1/14/2005, SECOND PRACTICE FINAL 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-D aw 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 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.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 no nzero vectors. Then the vectors ~a +~b and ~a −~b alwayspoint in different directions.8)T FIf all the second-order partia l 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 par ameterization.13)T FThe equation ρ = 1 in spherical coordinates defines a cylinder.14)T FFo r 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 of~F (x, y) = h−y, xi along the counterclockwise orientedboundary of a region R is twice the area of R.16)T FThere is no surface for which both 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 a ll normal vectors are parallel toeach other must be part of a plane.20)T FA vector field~F = h P (x, y), Q(x, y)i is conservative in the plane if and onlyif Py(x, y) = Qx(x, y) for 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 variable.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 graph?-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 for the plane Σ passing 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 .9 9) 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 g eometrically the set of all ~v, including zero, that are perpendicularto this vector ~u.b) Consider a function f of three variables. 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 function g(x, y). Explain the relation betweenthe gradient of g and the gradient of f. Especially, how do you relate the or tho gonality 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 the 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 count erclockwise. What isRC~F · d~r? Explain whyyour
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