1/14/2005, THIRD PRACTICE FINAL Math 21a, Fall 2005Name:MWF9 Ivan PetrakievMWF10 Oliver KnillMWF10 Thomas LamMWF10 Michael ScheinMWF10 Teru YoshidaMWF11 Andrew DittmerMWF11 Chen-Yu ChiMWF12 K athy PaurTTh10 Valentino TosattiTTh11.5 Kai-Wen LanTTh11.5 Jeng-Daw Yu• Please mark the box to the left which lists your section.• Do not detach pages from this exam packet or unstaplethe packet.• Show your work. Answers without reasoning can notbe given credit except for the True/False and multiplechoice problems.• Please write neatly.• Do not use notes, books, calculators, computers, or otherelectronic aids.• Unspecified functions are a ssumed to be smooth and de-fined everywhere unless stated otherwise.• You have 180 minutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 1011 1012A 1013A 1014A 1012B 1013B 1014B 10Total: 140Problem 1) True/False questions (20 po ints)1)T FFo r any two nonzero vectors ~v, ~w the vector ((~v × ~w) ×~v ) ×~v) is parallel to~w.2)T FThe cross product satisfies the law (~u ×~v) × ~w = ~u × (~v × ~w).3)T FIf the curvature of a smooth curve ~r(t) in space is defined and zero for allt, then 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 FFo r every c, the function u(x, t) = (2 cos(ct) + 3 sin(ct)) sin(x) is a solutionto the wave equation utt= c2uxx.6)T FThe length of the curve ~r(t) = (t, sin(t)), where t ∈ [0, 2π] isR2π0q1 + cos2(t) dt.7)T FLet (x0, y0) be 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)/√8 at the p oint (1, 1, 1).9)T FAssume~F is a vector field satisfying |~F (x, y, z)| ≤ 1 everywhere. For everycurve C : ~r(t) with t ∈ [0, 1], the line integralRC~F ·~dr is less or equal thanthe arc length of C.10)T FLet~F be a vector field which coincides with the unit normal vector~N foreach point on a curve C. ThenRC~F ·~dr = 0.11)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.12)T FIf a nonempty quadric surface g(x, y, z) = ax2+ by2+ cz2= 5 can becontained inside a finite box, then a, b, c ≥ 0.13)T FIf div(~F)(x, y, z) = 0 for all (x, y, z), then curl(~F) = (0, 0, 0) for all (x, y, z).14)T FIf in spherical coordinates the equation φ = α (with a constant α) definesa plane, then α = π/2.TF PROBLEMS FOR REGULAR AND PHYSICS SECTIONS:15)T FThe divergence of the gradient of any f(x, y, z) is always zero.16)T FFo r every vector field~Fthe identity grad(div(~F)) =~0holds.17)T FFo r every function f, one has div(curl(grad(f))) = 0.18)T FIf~F is a vector field in space then the flux of~F through any closed surfaceS is 0.19)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.20)T FFo r every function f(x, y, z), there exists a vector field~F such that div(~F ) =f.TF PROBLEMS FOR BIOCHEM SECTIONS:21)T FTossing 3 unbiased coins, t he possible numbers of heads appearing are 0, 1, 2,and 3. Therefore each of these events has probability 1/4.22)T FTwo events A, B for which P (B) > 0 are independent if and only ifP (A|B) = P (A). .23)T FFo r two independent random variables X, Y one has the following identitiesfor the variance D(X) − D(Y ) = D(X − Y ).24)T FLet A, B be arbitrary events. If P (A|B) = P (B|A) then P (A) = P (B).25)T FThe probability that f r om 10 random coins all 6 show ta il is smaller thanthe probability that 5 show tail.26)T FIf you throw 2 dice and you know the first one shows the number 1, thenthe chance t hat the second one shows 1 is less than 1/6.Problem 2) (10 po ints)Match the equations with the objects. No justifications are needed.I II III IVV VI VII VIIIEnter I,II,III,IV,V,VI,VII,VIII here Equationg(x, y, z) = cos(x) + sin(y) = 1y = cos(x) − sin(x)~r(t) = (cos(t), sin(t))~r(u, v) = (cos(u), sin(v), cos(u) sin(v))~F (x, y, z) = (cos(x), sin(x), 1)z = f(x, y) = cos(x) + sin(y)g(x, y) = cos(x) − sin(y) = 1~F (x, y) = (cos(x), sin ( x))Problem 3) (10 po ints)Mark with a cross in the 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, mark the fieldswhich are incompressible (that is if div (~F )(x, y, z) = 0 for all (x, y, z)). No justificationsare needed.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) (10 po ints)Let E be a parallelogram in three dimensional space defined by two vectors ~u and ~v.a) (3 p oints) Express the diagonals of the parallelogram a s vectors in terms of ~u and ~v.b) (3 points) What is the relation between the length of the crossproduct of the diagonalsand the a r ea of the parallelogram?c) (4 points) Assume t hat the diagonals are perpendicular. What is the relation b etweenthe lengths of the sides of the parallelogram?Problem 5) (10 po ints)Find the volume of the largest rectangular box with sides parallel to the coordinate planesthat can be inscribed in the ellipsoidx24+y29+z225= 1.Problem 6) (10 po ints)EvaluateZ80Z2y1/3y2ex2x8dxdy.Problem 7) (10 po ints)In this problem we evaluateR RD(x−y )4(x+y )4dxdy, where D is the triangular region bounded bythe x and y axis and the line x + y = 1.a) (3 points) Find the region R in the uv-plane which is transformed into D by the changeof variables u = x − y, v = x + y. (It is enough to draw a carefully labeled picture of R.)b) (3 points) Find the Jacobian∂(x,y)∂(u,v)of the transformation (x, y) = (u+v2,v−u2).c) (4 points) EvaluateR RD(x−y )4(x+y )4dxdy using the above defined change of variables.Hint. The general topic of change o f variables does not appear in this year. To solvethe problem nevertheless, we give the formula∂(x,y)∂(u,v)= xyyv− xvyufor the Jacobean. Theintegral in c becomes thenR RRu4/v4dudv. The region R is the triangle bounded by theedges (0, 0), (1, 1), (−1, 1).Problem 8) (10 po ints)a) (3 p oints) Find all the critical points of the function f(x, y) = −(x4− 8x2+ y2+ 1).b) (3 points) Classify the critical points.c) (2 points) Locate the local and absolute maxima of f.d) (2 points) Find the equation for the tangent plane to the g raph of f at each
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