7/14/2011 FIRST HOURLY PRACTICE II Maths 21a, O.Knill, Summer 2011Name:• Start by writing your name in the above box.• Try to answer each question on the sam e page as the question is asked. If need ed , usethe back or the next empty page for work. If you need additional paper, write your nameon it.• Do not detach pages from this exam packet or unstaple the packet.• Please write neatly. Answers which a re illegi b l e for the grader can not be given credit.• No notes, boo ks, calcu l at or s, comp u t er s , or other electronic aids can be allowed.• You have exactly 90 minutes to complete your work.1 202 103 104 105 106 107 108 109 1010 10Total: 1101Problem 1) (20 points) No justifications are needed.1)T FThe vector projection of h2, 3, 4i onto h1, 0, 0i is h2, 0, 0i.2)T FThe triple scalar product between three vectors is zero if and only if two ofthe vectors are parallel.3)T FThere are two vectors ~v an d ~w so that the dot product ~v · ~w is equal to thelength of the cross product |~v × ~w|.4)T FThe distance between two spheres of radius 1 whose centers have distanc e10 is 8.5)T FIf two vectors ~v and ~w are both parallel and perpendicula r , then one of thevectors must be the zero vector.6)T FThe curvature κ(~r(t)) is a lways smaller or equal than the length |~r′′(t)| ofthe acce ler a t io n vector ~r′′(t).7)T FThe cu r ve ~r ( t ) = hcos(t) sin(t), sin(t) sin(t), cos(t)i is located on a sphere.8)T FThe su r face x2+ y2+ z2= 2z is a sphere.9)T FThe length of the vector h4, 2, 4i is an integer.10)T FThe cu r vature of the curve h2 cos(t3), 2 sin(t3), 1i is co n st ant 2.11)T FThe graph of the funct i on f(x, y) = x2−y2is called an elliptic paraboloid.12)T FThe equ a t io n φ = 3π/2 in spherical coordinates defines a p l an e .13)T FThe vector h1, 2, 3i is perpendicular to the plane 2x + 4y + 6z = 4.14)T FThe cr oss product between h2, 3, 1i and h1, 1, 1i is 6.15)T FThe curve ~r ( t ) = hcos(t), t2, sin(t)i, 1 ≤ t ≤ 9 and the curve ~r(t) =hcos(t2), t4, sin(t2)i, 1 ≤ t ≤ 3 have th e same length.16)T FIf a stone falls for 3 seconds from height z = h to the ground z = 0 withgravitational acceleration −10 then the height is 30 meters.17)T FThe point (1, −1, 1) has the spherical coordinates the form (ρ, θ, φ) =(√3, π/4, π/4).18)T FThe point (1, −1, 1) has the cylindrical coordinates the form (r, θ, z) =(√3, π/4, 1).19)T FThe distance between two parallel lines in space is the distance of a pointon one line to the other line.20)T FFor two nonzero arbitrary vectors ~v and ~w the identity Proj~v(~v × ~w) =~0holds.Total2Problem 2) (10 points) No justifications are needed in this problem.a) (2 points) Match contour maps with functions f(x, y). Enter O, where n o match.I II IIIFunction f(x, y) = Enter I,II,IIIx2+ y2x2− y2x2− yx − y2b) ( 3 points) Match the graphs with the fu n ct i ons f(x, y). Enter O, wh er e no match.I II IIIFunction f(x, y) = Enter I,II,IIIx − y|x| − yx2− y2x2y2x − y3c) (2 points) Match the curves with their parametrizations ~r(t). Enter O, where no match.I II IIICurve ~r(t) = Enter I,II,IIIht, t2iht4, 1 + 2t4ih−t sin(t), t cos(t)ihsin(t), cos(t)id) ( 3 points) Match level surfaces with definition g(x, y, z) = 0. Enter O, where no match.I II IIIFunction g( x, y, z) = Enter I,II,II Ix2+ y2− z2x2− y2− 1x2− y2− zx2+ y2− zx − y + zProblem 3) (10 points) No justifications are needed in this problem3a) ( 5 points) Matching traces with surfaces.xy-trace yz-tracexz-tracexy-trace yz-tracexz-trace3A BC DThe figures above show the xy-trace,(the intersection of the surface withthe xy-pl an e ), the yz-trace (the in-tersection of the surface with the yz-plane), and the xz-trace (the intersec-tion of the surface with the xz-plane).Match the following equat i ons withthe t r aces . No justifications required.Enter A,B,C,D,E,F here Equationx2+ y2− (z − 1/3)2= 0x2− y2+ z = 0x2+ y2− z2− 1 = 0x2+ y2− z = 13b) ( 5 points) Matching parametrized surfaces.I II III IVMatch the para-metric surfaceswith their param-eterization. Nojustifications areneeded.Enter I,II,III,IV here Parametrization~r(u, v) = hu, v, v2− u2i~r(u, v) = hcos(u) sin(v), 2 sin(u) sin(v), cos(v)i~r(u, v) = h(v2+ 1) cos(u), (v2+ 1) sin(u), vi~r(u, v) = hu, 3, viProblem 4) (10 points)We want to find the distance between the lines x = y = z and (x−1)/2 = (y −2)/3 = (z −4)/4.4a) (4 points) Find a parametrization for each of the two lin es.b) ( 6 points) Find the distance between the two lines.Problem 5) (10 points)At the independence day celebrat i on on July 4, 2010 in Boston, two rockets were launchedat the same time. Their paths follow parabola:~r(t) = ht, t, 5 − t2i ,~R(t) = h2 − t, t, 4 + t − t2i .a) (3 points) They collide at some time t = t0. Fin d this time.b) ( 4 points) Compute the two velocity vectors ~r′(t) a n d~R′(t) a t t = t0.c) (3 points) Determine t h e cos of the angle of intersection between the curves at th e impactpoint.Problem 6) (10 points)An octahedron has 4 vertices A = (−1, −1, 0), B = (1, −1, 0) , C = (1, 1, 0) , D = (−1, 1, 0) inthe xy plane. Two other vertices are at E = (0, 0, a) and F = (0, 0, −a).a) (4 points) For which positive value of a is the distance between A and F equal to 2 and thesolid a regu l ar octahedron?b) ( 6 points) Find the distance between A and t h e line connectin g the points B and F .Problem 7) (10 points)5Let ~v = h3, 4, 5i, ~w = h1, 1, 1i. Compute the following expressions:a) (2 points) the area of the parallelogram spanned by ~v and ~w,b) ( 2 points) the vector (~v × ~w) × ~w,c) (2 points) the scalar ~v · ~w ,d) ( 2 points) the vector Proj~v(~w),e) (2 points) cos(α), where α is the angle between ~v and ~w.Problem 8) (10 points)a) (7 points) Find the arc length of the curve~r(t) = hcos(t2), sin(t2), t2ifrom 0 ≤ t ≤ 3.b) ( 3 points) Find the unit tangent vector~T (t) to ~r(t) at time t =qπ/2.Problem 9) (10 points)A stunt man wants to jump from the golden gate bridge from 8 0meters starting at ~r(0) = h0, 0, 80i. He moves from the plat for mwith initial velocity ~r′(0) = h0, 1, 0i . By the way, the swiss OliverFavre holds the record of jump i n g from 54 meters.a) (2 points) How long does the diver fall, if the acceleration ish0, 0, −10i.b) ( 3 points) Find the trajectory ~r(t) of th e stunt man.c) …
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