1/13/2002 FINAL EXAM Math 21aName:MWF9 Sasha BravermanMWF10 Ken ChungMWF10 Jake RasmussenMWF10 WeiYang QuiMWF10 Spiro KarigiannisMWF11 Vivek MohtaMWF11 Jake RasmussenMWF12 Ken ChungTTH10 Oliver KnillTTH11 Daniel Goroff• Start by printing your name in the above box and checkyour section in the box to the left.• Try to answer each question on the same page as thequestion is asked. If needed, use the back or next emptypage for work. If you need additional paper, write yourname on it.• 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. Justify your answers.• No notes, books, calculators, computers or other elec-tronic aids are allowed.• You have 180 minutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 1011 1012 10Total: 130Which section-specific problem do youchoose? Check exactly one problem. Onlythis problem can be graded. If you don’tcommit yourself here, the first attemptedproblem (of 12a-12e) will be graded.12a12b12c12d12eProblem 1) TF questions (20 points) Circle the correct letter. No justifications are needed.All functions or vector fields are assumed to be smooth, which means that arbitrary many partial derivatives exist.T FThe distance from (1, 2, −1) to (3, −2, 1) is (−2, 4, −2).T FThe plane y = 3 is perpendicular to the xz plane.T FAll functions u(x, y) that obey ux= u at all points obey uy= 0 atall points.T FThe best linear approximation at (1, 1, 1) to the functionf(x, y, z) = x3+ y3+ z3is the function L(x, y, z) = 3x2+ 3y2+ 3z2T FIf f(x, y) is any function of two variables, thenR10R1xf(x, y) dydx =R10R1yf(x, y) dxdy.T FLet C = {(x, y) ∈ R2| x2+ y2= 1 } be the unit circle in theplane and F(x, y) a vector field satisfying |F| ≤ 1. Then −2π ≤RCF ·dr ≤ 2π.T FA vector field F = hP (x, y), Q(x, y)i is conservative in the plane ifand only if Py(x, y) = Qx(x, y) for all points (x, y).T FLet a and b be two nonzero vectors. Then the vectors a + b anda − b always point in different directions.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.T FIf a, b are vectors, then |a × b| is the area of the parallelogramdetermined by a and b.T FThe distance between two points A, B in space is the length of thecurve r(t) = A + t(B − A), t ∈ [0, 1].T FThe function f(x, y) = xy has no critical point.T FThe length of a curve does not depend on the chosen parameteri-zation.T FThe equation ρ = 1 in spherical coordinates defines a cylinder.T FFor any numbers a, b satisfying |a| 6= |b|, the vector ha −b, a + bi isperpendicular to ha + b, b − ai.T FThe line integral of F(x, y) = h−y, xi along the counterclockwiseoriented boundary of a region R is twice the area of R.T FA surface in space for which all normal vectors are parallel to eachother must be part of a plane.T FThere is no surface for which both the parabola and the hyperbolaappear as traces.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.T FWhen using spherical coordinates in a triple integral, one needs toinclude the volume element dV = ρ2cos(φ) dρdφdθ.x 4 =2Problem 2) (10 points)Match the equations with the curves. No justifications are needed.I IIIII IVEnter I,II,III,IV here Equationr(t) = ht2, t3− tir(t) = h|1 − |t||, |t − |t − 1||ir(t) = h2 sin(13t), cos(22t)ir(t) = ht sin(1/t), t|cos(1/t)|iProblem 3) (10 points)a) Find an equation for the plane Σ passing through the points P = (1, 0, 1), Q = (2, 1, 3) andR = (0, 1, 5).b) Find the distance from the origin (0, 0, 0) to Σ.3Problem 4) (10 points)The equation f(x, y, z) = exyz+z = 1+e implicitly defines z as a function z = g(x, y) of x 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) inthe disc {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 + 1 onthe disc D = {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 multipliers.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, f i in R3, write down a formula for thecosine 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 are perpendicular to this vector u.b) Consider a function f of three variables. Explain with a picture and a sentence what itmeans 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 convince askeptic 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 orthogonality of ∇fto 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 .4Problem 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 fieldF(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 contained inD and does not lie on C. Explain whyZCF · dr = 0 .b) Let C be the unit circle oriented counterclockwise. What isRCF · dr? Explain why youranswer shows that there is no function f for which F(x, y) = ∇f(x, y) everywhere except atthe origin (0, 0).Problem 11) (10 points)Let F(x, y) be a vector field in the plane given by the formulaF(x, y) = hx2− 2xye−x2+ 2y, e−x2+1√y4+ 1i .If C is the path which goes from from (−1, 0) to (1, 0) along the semicircle x2+ y2= 1, y ≥ 0,evaluateRCF · dr.(Hint: use Green’s Theorem.)5SECTION SPECIFIC PROBLEMS Math 21aPlease choose one of the
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