(II) CIRCLE THE CORRECT ANSWER: Which of the following vector fields is conservative?~F (x, y, z) = 2xyi + (x2+ 2yz)j + y2k ,~G(x, y, z) = 2xyzi + (x2+ xyz)j + xy2k .(a)~F is conservative but~G is not.(b)~G is conservative but~F is not.(c) Both~F and~G are conservative.(d) Neither of them is conservative.5(III) CIRCLE THE CORRECT ANSWER: ComputeRC2xds where C is the curve given by{C : x = t, y = t2, 0 ≤ t ≤ 2}.(a) 1/12(b) (√17 − 1)/6(c) (17√17 − 1)/6.(d) (√15 − 1)/12(e) (15√15 − 3)/6(f) None of the above.(IV) CIRCLE THE CORRECT ANSWER:If~F is a vector field which is the curl of a vector function~G then:(a)~F is a constant vector field.(b)~F is the zero vector field.(c)~F has zero divergence.(d)~F has zero curl.(e) None of the above.6Problem 5. (5+8+7 points)a) Consider the wave equation∂2u(x, t)∂t2= a2∂2u(x, t)∂x2.Which one of the following functions solves the equation?CIRCLE THE CORRECT ANSWER. NO PARTIAL CREDIT.a) u(x, t) = cos(x − at)b) u(x, t) = cos(ax − t)c) u(x, t) = sin a(x −t)d) u(x, t) = a sin(x −t)e) none of the above.b) Suppose a particle in moving in the plane such that its velocity has constant magnitude.Which of the following is true?CIRCLE THE CORRECT ANSWER. NO PARTIAL CREDIT.a) Its velocity is perpendicular to its distance from the origin,b) Its velocity is zero.c) Its velocity is perpendicular to its acceleration.d) Its acceleration is zero.e) all of the abovef) none of the abovec) The equation r = 2 cos θ (in polar coordinates) isa) A parabola that is convex upb) A parabola that is convex downc) A circle centered to the left of the origind) A circle centered to the right of the origine) A two leaved rosef) none of the aboveCIRCLE THE CORRECT ANSWER. NO PARTIAL CREDIT.7Problem 6.- (10+15+15 points) a) Find the tangent plane to the surface given parametricallyby the equations x = u3, y = u2− v2, z = v at the point (1, 0, 1).Answer:b) Consider the surface S given by z = x2+ 4y, 0 ≤ x ≤ 2, 0 ≤ y ≤ 2. ComputeRRS2xdS overthis surface.Answer:c) Consider again the surface S given in b) with the upward orientation. Find the flux ofF = 2xi + 2xj across S.Answer:8Problem 7.- (20 points) Use Stokes’s theorem to evaluateHCF · dr where F (x, y, z) = e−xi +exj + ezk where C is the boundary of the plane 2x + y +2z = 2 in the first octant and is orientedcounterclockwise as seen from above.Answer:9Problem 8.- (20 points) Use the divergence theorem to evaluate the flux of F = (x2+ y4)i +x3z3j +px2+ y2k through the surface S, where S is the boundary of the region E below theparaboloid z = 5 − x2− y2and above the xy-plane.Answer:10Problem 9. (5x4=20 points) The graphs I and II below represent 2 planar vector fields~F = P i + Qj and two oriented curves C1and C2. NO PARTIAL CREDIT.(i) Is the line integralIC1P dx + Q dy in I positive, negative or zero?(ii) Is the line integralIC2P dx + Q dy in II positive, negative or zero?(iii) Is the planar curl of the vector field at the origin in I positive, negative or zero?(iv) Is the planar curl of the vector field at the origin in II positive, negative or zero?(v) Is the vector field in I conservative or not?(I)C1–2–112y–2 –1 1 2x(II)C2–2–1012–2 –1 1