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HARVARD MATH 21A - First practice exam

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FIRST PRACTICE EXAM FIRST HOURLY Math 21a,Spring 2003Name:MWF10 Ken ChungMWF10 Weiyang QiuMWF11 Oliver KnillTTh10 Mark LucianovicTTh11.5 Ciprian Manolescu• 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 work 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)Circle for each of the 20 questions the correct letter. No justifications are needed. Your scorewill be C −W where C is the number of correct answers and W is the number of wrong answers.T FThe vectors h3, −2, 1i and h−6, 4, 2i are parallel.T FThe length of the vector h3, 4, 0i is 25.T FFor any two vectors, ~v × ~w = ~w × ~v.T FThe vectors h1, 1i and h1, −1i are orthogonal.T FFor any two vectors ~v, ~w one has |~v + ~w|2= |~v|2+ |~w|2.T FThe surface x2− y2+ z2= 1 is a one-sheeted hyperboloid.T FThe set of points which have distance 1 from a line is a cylinder.T FIf |~v × ~w| = 0 for all vectors ~w, then ~v = 0.T FAny nonempty intersection of two planes is always a line.T FIf ~u and ~v are orthogonal vectors, then (~u × ~v) × ~u is parallel to ~v.T FTwo nonparallel lines in three dimensional space always intersectin a point.T FEvery vector contained in the line ~r(t) = h1 + 2t, 1 + 3t, 1 + 4ti isparallel to the vector (1, 1, 1).T FIf in spherical coordinates a point is given by (ρ, θ, φ) =(2, π/2, π/2), then its rectangular coordinates are (x, y, z) =(0, 2, 0).T FIf the velocity vector ~r0(t) of the planar curve ~r(t) is orthogonal tothe vector ~r(t) for all times t, then the curve is a circle.T FEvery point on the sphere of radius ρ is determined alone by itsangle φ from the z axis.T FThe equation r = 3 in cylindrical coordinates is a sphere.T FThe set of points which satisfy x2−2y2−3z2= 0 form an ellipsoid.T FA surface which is given as r = 2 + sin(z) in cylindrical coordinatesstays the same when we rotate it around the z axis.T FIf ~v × ~w = h0, 0, 0i, then ~v = ~w.T FThe curvature of a circle of radius r is equal to 1/(2πr).- =2Problem 2) (10 points)Match the equation with their graphs and justify briefly your choice.24681024681000.511.52246810-4-2024-4-202400.51-4-2024I II-2-1012-2-1012-1-0.500.51-2-1012-1-0.500.51-1-0.500.51012345-1-0.500.51III IVEnter I,II,III,IV here Equation Short Justificationz = sin(3x) cos(5y)z = cos(y2)z = log(x)z = x/(x2+ y2)3Problem 3) (10 points)Match the equation with their graphs 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= 04Problem 4) (10 points)Given the vectors ~v = h1, 1, 0i and ~w = h0, 0, 1i and the point P = (2, 4, −2). Let Σ be theplane which goes through the origin and contains the vectors ~v and ~w.a) Determine the distance from P to the origin.b) Determine the distance from P to the plane Σ.Problem 5) (10 points)Boba Fett is flying through the air when his rocket packmalfunctions and sends him spinning out of control. At timet = 0, he is at the point P0= (0, 0, 27) and moving withvelocity ~v = h10, 0, 0i. While he is in the air, his acceler-ation is given by ~a(t) = hπ2sin πt, π2cos πt+2t, −6ti for t ≥ 0.1. For t ≥ 0, find Boba’s position as a function of time.2. The ground is represented by the xy plane. At whattime does Boba hit the ground? What are the x and ycoordinates of the point, where he hits?Problem 6) (10 points)a) Calculate the unit tangent vector T , the unit normal vector N as well as the binormal vectorB for the curve ~r(t) = ht, cos(t), t2i at the point t = π.b) Verify that for a general curve the formuladdt~B(t) =~T (t) ×~N0(t) holds.Hint. Here are the formulas for the unit tangent vector, the unit normal vector as well as thebinormal vector.~T (t) = ~r0(t)/|~r0(t)|~N(t) =~T0(t)/|T0(t)|~B(t) =~T (t) ×~N(t).Problem 7) (10 points)a) Identify the surface whose equation is given in spherical coordinates as θ = π/4.b) Identify the surface whose equation is given in spherical coordinates as φ = π/4.c)Identify the surface, whose equation is given in cylindrical coordinates by z = r2. Eithername it or sketch the surface convincingly.5Problem 8) (10 points)Let ~r(t) be the space curve ~r(t) = (t2, sin(3πt), cos(5πt)).a) Calculate the velocity, the acceleration and the speed of ~r(t) at time t = 1.b) Write down the length of the curve from t = 1 to t = 10 as an integral. You don’t have toevaluate the integral.c) The curve t 7→ ~r(t) = (t3, 1 − t, 1 − t3) lies in a plane. What is the equation of this plane?Problem 9) (10 points)Let S be the surface given byz2=x24+ y2.a) Sketch the surface S.b) Let (a, b, c) be a point on the surface S. Find a parametric equation for the line that passesthrough (a, b, c) and lies entirely on the surface


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HARVARD MATH 21A - First practice exam

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