10/18/2006 FIRST HOURLY SECOND PRACTICE Math 21a, Fall 2006Name:MWF 9 Chen-Yu ChiMWF 10 Janet ChenMWF 10 Sug Woo ShinMWF 10 Jay PottharstMWF 11 Oliver KnillMWF 11 Kay Wen LanMWF 12 Valentino TosattiTTH 10 Gerald SacksTTH 10 Ilia ZharkovTTH 11 David HarveyTTH 11 Ilia Zharkov• Start by printing your name in the above box and checkyour section in the box to the left.• Do not detach pages from this exam packet or unstaplethe packet.• Please write neatly. Answers which are illegible for thegrader can not be given credit.• No notes, books, calculators, computers, or other elec-tronic aids can be allowed.• You have 90 minutes time to complete your work.• The hourly exam itself will have space for work on eachpage. This space is excluded here in order to save print-ing resources.1 202 103 104 105 106 107 108 109 10Total: 1001Problem 1) TF questions (20 points)Mark for each of the 20 questions the correct letter. No justifications are needed.1)T FThe vectors h3, −2, 1i and h−6, 4, −2i are parallel.2)T FIf |~v × ~w| = 0 then ~v =~0 or ~w =~0.3)T FThe surface z2+ 4y2= x2+ 1 is a two sheeted hyperboloid.4)T FThe surface 4x2− 4x + y2− 2y − 120 = −z2is an ellipsoid.5)T FThe parametrized lines ~u(t) = h1 + 2t, 2 −5t, 1 + ti and ~v(t) = h3 −4t, −3 +10t, 2 − 2ti are the same line.6)T FThe surface sin(x) = z contains lines which are parallel to the y-axis.7)T FIf ~u ·~v = 0, ~v · ~w = 0 and ~v is not t he zero vector, then ~u · ~w = 0.8)T FThe curva t ure of a curve dep ends upon the speed at which one travels uponit.9)T FTwo lines in space that do not intersect must be parallel.10)T FA line in space can intersect an elliptic parabo lo id in 4 points.11)T FIf ~u ×~v = 0 and ~u ·~v = 0, then one of the vectors ~u and ~v is zero.12)T FIf the velocity vector ~r′(t) and the acceleration vector ~r′′(t) of a curve ar eparallel a t time t = 1, then the curvature κ(t) of the curve is zero at timet = 1.13)T FIf the speed of a parametrized curve is constant over time, then the curvatureof the curve ~r(t) is zero.14)T FThe length of the vector projection of a vector ~v onto a vector ~w is alwaysequal to the length of the vector projection of ~w onto ~v.15)T FA quadric ax2+ by2+ cz2= 1 is contained in the interior of a spherex2+ y2+ z2< 100, then the constants a, b, c are all positive and the quadricis an ellipsoid.16)T FThere is a hyperboloid of the form ax2+ by2− cz2= 1 which has a tracewhich is a parabola.17)T FThe set of points in space which have distance 1 from the line x = y = zform a cylinder.18)T FThe velocity vector of a parametric curve ~r(t) always has constant length.19)T FThe volume of a parallelepiped spanned by ~u, ~v, ~w is |(~u ×~v) × ~w|.20)T FThe equation x2+ y2/4 = 1 in space describes an ellipsoid.2Problem 2a) (2 points)Match the equation with their graphs. No justifications are needed.I IIIII IVEnter I,II,III,IV here Equationz = sin(5x) cos(2y)z = cos(y2)z = e−x2−y2z = ex3Problem 2b) (5 points)Match the contour maps with the corresponding functions f(x, y) of two variables. No justifi-cations are needed.I II IIIIV V VIEnter I,II,III,IV,V or VI here Function f(x, y)f(x, y) = sin(x)f(x, y) = x2+ 2y2f(x, y) = |x| + |y|f(x, y) = sin(x) cos(y)f(x, y) = xe−x2−y2f(x, y) = x2/(x2+ y2)4Problem 3) (10 points)a) (7 points) Find a parametric equation for the line which is the intersection of the two planes2x − y + 3z = 9 and x + 2y + 3z = −7.b) (3 points) Find a plane perpendicular t o bot h planes and which passes through the pointP = (1, 1, 1).Problem 4) (10 points)Given the vectors ~v = h1, 1, 0 i and ~w = h0, 0, 1i and the point P = (2, 4, −2). Let Σ be theplane which goes through the origin (0, 0, 0) and which contains the vectors ~v and ~w. Let S bethe unit sphere x2+ y2+ z2= 1.a) (6 po ints) Compute the distance from P to the plane Σ.b) (4 points) F ind the shortest distance from P to the sphere S.Hint for b): Find first the distance from P to the origin O = (0, 0 , 0).Problem 5) (10 points)a) (6 points) Find an equation for the plane through the points A = (0, 1, 0), B = (1, 2, 1) andC = (2, 4, 5).b) (4 points) Given an additional point P = (−1, 2, 3), what is the volume of the tetrahedronwhich has A, B, C, P among its vertices.A useful fact which you can use without justification in b): the volume of the tetrahe-dron is 1/6 of the volume of the parallelepip ed which has AB, AC, and AP among its edges.Problem 6) (10 points)5The parametrized curve ~u(t) =< t, t2, t3> (known as the ”twisted cubic”) intersects theparametrized line ~v(s) = h1 + 3s, 1 − s, 1 + 2si at a point P . Find the angle of intersection.Problem 7) (10 points)Let ~r(t) be the space curve ~r(t) = (log(t), 2t, t2), where log(t) is the natural logarithm (denotedby ln(t) in some textbooks).a) What is the velocity and what is the acceleration at time t = 1?b) Find the length o f the curve fro m t = 1 to t = 2.Hint: you should end up with a final integral which does not involve any square roots andwhich you can solve.Problem 8) (10 points)A planar mirror in space contains the point P = (4, 1, 5) and is perpendicular to the vector~n = h1, 2, −3i. The light ray~QP = ~v = h−3, 1, −2i with source Q = (7, 0, 7) hits the mirrorplane at the point P .a) (4 po ints) Compute the projection ~u =~P~n(~v) of ~v onto ~n.b) (6 points) Identify ~u in the figure and use it to find a vector parallel to the reflected ray.6MirrorPQProblem 9) (10 points)We know the acceleration ~r′′(t) = h2, 1, 3i+ th1, −1, 1i and the initial position ~r(0) = h0, 0, 0iand initial velocity ~r′(0) = h11, 7, 0i of an unknown curve ~r(t). Find ~r(6).Problem 10) (10 points)Intersecting the elliptic cylinder x2+ y2/4 = 1 with the plane z =√3x gives a curve in space.a) (3 po ints) Find the parametrization of the curve.b) (3 points) Compute the unit tangent vector~T to the curve at the point (0, 2 , 0).c) (4 points) Write down the arc length integral and evaluate the arc length of the
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