8/11/2011 FINAL EXAM PRACTICE IV Maths 21a, O. Knill, Summer 2011Name:• Start by printing your name in the above box.• Try to answer each question on the same page as the question is asked. If needed, usethe back or the next empty page for work.• Justify your answers. Answers without derivation can not be given credit.• Do not detach pages from this exam packet or un s t ap l e the packet.• Please wr i t e neatly. Answers which are il l egi b l e for the grader can not be given credit.• No not es, books, calcu l at or s , computers, or other electronic aids can be allowed.• You h ave 180 minutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 1011 1012 1013 1014 1015 10Total: 1601Problem 1) ( 20 points)1)T FFor any two nonzero vectors ~v, ~w the vector ~v − ~w is perpendicular to ~v × ~w.2)T FThe cros s product satisfies the law (~u ×~v) × ~w = ~u × (~v × ~w).3)T FIf the curvature of a smooth curve ~r(t) in sp ace is defined and zero for allt, th e n the curve is part of a line.4)T FThe curve ~r(t) = (1 − t)A + tB, t ∈ [0, 1] connects the point A with thepoint B.5)T FFor every c, the function u(x, t) = (2 cos(ct) + 3 sin(ct)) sin(x) i s a solutionto th e wave equation utt= c2uxx.6)T FThe arc length of ~r(t) = (t, sin(t) ) , t ∈ [0, 2π] isR2π0q1 + cos2(t) dt.7)T FLet (x0, y0) b e the maximum of f(x, y) under the constraint g(x, y) = 1.Then fxx(x0, y0) < 0.8)T FThe function f(x, y, z) = x2− y2− z2decreases in the direction(2, −2, −2)/√12 at the point (1, 1, 1).9)T F~F is a vector field for which |~F (x, y, z)| ≤ 1. For every curve C : ~r(t) witht ∈ [0, 1], the line integralRC~F ·~dr is ≤ the arc length of C.10)T FLet~Fbe a vector field and C is a curve which is a flow line, thenRC~F·~dr> 0.11)T FThe di vergence of the gradient of any f(x, y, z) is always zero.12)T FFor every function f, one has div(curl( gr a d ( f ))) = 0.13)T FIf for two vector fields~F and~G one has curl(~F ) = curl(~G), then~F =~G + (a, b, c), where a, b, c are constants.14)T FFor every vector field~F the identity grad(div(~F )) =~0 holds.15)T FIf a nonempty quadric surface g(x, y, z) = ax2+ by2+ cz2= 5 can becontained inside a finite box, then a, b, c ≥ 0.16)T FIf~F is a vector field in space then the flux of~F through any closed surfaceS is 0.17)T FIf div(~F )(x, y, z) = 0 for al l (x, y, z), then curl(~F ) = (0, 0, 0) for all (x, y, z).18)T FThe flux of the vector field~F (x, y, z) = (y + z, y, −z) through the boundaryof a solid region E is equal to the volume of E.19)T FIf in spherical coordinates the equation φ = α (with a constant α) definesa plane, then α = π/2.20)T FFor every function f(x, y, z), there exists a vector field~F such that div(~F ) =f.2Problem 2) (1 0 points)-2 -1 1 2-2-112-1 -0.5 0.5 1-1-0.50.51I II-1 -0.5 0.5 1-1-0.50.51-1 -0.5 0.5 1-1-0.50.51III IVFor the sign of t h e curl or divergence, where either + (positive), − (negative) or 0 for zero.The vector fields are considered on the square [−1/2, 1/2]x[−1/2, 1/2] in this problem.3Enter I,II,III,IV here Vector fi el d curl sign divergence signF (x, y) = (x, y2)F (x, y) = (1 − y, x)F (x, y) = (y − x, −y)F (x, y) = (−x, y3)Problem 3) (1 0 points)Mark with a cross in th e column below ”conservative” if a vector fields is conservative(that is if curl(~F )(x, y, z) = (0, 0, 0) for all points (x, y, z)). Similarly, ma r k the fieldswhich are incompressible (that is if div(~F )(x, y, z) = 0 for all (x, y, z)). No justifica ti o n sare nee d ed .Vectorfield conservative incompressiblecurl(~F ) =~0 div(~F ) = 0~F (x, y, z) = (−5, 5, 3)~F (x, y, z) = (x, y, z)~F (x, y, z) = (−y, x, z)~F (x, y, z) = (x2+ y2, xyz, x − y + z)~F (x, y, z) = (x − 2yz, y − 2zx, z − 2xy)Problem 4) (1 0 points)Let E be a parallelogram in three dimensional space defined by two vectors ~u and ~v.a) (3 points) Express the diagonals of t h e paralle lo gr am as vectors in terms of ~u and ~v.b) (3 points) What is the relation between the length of the cr os sp r oduct of the diagonalsand t h e area of the parallelogram?4c) (4 points) Assume that the diagonals are perpendicular. What is the relation betweenthe len g t h s of the sides of the parallelogram?Problem 5) (1 0 points)Find the volume of the largest rectangular box with sid e s parallel to the coordinate planesthat can be inscribed in the ellipsoidx24+y29+z225= 1.Problem 6) (1 0 points)EvaluateZ80Z2y1/3y2ex2x8dxdy.Problem 7) (1 0 points)EvaluateR RD2xyx2+y2dxdy, where D is the intersection of the annulus 1 ≤ x2+ y2≤ 2 withthe secon d quad r a nt {x ≤ 0, y ≥ 0 }.Problem 8) (1 0 points)a) (3 points) Find all the cri t i cal points of the fun ct i on f (x, y) = −(x4− 8x2+ y2+ 1).b) (3 points) Classify the critical points.c) (2 points) Locate the local and ab s ol u t e maxima of f.d) (2 points) Find the equation for the tangent plane to the graph of f at each absolutemaximum.5Problem 9) (1 0 points)Find t he areaR RR1 dxdy of the 10 legged ”sea star” R, enclosed by the polar curver(θ) = 2 + sin(10 θ) ,where θ ∈ [0, 2π]. The photo to the right shows a real sea st ar .Problem 10) ( 10 points)Find the volume of the intersection of the interior of the one sided hyperboloid x2+y2−z2≤1 with the solid ball enclosed by the sphere x2+ y2+ z2≤ 9.-4-2 0 24-4-20246Problem 11) ( 10 points)Let the curve C be parametrized by ~r(t) = (t, sin t, t2cos t) for 0 ≤ t ≤ π. Let f (x, y, z) =z2ex+2y+ x2and~F = ∇f. FindRC~F · d~r.Problem 12) ( 10 points)a) Find the linear approximation L(x, y) of f(x, y) =√4 + 2x2+ 4y2at the point (x, y) =(2, 1).b) Fin d the equation for the tangent line to the level curve of f(u, v) at (2, 1).Problem 13) ( 10 points)Find the line integral of th e vector field~F (x, y) = hx30+ y, y50+ xi along the path~r ( t ) = h4 sin(π sin(t)) + sin(10t), ti with 0 ≤ t ≤ π/2.12 340.250.50.7511.251.5Problem 14) ( 10 points)7Evaluate the line integral of the vector field~F (x, y) = (y2, x2) in the clockwise di r ect i onaround the triangle in the xy-p l ane defined by …
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