1/14/2005, FINAL EXAM Math 21a, Fall 2005Name:MWF9 Ivan PetrakievMWF10 Oliver KnillMWF10 Thomas LamMWF10 Micha el ScheinMWF10 Teru YoshidaMWF11 Andrew DittmerMWF11 Chen-Yu ChiMWF12 Kathy 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 fo r the True/False and multiplechoice problems.• Please write neatly.• Do not use notes, books, calculators, computers, or otherelectronic a ids.• Unspecified functions are assumed 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 1011A 1012A 1013A 1011B 1012B 1013B 10Total: 140Problem 1) True/False questions (20 points)1)T FThe projection vector proj~v(~w) is parallel to ~w.2)T FAny parametrized surface S is a graph of a function f (x, y).3)T FIf the directional derivatives D~v(f)(1, 1) and D~w(f)(1, 1) are both 0 for~v = h1, 1i/√2 and ~w = h1, −1i/√2, then (1, 1) is a critical point.4)T FThe linearization L(x, y) of f (x, y) = x + y + 4 at (0, 0) satisfies L(x, y) =f(x, y).5)T FFo r any function f(x, y) of two varia bles, the line integral of the vector field~F = ∇f on a level curve {f = c} is always zero.6)T FIf~F is a vector field of unit vectors defined in 1/2 ≤ x2+ y2≤ 2 and~F istangent to the unit circle C, thenRC~F · d~r is either equal to 2π or −2π.7)T FIf a curve C intersects a surface S at a right angle, then at the point ofintersection, the tangent vector t o the curve is parallel to the normal vectorof the surface.8)T FThe curvature of the curve ~r(t) = hcos(3t), sin(6t)i at the p oint ~r(0) issmaller than the curvature of the curve ~r(t) = hcos(30t), sin(60t)i at thepoint ~r(0).9)T FAt every point (x, y, z) on the hyperb oloid x2+ y2− z2= 1, the vectorhx, y, −zi is t angent to the hyperboloid.10)T FThe set {φ = π/2, θ = π} in spherical coordinates is the negative x axis.11)T FThe integralR10R2π0Rπ0ρ2sin2(φ) dφ d θ dρ is equal to the volume of the unitball.12)T FFo ur points A, B, C, D are located in a single common plane if (B − A) ·((C − A) × (D − A)) = 0.13)T FIf a function f(x, y) has a local maximum at (0, 0), then the discriminantD is negative.14)T FThe integralRx0R1yf(x, y) dx dy represent s a double integral over a boundedregion in the plane.15)T FThe following identity is true:R30R2x0x2dydx =R60R3y /2x2dx dyTF problems 16-20 are for regular and physics sections only:16)T FThe integralRRScurl(~F) ·~dS over the surface S of a cube is zero for all vectorfields~F .17)T FA vector field~F defined on three space which is incompressible (div(~F ) = 0)and irrotational ( curl(~F ) = 0) can be written as~F = ∇f with ∆f = ∇2f =0.18)T FIf a vector field~F is defined at all points of three-space except the origin,and curl(~F ) =~0 everywhere, then the line integral of~F around the circlex2+ y2= 1 in the xy-plane is equal to zero.19)T FThe identity curl(grad(div(~F ))) =~0 is true for all vector fields~F (x, y, z ) .20)T FIf~F = curl(~G), where~G = heex, 5xz5, sin yi, then div(~F (x, y, z ) ) > 0 for all(x, y, z).TF problems 21-25 are for biochem sections only:21)T FThe expected value of the sum of two random variables is the sum of theirexpected values.22)T FLet X be a random variable. Suppose that we know both the expectationE(X) and and varia nce D(X). Does this information determine E(X2−5X + 4)?23)T FIf A and B ar e two events and B has positive probability, then P (A|B) isalways less than or equal to P (A).24)T FThe function Φξ=11+x2is the distribution function of some random variableξ.25)T FSuppose you throw two fair dice. The probability that the sum of theirupturned faces is 11 is 2/36.Problem 2) (10 points)Match t he equations with the space curves. No justifications are needed.I IIIII IVEnter I,II,III,IV here Equation~r(t) = ht2, t3− t, ti~r(t) = h|1 − |t||, |t − |t − 1||, ti~r(t) = h2 sin(5t), cos(11t), ti~r(t) = ht sin(1/t), t|cos(1/t)|, tiProblem 3) (10 points)Match t he equations with the objects. No justifications a re needed.I II III IVV VI VII VIIIEnter I,II,III,IV,V,VI,VII,VIII here Equation~r(s, t) = h(2 + cos(s)) cos(t), (2 + cos(s)) sin(t), sin(s)i~r(t) = hcos(3t), sin(5t)ix2+ y2− z2= 1~F (x, y, z) = h−y, x, 1ix2+ y2+ z2≤ 1, x2+ y2≤ z2, z ≥ 0z = f(x, y) = x2− yg(x, y) = x2− y2= 1~F (x, y) = h−y, xiProblem 4) (10 points)a) Find an equation for the plane Σ passing through the points ~r( 0),~r(1 ), ~r(2), where~r(t) = ht2, t4, ti.b) Find the distance between the point ~r(−1) and the plane Σ found in a).Problem 5) (10 points)A vector field~F (x, y) in the plane is given by~F (x, y) = hx2+5, y2−1i. Find all the criticalpoints of |~F (x, y)| and classify them. At which point or points is |~F (x, y)| minimal?Hint: Extremize f(x, y) = |~F (x, y)|2.Problem 6) (10 points)A house is situated at the point ( 0, 0) in the middle of a mountainous r egion. The altitudeat each point (x, y) is given by the equation f(x, y) = 4x2y +y3. There is a pathway in theshape of an ellipse around the house, on which the (x, y) coordinates satisfy 2x2+ y2= 6.Find the highest and lowest points in t he closed region bounded by the path.Problem 7) (10 points)a) (4 points) Where does the tangent plane at (1, 1, 1) to the surface z = ex−yintersectthe z axis?b) (4 points) Estimate f (x, y, z) = 1 + log(1 + x + 2y + z) + 2√1 + z at the point(0.02, −0.001, 0.01).c) (2 points) f (x, y, z) = 0 defines z as a function g(x, y) of x and y. Find the partialderivative gx(x, y) at the point (x, y) = (0, 0).Problem 8) (10 points)Fo r each of the following quantities, set up a double or triple integral using any coordinatesystem you like. You do not have to evaluate the integrals, but the bounds of each singleintegral must be specified explicitly.1. (3 points) The volume of the tetrahedron with vertices (0, 0, 0), (3, 0, 0), (0, 3, 0) and(0, 0, 3).2. (4 points) The surface area of the piece of the paraboloid z = x2+ y2lying in theregion z = x2+ y2, where 0 ≤ z ≤ 1.3. (3 points) The volume of the solid bounded by the planes z = −1, z = 1 and theone-sheeted hyperboloid x2+ y2− z2= 1.Problem 9) (10 points)A region R in the xy-plane is given in polar coordinatesby r(θ) ≤ θ for θ ∈ [0, π/ 2]. Find the double integralZZRsin(√x2+ y2)√x2+
View Full Document
Unlocking...