7/14/2011 FIRST HOURLY Maths 21a, O.Knill, Summer 2011Name:• Start by writing your name in the above box.• Try to answer each questi on on the same 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.• Except for problems 1-3 give details.• Please write neatly. Answers which are illegible for the grader can not be given credit.• No notes, books, calculators, computers, or other electronic aids can be allowed.• You have exactly 90 minutes t o complet e your work.1 202 103 104 105 106 107 108 109 10Total: 1001Problem 1) ( 20 poi nts) No justifications are needed.1)T FThe len gt h of the vector ~v ×~v is 3, if ~v = h1, 1, 1i.2)T FFor any 4 points A, B, C, D in space, there is a plane which goes throughthese points.3)T FFor any two points P, Q in space which have positive distance there is exactlyone pl ane which has equal distance to both points.4)T FThe gra p h of a function f(x, y) can be parametrized as ~r( x, y, z) = z −f(x, y).5)T FThe grap h of the function f(x, y) = x2− y2is called a hyperboloid.6)T FThe equa t io n ρ cos(θ) sin(φ) = 1 in spherical coordinates defines a plane.7)T FThe vector h1, 2i is perpendicular to the line x + 2y = 2.8)T FThe tri p l e scalar product between the vectors~i, h0, 2, 0i and h1, 1, 1i is 2.9)T FThe two parametrized curves ~r(t) = ht, t2, t6i, 0 ≤ t ≤ 2 and~R(t) =ht2, t4, t12i, 0 ≤ t ≤ 2 have the sam e curvature at t = 1.10)T FThe point (x, y, z) = (1, 1, 1) h as the sp h er i cal coordinates (ρ, θ, φ) =(√3, π/2, π/4).11)T FThe distance between the cylinders x2+ z2= 1 and x2+ (y −3)2= 1 is 1.12)T FIf P is the projection onto t h e line spanned by a vector ~w then P( ~w) = ~w.13)T FThe vector projection of h2, 3, 1i onto h1, 1, 1i is parallel to h1, 1, 1i.14)T FThe triple scalar product has the property ~u·(~v ×~u) = 0 for any two vectors~u,~v.15)T FFor any two vectors ~v and ~w, the length of the cross p r oduct ~v × ~w is equalto th e length of ~v + ~w.16)T FIf the domain of a function is the entire plane, t h en the range of the functionis the entire line or a half line.17)T FThere are two uni t vectors ~v an d ~w which have the property that ~v · ~w = π.18)T FIf we travel with u n i t speed on a street, then the curvature κ(~r(t)) of thestreet i s bou n ded by the magnitude of the acceleration |~r′′(t)|.19)T FThe curve ~r ( t ) = hcos(2t) sin(3t), sin(2t) sin(3t), cos(3t)i is located on asphere.20)T FThe sur face x2− y2+ z2= 4z − 3 is a one sheeted hyperboloid.Total2Problem 2) ( 10 poi nts) No justifications are needed in this problem.a) (2 points) Match the graphs of the functions f (x, y). Ente r O, if there is no match.I II IIIFunction f(x, y) = Enter O,I, II or IIIy3sin(x) + sin(y)x2− y2x4− y4x + yb) (3 points) Match the space curves with their parametrizations ~r(t). Enter O, if there is nomatch.I II IIIParametrization ~r(t) = O, I,II,III~r(t) = h co s( 2t ) , sin(3t), ti~r(t) = h | t | , |t|, t2i~r(t) = h s in ( 2t ) /2 , sin2(t), cos ( t ) i~r(t) = h 1 + 2t, 2 + 3t, tic) (2 points) Match th e functions g with the level surface 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+ z = y2g(x, y, z) = x2− y2= 1g(x, y, z) = x2− y2+ z2= 0g(x, y, z) =x22+ 2y2+ z2= 1d) (3 poi nts) 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) = ht − 1, s, t + si~r(s, t) = ht, s cos(t), s sin(t)i~r(s, t) = hs, s cos(t), s sin(t)iProblem 3) ( 10 poi nts) No detailed justifications are needed.a) (6 points) We are given a point P , a line L determined by A and ~v a li n e M deter m i n ed byB a n d ~w an d a plane Σ. Match the distances with distance formulas and possible explanations:3distance the formula involves (A-C) and can be explained by (U-W)d(P, Σ)d(L, M)d(P, L)A) triple scalar produ ctB) dot p r oduct a n d no cross productC) cro ss product on l yU projection of a connection vectorV volume of paral l el ep i ped d i v ided by base areaW area of parallelogram divided by base lengthb) (4 points total) Edwin Abbott’s novella ”Flatland, A Romanceof M any dimensi on s” deals with ”flatlanders” who can not perceivethree dimensional objects. By looking at various traces f = c aflatlander still can see a three dimensional object. For example, athree d i m en s io n al spher e is visualized as follows.Edwin Abbott: 1838-1926b1) (2 points) We are in flatland, where multivariable calculus students only work with 2 di-mensions. To get a feel for the third dimen si on the following sequence of generalized tracesare shown with a frame around each picture. They illustrate t h e graph of a fu n ct i on of twovariables. Which surface is under consideration?b2) (2 poi nts) In the first midterm exam of that multivariable calculus course, the class has tofind t h e name of the surface of the following sequence of traces. Can you do it ?Problem 4) ( 10 poi nts)4Find the equation ax + by + cz = d of a plane Σ perpen -dicular to th e planes x + y −z = 1 and x + 2y −3z = 0so that the surface Σ contains the point P = (2, 1, 2).Problem 5) ( 10 poi nts)In thi s probl em we find some parametrizations of surfaces which is of the form~r(u, v) = hx(u, v), y(u, v), z(u, v)i .a) (2 points) Parametrize the paraboloid z = x2− y2.b) (3 poi nts) Parametrize (the entire!) ellipsoid (x − 1)2+(y−2)24+ z2= 1.c) (2 points) Parametrize the plane x + y + z = 3.d) (3 poi nts) Parametrize the cylinder x2+ z2= 1.Problem 6) ( 10 poi nts)5a) (6 points) Find the velocity vectors ~r′(π/2) and~r′(3π/2). Now find the angle between these vectorsat th e point, where the curve~r(t) = h co s( 3t ) , sin(2t)iself-intersects at t h e origin .b) (4 poi nts) Fi n d ~r′(π/2), ~r′′(π/2) and ~r′(π/2) ×~r′′(π/2). What is the curvature |~r′(π/2) ×~r′′(π/2)|/|~r′(π/2)|3at (0, 0)?Problem 7) ( 10 poi nts)Compute the foll owing expressions :a) (2 points) …
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