8/5/2010 FINAL EXAM PRACTICE II Maths 21a, O. Knill, Summer 2010Name:• Start by printing your name in the above box.• Try to answer each question on the same page as the question is asked. If needed, usethe back or the next empty page for work.• Do not detach pages from this exam packet or unstaple the packet.• Please try to write neatly. Answers which are illegible for the grader can not be givencredit.• No n ot es, books, calculators, computers, or other electronic aids are allowed.• Problems 1-3 do not r equ i r e any justifications. For the rest of the p r ob l em s you have toshow your work. Even correct answers without derivation can not be given credit.• You have 180 m i nutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 1011 1012 1013 1014 10Total: 1501Problem 1) (20 points)1)T FThe q u adratic surface x2+ y − z2= −5 is a hyperbolic paraboloid.2)T FThere are vectors ~u and ~v such that |~u × ~v| > |~u||~v|.3)T FR2π0R50r dθ dr is the area of a disc of radius 5.4)T FIf a vector field~F (x, y) satisfies curl(F )(x, y) = Qx− Py= 0 for all points(x, y) in t h e plane, then~F is a gradient field.5)T FThe jerk of a parameterized curve ~r(t) = hx(t), y( t ) , z(t)i is p ar al l el to theacceleration if the curve ~r(t) is a line.6)T FThe curvature of the curve ~r(t) = h3 sin(t), 0, 3 cos(t)i is twice the curvatureof t h e curve ~s( t ) = h6 + 6 sin(t), 6 cos(t), 0i.7)T FThe curve ~r(t) = hsin(t) , t2, cos(t)i for t ∈ [0, 10π] is located on a cylinder.8)T FIf a function f(x, y) has the property that fx(x, y) is zero for all x, y, thenf is the constant function.9)T FIf the unit tangent vector~T (t) of a curve ~r(t) is always parallel to a planeΣ, t h en the curve is contained in a plane parallel to Σ.10)T FIf (x0, y0) is an extremum of f(x, y) under the constraint x2+ y2= 1, thenthe same point is an extremum of 10f(x, y) under the same con st r a int.11)T FAt a critical point (x0, y0) of a function f (x, y) for which fxx(x0, y0) > 0,the critical point is always a minimum.12)T FIf a vector field~F (x, y) is a gradient field , and C is a closed curve whichlooks like a figure 8, thenRC~F ·~dr is zero.13)T FIf C is part of a level curve of a function f(x, y) a n d~F = hfx, fyi is thegradient field of f, thenRC~F ·~dr = 0.14)T FThe divergence of the grad i ent vector field~F (x, y, z) = ∇f(x, y, z) is alwaysthe zero fun ct i on .15)T FThe line integral of the vector field~F (x, y, z) = hx, y, zi along a line segmentfrom (0, 0, 0) to (1, 1, 1) is 3/2.16)T FThe area of a region G can be expressed as a line integral along its bound-ary.17)T FThe flux of the vector field~F (x, y, z) = hx, y, −zi through the bounda r y Sof a solid ellipso id E is equal to the volume the ellipsoid.18)T FIf~F is a vector field in space and S is a torus surface, then the flux ofcurl(~F ) th r o u gh S is 0.19)T FIf the divergence and the curl of a vector field~F are both zero, then it is aconstant field.20)T FFor any function f, the curl of~F = grad ( f ) is the zero field h0, 0, 0i.2Problem 2) (10 points)a) ( 4 points) Match the regions with the corresponding double integralsa0.00.20.40.60.81.00.20.40.60.81.0b0.00.20.40.60.81.00.20.40.60.81.0c0.00.20.40.60.81.00.20.40.60.81.0d0.00.20.40.60.81.00.20.40.60.81.0Enter a,b,c,d FunctionR10Rxx/2f(x, y) dydxR10Ry0f(x, y) dxdyR10Rx/20f(x, y) dydxR10R1y/2f(x, y) dxdyb) (6 points) Match the parametrized or implicit surfaces with their definitionsA B CD E FEnter A-F here Function or parametriza t io n~r(u, v) = hcos(u), sin(u), vi~r(u, v) = hu − v, u + 2v, 2u + 3vix2+ y2/3 + z2/3 = 1~r(u, v) = h(sin(v) + 1) cos(u), (sin(v) + 1) sin(u), viz − x + sin(xy) = 0x2+ y2− z2= 03Problem 3) (10 points)a) ( 4 points) Match the vector fields and curves with the corresponding line integralI IIIII IVEnter I,II,III,IV Line integralR2π0hcos(t), sin(t)i · h− sin(t), cos(t)i dtR2π0h−t, t2i · h1, 1i dtR2π0ht2, ti · h1, 2ti dtR2π0h−3 sin(t), 3 cos(t)i · h− sin(t), cos(t)i dtb) (6 points) Fill in from following choice: ”arc length”, ”surface area”, ”chain rule”,”volume of parallelepiped”, ”area of par al le lo gr am ” , ”line integral”, ”flux integral”, ”cur-vature”.Formula Name of formula or rule or theoremR RR|~ru× ~rv| d ud vddtf(~r(t)) = ∇f(~r(t)) · ~r′(t)Rba|~r′(t)| dt|~r′(t)×~r′′(t)||~r′(t)|3|~u · (~v × ~w)|R10R10~F (~r(u, v)) · (~ru× ~rv) d ud v4Problem 4) (10 points)Given the line x − 1 = y − 2 = z − 3 and the point P = (8, 4, 5). Find the equationax + by + cz = dof t h e plane which contains the line and the point.Problem 5) (10 points)Find all the criti cal points of the function f(x, y) = y3− 3y2+ 4x + x2− 3 and classifythem by telling whether they are local ma xi m a, local minima or saddle points.Problem 6) (10 points)The hyperbolic paraboloid x2− y2− 3z = 0 contains the point P = (1, 1, 0) and the pointQ = (3, 0, 3). Find the tangent planes to the surface at P and Q and find a parametrization~r(t) of the line of intersection of these two planes.Problem 7) (10 points)A water reservoir in Burlington , MA (the map t o the right i s centered there) is boundedby a solid cylinder x2+ y2≤ 1. It has as the roof the cone x2+ y2= (z − 6)2and isbounded fro m below by the xy-plane z = 0. What is t h e volume of the reservoir?5Problem 8) (10 points)Find the maxim a and minima of the function f(x, y) = x2− y2on the parabola x + y2= 1using the Lagrange multiplier method.Problem 9) (10 points)Compute the surface area of the surface ~r(u, v) = hu3, v3, u3− v3i parametrized so th at(u, v) is in the unit disc.Problem 10) (10 points)Evaluate the following double integralZ20Z1x/2cos(y2) dy dx .Problem 11) (10 points)Find the value of t h e line integralZC~F (~r(t)) · ~r′(t) dt ,where~F (x, y) = hy + sin(cos(x)), −2xi and C is the boundary of the unit circle traversedin the cou nter clockwise direction.Problem 12) (10 points)Find the value of t h e flux integralZ ZScurl(~F )(~r(u, v)) · ~ru× ~rvdudvwhere~F (x, y, z) = h−y, x, zi and S is the part of the two-sheeted hyperboloid x2+y2−z2=−1 which satisfies 1 < z < 2 and which is oriented so that the normal vector pointsdownwards on S.6Problem 13) (10 points)Let E be the solid which is bounded on the si d e by t h e cone S1: x2+ y2= z2, 0 < z < 1and on top by the disc S2= x2+ y2≤ 1, z = 1. Let~F (x, y, z) = h1 …
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