PRACTICE EXAM FIRST HOURLY Math 21a, Fall 2005Name:MWF9 Ivan PetrakievMWF10 Oliver KnillMWF10 Thomas LamMWF10 Michael ScheinMWF10 Teru YoshidaMWF11 Anderew DittmerMWF11 Chen-Yu ChiMWF12 K athy PaurTTh10 Valentino TosattiTTh11.5 Kai- Wen LanTTh11.5 Jeng-Daw Yu• 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 wor k on eachpage. This space is excluded here in order to save print-ing resources.1 202 103 104 105 106 107 108 109 10Total: 100Problem 1) TF questions (20 points) No justifications needed1)T FThe length of the sum of two vectors is always the sum of the length of thevectors.2)T FFor any three vectors, ~v × (~w + ~u) = ~w ×~v + ~u ×~v.3)T FThe set of points which satisfy x2+ 2x + y2− z2= 0 is a cone.4)T FThe functions√x + y − 1 and log(x + y − 1) have the same domain ofdefinition.5)T FIf P, Q, R are 3 different points in space that don’t lie in a line, then~P Q×~RQis a vector orthogonal to the plane containing P, Q, R.6)T FThe line ~r(t) = (1 + 2t, 1 + 3t, 1 + 4t) hits the plane 2x + 3y + 4z = 9 at aright angle.7)T FThe graph of f(x, y) = cos(xy) is a level surface of a function g(x, y, z).8)T FFor any two vectors, ~v × ~w = ~w ×~v.9)T FIf |~v × ~w| = 0 for all vectors ~w, then ~v =~0.10)T FIf ~u and ~v are orthogonal vectors, then (~u ×~v) ×~u is parallel to ~v.11)T FEvery vector contained in the line ~r(t) = h1 + 2t, 1 + 3t, 1 + 4ti is parallelto the vector h1, 1, 1i.12)T FThe curvature of the curve 2~r(4t) at t = 0 is twice the curvature of thecurve ~r(t) at t = 0.13)T FThe set of points which satisfy x2− 2y2− 3z2= 0 form an ellipsoid.14)T FIf ~v × ~w = (0, 0, 0), then ~v = ~w.15)T FThe set of points in space which have distance 1 fr om a line form a cylinder.16)T FEvery vector contained in the line ~r(t) = h1 + 2t, 1 + 3t, 1 + 4ti is parallelto the vector h1, 1, 1i.17)T FThe equation x2+ y2/4 = 1 in space describes an ellipsoid.18)T FFor any three vectors ~a,~b and ~c, we always have (~a ×~b) ·~c = −(~a ×~c) ·~b.19)T FThe set of points in the xy-plane which satisfy x2−y2= −1 is a hyperbo la .20)T FTwo nonzero vectors are parallel if and o nly if their cross product is~0.Problem 2) (10 points)Match the contour maps with the corresponding functions f(x, y) of two variables. No justifi-cations are needed.I II IIIIV V VINon-graphical description:I) shows level curves which are symmetric with respect to the y axes, on bot h sides, there areconcentric lense shaped closed curves.II) shows concentric elliptic level curvesIII) shows a family of vertical linesIV) shows many a periodic pattern of level curves, where each pattern contains in the centercircle like curves which become more and more square likeV) shows diamond shaped level curves with cornersVI) shows a family of lines which meet in the cent er.Enter 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)Problem 3) (10 points)Match the equation with the pictures and justify briefly your choice.I II IIIIV V VIEnter I,II,III,IV,V,VI here Equation Short explanationx4+ y4+ z4− 1 = 0−x2+ y2− z2− 1 = 0x2+ z2= 1−y2+ z2= 0x2− y2+ 3z2− 1 = 0x2− y −z2= 0Nongraphical description:Figure I) shows two shells, where each shell looks like a bowlFigure II) shows a surface which looks like the front part of a human neck cut on the to p belowthe chin and at the bottom at the upper part of the shoulderFigure III) shows two planes which cross each other ort hogonallyFigure IV) shows a tube like surface which is narrow in the middle, and wide at the ends andwhich has circular cross sectionsFigure V) shows a tube like surface with circular cross sectionFigure VI) shows a rounded cub eProblem 4) (10 points)a) (6 points) Find a parameterization o f the line of intersection of the planes 3x − 2y + z = 7and x + 2y + 3 z = −3.b) (4 point s) Find the symmetric equationsx − x0a=y − y0b=z − z0crepresenting that line.Problem 5) (10 points)a) (4 points) Find the area of the parallelogram with vertices P = (1, 0, 0) Q = (0, 2, 0),R = (0, 0, 3) and S = (−1, 2, 3).b) (3 po ints) Verify that the triple scalar product has the property [~u+~v, ~v+ ~w, ~w+~u] = 2[~u,~v, ~w].c) (3 points) Verify that the triple scalar product [~u,~v, ~w ] = ~u · (~v × ~w) has the property|[~u,~v, ~w]| ≤ ||~u|| · ||~v|| · ||~w||Problem 6) (10 points)Find the distance between the two lines~r1(t) = ht, 2t, −tiand~r2(t) = h1 + t, t, ti .Problem 7) (10 points)Find an equation for the plane that passes through the origin and whose normal vector isparallel to the line of intersection of the planes 2x + y + z = 4 and x + 3y + z = 2.Problem 8) (10 points)The intersection of the two surfaces x2+y22= 1 and z2+y22= 1 consists of two curves.a) (4 p oints) Parameterize each curve in the form ~r ( t) = (x(t), y(t), z(t)).b) (3 point s) Set up the integral for the arc length of one of the curves.c) (3 points) What is the a r c length o f this curve?Problem 9) (10 points)Find the curvature κ(t) of the space curve ~r( t) = h−cos(t), sin(t), −2ti.Hint. Use one of the two formulas for the curvatureκ(t) =|~T′(t)||~r′(t)|=|~r′(t) ×~r′′(t)||~r′(t)|3,where~T (t) =
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