8/11/2011 FINAL EXAM PRACTICE I Maths 21a, O. Knill, Summer 2011Name:• 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 un s t ap l e the packet.• Please try to write nea t ly. Answers which are illegible for the grader ca n not be givencredit.• No not 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 h ave 180 minutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 1011 1012 1013 10Total: 1401Problem 1) ( 20 points)1)T FThe line ~r(t) = ht, t, −ti is is contained in the plane x + y + z = 1.2)T FThe qu ad r a ti c surface x2+ y2= z2is an ellipti c paraboloid.3)T FIf~T (t),~B(t),~N(t) are the unit tangent, n or m a l an d b i n or m a l vectors of acurve with ~r′(t) 6= 0 everywhere, then~T (t) ·~B(t) ×~N(t) is always equal to1 or −1.4)T FIf |~u ×~v| = 0, then Proj~v(~u) = ~u.5)T FThere is a vector field~F (x, y) which has the property curl(~F ) = div(~F ) = 1.where curl(~F )(x, y) = Qx(x, y) − Py(x, y) and div(~F )(x, y) = Px(x, y) +Qy(x, y).6)T FThe acceleration vector ~r′′(t) = hx(t), y(t)i is always in the plane spannedby the vector ~r(t ) and the velocity vector ~r′(t).7)T FFor every curve on the unit sphere, the curvature is constant and equ a l to1.8)T FIf a smooth fu n c ti o n f(x, y) has n o maximum nor minimum, then it doesnot have a critica l point.9)T FThe linearization L(x, y) of a cubic function f(x, y) = x3+y3is the functionL(x, y) = 3x2+ 3y2.10)T FIf~F (x, y) is a gradient field~F = ∇f and ~r(t) is a flow line satisfying~r′(t) =~F (~r(t)) th enddtf(~r(t)) = |F |2(~r(t)).11)T FIf f + g and f − g have a common critical point (a, b), then this point is acritical point of both f and g.12)T FAssume a vector field~F (x, y, z) is a gradient field, thenRC~F ·~dr = 0 whereC is the intersection of x2+ y2= 1 with z = 1.13)T FIf the flu x of vector field is zero through any surface S i n space, then thedivergence of the field is zero everywhere in space.14)T FThe curl of a gradient field~F (x, y, z) = ∇f(x, y, z) is z er o, if f(x, y, z) =√x10+ y10z2.15)T FThe line integral of the curl of a vector field~F (x, y, z) = hx, y, zi along acircle in th e xy− plane is zero.16)T FFor a solid E which is rotationally symmetric around the z-axes, the integralRRRE√x2+ y2dxdydz is equal to the volume of the solid.17)T FThe curvature of th e curve ~r(t) = h1+2 cos(1+t), 1+2 sin ( 1+t) i is constantequal to 1/2 everywhere.18)T FThe directional derivative of f(x, y, z) = div(~F (x, y, z)) of the divergence ofthe vector field~F = hP, Q, Ri in the directi on ~v = h1, 0, 0i is Pxx+Qxy+Rxz.19)T FR2π0R2π0r d θ dr =R2π0R2π01 dxdy.20)T FThe set {φ = π, ρ > 0 } in spherical coordinates is the negative z-axis.2Problem 2) (1 0 points) No justifications are necessary.a) (4 points) Match the vector fields with the definitions1 23 4Enter vector field1-4~F (x, y) = hx + y, x − yi~F (x, y) = h0, xi~F (x, y) = h−y, xi~F (x, y) = hx, 0ib) (3 points) Match the p a r t ia l differential equation s (PDE’s) with their names1) Wave equa t io n2) Heat equati on3) Tran s port equation4) Bu rg er s equationEnter 1-4 PDEut− uxx= 0utt− uxx= 0ut− ux= 0c) (3 points) Match the cu r ves1234Enter 1-4 Parametrized curve~r(t ) = hcos(4t), sin(7t ) i~r(t ) = h√t sin(t),√t cos(t)i~r(t ) = h|cos(4t)|, |sin ( 7 t) | i~r(t ) = ht3, t4i3Problem 3) (1 0 points) No justifications are necessarya) (6 points) Check the boxes which apply. Leave the other boxes empty. The expr essi on”involves XYZ” means that the formulation of the statement contains the object XYZsomewhere.Statement involves involves involvesa curve a surfa ce a vector fieldStokes theoremDivergence theoremLagrange equationsFund. theorem line integralsSurface area formulaCurvature formulab) (4 points) Match the objects with their definitions1 23 4Enter 1-4 object definition~r(t ) = hcos(3t), sin(t ) , cos(2t)icos(3x) + sin( y) + cos(2z) = 1~r(t , s) = hcos(3t), sin(s), c os( 2t ) i~F (x, y, z) = hcos(3x), sin(y), cos(2z)i4Problem 4) (1 0 points)a) (5 Points) Write d own a parametri za t io n ~r(t) of the linewhich is perpendicular to the plane x + 2y + z = 0 and wh i chpasses th r o u gh the origin.b) (5 points) Fi n d the distance of this line t o the point (3, 4, 5).Problem 5) (1 0 points)Find t he place where the curlf(x, y) = curl(~F )(x, y) = Qx(x, y) − Py(x, y)of the vector field~F (x, y) = hP (x, y), Q(x, y)i = hx + y2+ y, x2y + 2x + y2iis maxi m a l und er the constraint that the divergenceg(x, y) = div(~F )(x, y) = Px(x, y) + Qy(x, y) .is equal to 1. Find the functions f, g and so lve the problemusing Lagrange.Problem 6) (1 0 points)a) (5 points) Find the surface area of th e surface~r(s , t) = hs cos(t), s sin ( t) , tiwith 1 ≤ s ≤ 2, 0 ≤ t ≤ 4π.b) (5 points) Find the arc length of the cur ve~r( t ) = hcos(t), sin(t) , tiwith 0 ≤ t ≤ 4π.Hint. You can use without derivation the during l ect u r e derived integralR√x2+ 1 dx =5(x√x2+ 1 + arcsinh(x))/2 and you can leave t er m s like arcsinh(2).Problem 7) (1 0 points)Find th e volume of the solid given in spherical coordinates asρ(φ, θ) ≤ cos2(φ) .with 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π.Problem 8) (1 0 points)Find t he fluxRRS~F ·~dS of the vector field~F (x, y, z) = hx3, y3, z + (1 −x2− y2)(1 −z2)ithrough the boundary S of the solid cylinderE : x2+ y2≤ 1, z2≤ 1 .The boundary of S of the solid E is oriented outwards asusual.Problem 9) (1 0 points)Where on the sphere is the functionf(φ, θ) = sin(φ) + sin ( θ)extremal? Fi n d all maxima, minima and s ad dle points of fas well as the global maxima and minima.Remark. The var i ab l es φ, θ are the usual spherical coordi-nates variables. You are welcome of course to write f ( x , y) =sin(x) + sin( y) and look for solutions 0 …
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