7/14/2011 FIRST HOURLY PRACTICE I 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 addi t io n al paper, write your nam eon i t .• Do not detach pages from this exam p acket or unst a p l e the packet.• Please write neatly. Answers which are i l leg i b l e for the grader can not be given credit.• No no t es, books, calculators, computers, 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 10Total: 1001Problem 1) (20 points) No justifications are needed.1)T FThe l en g th of the vector h1, 1, 1i is equal to 3.2)T FAny two distinct points A, B in space determine a unique lin e which containsthese two points.3)T FFor any two non-intersecting li n es L, K, there is exactly one point P whichhas equal d i st ance to both li n es.4)T FThe graph of f (x, y) is a surface in space which is equal to th e level surfaceg(x, y, z) = 0 of g(x, y, z) = f(x, y) − z.5)T FThe graph of the function f(x, y) = x2−y2is called an elliptic paraboloid.6)T FThe eq u at i on ρ cos(θ) = 1 in spherical coordinates defines a plane.7)T FThe vector h1, 2, 3i is perpendicular to the plane x + 2y + 3z = 4.8)T FThe cr os s product between the vectors h1, 2, 3i and h1, 1, 1i is 6.9)T FThe two parametrized curves ~r(t) = ht, t2, t6i, 0 ≤ t ≤ 1 and~R(t) =ht2, t4, t6i, 0 ≤ t ≤ 1 have the same arc len g t h .10)T FThe point (1, 0, 1) ha s the spherical coordinates (ρ, θ, φ) = (√2, 0, π/4).11)T FThe distance between two parallel planes is the distance of any point on oneplane to the other plane.12)T FProj~w(~v × ~w) =~0 ho lds for all nonzero vectors ~v, ~w.13)T FThe vector projection of h2, 3, 4i onto h1, 0, 0i is h2, 0, 0i.14)T FThe triple scalar product ~u · (~v × ~w) between three vectors ~u,~v, ~w is zero ifand only if two or more of the 3 vectors are parallel.15)T FThere are two vectors ~v and ~w so tha t the dot product ~v · ~w is equal to thelength of t h e cross product |~v × ~w|.16)T FTwo cylinders of radius 1 whose axes are lines L, K have distance d(L, K) =3 have d i st an ce 2.17)T FThere are two unit vectors ~v and ~w that are both parallel and perpendicular.18)T FAssuming the curvature to exist for all time, the curvature κ(~r(t)) is alwayssmaller than or equal to |~r′′(t)|/|~r′|2.19)T FThe cu r ve ~r(t) = hcos(t) sin(t), sin(t) sin(t) , sin(t)i is located on a sp here.20)T FThe su r fa ce x2+ y2+ z2= 4z − 3 is a sphere of radius 1.Total2Problem 2) (10 points) No justifications are needed in this problem.a) ( 2 points) Match the graphs of the functio n s f(x, y). Enter O, if there is no match.I II IIIFunction f(x, y) = Enter O,I,II or IIIx2||x| − |y||x2+ y2x/(1 + y2)b) (3 points) Match the space curves with their parametrizations ~r(t). Enter O, if there is n omatch.I II IIIParametrization ~r(t) = O, I, II,III~r(t) = ht cos(t), t sin(t), ti~r(t) = hcos(t), sin(t), ti~r(t) = hsin(t), cos(t), 0i~r(t) = hsin(3t), cos(2t), cos(t)ic) (2 points) Match the functions g with the level sur face g(x, y, z) = 1. Enter O, where nomatch.I II IIIFunction g(x, y, z) = 1 O, I,II,IIIg(x, y, z) = x2− y2− z = 1g(x, y, z) = x − y2= 1g(x, y, z) = x2− y2+ z2= 1g(x, y, z) = x2+ y2+ z2= 1d) ( 3 points) Match the surface with the parametrization. Enter O, where no match.I II IIIFunction g(x, y, z) = O,I,II,III~r(s, t) = ht, s, tsi~r(s, t) = ht2+ s2, s, ti~r(s, t) = ht, t cos(s), t sin(s)i~r(s, t) = ht, s, tiProblem 3) (10 points) No justifications are needed in this problem.a) ( 2 points) Translate from polar to Cartesian coordinates or back:3Polar coordinates (θ, r) = Cartesian coordinates (x , y) =(π/2, 1)(1, 1)b) ( 2 points) Tr an s la t e from spherical to Cartesian coordinates or back:Spherical coordinates (θ, φ, ρ) = Cartesian coordinates (x, y, z) =(π/2, π/2, 1)(1, 1, 1)c) (3 points) Match the curves given in polar coordinates. Enter O, if there is no match.I II IIISurface Enter I,II,II, Or = π/4r = −θθ = π/4r = θd) ( 3 points) Match the surfaces given in spherical coordinates. Enter O, if there is n o match.I II IIISurface Enter I,II,III,Oρ = π/4φ = π/4θ = φθ = π/4Problem 4) (10 points)a) (2 points) Given two points A = (1, 2, 3) and B = (4, 5, 6). Find the midpoint M betweenthese two points.b) (5 points) Find the equation ax + by + cz = d of the plane for which every point has equaldistance to both A and B.c) (3 points) Write down a parametrization ~r(t) = hx(t), y(t), z(t)i of the line containing bothA and B.4Problem 5) (10 points)In this problem you have to find parametrizations of surfaces. The parametrization should havethe for m~r(u, v) = h x( u, v), y(u, v), z(u, v)i .a) ( 2 points) Parametrize the paraboloid z = − x2− y2.b) ( 2 points) Parametrize the ellipsoid x2+ y2/4 + z2/9 = 1.c) (2 points) Parametrize the plane x = y.d) ( 2 points) Parametrize the cylinder x2+ z2= 9.e) (2 points) Parametrize the cone x2+ y2= z2.Problem 6) (10 points)In the soccer world championship of 2010, theEnglish team scored a goal against the Germanteam which the referee did not see. Assume the ballfollowed the li n e x = y = (z − 1)/3 and that thereferee was at position R = (3, 2, 1).Remark: The incidence was a Wimbleton revanche and confirmeda word of wisdom of former player Gary Lineker who said a coupleof years ago: ”Soccer is a g a me for 22 people that run around, playthe ball, and one referee who ma kes a slew of mista kes, and in theend, Germany always wins.”a) ( 4 points) Find a parametrization of the line L which the ball followed.b) ( 6 points) Find the minimal distance of L from the point R.Problem 7) (10 points)Compute the following expressions:a) ( 2 points) the length of the vector h1, 1, 2i,b) ( 2 points) the cross product h 1, 1, 2i × h2, 2, 1i,c) (2 points) the dot product h1, 1, 4i · h2, 4, 2i,5d) ( 2 points) the projection Projh1,1,2i(h1, 2, 3i),e) (2 points) the angle between h1, 1, 0i and h0, 1, 1i.Problem 8) (10 points)a) ( 3 points) Find the unit tangent vector~T (t) of the curve~r(t) = …
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