7/14/2011 FIRST HOURLY PRACTICE III 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 ~v = h1, 3, 5i is perp en d i cu l a r to the plane x + 3y + 5z = 1.2)T FThe set of points which satisfy x2− y2+ z2− 2z + 1 = 0 forms a doublecone.3)T FThe set of points in R3which have distance 1 from a point form a cyl inder.4)T FThe su r face −x2+ y2+ z2= 1 is called a one-sheeted hyperboloid.5)T FThe two vectors h2, 3, 0i and h6, −4, 5i are orthogonal to each o t h er .6)T FTwo nonzero vectors are parallel if and only if their dot product is 0.7)T FThe cr oss product is associative: ~u ×(~v × ~w) = (~u ×~v) × ~w.8)T FEvery vector contained in the line ~r(t) = h1 + 2t, 1 + 3t, 1 + 4ti is p a ra l lelto t h e vector (2, 3, 4).9)T FThe l inex−12=y−13=z−14hits the pl an e 2x + 3y + 4z = 9 at a right angle.10)T FTwo planes ax + by + cz = d and ux + vy + wz = e intersect in a line if|ha, b, ci × hu, v, wi| > 0.11)T FThe equations x − 2 = y − 3 = z − 4 describe a line which contains thevector h1, 1, 1i.12)T FIn sp herical coordinates, the equation cos(φ) = sin(φ) defines a cone.13)T FA point with spherical coordinates (ρ, θ, φ) = (1, π/2, π/4) has cylindercoordinates (r, θ, z) = (1/√2, π/2, 1/√2).14)T FIf in rectangular coordinates, a point is given by (1, 0, −1), then its sphericalcoordinates are (ρ, θ, φ) = (√2, π/2, 3π/4).15)T FThe volume of a parallelepiped spanned by (1, 0, 0), (0, 1, 0) and (0, 1, 1) isequal to 2.16)T FThe vector projection of the vector (2, 4, 5) onto the vector (1, 1, 0) is(3/2, 3/2, 0).17)T FIf g(x, y, z) = 0 is a surface given implicitly, then ~r(u, v) =hu, v, g(u, v, g(u, v, 1))i is a paramet r i zat i on of the surface.18)T FIf z = g(x, y) is a graph th e n ~r(u, v) = hu, v, g(u, v)i is a parameterizationof the surface.19)T FThe d i st ance from the point P = (1, 1, 1) to the x axes is√2.20)T FThe d i st ance between the point P = (1, 1, 1) and the xy plane is√2.Total2Problem 2) (10 points)a) (3 points) Match the contour maps with the corresponding fun ct i ons f(x, y) of two variables.Enter O if no fig u r e matches. No justifications are needed.I II III IVEnter I,II,III,IV or O Function f(x , y)f(x, y) = x cos(y)f(x, y) = 3x2+ 4y2f(x, y) = cos(x)Enter I,II,III,IV or O Function f(x , y)f(x, y) = x2− y2f(x, y) = |xy|f(x, y) = |x| − |y|b) ( 4 points) Match the quadrics with the functions. No justifications are necessary.abcd3Enter a,b,c,d here Equationx + y2− z2− 1 = 0y2− z2+ 1 = 0Enter a,b,c,d here Equationx2+ y2− z2+ 1 = 0x2+ y2− z2− 1 = 0c) (3 points) Match the equation with their graphs. No justifications are necessary.A B CEnter A,B,C here Equation Enter A,B,C here Equation Enter A,B,C here Equationz = e−x2−y2z = x cos(x) z = x − y4Problem 3) (10 points)a) (5 points) Surfaces z = f(x, y) which are graphs can be written implicitly as g(x, y, z) = 0,parametrized as ~r(u, v). For example, z = log(xy) is given by g(x, y, z) = 0 with g( x , y, z) =z −log(xy) or parametrized as ~r(u, v) = hu, v, log(uv)i. Complete the following table by filli n gin t h e choices A − J below. No justifications are needed in this problem.z = f(x, y) for g(x, y, z) = 0 ~r(u, v) = hx(u, v), y(u, v), z(u, v)ihv cos(u), v sin(u), vix + y − 2z = 0f(x, y) = x2− y2hcos(u) sin(v), sin(u) sin(v), cos( v)i, v < π/2z − sin( xy) = 0f(x, y) = xA) f (x, y) = x − yB) f (x, y) = x2+ y2C) z − x2− y2D) h 1 + u + v, 1 + u − v, uiE) z − x2+ y2.F) f (x, y) =√1 − x2− y2G) hu , v, u2− v2iH) x2+ y2+ z2− 1 = 0I) h u, v, uiJ) z − x = 0b) ( 5 points)Quantity Check if it depends on parametr i zat i on of ~r Is a vectorCurvature of ~r(t)Arc length of ~r(t) from 0 to 1Acceleration of ~r(t)Jerk of ~r( t )Speed of ~r(t)Unit tangent of ~r(t)Normal of ~r ( t)Binormal of ~r(t)∇f(~r( t ) ) ·~r′(t)~r′(t) ×~r′′(t)Problem 4) (10 points)A billar d ball starts at A = (1, 1, 0), travels along the vector ~u = h2, −2, 0i to other point Bwhere it bounces off an ot h er ball. It travels from there alon g the vector ~v = h−3, 4, 0i to athird point C, where it bounces off a wall, rolling along the vector ~w = h1, 1, 0i to its finaldestination D. In other words, you know A,~AB = ~u,~BC = ~v and~CD = ~w.a) ( 5 points) What are the coordinates of t h e point D?5b) ( 5 points) Find the total distance traveled by the ball along the path A, B, C, D.Problem 5) (10 points)a) (5 points) Find the symmetric equation of the line which contains t h e p oi nt P = (3, 4, 1)and the point Q = (5, 5, 5).b) (5 points) What is th e equation of the plane perpendicula r to the lin e in a ) which passesthrough the point P = (3, 4, 1)?Problem 6) (10 points)We look at a polyhedron which has the shape of a scaled octahedron. Its vertices are A =(1, 1, 0), B = (−1, 1, 0), C = (−1, −1, 0), D = (1, −1, 0), E = (0, 0, 1), F = (0, 0, −1).a) (5 points) Parametrize the line L passing through A,E and the line K passing through B,F.b) ( 5 points) Find the distance between these two lines L and K.Problem 7) (10 points)The plane 3x + y + 2z = 6 cuts ou t a triangle T from the octant x > 0, y > 0, …
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