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HARVARD MATH 21A - FIRST HOURLY FIRST PRACTICE

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10/18/2006 FIRST HOURLY FIRST PRACTICE Math 21a, Fall 2006Name:MWF 9 Chen-Yu ChiMWF 10 Janet ChenMWF 10 Sug Woo ShinMWF 10 Jay PottharstMWF 11 Oliver KnillMWF 11 Kai-Wen LanMWF 12 Valentino TosattiTTH 10 Gerald SacksTTH 10 Ilia ZharkovTTH 11 David HarveyTTH 11 Ilia Zharkov• 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, o r o t her elec-tronic aids can be allowed.• You have 90 minutes time to complete your work.• The hourly exam itself will have space for work on eachpage. This space is excluded here in order to save print -ing resources.1 202 103 104 105 106 107 108 109 1010 10Total: 1101Problem 1) TF questions ( 20 points) No justifications needed1)T FThe length of t he sum of two vectors is a lways 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) = h1 + 2t, 1 + 3t, 1 + 4ti hits the plane 2x + 3y + 4z = 9 at aright a ngle.7)T FThe graph of f(x, y) = cos(xy) is a level surface of a f unction 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 FThere is a quadric ax2+by2+c z2+dx+ey+fz = e which is a hyperbola whenintersected with the plane z = 0, which is a hyperbola when intersected withthe plane y = 0 and which is a parabola when intersected with x = 0.13)T FThe curvature of the curve 2~r( 4 t) at t = 0 is twice the curvature of thecurve ~r(t) at t = 0.14)T FThe set of points which satisfy x2− 2y2− 3z2= 0 form an ellipsoid.15)T FIf ~v × ~w = (0, 0, 0), then ~v = ~w .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 FTwo nonzero vectors are parallel if and only if their cross product is~0.18)T FThe vector ~u × (~v × ~w) is always in the same plane together with ~v and ~w.19)T FThe line ~r(t) = h1 + 2t, 1 + 2t, 1 −4ti hits the plane x + y + z = 9 at a rightangle.20)T FThe intersection of the ellipsoid x2/3+ y2/4+ z2/3 = 1 with the plane y = 1is a circle.2Problem 2a) (10 points)Match the cruves with their parametric definitions.I IIIII IVV VIEnter I,II,III,IV,V or VI here Parametric equation for the curve~r(t) = ht, sin(1/t)ti~r(t) = ht3− t, t2i~r(t) = ht + cos(2t), sin(2t)i~r(t) = h|sin(2t)|, cos(3t)i~r(t) = h1 + t, 5 + 3ti~r(t) = h−t cos(t), 2t sin(t)i3Problem 2b) (4 points)Match the equations with the surfaces.I II IIIIV V VIEnter I,II,III,IV,V,VI here Equationx2− y2− z2= 1x2+ 2y2= z22x2+ y2+ 2z2= 1x2− y2= 5x2− y2− z = 1x2+ y2− z = 14Problem 3) (10 points)a) (6 points) Find a parameterization of the line of intersection of the planes 3x − 2 y + z = 7and x + 2y + 3z = −3.b) (4 points) Find the symmetric equationsx − x0a=y − y0b=z − z0crepresenting t hat line.Problem 4) (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 points) 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 5) (10 points)Find the distance between the two lines~r1(t) = ht, 2t, −tiand~r2(t) = h1 + t, t, ti .Problem 6) (10 points)5Find 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 7) (10 points)The intersection of the two surfaces x2+y22= 1 and z2+y22= 1 consists of two curves.a) (4 points) Parameterize each curve in the form ~r(t) = (x(t), y(t), z(t)).b) (3 points) Set up the integral for the arc length of one of the curves.c) (3 points) What is the arc length of this curve?Problem 8) (10 points)a) (6 points) Find the curvature κ(t) of the space curve ~r(t) = h−cos(t), sin(t), −2ti at thepoint ~r(0).b) (4 points) Find the curvature κ(t) of the space curve ~r(t) = h−cos(5t), sin( 5 t), −10ti at thepoint ~r(0).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) = ~r′(t)/|~r′(t)|. The curvatures in b) can be derived from the curvature in a).There is no need to redo the calculation, but we need a justification.Problem 9) (10 points)For each of the following, fill in the blank with < (less than), > (greater than), or = (equal).6Justify your answer completely.1.The arc length of thecurve parameterized by~f(t) = hcos 2t, 0, sin 2ti,0 ≤ t ≤ π.The arc length of thecurve para meterized by~g(u) = h3, 2 cos u2, 2 sin u2i,0 ≤ u ≤√π.2.The arc length of thecurve parameterized by~f(t) = ht2, 2 cos t, 2 sin ti,0 ≤ t ≤ 2π.The arc length of thecurve para meterized by~g(u) = hu4, 2 cos u2, 2 sin u2i,0 ≤ u ≤ 2π.3.The arc length of the curve pa-rameterized by~f(t) = h1 +3t2, 2 − t2, 5 + 2t2i, 0 ≤ t ≤ 1.The arc length of thecurve para meterized by~g(u) =D12u2, u,2√23u3/2E,0 ≤ u ≤ 2.4.The arc length of thecurve parameterized by~f(t) = hsin t, cos t, ti, 1 ≤ t ≤ 5.The arc length of thecurve para meterized by~g(u) = hu sin u, u cos u, ui,1 ≤ u ≤ 5.Problem 10) (10 points)Given the plane x + y + z = 6 containing the point P = (2, 2, 2). Given is also a second pointQ = (3, −2, 2).a) (5 points) Find the equation ax + by + cz = d for the plane through P and Q which isperpendicular to the plane x + y + z = 6.b) (5 points) Find the symmetric equation for the intersection of these two


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HARVARD MATH 21A - FIRST HOURLY FIRST PRACTICE

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