5/23/2006, SECOND PRACTICE FINAL Math 21a, Spring 2006Name:MWF 10 Samik BasuMWF 1 0 Joachim KriegerMWF 1 1 Matt LeingangMWF 1 1 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, boo ks, calculators, computers, or otherelectronic aids.• Unspecified functions are assumed to be smooth a nd de-fined everywhere unless stated otherwise.• You have 180 minutes time to complete your work.• The iochem 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)1)T FThe distance from (1, 2, −1) to (3, −2, 1) is (−2, 4, −2).2)T FThe plane y = 3 is perpendicular to the xz plane.3)T FAll functions u(x, y) that obey ux= u at all points obey uy= 0 at all points.4)T FThe best linear approximation at (1, 1, 1) to the function f(x, y, z) = x3+y3+ z3is the function L(x, y, z) = 3x2+ 3y2+ 3z25)T FIf f(x, y) is any function of two variables, thenR10R1xf(x, y) dydx =R10R1yf(x, y) dxdy.6)T FLet C = {(x, y) ∈ R2| x2+ y2= 1 } be the unit circle in the plane and~F (x, y) a vector field satisfying |~F | ≤ 1. Then −2π ≤RC~F · dr ≤ 2π.7)T FLet ~a and~b be two nonzero vectors. Then the vectors ~a +~b and ~a −~b alwayspoint in different directions.8)T FIf all the second-order partial derivatives of f (x, y) vanish at (x0, y0) then(x0, y0) is a critical point of f.9)T FIf ~a,~b are vectors, then |~a ×~b| is the area of the parallelogram determinedby ~a and~b.10)T FThe distance between two points A, B in space is the length of the curve~r(t) = A + t(B − A), t ∈ [0 , 1].11)T FThe function f(x, y) = xy has no critical point.12)T FThe length of a curve does not depend on the chosen parameterization.13)T FThe equation ρ = 1 in spherical coo rdinates defines a cylinder.14)T FFor any numbers a, b satisfying |a| 6= | b|, the vector ha −b, a + bi is perpen-dicular to ha + b, b − ai.15)T FThe line integral o f~F (x, y) = h−y, xi along the counterclockwise or ientedboundary of a region R is twice the area of R.16)T FThere is no surface for which bo t h the parabola and the hyperbola appearas traces.17)T FIf (u, v) 7→ ~r(u, v) is a parameterization for a surface, then ~ru(u, v)+~rv(u, v)is a vector which lies in the tangent plane to the surface.18)T FWhen using spherical coordinates in a triple integral, one needs to includethe volume element dV = ρ2cos(φ) dρdφdθ.TF PROBLEMS FOR REGULAR AND PHYSICS SECTIONS:19)T FA connected surface in space for which all normal vectors are parallel toeach other must be part of a plane.20)T FA vector field~F = hP (x, y), Q(x, y)i is conservative in the plane if and onlyif Py(x, y) = Qx(x, y) fo r all points (x, y).TF PROBLEMS FOR BIOCHEM SECTIONS:21)T FSuppose X and Y are two random variables such that E[X] > E[Y ]. Is italways the case that P[X > Y ] > 1/2?22)T FIf φ is the density function of a random variable χ, thenRφ(x) dx is theexpectation Eχ of the random varia ble.Problem 2) (10 points)-2-1 12-0.20.20.40.60.81We have a function u(t, x) which is a solution to partialdifferential equation. In all cases, we have u(0, x) =e−x2. The picture to the left shows this function u(0, x).Which partial differential equation is involved, when yousee the function u(1, x) as a gra ph?-2-1 12-0.20.20.40.60.81-2-1 12-0.20.20.40.60.81I II-2-1 12-0.20.20.40.60.81-2-1 12-0.20.20.40.60.81III IVEnter I,II,III,IV here Equationut(x, t) = ux(x, t)ut(x, t) = uxx(x, t)utt(x, t) = uxx(x, t)ut(x, t) = −ux(x, t)Problem 3) (10 points)a) Find an equation f or the plane Σ pa ssing through the points P = (1, 0, 1), Q = (2, 1, 3)and R = (0, 1, 5).b) Find the distance from the origin O = (0, 0, 0) to Σ.c) Find the distance from the point P to the line through Q, R.d) Find the volume of the parallelepiped with vertices O, P, Q, R.Problem 4) (10 points)The equation f(x, y, z) = exyz+ z = 1 + e implicitly defines z as a function z = g(x, y) ofx and y.a) Find formulas (in terms of x,y and z) for gx(x, y) and gy(x, y).b) Estimate g(1.01, 0.99) using linear approximation.Problem 5) (10 points)Find the surface area of the surface S parametrized by ~r(u, v) = hu, v, 2 +u22+v22i for(u, v) in the disc D = {u2+ v2≤ 1 }.Problem 6) (10 points)Find the local and global extrema of the function f(x, y) = x3/3 + y3/3 −x2/2 −y2/2 + 1on the disc {x2+ y2≤ 4 }.a) Classify every critical point inside the disc x2+ y2< 4.b) Find the extrema on the boundary {x2+ y2= 4} using the method of Lagrange multi-pliers.c) Determine the global maxima and minima on all of D.Problem 7) (10 points)a) Given two nonzero vectors ~u = ha, b, ci and ~v = hd, e, fi in R3, write down a formulafor the cosine of the angle between them. Find a nonzero vector ~v that is perpendicular to~u = h3, 2, 1i. Describe geometrically the set of all ~v, including zero, that a r e perpendicularto this vector ~u.b) Consider a function f of three varia bles. Explain with a picture and a sentence whatit means geometrically that ∇f(P ) is perpendicular to the level set of f through P .c) Assume the gradient of f at P is nonzero. Write a few sentences that would convincea skeptic that ∇f(P ) is perpendicular to the level set of f at the point P .d) Assume the level set of f is the graph of a f unction g(x, y). Explain the relation betweenthe gra dient of g and the gradient of f. Especially, how do you relate the orthogonality of∇f to the level set of f with the orthogonality of ∇g to the level set of g?Problem 8) (10 points)Let R be the region inside the circle x2+ y2= 4 and above the line y =√3. EvaluateZ ZRyx2+ y2dA .Problem 9) (10 points)A region W in R3is given by the relationsx2+ y2≤ z2≤ 3(x2+ y2)1 ≤ x2+ y2+ z2≤ 4x ≥ 01. Sketch t he region W .2. Find the volume of the region W .Problem 10) (10 points)Consider the vector field~F (x, y) = h−yx2+ y2,xx2+ y2idefined everywhere in the plane R2except at the origin.a) Let C be any closed curve which bounds a region D. Assume that (0, 0) is not containedin D and does not lie on C. Explain whyZC~F · d~r = 0 .b) Let C be the unit circle oriented counterclockwise. What isRC~F · d~r? Explain whyyour answer
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