5/23/2006, FIRST PRACTICE FINAL Math 21a, Spring 2006Name:MWF 10 Samik BasuMWF 10 Joachim KriegerMWF 11 Matt LeingangMWF 11 Veronique GodinTTH 10 Oliver KnillTTH 115 Thomas Lam• 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 assumed to be smooth and de-fined everywhere unless stated otherwise.• You have 180 minutes time to complete your work.• The Biochem section can ignore problems with vectorfields and line integrals.1 202 103 104 105 106 107 108 109 1010 1011 1012A 1013A 1014A 1012B 1013B 1014B 10Total: 140Problem 1) True/False questions (20 points)Mark for each o f the 20 questions the correct letter. No justifications are needed.1)T FThe length of the curve ~r(t) = hsin(t), t4+ t, cos(t)i on t ∈ [0 , 1] is the sameas the length of the curve ~r( t) = hsin(t2), t8+ t2, cos(t2)i on [0, 1].2)T FThe parametric surface ~r(u, v) = (5u −3v, u −v −1, 5u −v − 7) is a plane.3)T FAny function u(x, y) that obeys the differential equation uxx+ ux−uy= 1has no local maxima.4)T FThe scalar proj ection of a vector ~a onto a vector~b is the length of the vectorprojection of ~a onto~b.5)T FIf f(x, y) is a function such that fx− fy= 0 then f is conservative.6)T F(~u ×~v) · ~w = (~u × ~w) ·~v for all vectors ~u,~v, ~w.7)T FThe equation ρ = φ/4 in spherical coordinates is half a cone.8)T FThe function f(x, y) =(xx2+y2if (x, y) 6= (0, 0)0 if (x, y) = (0, 0)is continuous at everypoint in the plane.9)T FR10Rx01 dydx = 1/2.10)T FLet ~a and~b be two vectors which are perpendicular to a given plane Σ.Then ~a +~b is also perpendicular to Σ.11)T FIf g(x, t) = f(x−vt) for some function f of one variable f(z) then g satisfiesthe differential equation gtt− v2gxx= 0.12)T FIf f(x, y) is a continuous function on R2such thatR RDf dA ≥ 0 for anyregion D then f(x, y) ≥ 0 for all (x, y).13)T FAssume the two functions f(x, y) and g(x, y) have both the critical point(0, 0) which a r e saddle point s, then f + g has a saddle point at (0, 0).14)T FIf f(x, y) is a function of two variables and if h(x, y) = f(g(y), g(x)), thenhx(x, y) = fy(g( y), g(x))g′(y).15)T FIf we rotate a line around the z axis, we obtain a cylinder.16)T FIf u(x, y) satisfies the transport equation ux= uy, then the vector field~F (x, y) = hu(x, y), u(x, y)i is a g radient field.17)T F3 grad(f ) =ddtf(x + t, y + t, z + t).18)T FIf a vector field~F is defined at all point s in three-space except the originand curl(~F) =~0 everywhere, then the line integral o f~F around any closedpath not passing through the origin is zero.TF Problems for regular sections:19)T FIf~F is a vector field in space and f is equal to the line integral of~F alongthe straight line C from (0, 0, 0) to (x, y, z), then ∇f =~F .20)T FThe line integral of~F (x, y) = (x, y) along an ellipse x2+ 2y2= 1 is zero.21)T FThe identity div(gr ad(f)) = 0 is always true.TF Problem for probability theory sections:22)T FInside a bag is are two coins, one coin has both sides heads and one coinis normal (one head and one tail). I randomly pick one of the coins andrandomly look at one side, seeing a head. Is the probability that the otherside of the same coin is a tail equal to 1/2?23)T FIf X and Y are independent random variables, then D(X + Y ) = D(X) +D(Y ).24)T FThe expectation o f the product of two random variables is always the prod-uct of the expectations..Problem 2) (10 points)Match the equations with the curves. No justifications are needed.I IIIII IVEnter I,II,III,IV here Equation~r(t) = (sin(t), t(2π − t))~r(t) = (cos(5t), sin(7t))~r(t) = (t cos(t), sin(t))~r(t) = (cos(t), sin(6/t))Problem 3) (10 points)In this problem, vector fields F are written as F = hP, Qi. We use abbreviations curl(F ) =Qx−Pyand div(F ) = Px+Qy. When stating curl(F )(x, y) = 0 we mean that curl(F )(x, y) =0 vanishes for all (x, y). The statement curl(F ) 6= 0 means that curl(F )(x, y) does notvanish for a t least one point (x, y).The same remark applies if curl is replaced by div. Check the box which match theformulas of the vector fields with the corresponding picture I,II,III or IV. Mark also theplaces, indicating the vanishing or not vanishing of curl and div. In each of the four lines,yo u should finally have circled three boxes. No justifications are needed.Vectorfield I II III IV curl(F ) = 0 curl(F ) 6= 0 div(F ) = 0 div(F ) 6= 0~F (x, y) = (0, 5)~F (x, y) = (y, −x)~F (x, y) = (x, y)~F (x, y) = (2, x)I IIIII IVProblem 4) (10 points)a) Find the scalar projection of the vector ~v = (3, 4, 5) onto the vector ~w = (2, 2, 1).b) Find the equation of a plane which contains the vectors h1, 1, 0i and h0, 1, 1i and con-tains the point (0, 1, 0).Problem 5) (10 points)a) (5 points) Find the surface area of the ellipse cut from the plane z = 2x + 2y + 1 bythe cylinder x2+ y2= 1.b) (5 points) Find the arc length of the plane curve ~r(t) = (sin(t)et, cos(t)et) for t ∈ [0, 2π].Problem 6) (10 points)a) If u( x, y) and v(x, y) are two functions, then (uv)xx= uxxv + 2uxvx+ uvxx.b) The identity ∆(uv) = ∆uv + u∆v + 2∇u · ∇v holds.c) Assume u and v satisfy the Laplace equation ∆u = uxx+ uy y= 0 and ∇u ·∇v = 0 thenuv satisfies the Laplace equation.Problem 7) (10 points)Let f(x, y, z) = 2x2+ 3 xy+2y2+z2and let R denote the region in R3, where 2x2+ 2 y2+z2≤1. Find the maximum and minimum values of f on the region R and list all points, wheresaid maximum and minimum values are achieved. Distinguish between local extrema inthe interior and extrema on the boundary.Problem 8) (10 points)Sketch the region of integration of the following iterated integral and then evaluate theint egral:Zπ0 Z√π√zZx0sin(xy)dydx!dz .Problem 9) (10 points)Evaluate the line integralZC~F ·~dr ,where C is the planar curve ~r(t) = (t2, t/√t + 2), t ∈ [0, 2] and~F is the vector field~F (x, y) = (2xy, x2+ y). Do this in two different ways:a) by verifying that~F is conservative and replacing the path with a different path con-necting (0, 0) with (4, 1),b) by finding a potential U satisfying ∇U =~F .Problem 10) (1 0 points)Evaluate the line integralRC~F ·~dr, where~F = (x +exsin(y), x + excos(y)) and C is the rig ht handed loopof the lemniscate
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