8/5/2010 FINAL EXAM PRACTICE I 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 l i n e ~r(t) = ht, t, ti is perpendicular to the plane x + y + z = 10.2)T FThe q u adratic surface −x2+ y2+ z2= −1 is a one sheeted hyperboloid.3)T FThe relation |~u × ~v| = |~u · ~v| is only possible if at least one of the vectors ~uand ~v is the zero vector.4)T FRπ/20R10r3dθ dr =R10R10x2+ y2dxdy.5)T FIf a vector field~F (x, y) satisfies curl(~F )(x, y) = Qx(x, y) − Py(x, y) = 0 anddiv(~F )(x, y) = Px(x, y) +Qy(x, y) = 0 for all points (x, y) in the plan e, then~F is a constant field.6)T FThe acceleration vector ~r′′(t) = hx(t), y(t)i, the velocity vector ~r′(t) and~r′(t) × ~r′′(t) form three vectors which are mutually perpendicular.7)T FThe curvature of the curve ~r(t) = hsin ( 2t ) , 0, cos(2t)i is equal to the curva-ture of the curve ~s(t) = h0, cos(3t), sin ( 3t ) i .8)T FThe space curve ~r(t) = ht sin(t), t cos(t), t2i for t ∈ [0, 10π] is located on acone.9)T FIf a smooth funct i on f(x, y) has a global maximum, then this maximum isa cri t i cal point.10)T FIf L(x, y) is the linearization of f(x, y) and ~s(t) is the line tangent to thecurve ~r(t) at t0. The n d/dtL(~s(t)) = d/dt f (~r(t)) at the time t = t0.11)T FIf~F is a gradient field and ~r(t) is a flow line defined by ~r′(t) =~F (~r(t)),then the l ine integralR10~F ·~dr is either positive or zero.12)T FIf we extremize the function f(x, y) un d er the constraint g(x, y) = 1, andthe functions are t h e same f = g, we have infinitely many extrema.13)T FIf a point (x0, y0) is a critical point of f(x, y) under the constraint g(x, y) =1, then it is also a critical point of the function f(x, y) withou t constraints.14)T FIf a vector field~F (x, y) is a gradient field, then any line integral along anyellipse is z er o.15)T FThe flux of an irrotational vector field is zero through any surface S inspace.16)T FThe divergence of a gradient field~F (x, y) = ∇f(x, y) is zero.17)T FThe line integral of a vector field~F (x, y, z) = hx, y, zi along a circle in thexy− plane is zero.18)T FFor any solid E, the moment of inertiaRRREx2+ y2dxdydz is always largerthan the volumeRRRE1 dxdydz.19)T FThe c u r vature of a circle is always larger than the acceleration.20)T FThe di r ect i onal der i vative of div (~F (x, y)) of the divergence of the vectorfield~F = hP, Qi in th e directi on ~v = h1, 0i is Pxx+ Qxy.2Problem 2) (10 points) Match objects with definitions. No justifications necessary.Match the objects with their definitions1 23 45 678Enter 1-8 or 0 i f no match Object definition~r(t) = h(2 + cos(10t)) c os ( t ) , (2 + cos(10t)) sin(t), sin(1 0t ) i~F (x, y, z) = h−y, x, 2i~r(t, s) = h(2 + cos(s)) cos(t), (2 + cos(s)) sin(t), sin(s)i{(x, y, z) | sin(x2) − cos(y2) = 1 }~F (x, y) = hx − y2, y − x2ixyz = 0x2+ y2− z2= 1{(x, y) | s in(x2sin(x))y + sin(y − x ) = c }~r(t) = hsin(t) + cos(5t), cos(t) + cos(6t)i3Problem 3) (10 points)a) (4 points) Check every box to the left, for which t h e missing part to the right is ∇f(1, 2).The function f(x, y) i s an arbitrary nice function like for exampl e f(x, y) = x − yx + y2.The curve ~r(t), wherever it appear s, parametrizes the level curve f(x, y) = f(1, 2) and hasthe property that ~r(0) = h1, 2i.Check Topic StatementLinearization L(x, y) = f(1, 2)+ ·hx − 1, y − 2iChain ruleddtf(~r(t))|t=0=·~r′(0)Steepest descent f decrea ses at (1, 2) most in the direction ofEstimation f(1 + 0. 1, 1.99) ∼ f(1, 2)+ ·h0.1, −0.01iDirectional derivative D~vf(1, 2) = ·~vLevel curve of f throu gh (1, 2) has the form ·hx − 1, y − 2i = 0Vector projection of ∇f (1, 2) onto ~v is ~v(~v· )/|~v|2Tangent line of ~r(t) at (1, 2) is parametrized by~R(s) = h1, 2i + sb) (3 points) The surfaces are given either as a parametrization or implicit l y. Match them.Each surface matches one definition.A B CD E FEnter A-F here Function or parametrizatio n~r(u, v) = hu2, v2, u2+ v2i~r(u, v) = h(1 + sin(u)) cos(v), (1 + sin(u)) sin(v), ui4x2+ y2− 9z2= 1x − 9y2+ 4z2= 1~r(u, v) = hu, v, sin(u2+ v2)i4x2+ 9y2= 1c) ( 3 points) Match the solids with t h e tripl e integrals:4Enter A-D 3D integral computing volumeR2π0Rπ/40R1/ cos(φ)0ρ2sin(φ) dρdφ d θRπ0Rππ/2Rsin(φ)0ρ2sin(φ) dρdφ d θRπ0Rππ/2R10ρ2sin(φ) dρd φ d θR2π0Rπ0Rθ0ρ2sin(φ) dρdφ d θA B C DProblem 4) (10 points)We want to determine whether the distance of thesphere S of radius 1 centered at P = (1, 2, 3) to theplane E : x + y + z = 1 is larger than the distance ofthe same sph e r e to the line L : x + y = y + z = x + z.a) (5 points) Find the distance from the sphere S tothe plane E.b) (5 points) F ind the distance from the sphere S tothe line L.Problem 5) (10 points)Where does the vector field~F (x, y) = hP, Qi = hy(x3− 3x), x(y3− 3y)ihave maximal or minimal curlf(x, y) = curl(~F )(x, y) = Qx(x, y) − Py(x, y) .a) (8 points) Find all extr em a and determine whetherthey are m axi m a , min im a or saddle points.b) (2 points) Is there a global maximum of f(x, y)?5Problem 6) (10 points)A sprinkler at position (0, 0, 1) throws out water wit hconstant speed and elevation angle 45 degrees. Thewater is under constant gravitational accelerationh0, 0, −10i.a) (5 points) Fin d the trajectory ~r(t), if the init i alvelocity is ~r′(t)|t=0= hcos(θ), sin(θ), 1i and write downthe formula for the arc l en gt h from t = 0 to t = 1. Youdo not have to star t evaluating the i ntegral.b) (5 points) All the trajecto r ies together form a surface~r(θ, t). Parametrize this surface …
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